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V 41
and its inverse
U are now smooth maps. (3) Obviously the composition of two smooth maps is again smooth. Examples. (1) F : B ^ S 1 , ^ ) = (cosMini)(2) F : S2n+X -> P n ( C ) , F((z0,.. .,*„)) = *((*„,..., zn)) = [z0,..., zn\. (3) Let ip : 5 2 \{'(0,0,1)} -»• ffi2 be given by the stereographic projection from the 2-sphere to the plane {x3 = 0} C K3, and ip0 : {[zo>zi] 6 IPi(C)|2o ^ 0} = U0 -*• C the usual chart [^0,^i] >-> f1- Then the map F : P i ( C ) -> 5 2 defined by F([0,1]) = '(0,0,1) and F\u0 = ip'1 o i? o ^ , where i? : C —>• R 2 , w i-^ t(Re(w),Im(w)), is a smooth map from the smooth (real!) manifold underlying the complex manifold P i ( C ) to the 2-sphere. Definition. Let M and N be smooth manifolds and F : M -* N a smooth map. Then F is called a "diffeomorphism" if F is bijective and F~l is smooth. Examples. (1) Let A be in GL(m,R) a diffeomorphism. N a diffeomorphism. Then V C Rn be a coordinate chart with domain U, an open set in Q. Then there is an induced vector bundle isomorphism $ = (Ttp)* : T*V ->• T*U over ip~l : V ->• U defined as follows: * l r ; v = ( V ' W V ) * : T ^ = (TXV)* -> ( V ^ t / ) * = T;_ 1 ( x ) £/ Furthermore we trivialize T*V as usual: let a: = t(x\,..., 2/ == '(j/i> • • • i 2/n) i n J£n> then we set V = tp(U) C Rm with U open in M and such that V = Pn) : U -> :=moV. 143 Aidx1 M, differentiable, U open sets ofMn}. 2. (Diff(M),{4>}). The group of diffeomorphism of a manifold M, with plots taken as all the functions {
and F = TA : Mm -^ K m . Then F is
(2) The map F : F i ( C ) -> S2 defined in the preceding Example (3) of smooth maps is a diffeomorphism. Definition. Let M be a smooth manifold of finite dimension and TV a subset of M. We call N a "closed submanifold of M" if N is closed in M and if for each pin N there is an admissible chart U(p) ^rVC R m such that
Examples. (1) Let O be open in Rm with the subspace topology induced from the usual metric topology on W1 and let pi and p 2 be distinct points in O. Then there exists Ci and e2 in K >0 such that Btk(pk) = {x G R m | \\x - pk\\ < ek} are contained in Cl and disjoint, i.e. 0 is Hausdorff. (2) Let M = (E\{0})U{pi,p 2 } and let T be defined as follows: a subset U' C M contained in M\{0} is open if and only if it is open as a subset of R\{0} C 1R with its usual subspace topology, and a subset U C M containing Pi or P2 (or both of them) is open if and only if (UC\ (R\{0}) is open in M\{0} and there is a e > 0 such that B e (0)\{0} is contained in U n (E\{0})). This defines a non-Hausdorff topological space that allows for a smooth atlas. Let us also recall that a subset B C T of the topology of a topological space (M, T) is called a "basis of the topology" if for each U in T there is a family {U\\X E A } c B such that LUGA^A = U. Furthermore we call a topological space "second countable" if its topology has a countable basis. Examples. (1) The set B = {B1/n(q) \ n € N^ 1 , q € Q71} is a countable basis of the usual topology of Rm. (2) A non-countable set M equipped with the discrete topology (i.e. T is the power set V{M)) is not second-countable. (3) There are non-second countable Hausdorff topological spaces that have a smooth atlas (see [Sp]). In order to avoid lengthy statements in the sequel of the text - unless it is explicitely stated otherwise - we will assume that a "manifold" is always a finite dimensional, smooth, Hausdorff and second countable manifold. On the other hand we will neither assume that a manifold is connected nor that it is pure-dimensional. Bibliographical remarks. Beside [Sp] quoted above we recommend [AMR],[Bo] and [Lan] for further reading on the foundations of differ•entiable manifolds. 2.2
Lie groups and smooth
actions
Definition. A "Lie group" is a manifold that carries a group structure such that the map GxG-+G,(g,h)>-* is smooth. 43
g-h-1
Remark. Equivalently one can ask for the smoothness of the following two maps: / : G -» G,g n- 5 _ 1 and M : G x G -¥ G,(g,h) •->• # • h. Examples. (1) Let V be a real respectively complex vector space of finite dimension. Then the addition "+" of vectors makes (V, +) into an abelian smooth respectively complex-analytic Lie group. (2) Let T be a countable group. Then the discrete topology makes F a zerodimensional manifold and it follows that T is a zero-dimensional Lie group. (3) Considering S1 as a subgroup of C\{0} and S3 as a subgroup of H\{0} (H denotes the skew-field of quaternions) give rise to Lie group structures on these spheres. (4) For K = E and C, the "general linear group" GL(n,K) is open in Mat (n x n,K) and matrix multiplication induces a group structure on this set. The usual formulas for multiplication and the invese of a matrix show that GL(n,R) is a smooth real Lie group and GL(n,C) a "complex-analytic Lie group". Analogously GLK(V) is a smooth respectively complex-analytic Lie group if V is a finite dimensional vector space over K = E respectively over K = C. (5) If V is a finite dimensional K-vector space (K = E or C) then the "special linear group" SL(V) = SLK(V) = { T e GLK{V) | det(T) = 1} is a closed submanifold of GL^(V) and a Lie group. If V is furthermore equipped with a symplectic form w then the some assertions hold for Sp(V, to). (6) The sets 0(n,K) - {A £ Mat(n x n,K) | lO • O - E„} are closed submanifolds of Mat(n x n, K) and Lie groups, the so-called "real (respectively complex) orthogonal groups". (7) The sets U(n) = {A e Mat (n x n,C)|*A • A = E^} and SU(n) = {A e U(n)\ det(yl) = 1} are closed submanifolds of GL(n, C) (looked upon as a real manifold) and Lie groups. Definition. Let G be a Lie group and H a subset of G. We call H a "closed Lie subgroup" if H is a closed submanifold of G and a subgroup of G. Exercises. (1) Show that a closed Lie subgroup is a Lie group. (2) Show that SL(n, E) and 0(n, E) are closed Lie subgroups of GL(n, E). Definition. Let Gi and G 2 be two Lie groups. A map F : G\ -> G 2 is called 44
a "Lie group homomorphism" if F is smooth and a group homomorphism. A Lie group homomorphism F is called a "Lie group isomorphism" if F is bijective and F~l is a Lie group homomorphism. We call two Lie groups Gi and G2 "isomorphic (as Lie groups)" if there exists a Lie group isomorphism F : Gi -»• G 2 . Example. The following Lie groups are isomorphic: 5 0 ( 2 , E) = 0(2, R) n SL(2,1).
5 1 ,C/(1) and
Definition. Let M be a manifold and G a Lie group. A smooth map 1? : G x M -» M is called a "smooth (left)-action (of G on M)"if the following two conditions hold (i) i?(e,p) = p for all pin M (e denotes the neutral element of the group G), (ii) d(g,fl(h,p))
= d(g-h,p) for all g,h in G and all p in M.
Remarks. (1) The map M -» M,p t-t tf(g,p) for fixed g is often denoted by #ff. Since i9p o iJ9_i = Idjvf it follows that ??s is a diffeomorphism for each g in G. Another useful notation is gp for # s (p). (2) A right-action is defined similarly replacing condition (ii) by (ii') •d(g,i9(h,p)) = §(h-g,p) for all g,h in G and all p in M. Examples. (1) GL(n,E) x i C ->R",tf(.4,:E)
=TA(x)=A-x.
(2) GL(n,R) x G t ( l " ) -> G fc (E"),i?(A, VK) =TA{W). (3) ^ ( n , E) x £ ->• £, where £ = {W C G„(E 2n )|W r is Lagrangian} and -d as in (2). (4) Let G be a Lie group and M = G, then tfL : G x M -> M,dL(g)(h) = g • h and tfR : G x M -> M,$R(g)(h) = h-g~1 are smooth transitive actions. (5) i > : Z 2 x S m ^ S m ,i9(0,a:) = x,0(l,x) (6) i 9 : Z n x E ™ -> E",t9(a,a;) = x + a.
= -x.
(7) d : GL(n + 1, K) x P„(K) -»• P n (K),^(A, [z]) = [A • z] for K = E and C. (The line generated by z is as usual denoted by [z].) Definition. Let i 9 : G x M - ) - M b e a smooth action. (1) The action is called "transitive" if for each pair p, q in M there is at least one g in G such that g • p = 3. (2) The "fixed point set of an element g in G" is the set {p e M|p • p = p} 45
and the "fixed point set of G" is the set MG = f) {p g M\g • p = p} — {p e g€G
M\g-p = p\/geG}. (3) The "G-orbit through p" is the set {g • p\g G G} C M (for p a point in M). (4) The "isotropy group (or stabilizer) of p (under the G-action)" is Gp = {g€G\g-p = p}. (5) The "orbit space" is the space of equivalence classes M/~ = {\p]\p € M}, where p ~ q if and only if there exists a [p] is called the "canonical projection" and is often denoted by n. Exercise. Go through the examples of actions (1) - (7) above and determine the fixed point sets of the group, the G-orbits, the isotropy groups and (set-theoretically) the orbit spaces of these actions. Definition. Let T be a discrete Lie group, M a manifold and d : T x M —> M a smooth action. We call the action "free" if the fixed point set is empty for all 7 in r \ { e } . We call the action "properly discontinuous" if for all K compact in M the set {7 £ T\^(K) (1K ^ 0} is finite. Proposition. Let fi : T x M -> M be a properly discontinuous and free action. Then the orbit space M/^ carries a unique differentiable structure such that the projection IT : M —> M/~ is a local diffeomorphism. Remark. A smooth map F : M —>• N is a "local diffeomorphism" if for each p in M there is an open set U containing p such that F\u • U ^ F(U) is a diffeomorphism. Proof of the proposition. Let T be the quotient topology on M / ~ , i.e. U is in T if and only if n~1(U) is open in M. Since M is a manifold and T is countable it follows easily that T has a countable basis. Let now x be in M and W -^ V be a coordinate chart such that Ba(v»(a;)) C V. Let W1/n - ^ - ' ( B i / n ^ W ) and Kn = W^~ for n > 1. (The notation Be (p) denotes the e-ball around a point p in E m .) Then the Kn are compact and contained in W. Consider now a 7 ^ e such that j(Ki) D K\ ^ 0. Since K\ is compact and 7 has no fixed-point it follows easily that there is a N(-f) > 1 such that j(Kn) n Kn = 0 for n > N(j). Since the set {7 € T\j(Ki) C\Ki ^ %} is finite there is a TV > 1 such that KN fl J(KN) = 0 for all 7 jt- e. Thus n\w1/N • WI/JV -> ""(^l/w) is a homeomorphism. 46
Starting with any atlas we can now construct an atlas {(Wx, $x)\x € M} such that W x ri7(W x ) - 0 for all 7 ^ e (and for all x e M). Since ir(Wx) = : Ux is an open set in M / ~ containing ir(x) we have an open covering {t/x|:c € M} for M / ^ and homeomorphisms
:
^
- ^ *x(Vr x ).
Let now y ^ x such that Ux f~l £/y contains a point 7r(z) (with z € Wx.) Then by construction of the Wp for p in M the connected component of (n\w*)~1(Ux fl f/y) containing z is in Wx (~l 7(C/y) for a unique 7 in T. It follows that the coordinate changes of the topological atlas {(Ux,
carries the discrete topology and thus our assumption was wrong and T acts properly discontinously by i? = i? L |r on M = G. The quotient Mj~ being the set {Tg\g E G} of left restclasses, we denote it sometimes by T/G as in abstract group theory. The corresponding quotient for the action t? fl |r is then of course denoted by G/T. As an illustration we give some concrete examples. (3.1) Let i ? b e a unital ring and a,b,c £ R
GR =
Then GR is a group, and GR respectively Gc is a real Lie group respectively a complex-analytic Lie group. Examples of discrete Lie subgroups are Gz C GR and Gz[i] C Gc, and G Z \ G R respective Gz[i\\Gc are compact smooth respective complex-analytic manifolds. (3.2) Let R be as in (3.1) and NR ~ GR x R = ) ( ( 0 1 c ),d)
a,b,c,deR},
the product of the group GR, defined above in (3.1) with the abelian group (R, +). Then Nz is a discrete subgroup of N R , and NZ\N$L and NR/NZ are compact smooth real four-dimensional manifolds. Bibliographical remarks. A good reference for the elementary theory of quotients by discrete groups is [Bo]. A lot of material on differentiable actions of compact Lie groups can be found in the classic [Bre]. 2.3
Vector
bundles
Definition. A "smooth (real respectively complex) vector bundle of rank r (over a manifold M)" is a manifold E together with a smooth projection p : E -> M such that for each x in M the fiber p~l{x) =: Ex has the structure of a r-dimensional (real respectively complex) vector space and such that the following condition of "local triviality" is satisfied: for each x in M there is an open set U containing x and a diffeomorphism ^a : p~x{U) -4- U x W such that pr\ o vj/y = p and ^u\Ex : Ex -> {x} x Kr is a K-vector space isomorphism for each x 6 U. (K equals E or C here). Remark.
We will often speak of the "vector bundle E A- M" in order to 48
have a short notation including the projection map p. Furthermore E will sometimes be called the "total space" and M the "base" of the vector bundle. Examples. Let K be R or C. (1) The product manifold M x W with p = pr\ is a vector bundle, the socalled "trivial vector bundle of rank r over M". (2) Let i? : Z x (E x E) -»• E x R,0(n,(x,v)) = (x + n, ( - l ) n • v) and E = (E x E ) / ^ A R/Z,p([x,v]) = [x]. Then E is a vector bundle over E/Z = S 1 . (3) Let H = {([z],u) G FTO(K) x K m + > € ( ( Z ) ) K } - Then H is a closed submanifold of PTO(K) x K m + 1 and the fibers of the projection p : H —> P m (K),p([2;],u) = [z] are the one-dimensional K-vector spaces p _1 ([^]) = {(W,V)|I;G((Z))K}(S((Z))KCK™+1).
"Local trivializations" are obtained as follows: let Uj Fm(K)\zj ^ 0} and Xj : E/j x IK -> p _ 1 ( ^ j ) be defined by
=
{[z] £
Z
Xji[z],X)=({z],X.(
f,...,Z-t±,l,ZJ±^,...,Z-^)).
\
Zj
Zj
Zj
Zj
I
a
It is easy to check that Xj is diffeomorphism, that is "fiber-wise linear" and thus \£j = xj1 1S a local trivialization of if A P m (K) over Uj (for j = 0,...,m). Definitions. (1) Let .E A s : M —>• F vector space by s : M -> Ex=p-1(x).
M a vector bundle. A "smooth section of En is a smooth map such that p o s = I d ^ . The set of smooth sections of E form a which is denoted by Tc°°(M,E). The "zero-section" is defined E,s(x) = 0X, where 0^ is the zero-element of the vector space
(2) Let S i ^ Mi and £ 2 ^ M 2 be vector bundles and let / : Mx ->• M 2 be a smooth map. A smooth map F : E\ ->• F 2 is called a "smooth vector bundle homomorphism (over / ) " if F{[E{)X) C {E2)f(x) for all x in M : and F ^ d ) , . : ( F i ) i -*• (E2)f(x) is a linear map for all x in M\. (Sometimes we refer to these properties by saying that F is "fiber-preserving" and "fiber-wise linear".) If / is a diffeomorphism and F\(El)x is an isomorphism for all x in M the map F is called a "smooth vector bundle isomorphism (over / ) " . (3) Two vector bundles over the same base manifold M are called "isomorphic" if there exists a smooth vector bundle isomorphism over / = Idj^. 49
A vector bundle is called "trivializable" or shortly "trivial" if it is isomorphic to the trivial bundle of the same rank. Remark. The vector bundle E of Example (2) is non-trivial. Definition. Let E A M be a vector bundle of rank r and {Ua\a G A} an open covering of M such that $a : p _ 1 (Ua) - > l / a x K r are local triviahzations of E. Then the maps gap : Uap = Ua f) Up ->• GL(W) defined by *a°*pl
:Ua0xK~
-*UapxVr,*aoVpl(x,v)
=
(x,ga{,(x)-v)
are called the "transition functions of E with respect to the local triviahzations {Ua\a G 4 } " . Remark. ties" :
The family {gap\a,/3 G A} fulfills the following "cocycle identi-
(1) 9aa(x) = 1 for all x in Ua, (2) (gapix))'1
= 90a(x) for all x in Uap,
(3) gap{x) • 9p7(x) • gla{x)
= 1 for all x in Uap1 =
Proposition. Let M be a manifold.
UanUpnU^.
Then
(i) If {Ua\a € A} is an open covering of M and gap : Uap —> GL(Er) is a family of maps fulfilling the cocycle identities given above then the set (lJ Q , € j 4 ([/ a x K r )J / ~ , where (x,v) in Up x Kr and (y,w) in Ua x W are equivalent if and only if x — y £ Uap and w = gap(x) • v, is a vector bundle of rank r over M. (ii) If E —> M is a vector bundle of rank r and if {gap\a,fl € A] are its transition functions with respect to a trivializing open cover {Ua\a G .4} then E is isomorphic as a vector bundle to ( U a e A ^ "
x
K r )j/~.
(Hi) Given the situation of (ii) then a section s of E defines (and is defined by) a family of smooth maps sa : Ua —• W (a G A) such that sa(x) = gap{x)-sp(x) for all x in Uap and for all a, (3 in A. Proof. Exercise.
•
Remarks. (1) Using the transition functions we can apply pointwise the usual operations from multilinear algebra to construct new vector bundles from given ones. Let E and F be vector bundles then E © F with fiber (E © F)x = Ex © Fx is called the "direct sum of the vector bundles E and F". Similarly one defines E <8> F, Horn (E, F), and, if F C E is a "subbundle", E/F. Furthermore we 50
have the bundles E* with fiber {E*)x = (Ex)* and ®kE*,SkE*
and KkE* for
(2) If / : M —> N is a smooth map and p : E —> N a vector bundle we have the "pullback of E under / " (or simply the "pullback bundle") with total space f*E={(x,l)eMxE\f(x)=p(l)}, a closed submanifold of M x E, and projection f*p : f*E -> M given by (f*p)(x,l)=x. Bibliographical remarks. We recommend [BT] to get some intuition for vector bundles and [AMR] for many technical details. 2.4
The tangent
bundle
Definition. Let M be a manifold and p in M. (1) A "curve at p" is a smooth map 7 : / —> M, where / is an open intervall in R containing 0, such that 7(0) = p. (2) Two curves 71 and 72 at p are called "tangent with respect to the chart (U,V)" if p i s in U and ±\Q(
&\o(
Lemma. Let (U,tp), (U',tp') be charts on a manifold M and p in U (~) V. Then two curves 71 and 72 are tangent with respect to (U, ip) if and only if they are tangent with respect to (U',
o l2)(t)
= j
t
\ y o 7 2 )(i).
Exchanging the roles of ip and tp' we arrive at the assertion. m
n
D
Remark. Let V be open in E , V open in R and * : V -»• V a smooth map. We recall that the "derivative of \?" in a point x in V is given by the unique linear map Dx$ : Rm -> E n such that for to £ R m \ { 0 } lim
IMKo Definitions.
||^(a; + « ; ) - $ ( a ; ) - ( D a ^ ) ( u ; ) H ||w||
_Q
Let M be a manifold and let p be a point in M. 51
(1) Two curves 71 and 72 at p are called "equivalent" if 7i and 72 are tangent with respect to one (and thus every) chart. (2) The space {[7]p | 7 curve atp} of equivalence classes of curves at p is called the "tangent space to M at p" and is denoted by TPM. (3) The disjoint union \Jp€MTpM is called the "tangent bundle of M " and is denoted by TM. (4) Let AT be a manifold and / : M —>• N a smooth map. The map Tpf : TPM -> T/(P)JV, [y]p >-»•[/ o 7]/(p) is called the "tangent of / at p " . Remarks. (1) The tangent of a map / at p is well-defined. Proof as an exercise.) (2) Despite the terminology it is not obvious that the tangent bundle is a vector bundle! (The proof will be given in the course of this section.) (3) Given a smooth map / : M —> N we have set-theoretically the "tangent of/" Tf:TM^
TN, (T/)([ 7 ] p ) = (T P /)([ 7 ] P ).
Denoting the projection TM —¥ M, [7]p i-> p by PTM (and analogously for N) we obviously have PTN ° Tf = / o PTM • Lemma. Let L,M,N smooth maps. (i)ThenT(fog)
=
be manifolds and f : M —> N and g : L -+ M be (Tf)o(Tg).
(ii) If M = TV" and / = I d M then Tf = TUM = ldTM(hi) If / is a diffeomorphism then Tf is bijective and (Tf)'1 Proof. Exercise.
=
T(f~1).
Lemma. Let V be open in Rm,x in V and w in W71. Let us set jw(t) x + tw. Then (i) | T V ] X = [7IL>"]X if and only if w' — w" in E m , and
• =
(ii) Xx '• IRm ->• TXV, w H> [yw]x is bijective. Remark. To be completely unambiguous one should note the curve t t-> x + tw by j W t X - In order to simplify the notation we will stick to 7™ especially if we consider the equivalence classes [yw]x, where no danger of confusion can arise. Proof of the lemma. Ad(i). Let us assume that [yw']x = bv'Jz- Then 0 = ^ 52
(lw'(t) —"fw"(t)) =
w' — w". Thus the first assertion is proven. Ad(ii). By (i) Xx is injective. Let now 7 be any curve at x and w = 7(0). It follows that ft
(7(f) - 7™(<)) = 0, i.e. [7]* = [yw]x.
•
Proposition. Let V be open in E m . Then (i) TXV is a vector space with a basis given by [jek]x for k = 1 , . . . , m. vector ek is of course the k-th unit vector in Rm.)
(The
(ii) TV carries naturally the structure of a trivial vector bundle over V via the bijection Xv = V x E ro -»• TV,Xv(x,w)
= Xx(w) = [jw]x,
fulfilling pxv °Xv - W\ • (Hi) IfV is open in Rnandf:V-> V is a smooth map then Tf :TV -> TV is a smooth vector bundle homomorphism over f. Identifying TV with V x E m and TV with V x E n we have {Tf)(x,w) = (Txf)(w) = (a:, (Dxf)(w)). (iv) If a map f as in (Hi) is a diffeomorphism then Tf is a vector bundle isomorphism. Proof. The first assertion follows immediately from the preceding lemma. Let xv '• V x Mm -*• TV be given by xv(x,w) = [yw]x then we have Xv ° PTV = Idy ° pn and thus TV carries canonically the structure of a smooth real vector bundle of rank m, proving (ii). Since (Tf)[jw]x = [f o iw]S(x) and ft (/ o 7W)(0) = {Dxf){w) we arrive at {Tf)[lw]x =
b(Dxf)(w)]f{xy
This implies with <2v = ( x v ) - 1 and *' v , = {xv)~l
that
*V' ° Tf ° (*v) _ 1 : V x l m -> V x H" is the map (x,w) H^ (f(x),(Dxf)(w)). Obviously this map is a smooth vector bundle homomorphism over the map / : V —> V. The last assertion follows from (iii) and the properties we have shown for Tf as a set-theoretic map. • R e m a r k . Let V be open in Rm, V open in W1 and / : V ->• V smooth. Denoting the canonical basis of E m by { e i , . . . , em} and the canonical basis of E™ by { e i , . . . , en} we have for each x in V a matrix Ax in Mat(n x m, E) n
such that (Txf)(ej)
= YHAx)ijU-
Writing / = t{h,...,fn)
i=l
53
with n scalar
functions /» it follows that (Ax)ij = §£:(x), i.e. Ax is the Jacobi matrix of / in x. (Fill in the details of the computation as an exercise.) Definition. Let M be a manifold and p : U -> V C E m a chart. We call the bijection Tip : TU —> TV the "natural bundle chart (associated to the chart (U,
•• (T<pa)(TUa0) -> (T
a
Since T(ipp o ip~ ) ^ diffeomorphism, it follows that the atlas T21 on TM is smooth, and furthermore that PTM is a smooth map. Since TVa is isomorphic as a vector bundle over Va to the trivial vector bundle Va x R m via the vector bundle isomorphism (Xa) _ 1 : = (Xv„)_1> f° r a given a, TUa inherits the structure of a vector bundle from TVa. (We denote the map Va x E m —> TVa,(x,w) i-> [yw]x again by xva-) Using the fact that the transition maps T(
o Xa o T<pa : TUa
are local trivializiations of TM over Ua •
^UaxRm •
Exercise. Show the last assertion of the proposition, i.e. the independence of this construction of the chosen admissible atlas given a fixed differentiable structure on M. Proposition. Let M and N be manifolds and f : M —>• N a smooth map. Then (i) Tf : TM ->• TN is a smooth vector bundle homomorphism over f, and 54
(it) if ipa = (a:?,... ,x%) : UQ ->• Va C Rm respectively ^/s = (j/f, • • •, vi) • U'a -> Vjj C K" are Zoca/ coordinates nearp in M respectively f{p) in N, then the map Tpf : TPM —>• Tf^N is given by the "Jacobi-matrix of f in the local coordinate systems", i.e. setting *(/l, • • • , In) = f = 1>p ° / o V'1 : cpa(Ua 0 f-^U'p)) m
anrf denoting the canonical basis of WL and E { c i , . . . , e n } we /lave T ^ o T / o T V " 1 = Tf
i=i
n
-> K"
again by { e i , . . . , e m } and and
i
where, as usual, [7^] is the tangent vector in x represented by 7 e i (t) = x+te and analogously for [nei] • Proof. Exercise.
•
Remark. Obviously the basic theorems of differential calculus in several variables can now be applied to smooth manifolds and yield for example the following Proposition. Let f : M —t N be a smooth map and q in N. If the rank of Tpf is either maximal for all p in f~l (q) or constant on a neighborhood of f_1(q) in M then f_1(q) is a (smooth) closed submanifold of M. Proof. Exercise using the rank theorem in local coordinates. • Bibliographical remarks. Beside the books mentioned at the end of Section 2.1 we would like to recommend [BroJa] and [Bro] - the latter being unfortunately available only in german language. 2.5
Vector fields on
manifolds
Definitions. Let M be a manifold. (1) The vector space of real-valued smooth functions on M is denoted by £(M) = C°°(M,R). (2) Let Uf and Ug be open neighborhoods of p in M and / 6 £(Uf),g e £(Ug). We say " / and g are equivalent in p", f ~ p g, if and only if there is an open neighborhood V of p such that V C Uf n Ug and f\y = g\v- The set of equivalence classes {/ :Uf -¥ M\p e Uf, Uf open in M, / smooth } / ^ p 55
is denoted by £P(M) and an equivalence class [/ : Uf -> M]p is denoted by / and called a "germ of a smooth function in p". ~P
Proposition. Let M be a manifold and p in M. Then (i) £P(M) is a commutative, associative, unital M.-algebra, and (ii) the vector space Der{£p{M)) := {Dp: £P(M)->R\DP
isR-linear
andDp( f • g ) = Dp( / )-g(p)+f(p)-Dp( n
t~*jp f^*p
^p
g )} f
~*jp
of its "(scalar-valued) derivations" is U.-linearly isomorphic to TPM. Proof. The first assertion follows from the observation that £ (U) is a commutative, associative, unital E-algebra for each open set U in M. Since £{Ua) —»• £(Va), f \-> f o ip-1 is a E-algebra isomorphism for each chart tpa : Ua ->• Va C E m , we can assume that M = E m and p = 0. The map ToK ™
J£> Der (£ 0 (E r o )), *o([7]o)(,/ 0 ) = | | Q ( / < " > ) ( * )
is well-defined and an injective E-vector space homomorphism. In order to show that So is surjective we use the "Fundamental lemma" below to develop a smooth function / near 0 in E m as follows
f(x) =
f(0)+J2fj(x)-xj, 3= 1
where fj is a smooth function near 0 fulfilling fj (0) = g£- (0). Given now £>o in Der (£ 0 (E m )) we set a,j = D0(XJ) e l , u ; = Y^f-i ajeji a n d Jw{t) = t • w as usual. Then we have ,
m
« -
m
Lemma ("Fundamental lemma"). Lei C be an open neighborhood of 0 in E m and / : U —• E a smooth function. Then there are smooth function fj defined near 0 (/or j = 1 , . . . , m) such that
fM = f-(O) and m
/(*) = /(0) +£/,-(*)• s; for all x in a ball Be (0) for e > 0 sufficiently small. 56
Proof. Let e > 0 such that the ball Be(0) is contained in U. For all x in Be(0) we have
Obviously fj(x) := J0 -§£-(t • x)dt is a smooth function on B£(0) and /,(0) =
Corollary.
n
'
&(0).
m
Let V be open in R TXV - S Der(£x(V)),
and xinV. Sx([7]x)(
Then the isomorphism
f ) = -£ ( / o 7 ) ( t ) ~x at o
maps [7efc]x ^o i/ie partial derivative g^- . Proof. Exercise.
D
Definition. Let M be a manifold. A "vector field (on M ) " is a smooth section of the tangent bundle TM of M. We denote the E-vector space of all vector fields by X(M) = TC^(M,TM). Proposition. Let V be open in E m and let Der(£(V)) = {D: £(V) -> £{V)\D is E-linear and D(f-g) =
D(f)g+f-D(g)}
be the space of all "derivations of £(V)". Then the maps Sx : TXV -> Derx(£x(V)) yield a R-vector space isomorphism S : X(V) —> Der(£(V)). Proof. Identifying TV via \v with V x Rm, a vector field X on V is given by a smooth map w : V -¥ Rm (X(x) = [7™(x)]x in TXV). The map 5X takes X{x) to Y^jLi wj(x)~si~ > where w — t{wi,... ,wm). Obviously 3
X
Wj ^ . is an element of Der(£(V)), since the functions vjj are smooth, and furthermore we observe that the map 5 : X(V) —¥ Der(£(V)), S{X) = Dx is injective. Let now D be in Der(£(V)), then applying D to germs of smooth functions yields a map Dx : £X(V) -> E , D x ( / ) = (D(f))(x), f**>X
which is an element of Devx(£x(V)). Let now the functions Wj be defined by vjj(x) = DX(XJ) (XJ being again the j - t h coordinate o n V C E m ) . Since DX(XJ) = D(XJ)(X), these functions are smooth and thus define a vector field 57
X on V. It follows that for / in £{V)
by applying the fundamental lemma to functions defined near x. Since D is uniquely determined by the family {£>x|a; € V} it follows that D — Dx = 8(x) i.e. S is an isomorphism. • Corollary. Let M be a manifold. Then there is a R-vector space isomorphism S : X(M) —*• Der{£{M)) such that in all charts (V,(p) of M 6 is given as in the preceding proposition. Proof. Since a vector field as well as a derivation of £ (M) are uniquely determined by their restrictions to the chart domains in an atlas of M, the map 5 is well-defined and injective. Given now D in Der (£(M)) we construct Xa on Ua for an atlas {{Ua,ipa)\ct £ A} as in the proof of the preceding lemma. It remains only to show that the Xa define a global vector field, i.e. a section of TM. This easily follows from the transformation properties of the partial derivatives defined on coordinate charts and the definition of the bundle charts Tipa of TM. (Details as exercise.) • Remark. The last proposition explains why one speaks of a "vector field on M, locally given by $Z£=i wfgfs-", where (pa = ( x f , . . . , x ^ ) are local coordinates and w? : Va = ipa(Ua) —> R are smooth functions. Definition. Let K be a field and g be a K-vector space with a map [, ] : g x g ->• g. We call (g, [,]) a "(K-)Lie algebra" if the following conditions are satisfied: (1) [,] is K-bilinear, (2) [,] is anti-symmetric, (3) [, ] fulfills the "Jacobi identity": [u, [v, w]] = [[it, v],w] + [v, [u, w]]
for all u,v,w
in g.
Example (and exercise). Let A be an associative K-algebra. Then [S,T] := S T -T • S defines a K-Lie algebra structure on A. Thus for a K-vector space E the algebra A — EndjK(E) is a K-Lie algebra. Remark. Originating in the above example, the name "commutator" for the map [, ] : g x g -+ g is frequently used, even if g is not constructed from 58
an associative algebra. Definition. Let (g, [,]) be a K-Lie algebra and I) C g a subset. We call f) a "Lie subalgebra (of g over K)" if f) is a K-subspace of g such that [£, rj\ is in f) for all £,?7 in f). Lemma. Let A be a K-algebra and Der(A) := {D £ EndK(A)\D(a • b) = D(a) • b + a • D(b) V a,b £ ^4}. Then Der(A) is a Lie subalgebra of (End^(A), [, ]), the space of K-linear vector space endomorphisms of A with the commutator [, ] coming from the associative composition of endomorphisms (as in the preceding example). Proof. Direct calculation (exercise).
•
Corollary 1. Let M be a manifold. Then the space Der{£{M)) of the associative W-algebra £{M) is a Lie subalgebra of (Endu(£(M)), [,]) . Proof. Follows directly from the preceding lemma. • Corollary 2. Let M be a manifold. Then the M.-vector space X(M) naturally carries the structure of a Lie algebra induced from Der(£(M)). Proof. Let X,Y be in X(M) and D = [DX,DY] in Der(£(M)). Since S : X(M) -> Der(£(M)) is an isomorphism there is a unique Z in 3C(M) such that Dz = 6{Z) = [DX,DY]. The bilinear map [X,Y]:=6-H[6(X),6(Y)}). obviously defines a Lie bracket. Definitions.
•
Let M be a manifold and X in X(M).
(1) Let / be a connected open neighborhood of 0 in R A smooth curve 7 : / —>• M is called a "(local) integral curve of X with initial condition p (in M)" if the following conditions are satisfied -K*) = ^ ( * ) = m 7 ) ( ^ | t ) = * ( 7 ( t ) )
V i G /
and
7(o) = P .
(2) Let ft be an open set i n l x M containing {0} x M . A smooth map ipx : 0. -»• M is called a "local flow of the vector field X " if for each p in M the curve t i->
Ip in M. containing 0 such that ipx (p) is defined on Ip and cannot be extended beyond Iv. The set fi := {(t,p) 6 E x M\t £ Ip} is open in E x M and the map
vfivfip)),
in short
"cpx = Idu
and >px+s = yx o yx."
Proof. The curves t t-> fx+s(p) and t i->
be as in the theorem, t in E and U be open Then ipx : U -> M is a diffeomorphism onto
Proof. Obviously t £ Ip for all p'mU. By Corollary 1 we have for p in U and s in [0, t] the equality
and V*t = ( y f y - 1 :¥>?(£/)->tf.
•
Definition. Let M be a manifold and X in X(M). The "support of X" is the closed subset of M defined by supp (X) = {p G M\X(p) 60
? 0}.
Corollary 3. Let M be a manifold and X a vector field on M with compact support. Then Q = R x M, i.e. the "flow of X is global". The map ipx : R x M ->• R is then a smooth action of the Lie group (R,+) on M. Proof. Let p be in M\supp (X) then ipx (p) = p, i.e. the unique maximal integral curve is denned for all t in R. Since fl is open in M there is for each p in M some ep > 0 and an open neighborhood Up of p in M such that (—ep, ep) x Up is in Q,. Covering the compact set K = supp (X) by {K n Up\p 6 K} we find €o > 0 such that (—eo, eo) x M is in fi. Let now e m a x > eo > 0 be the supremum of all e > 0 such that (—e, e) x M C fl. Assuming that e is not +oo there is a p in M such that Ip is at least unilaterally bounded. Without loss of generality we may assume that Ip C (—oo,3 • ema.K/2). Setting j(t) = ^ ^ ( V m O ' ) ) for t in [0,2e max ) we see that the assumption on Ip is wrong and therefore e m a x = +oo, i.e. the flow is global. The two preceding corollaries now imply that
N0 o n M " : •8 : N0 x M -> M,i?(n,p) = (jpo • • • o
Fi]-f piJ = — -r— with initial condition dqj
(q°,p°)
in
M
are equivalent to 7(<)=*ff(7(*)) for the vector field XH{q,p)
and
l(0) = (q°,p°)
= £?=1 ( ^ ^ - - | £ ^ - ) .
Proof. Exercise.
•
Remark. The vector field XH in the last proposition is called the "Hamiltonian vector field" associated to the Hamilton function H. Bibliographical remarks. As at the end of the previous section, plus a solid reference on ordinary differential equations as [Ar2]. 2.6
Differential forms and the Lie derivative
Definitions. Let M be a manifold and TM its tangent bundle. (1) The vector bundle (TM)* =: T*M p^f M is called the "cotangent bundle of M" and the vector space (PT*M)~1{X) = (TXM)* =: T*M the "cotangent space in x" (x G M). (2) A section of T*M is called a "(differential) one-form on M" and the vector space of all its sections is denoted by £1(M) = r c ~ ( M , T * M ) . (3) Analogously we define for k > 1 the bundles h.kT*M := Ak(TM)* and call their sections "(differential) fc-forms on M". The section spaces are denoted by £k(M) := Tc°°(M,AkT*M). (4) The space of smooth functions S(M) = C°°(M, R) is also called the space of "0-forms", £°(M) := B(M). Remark. If m = dimK M, then kkT*M 62
— {0} for all x in M and all k > m,
implying notably that there are no (non-trivial) fc-forms with k > d i m ^ M . Definition. Let V be open in Rm with coordinates (xi,... ,xm) and let a vector field X be identified with the associated derivation 8(X) = T,T=iaJ~^ ( a i e £(v))- T h e differential one-form dxj € £X{V) is defined by {dxj)p(-^-\ ) — 5j:k for all p in V and thus dxj(X) = a,j. Lemma. Let V be open in W1 and pinV. (i) T*V = (({dxi)p,..., k
(ii) K T;V
Then
(dar m )p})) R , and
= (({dii! A . . . A dxik)p\l
< »i < • • • < *fc < « } ) ) R . 0
Furthermore the section space X(V),£ (V) S(V) -modules with module basis as follows
and Sk{V)
for k > 1 are free
^3£(K) = (({^r,...,^}))£(v), (iv)£°(V) k
(v)£ (V)
=
(({l}))£{v),and
= (({dxhA---Adxik\l
< • • • < ik
Proof. Exercise.
• k
Definition. Let M be a manifold, p in M, v in TPM and r) in K T*M {k > 1). Then the "contraction of n and 77" is the (fc — l)-form V^TJ = ivt] € Ak~1T*M defined as follows: (ivv)(vi,...,vk-i)
'•= r](v,vi,...,vk-i)
for all
vi,...,vk-i
in
TpM.
Exercise. Check that ivr) is multilinear and alternating, i.e. in Ak~1Tp*M. Lemma. Let M be a manifold, X in X(M) contraction X-jn = ixr] defined by (ixv)p '•= ixvr)v is a smooth differential (k — \)-form, Proof. Exercise.
for all p
and rj in £k(M),
then the
in M
i.e. ixf] is in
£k~1(M). •
k
Remark. On the space A*(T*M) = 0 A (T*M)
one has the multiplication
fc>0
"A" of exterior algebras. This is easily globalized as follows. Lemma. Let M be a manifold and-q e £k(M),n
e £l(M).
(77 A fi)p :— r)p A pbp for all p in M 63
Then the formula
defines a smooth (fc + l)-form n A fi on M. Furthermore the space of sections of the vector bundle A*T*M := ® AkT*M is canonically isomorphic to ® Ek{M) and this space together fc>0 fc>0
with the wedge-product is a super-commutative, over the ring £{M). Proof. Exercise.
associative, unital algebra •
Remark. Given a smooth map / : M -> N between two manifolds M and N, and a point p in M we have the tangent of / in p, which we will denote also by (/») p , i.e.
(f.)P~Tpf:TvM-+TmN. Since Tpf is linear we have induced maps ®kTJ,.N AkT*{p)N -»• Akr;M.
-» ®kT*M
Definition. Let / : M -+ N be smooth and n in £k(N). f*n is defined by
Then the fc-form
(f*v)P(vi,.
..,vk)
and
:= Vf(P)((f*)p(vi), • • •, (/*)P(ufc)) V p e Af, Vwi,...,w fe e T p M
is called the "pull-back of 77 by / " . Exercise. Check that f*n is a smooth fc-form on M. Remark. It is important to note that the anologous construction on a vector field does not always work, i.e. {f*X)p = (f*)PXp G Tf^N for p in M and X in X(M), but this is in general only a section of the bundle f*(TN) -»• M and does not necessarily define a vector field on N. Example. Let M = N = R and f(x) = x2. Then ( T p / ) ( a ( a ; ) ^ | p ) G Tf{p)R defines a vector field on R if and only if a(x) = 0 for all x in E, i.e. if and only if the vector field X = a(x)-j^ is everywhere zero. Lemma. Let L, M, N be manifolds and g : L ->• M and f : M -»• N be smooth. Furthermore let n G £k (N) and \x in £l (TV). Then
WrfoA/i) = (/*i7)A(/», (U) f*ip = xp o / for ip in
£°(N),
(Hi) f* : £*(N) -> £*(M) is anR-linear (over R) fulfilling f*W • V) = ( / » • (f*v) 64
even homomorphism of superalgebras for all vb in £°(N) ,
(iv)(fog)*=g*of*:£*(N)^£*(L). Proof. Exercise.
•
Definition. Let M be a manifold, X a vector field on M and
for all p in M.
Proposition. Lei M, X, y>* be as in the preceding definition and let n, 7/ be in £k(M) and p, in £l(M) with k,l > 1. TTien ('ij Cxr] is in £k(M), (ii) £x(A • n) = A • £x?7 /or A in R, (Hi) Cx(v + V') = £xr} +
Cxn',
(iv) CX(V A ju) = (£*•»?) A A* + r? A ( £ X M ) -
Proo/. Ad(i). Since the local flow ypx is a smooth map from its domain of definition ftClxMtoMit easy to deduce that CxV exists in every point p of M and that (Cxrl)p i s m Ak(TpM)*, i.e. CxV is a section of AkT*M. Since differentiability of a section is a local condition we may assume that M = V is an open subset of JRm. The fact that £k{V) = {({dxh A • • • A xik\ix < • • • < ik}))s(V) implies that CxV is a smooth section if and only if (Cxw)(Xi,... ,Xk) is a smooth function for all X\,..., Xk in X(V). For p near a fixed point po in V we have ((Cxv)(Xu • • •, Xk))(p) = ^\0((
- fL (vw((^)'),(^(P)).-((rf).),(^(P)))) == J \ F^ t 0 ~ dt 0 and for fixed X,Xi,... ,Xk,n the function F(t,p) is defined for small t and p near p 0 and is smooth in both variables. Thus (Cxy){Xi,..., Xk) is a smooth function in p near p0 for all po in V, and the first assertion is proven. Assertions (ii) and (iii) follow directly from the E-linearity of the maps The last assertion can be derived from the formula (
65
one real variable.
D
Lemma. Let M be a manifold, X in 3t(M) and f in £°(M) = £{M). Cxf
= X{f),
i.e.
Cxf =
Then
Dx(f),
where Dx is the derivation of £{M) canonically associated to X. If furthermore 77 is in £k{M) with k > 1 we have Cx{f • v) — (£xf)'
V+
/ " (CxT,).
Proof. Let p be in M, then
(CxfKp) - i\MTf)r
= || 0 (») = !|„/
i.e. (Cxf)(p) is the derivative of / in the direction ^\0fx(p) = X(p). Since this is the very definition of (Dx(f))(p) the first part is proven. The second part follows as sketched in the proof of Assertion (iv) of the preceding proposition. D Definition. Let M be a manifold and X and Y in X(M). The "Lie derivative of Y with respect to X " is defined as follows ( - C x n ( p ) ~ | | 0 { ( ( ^ ) * ) v x ( p ) ( ( v f ( p ) ) ) } for all p in M. Remark. Since the flow tpx is smooth CxY is a well-defined section of TM. Its smoothness will follow a fortiori from the next proposition. Definition. Let M' and M" be manifolds and F : M' -¥ M" a diffeomorphism. For a vector field Z on M' we define its "push-forward" as follows (F*Z)q := (TF-i{q)F)(ZF-Hq))
= ( F , ) F - I ( , ) ( ^ F - I ( , ) ) for all 9 in M " .
Remark. Considering a fixed point 5 = ^(p) the above formula is just the tangent of the map F. In the case here considered when F is a diffeomorphism, F*Z is easily seen to be a smooth section of TM", i.e. F*Z is a vector field on M". Lemma. Let M',M",F and Z be as in the preceding definition, and let ipf be the flow of Z on M'. Then the flow of F*Z on M" is given by f*z = F o (p? o F-1. Furthermore, iff is in £{M") then ((F,Z)(f))(q) = (Z(f o F))(F" 1 (9)) for all q in M". ((F*Z)(f) denotes of course again DF*z{f)). Proof. Obviously we have F o ip% o F - 1 = H M » • Furthermore
jt\o(F°
so that the first assertion is proven. Let now / be in £(M") then ((F.Z)(/))(g) = | | o ( / o tf-z)(q) = ±\Q(f
dl
(foF){tf(F-1(q)))=ZF-i{q)(foF)
F-l)(q)
oFotfo =
(Z(foF))(F-1(q)).
n Proposition.
Let M be a manifold and X and Y in 3E(M). Then CXY
= [X,Y}.
Proof. Since X(M) is canonically isomorphic to the derivations of £(M) it is enough to show that (CxY)(f) = ([X,Y]){f) for all / in £{M). Let / be in £{M) and p a point in M. There is a e = e(p) > 0 and an open neighborhood U of p in M such that ip* | v : U —> Wt =
(CXY)P( f ) = -£ {((y>* ).) x , , ( > > f (P)))}( / ) = dt . { ( ( V - t ) ^ ) B ( / ) } . where we consider for every t the diffeomorphism F =
(£xY)p(n
=
jt\Q{(Y(fo^t))(tf(p))}.
In order to calculate this derivative with respect to t we introduce the smooth maps A : Ie := (-e,e) -» Je x Ie,A(t)
= (A1(t),A2{t))
:= (M)
and
V : Je x It -+ R, 0 here.) It follows that
icxr)r = > . * > „ , . (g(o,o,) • (^
= (X(Y(f))(p))
• 1 - (Y(X(f))(p))
• 1=
([X,Y](f))(p).
a Proposition. Let M be a manifold, X a vector field on M and r\ a k-form on M (with k>\). Then (CxvKXi,
...,Xk)=
CxiviXu...,
Xk))
-
k
- 2_^ v(Xi > • • • > Xj-i, £xXj,
Xj+i,...,
Xk)
3=1
for all vector fields Xi,..., Xk on M . Proof. For all (t,p) in fi, the domain of the local flow of X, we have
((tf)Hv(Xi,...,x2)))(p) = {((tpxty((tfyr1))(x1,...,xk)}(
(£x(ri(X1,...,Xk)))(p)
=
ft\o((
k
= {(£xr))(X1,...,Xk))(p)
+
J2v(Xi,...,Xj-1,£xXj,Xj+1,...,Xk)(p). 3=1
a Bibliographical remarks. Books on manifolds as quoted at the end of 2.1 plus [GEL]. 2.7
The exterior cohomology
Example. Yfj=i^j^~
derivative
of differential
forms
and de
Rham
Let fi be an open subset of R 3 . A vector field, given as K = with Kj in £(fi), is - especially in physics - often described
by the smooth vector-valued function K — t(Ki,K2,K3) : Q, ->• E 3 . The canonical Riemannian structure and orientation allow the definition of the usual operations of vector calculus (for if> in £(Cl) the expression djip denotes 68
here the partial derivative •*£. with j in {1,2,3}): * the "gradient" of a smooth real-valued function / on fi is defined as 3
grad/ = V/ = 2 > i / ) o Z 7 1 = 1
•?
orb y v/ = i(a1/,a2/,a3/), * the "curl" of a vector field K is given as r\
curlK = \7xK=
(d2K3-d3K2)
r\
— + (d3K1-d1K3)-OX\
r\
+ (d1K2-d2K1) OX2
— OX3
or as V x K, where V = '(gf-, gf-, gf-) and " x " is the vector product, * the "divergence" of a vector field L is 3
div L =
Y^djLj
or as V • L, where "•" is the scalar product. It is easily checked that div (curl K) = 0 and curl(grad/) = 0. Thus in order to decide, e.g., if a given force field K is conservative, i.e., a gradient field, we immediately find the necessary condition cui\(K) = 0. In order to translate the "exact sequence" (i.e., div o curl = 0 and curl o grad = 0)
s°(ii) ^4 x(n) ^4 i(n) -^ e°(n) into the language of differential forms we define the following £(f2)-module isomorphisms 3
r 2 : 3£(fi) - • £ 2 (ft),
3
^ ( ^ L , — )
= Lidx2 A 0(23 — L2dx\ A
Setting do := T\ O grad, d\ := T2 o curl o (n) 1, d3 := T3 o div o (r 2 ) * we get an exact sequence (dj+i o d^ = 0) as follows:
£°(n) A fx(n) A £2(0) A £3(n). Over first objective is to generalize this sequence to an arbitrary manifold. Definition. Let M be a manifold and r\ a differential fc-form on M (k > 0). The "exterior derivative d(r]) of 77" is denned by the following formula: fc+i
(d(r,)){X1,...,Xk+l):=^2(-l)^+1XMXi,...,Xj,...,Xk+1))
+ ^ ( - l ^ ' r / a ^ . ^ ] , ^ . • • • ,Xi,...
,Xjt..
.,Xk+i)
i<j
for all Xi,... ,Xk+i in X(M). (A hat "A" on a vector field means that the corresponding vector field is omitted.) Remarks. (1) The expression 0^(77) defines a smooth (k + l)-form on M. (Proof as an exercise.) (2) For / in £(M) = £°(M) and X in 3£(M) we have (d(f))(X) = X{f) = Dx(f), and for a in £l(M) and XUX2 in X(M) we have (d(a))(X1,X2)
= X1(a(X2))
- X2(a(X1))
- a([X 1 ; X 2 ]).
Let us recall that a fc-form r) on an open set V C W1 can be described as a finite sum J2 fjdxj, where J = (jfi,..., jk) is a multi-index of "length fc" (|J\ = k), ji < • • • < jk, and dxj = dx^ A . . . A dxjk and fj in £°(V). Lemma. Let f, fj be in £°(V) for V an open set in E m and ?? = X)
fjdxj
\j\=k
a differential k-form with k > 1. Then m
df (i) d(f) = "£, g^dxj, and
(ii)d(r))=
£
(d(fj))Adxj.
\J\=k
Proof. Ad (i). Since df is a l-form there are function gj on V such that d(f) = Y^jLi 9jdxj. We calculate
»-(i*«*)<;£;)-<«/»(£> = <£) - g . 70
Ad(ii). Since the operator d : £k(M) -» £k+1(M) is obviously E-linear we may assume that r/ = / d x i A . . . A dxk- Since d(r]) is a (A; + l)-form we know that 771
d{rf)=
^2 gidxi A . . . A dxk A dxt + ^ gj>,dxji, l=k+l |J'|=fc+l
where J ' = (j[,... ,j'k+1) {j'i.---.i*+i}- We have
^
with j[
' 9a;,-'
< ••• < j ' k + 1 and {l,...,fc}
£
)
9^,^90;,;'••'9xJ,'---'9,UiJ;
+ J2(-Dr+Sv(
9a:j'
9 9a;^
9 ' 9a;j' j' '• '• "• 'j adxj
' - - - >
9_
a9 a;,' '
a
\
' 9a;,'
/
•
,
-
'
= 0 since by [^fr, gf-] = 0 the second sum vanishes and since the first sum is zero by J'\{j'r} £ { 1 , . . . , k} for all r. Furthermore ,JI
d
wi 1
+
d
d
\
!\i+i
d
I i
d
d
dxi''
9a; i ' " "' dxk
1 H,H d f , d
<- >
k
V^
9
+
d d " 9a;fc ' 9a;;
9
£('<£-----£>) B « > H l £ - £ ] •£••••)•
The third sum vanishes termwise as above since the "coordinate vector fields commute", i.e. gf-;, g§H = 0 for all r and s. The first sum also vanishes termwise since {dxi A.. .Adxk)(-£^,..., Thus we find
g f - , . . . , ^ , ^ ) is zero for all / > fc.
9, = (_„«« (/) 9
and hence d{rf) = ^2 (-l)k+2—^-dxxA...Adxk OXi /=fc+l
Adxi = ^ I I I l=k+\ 71
-^-dxtAdxiA...Adxk OX[
= ] P -=—dxi A dx\ A . . . A dxk = (d(/)) A dsi A . . . A dxk1=1
D Lemma. Let V be open in R
m
and j in { 1 , . . . ,m}.
Then
d(xj) = dxj.
Proof. The right hand side is the 1-form uniquely determined by (dxj)[~) — 6jti for / = 1 , . . . , m. Let us compare this to the exterior derivative of the j-th coordinate function
a R e m a r k a n d definition. Since the preceding lemma shows that there is no difference between d(xj) and dxj from now we will simplify the notation and will not distinguish between them, i.e. dr]:=d(r])
for all n in
£*{M).
Proposition. Let M be a manifold of dimension m, n a k-form and /J, a I-form on M. Then (i) d(X • rj) = A • dn for A in R, (ii) d(n + /j,) = dn + dfi if k = I, (Hi) d(r] A fi) — (drj) A fj, + {-l)krj A dfi, (iv) dn = 0 if k > m, (v) d{dn) = 0. Proof. The first two assertions follow directly from the definition and the fourth one from the fact that there are no non-zero n-forms for n > m on a m-dimensional manifold. In order to show the remaining assertions we observe first that d is a "local operator", i.e. to calculate {dn)p it is enough to consider rj on an open neighborhood of p. We can therefore assume for the rest of the proof that M = V is an open subset of Rm. Furthermore we may assume that n = f dx^ A . . . A dxik and [i — g dxj1 A . . . A dxj,. 72
It follows d(rj A /J,) = d(fg dxix A . . . A dxtk A dxj1 A . . . A dxjt) = d{fg) A dxh A . . . A dxjt = (g{df) + f(dg)) A dxh A . . . A ctej, = (df A da;^ A . . . A do;^) A (g dxjx A . . . A dxj,) + + (-l)k(f dx^ A . . . A dxik) A (d# A dz^ A . . . A dxj,) = (dri)An+{-l)kriA{dn). Ad(iv). Let / be in £°(M) and XUX2 (d(df))(XuX2)
in £ ( M ) . Then
= X!((d/)(X 2 )) - X 2 ((d/)(X!)) (df)([X1,X2}) = X1(X2(f)) - X2(X1(f)) {XuX2)(f) = 0.
By localizing near a point p in M we can again assume that M = V, open in E m , and that rj = f dx^ A . . . A da;ifc. It follows that d{drj)) = d((df) A dx^ A . . . A da;^) = (d(df)) A dxij A . . . A da;ifc + (d/) A d(dx^ A . . . A d^,.) = 0 2
since d (/) = 0 and d(dxi) = 0.
•
Example (and exercise). Check that in the example at the beginning of this section the operators dj are the operator d in the different degrees (of differential forms.) Translate the assertions of the preceding proposition to formulas in vector analysis. Remark. Since £*{M) = ®k>Q£k(M) the operators d : £k{M) -*• £k+1(M) can be put together to build an operator d : £*(M) -> £*{M). The assertions (i) - (iii) can then be rephrased by saying that d is an odd super-derivation of the real super-algebra £ *(M). Proposition. Let M and N be manifolds, and dM o-nd djv the respective exterior derivatives. Let furthermore F : M —> AT be a smooth map then dM°F*
=F*odN.
Proof. Going to local coordinates we may assume that M = V is open in E m , jV = W is open in W1, and that the map F : M -> N is given by n scalar functions F = t{F1,..., Fn). Given a fc-form n on W we can assume that rj = g dy\ A . . . A dy^, where g is in £° (W) and yi,..., yk are the first k coordinate functions on W C E n . 73
We calculate (setting intermediately OIM = d^ = d for convenience): (dM o F*)(n) = dM{F*(gdyi A . . . A dyk)) = dM({g o F)dFx A . . . A dFk) = d{goF)AdFlA...A dFk
= ~[~[
d
j'=i
Vi
Y,Y,{p-°F)^dx lAdF1A...AdFk dx i
yj
= F*(dg) AdF1A...AdFk
= (F* o dN)(r)).
a Corollary. Let M be a manifold and X a vector field on M. following identity of operators holds on £* (M).
Then the
Cx ° d — do CxProof. Since {Cx1*}) — ^ preceding proposition.
(((P?)*v) the assertion follows easily from the •
Proposition ("Cartan's magic formula"). Let M be a manifold and X a vector field on M. Then the following identity of operators holds on £*{M): Cx = d o ix + ix ° d. Proof. The formula holds trivially for 0-forms, i.e. functions on M. Let now k > 1 and 77 a fc-form on M. Let furthermore Xi,..., Xk be in 3£(M) and set X0 = X. Then ((doix+ixod)(r,))(X1,...Xk)
=
= (d(r,(X,.. . ) ) ) ( X i , . . . , Xk) + ((drj)(X0,. • . ) ) ( * i , •
..,Xk)
k
= Y/(-i)i+1Mv(Xo,x1,...,xi,...,xk)) i=l
+ 5 3 (-iy+ir,(X0,[Xi,Xj],Xi,...,Xl,...,XJ,...,Xk)
+
k (_1)0+i+iXo(7?(Xi)
> Xk))
+
^(-ly^XiMXo, t=l
+^(-l^dXo.Ijili,
...,Xj,...,Xk)
0<j
74
...,Xi,...,xk))
(-iy+jv([Xi,x:],x0,...,xi,...,xj,...,xk)
+ Y^ 0
= X0(V(XU...,
Xk)) - J2v(Xi,-
• • ,Xi-i, [X0, Xj],Xj+i,.
..,Xk)
0<j
=
(Cxr1)(Xu...,Xk).
a Remark. Beside this "algebraic" proof of Cartan's formula there is also an "analytic" proof using the flow of the vector field X. We will not present it here but we will use the latter approach to get a stronger result for "timedependent vector fields" as a preparation for Moser's method to prove Darboux's theorem. Corollary. Let M be a manifold, X,Y in £ ( M ) and f in £°{M). rj in £k(M) one has (i)£fXr] = f-CxV + (df)A(i(X)r1). Furthermore the following identities hold on £*(M) (ii) [ £ x , i y ] = i[x,Y], (Hi) [CX,CY]
Proof.
Then for
= £{x,Y]-
Ad(i).
Cfxr, = d(V(fX,...))
+ (dv)(f -X,...)=d(f-
V(X,
= df A (V(X, . . . ) ) + / • d(V(X, ...)) + / •
...)) + / •
(dv)(X,...)
(dv)(X,...)
= (df) A (i(X)r,) + f-(doix+ix°d)(r1) = f-Cx'n + (df) A (i(X)r,). The second assertion follows from the explicit formula for CzV, Z € J £ ( M ) , p a differential form shown at. the end of Section 2.6. The third formula is then derived from the second and Cartan's magic formula. • Remark. If M is a manifold of dimension m then we have the sequence
£°(M) A£\M)A---A
£k(M) 4 £k+1{M) 4 • • • 4 £m(M) 4 0,
which is a "complex" in the sense that d o d = 0. Definition. Let M be a manifold, then for k > 1 the "fc-th de Rhamcohomology group of M" is defined as follows
„t „ „
ter(d:£k(M)-*£k+1(M))
„fc/,,™v 75
where ker and im denote the kernel and the image of the corresponding Rlinear maps. For k = 0 one defines H°dR(M) := H°R(M,R)
:= ker(d : £°(M) -+
E\M)).
Remarks. (1) A differential form rj such that drj — 0 is called "closed", and if there is a form /J, fulfilling d[i = rj we call rj "exact". The Rham cohomology groups measure therefore "how many closed forms on M are not exact." (2) Though traditionally called "cohomology groups" the spaces HdR(M) in fact R-vector spaces. (3) If the dimension of M is less or equal than m, then HdR(M) k>m+l.
are
= {0} for all
(4) A function / in £°(M) is in the kernel of d if only if / is locally constant. (Proof as an exercise.) Thus for a connected manifold M we have that
H°dR(M) s R. (5) Tensorizing the bundles AkT*M with the trivial complex bundle M x C ->• C we get smooth complex vector bundles A f e T*M® K C —> M. Its sections r c ° ° ( M , A f c T * M 0 R C ) are denoted by ££(M) and called "complex(-valued) differential fc-forms on M". The exterior derivative d can be extended to the spaces ££(M) by complex-linearity and we can, in complete analogy to the preceding definition, define HdR(M,
^ {curl-free force fields on ft} — { conS ervative force fields on ft}
Let now ft := R 3 \{'(zi,£2,0:3) £ W?\xi = 0 and x2 = 0} and K A direct calculation shows that V x K — 0, i.e. K is curl-free. 76
=
Assuming now that K = VV for a function V : Cl -> E, then the work along a path should depend only on the endpoints, i.e. in physicists' language
L
K-ds = V(b) - V(a)
>c
if C is a path from a to b. Defining now paths C£ for e = ± 1 by maps 7 e : [0,1] -» ft as follows
(
cos(7rf) e • sin(Trt)
we find 7 e (0) = I 0 ] ,7 e (l) = I 0 I and fc
W
W
Kds = 7re, i.e. K cannot be
a conservative field. Otherwise stated, the differential 1-form ax = ]Ci=i Kjdxj defines a non-zero class in H\R(Sl). In fact, one has H\R(p) = (([ai<-]))RIn order to prove the aforementioned fact as well as the "Poincare lemma" below we will first investigate the relation between homotopies and cohomology in general. Definitions. Let M and N be manifolds and A a closed submanifold of M. (1) For a smooth map F : [0,1] x M -> N we set Ft(p) := F(t,p). We call F a "(smooth) homotopy between the maps F0 and F\ (from M to TV)." (2) Let / : M —• TV and g : TV —»• M be smooth maps such that g o / is homotopic to I d ^ and fog homotopic to Idjv- We then say that "M and TV have the same homotopy type (in the C°°-sense)". (3) If M has the same homotopy type as a point we call M "contractible". (4) If i : A —> M is the inclusion map and r : M —> A is a smooth map that restricts to the identity on A, i.e. r o j = Id^, we call r a "retraction of M to A". If furthermore i o r : M -> M is homotopic to the identity of M we call r a "deformation retraction of M onto A." R e m a r k . A map F : [0,1] x M -» TV is smooth if F t is smooth for all £ and if £ H-> F(t,p) is smooth for all p in M, where smoothness in the boundary points is defined by taking appropriate (one-sided) differential quotients. Proposition. Let F : [0,1] x M -> TV be a smooth map. Then there exists a R-linear operator H : E*(N) ->• £*{M), lowering the degree of differential forms by one, that satisfies doH + Hod 77
=
F*-F*.
Remark. Such an operator H is called a "homotopy operator". Proof of the proposition. Let /x be in £l (N), p in M and vi,...,vi-i it : M -± [0,1] x M, it(p) = (t,p) and define (.H»p(i>i,...,t>;_i) := /
in TPM. We set for tin [0,1] :
[(F*lj:)(t,p)(g^
,(it)*Pvi,...,(it)*pVi-i)]dt.
(Since [0,1] x M is a product the injections it are obvious and we will omit the maps (it)* in the rest of the proof.) In order to show the asserted formula it is enough to consider an open neighborhood of a given point p in M, i.e. we can assume that M = V is open in K m . Thus H\x is determined by its values on the "coordinate fields": (#»x(
d dxh
d
;'
)
' 8x 1
= J [(F*u)it,x){
at
(t,x) dxix
dxi
(t,x)
(t,x)
)]dt,
where x is in V and i\ < • • • < ii-\. This local description of H[i immediately shows that HJJ, is smooth in a; as a "parameter-depending" integral. Let us now calculate the terms in the left hand side of the assertion, applied to a fixed fc-form rj on N (and for i\ < • • • < ik)'-
(d(Hrl))x(^-,...,^r)
=
dx
dx^
= B-v + '(^L(«'dxita
a dxi,
3= 1
d
Y,(-l)r+s(Hr])x(
•
)
)
+
d
dxi ' dxi^
•
•
•
)
= D-')-'/[^L((^)(|^ i
d dx*, ))
since the coordinate fields commute. On the other hand d
d
(^"•(^r--s:)=jft(J"*"-(5-^r-- £>]* rl
a a =1 -[(*(*••,)),(!, dt' dx^
a '
' dxik 78
)] dt
dt,
^(-'^/'[^{^•"(I-K:--^--^:)}]* since all commutator terms involve either [^,gf^] or [gf^, gf;] and thus vanish. We therefore arrive at
w^)+H(« ) .(^,....^-)-jr , [|{(F-,),(^....,^-)}] 1 i ( M^-F.-,),^
^->. D
Definition. Let M and iV be manifolds and / : M -> N be a smooth map. The "induced map on de Rham cohomology" is defined as follows r(M)~[f*v]
for
all classes
[n] in H*dR(N).
Remark. in HdR(M)
Given a class c in H%R(N), one easily verifies that the class f*c is independent of its representative n in £*(N).
Lemma.
Let M and N be manifolds and f : M —> N a smooth map. Then
(i) f* •• H*dR(N) -»• H*dR(M) is R-linear, (ii) HdR(M) is a super-commutative, tiplication given by
associative real super-algebra with mul-
[rj\A[fi\ := D?A/z], (Hi) f* is a algebra-homomorphism, i.e. [f*TJ\A[f*/jb] = f*\r) A fj] . (iv) If L is a further manifold and g : L —>• M is smooth then (fog)* = g* o / * on de Rham cohomology. (v) If M = N and f = IdM then f* = (IdM)* = IdH'dR{M) • Proof. Exercise.
•
We collect now several important applications of the last proposition (and the last lemma.) 79
Corollary 1. Let M and N be manifolds and f,g:M—>N maps that are homotopic. Then r=g*:
be two smooth
H*dR(N) -+ H*dR(M).
Proof. Let F : [0,1] x M -t N be a smooth map such that F0 = / and JF\ = g. For a class c in HdR{N) represented by a closed form 77 in £k(N) we find
+ #*?]
= [FSv] + [dff^] = [F^rj] = [f*r,] = f*[v] = f*c.
a Corollary 2. Let M and N be manifolds having the same homotopy type. Then HdR(M) and HdR(N) are isomorphic as R-algebras. Proof. Let / : M —»• N and g : TV —> M be smooth maps such that g o / is homotopic to M M and fog homotopic to Idjv- Then, by Corollary 1 / * °9* = (9° / ) * = ( M M ) * = l&H*dR(M) and 5*o/* = (/o5)* = (Idwr=Idff.Jl(Ar) and thus / * and 5* are mutually inverse isomorphism between HdR{M) H*dR{N).
and •
Corollary 3. Lei M be a contractible manifold. Then HdR(M) = K and ffdfl(M) = {0} for k>0. Proof. Since the cohomology of the zero-dimensional connected manifold consisting of one point is isomorphic to K in degree zero and trivial in all other degrees the assertion follows from Corollary 2. • Corollary 4. Let M be a ball 1BR(0) with radius 0 < R < 00 in Rm, then H°dR(M) S E and HkdR{M) = {0} for k>0. Proof. The map F : [0,1] x Bfl(0) -» B f i (0),F(*,a;) = (1 - *) • a; is a smooth homotopy from M to the origin 0 in IBfj(O) such that Ft(0) = 0 for all t. It follows easily that Bfl(0) is contractible and thus by Corollary 3 the de Rham cohomology of M — JBR(Q) is as asserted. D Corollary 5. Let M be a manifold and A a closed submanifold such there exists a deformation retraction r : M -» A from M onto A. Then r* : 80
^diii-A) —> HdR(M) is a R-algebra isomorphismProof. Follows directly from Corollary 2.
D
Corollary 6. Let M and N be manifolds and
H*dR{A).
Bibliographical remarks. As at the end of the last section plus the parts on de Rham cohomology in [BT], [J] and [KLJ. 2.8
Integration
of differential
forms
on
manifolds
Definitions. Let M be a connected manifold of dimension m. (1) An "orientation form on M" is an element ft in £m(M) such that ftp ^ 0 for all p in M, i.e. ((ft p )) R = A m (T p M)* for all p in M. (2) Let ft' and ft" be two orientation forms on M. Then "ft' and ft" are equivalent (as orientation forms)" if there is a smooth function / : M —>• K such that f(p) > 0 for all p in M and ft" — f • ft'. We write then Ls' J orientation ~ I " J orientation (3) An "orientation on M" is the class [ft]orientation °^ a n orientation form ft on M. (4) The manifold M is called "orientable" if there exists an orientation form on M. (5) An atlas 21 = {(Ua,ipa)\a € ^4} of an oriented manifold (M, [ft]orientation) *s c a u e d "positively oriented (with respect to the orientation [ft] 0 rientation)" i f f o r a11 a i n A (^1),Q = 5 a ( i < A . . . A < with a smooth strictly positive function ga : Va = <pa(Ua) —>• K. (The coordinates on Va C Mm are denoted by (xf,..., x%).) 81
Remarks. (1) An orientation form (respectively an orientation) on a m-dimensional manifold should be intuitively seen as a "smoothy varying orientation form (respectively an orientation) on TPM for all p in M". (2) A m-dimensional connected manifold M is orientable if and only if the smooth real line bundle AmT*M -)• M is trivializable. (3) An orientable connected manifold has exactly two orientations. (4) The notion of a positively oriented atlas is well-defined. (5) A manifold M is called orientable if each connected component of M is orientable. (6) Let M and N be manifolds and O M and fiyv orientation forms on M respectively N. A smooth map F : M —> N is called "orientation-preserving" if [F "AfJorientation = [^ M JorientationExamples. (1) The form dx\ A . . . A dxm on E m is called the "canonical orientation form on E m " . (2) Let m > 2 and let / : E m -> E, f(x) = f (||z|| 2 - 1) then { i 6 l m | f(x) = 0} = Sm and TpSm = {v £ TpRm \ (df)p{v) = 0} for all p in Sm-1. Furthermore U := Bi (0) = {x eRm\ f(x) < 0} and dU = {x G E m | f(x) = 0} = grn-i rp^g "( o u t W ard pointing) normal field on g m _ 1 " is defined as
Note that N is not a vector field on S1™-1 but the restriction of a vector field defined on an open neighborhood of 5 m _ 1 in E m to 5 m _ 1 , i.e. a section of (TK m )| s „.- 1 More concretely we have the formula N(p) = YlT=i PJ af~l • "-,et furthermore A := dx\ A . . . A dxm be the canonical orientation on E m and let us set H p := z;v(p)Ap = (*ivA)pL £,„_!• Then it is a non-vanishing (m—l)-form on 5 m _ 1 , in explicit terms m
Op = ( /__](—lyxjdxi A . . . A dxj A . . . A dxm)
.
The orientation [^] 0 rientation o n ^m~1 1S called the "canonical induced boundary orientation (with respect to 5 m _ 1 = dU and the orientation 82
lAJorientation) • (3) Let again m > 2 and P m _i(K) = P(R m ). Denoting the canonical projection E m \ { 0 } —> P(R m ),a; i-> [a;] by fr, we have a smooth surjective map of constant rank m — 1 denned by •K : 5 ™ - 1 ->P m _i(lR),
7r(z) := n(x) = [x].
For any (m - l)-form 0 on P m _ 1 (M) we have the pullback n*Q in ^ m - i ( 5 m - i ) g i n c e p m _ 1 (E) e* Sm/~, where i ~ y if and only if either y = x or y = -x =: T(X), a differential form r\ on 5 m _ 1 is the pullback of a form on P m _i(R) if and only if T*r) = r\. Obviously T*Q = ( - l ) m n for Q the orientation form on Sm~1 constructed in Example (2). It follows that P m _i(IR) is orientable for m an even integer with m > 2. Let now be m > 2 and odd. Assuming that P m _i(E) is orientable, the pull-back of an orientation form 0 is a multiple of 0 : 7T*0 = g • 0, with g a smooth function on 5 T O _ 1 . Since IT has constant rank equal to m — 1, n*Q is an orientation form on 5 m _ 1 and, after possibly changing 0 to (—0), we can assume that g > 0 on 5 m _ 1 since Cl is an orientation form. The identity g-n
= r*(g • il) = T*(g) • r*(ft) = (g °
T)(-SI)
implies that g is a strictly positive smooth function on 5 m _ 1 fulfilling g(—x) — —g{x) for all x in 5 m _ 1 . Since there are no such functions g it follows that P2n(K) is not orientable for n > 1. Definitions. M.
Let M be a manifold and 21 = {(Ua,(pa) \a £ A} a,n atlas of
(1) The atlas 21 is called "locally finite" if for all p in M the number of a in A such that p is in Ua is finite. (2) Let 21 be now a locally finite atlas. A collection of smooth functions {Xa | a G A} on M is called a "partion of unity (subordinate to 21)" if the following conditions are satisfied: (I) the values of \a are in [0,1] for all a in A, (II) The closed set suppxa = {x £ M\xa(x)
^ 0} is contained in Ua,
(III) For all p in M one has E a € / i X<*(p) = 1(Note that the sum in (III) is finite since 21 is locally finite.) Proposition. Let M be a manifold. Then there exists a locally finite atlas and for each locally finite atlas there exists a partition of unity. Remark.
We will not give a proof of the preceding proposition, but we 83
stress at this point that we included the conditions of Hausdorff and secondcountability in our definition of a manifold. These conditions assure the existence of partitions of unity. (See textbooks on manifolds as [AMR] for details.) As a first application we note the following Proposition. equivalent:
Let M be a m-dimensional
manifold. Then the following are
(i) M is orientable. (ii) M has a locally finite atlas that is positively oriented with respect to an orientation form fl on M. (Hi) M has a locally finite atlas {(Ua,<pa) \a £ A} such that the Jacobi determinants det ({faff)*) ire everywhere positive for all a,f3 in A. Proof. Let M be oriented by the orientation form ft and let 2t = {(Ua,ipa) \ct e ^4} be any locally finite atlas. We define tpa : Ua ->• Rm as follows: if (^~ 1 )*fi = gadxa A . . . A dx!^ and ga is everywhere positive, we set <pa -.= <pa. If ga is everywhere negative then we define tpa := v£ o (pa, where ^f(xa ,xa,... ,x^) := (—x^x",...^^). It follows that 21 = {(Ua, tpa) | c\ 6 A} is a positively oriented atlas (with respect to ft). Given any atlas 21 = {(Ua,(fia) \a € A} and ((p'1)*^ = gadxa/\.. .A dx^, we calculate (det((y> a/ ,).)) • dx{ A . . . A dxi =
° (Va)*(— • 9a dxa A . . . A dxam) 9a
p
1
ga°<Pa
= (— —=1){^rn = -j^-dx^...dxt ga°fa°iPp
P
9a\Va(})
Thus for a positively oriented atlas we have (det(<pa/g)„) > 0. It remains only to prove that (iii) implies (i). Given a locally finite atlas 21 = {(Ua,(pa) | a e A} such that (det((<pap)*)) > 0 and a partition of unity {Xa | ct € A} subordinate to 21 we define differential forms Cla := Xa • Va(^xi A . . . A dx^) on M. Since ila{p) — 0 for all but a finite number of a for each p in M, the differential m-form
ft := Y^ na a€A
84
is well-defined on M. Given p in Up we have (with Ap = {a £ A\p £ Ua})
= £ [(xa°^)'((^r»*'KA...A<))]wW a 6 A,,
= £
Xa(p)-(y^(«toi A - . - A d r ^ ) ) ^ ^ )
= [ £
Xaip) • (det((V7aj8)*p))j • dxf A . . . A dx&m
aeA,,
and thus fi is a nowhere-vanishing m-form on M, i.e. an orientation form (and 21 a positively oriented atlas with respect to fi). • Corollary. Let M be a connected manifold with m — dims M > 2 and U an open subset of M such that dU is a closed (m—1) -dimensional submanifold of M and such that dUD U (U denotes the interior of the closure of U) is empty. If M is oriented then dU has a "canonical induced boundary orientation". Proof. Let 21 = {(Ua,<pa)\a £ A} be a locally finite oriented atlas of M such that for N := dU we have (pa(Ua n N) = {(xf,... ,x%) £ Va = <Pa(Ua)\x? = 0} and <pa(UanU) = {(atf,...,a;») € Va\xf < 0}. Let B = {a £ A | Ua D N ^ 0} and * a : Ua D N -> E" 1 " 1 be defined by * Q ( p ) := (s/ia(p),---.y^-i(p))~(a;?(p).---.^(p))- ThenS = {(t/an7V,*a)|aGB} is a locally finite atlas for TV and it is easy to check that det((* a/ 3)«) > 0 for all a, ft in B. By the preceding proposition there exists an orientation form on N = dU such that B is positively oriented with respect to this "canonical boundary orientation". • Exercise. Show that the boundary orientation we constructed on 5 m _ 1 is a special case of the preceding corollary. Definition. Let V be open in E m and let A = g-dx\f\.. such that suppg is compact. Then gdxi... Jv
Jv
.Adxm be in £m(V)
dxn
where the right-hand side is defined by iterated integration (in the sense of the Lebesgue or the Riemann integral). Lemma. Let V and V" be open sets in W1, oriented by the standard orientation form ofRm. Let furthermore A be in Sm{V") with compact support, 85
and ip :V' —¥ V" an orientation-preserving
diffeomorphism.
Then
f
JV"
Proof. Using the transformation formula for multiple integrals we find /
A
= /
JV(V') = V"
g(y)dyi--,dym=
((det(p*)(x))-g(
J
JV
= / Jv
((dettp*)(x))-g(tp(x))dxiA.../\dxm
= / 9(
a Definition. Let M be an oriented ra-dimensional manifold and A a m-form with compact support. Then JM
J
apA
where 21 = {(Ua,(pa) \ a & A} is a, locally finite, positively oriented atlas for M and {\a | &• £ ^4} is a partition of unity subordinate to 21. Exercise. Show that JM A is well-defined, i.e. independent of the chosen locally finite, positively oriented atlas and the chosen subordinate partition of unity. (Hint: Use the preceding lemma.) Theorem ("Stokes' theorem"). Let M be a connected, oriented, Tridimensional manifold with m > 2, and r] in £m~l{M). Let furthermore U be an open subset of M such that its closure U is compact and its boundary dU is a smooth closed submanifold of M. Then drj=
rj.
JU
JdU
Proof The detailed derivation of this theorem can be found in many textbooks on manifolds. For a short proof see, e.g., [BT], pp. 31. • Remarks. (0) A purist whould introduce the injection j : dU -> M,j(p) Stokes' formula as follows:
f drj= ( JU
JdU
86
fin).
= p and write
It obviously follows that the integral of j*{rf) over boundary components of dimension strictly smaller than m — 1 vanishes since they have no non-zero (m — l)-forms. (1) Given our preparations the proof of Stokes' theorem is reduced to an ingenious reduction to the "Fundamental theorem of calculus": if F : R —t R smooth and a < b then F(b) - F(a) = Ja F'(x) dx. Though formally not included in the above formulation of Stokes' theorem it fits in the following sense: Let M = R, U = (a, 6) and r] = F in £°(M) = Sm-1{M). The outward pointing normal vectors in dU = {a, b} are then
N(a) = -4-dx
and N(b) =
d dx b
i.e. parallel to the positively oriented basis ^ in b and anti-parallel in a. Thus "the integral of the zero-form F over dU with respect to the boundary orientation" should be (—F(a)) + F(b), i.e. F(b) - F(a) = [ F= f dF= [ F'(x) dx = f F'(x) dx. JdU JU Ju Ja (2) The usual integral theorems known from vector calculus in R2 and R3 are special cases of Stokes' theorem. As an example we will give the following. Corollary (Gauss' theorem). Let V be open in R3 and K = t 3 (Ki, K2,K^) : V —> R be a smooth force field. Let furthermore be U open in V such that its closure U is compact and contained in V and such that dU is a smooth closed submanifold of V. Then we have for n = i/^A (with K = J2j=i KJW7
an
d ^
=
^ i A dx2 A dx3):
f d(iKA) = [ JU
iKA.
Jd dU
Proof. Obviously the assertion of the corollary is a special case of Stokes' theorem. • Remark. The interesting part of the corollary is given by a further translation into vector calculus. First, a direct calculation shows that d{iKA) = ( E ? = i ^f)'A = ( d i v (*0) • A = (V • K) • A. Secondly, in a point p of dU the two-form iKK restricted to TpdU is necessarily proportional to the canonical orientation (i/vA) p = ijv(P)Ap, where N(p) in TPV is uniquely fixed as the outward pointing normal vector such that ||AT(p)|| = 1, N(p) is orthogonal 87
to TpdU C TpV = TpW? and p + eN(p) is outside U for small e > 0. (Here ^ ( P ) = E ' = i ^ ( P ) a f - | p and TV = '(JVx, JV2,7V3) of course.) Let us remark that JV(p) = ( £ » = 1 H & ( p ) | 2 ) _ i • ( E ^ X & ( P ) • a f j l j if 5 is a local function near p such that {5 < 0} = U and {5 = 0} = dU and dg\du 7^ 0 as considered before in this section. We fix an ordered orthonormal basis {vi,v2} of TvdU such that (iwA)p(i)i,t)2) > 0 (and then equal to one in fact) and calculate: (**(„)ApXwi.wa) = Ap(/f(p),vi,W2) = AP({K(p) •
N(p))N(p),vuv2),
since {N(p),vi,v2} is an orthonormal basis of TPV = TPR3 and thus (iK^)P{vi,v2) = ((^•7V)(p))(i N n) p (t; 1 ,t;2),i.e. a^Q = (K-N)iNQ.. Interpreting N(i^il) as the "vectorial surface element dS" we arrive at a formulation of Gauss' theorem which is frequently found in the physics literature: f (V • K)dx! dx2 dx3= f K- dS. Ju JdU Proposition. Let M be a manifold and N a compact submanifold with an orientation. Then the integral over N defines a linear functional on H£R(M). Proof. For p in £k(M) let JN p. denote the value of fN i*N(p) defined by the orientation of N. (The inclusion map N <-» M is denoted by in here.) Let us assume that p. = drj for r\ in 8k~1{M). Then i*N(p) = d(i*N(r])) and with Stokes' theorem we will show that fN i*N{p) — 0. Let p be any point in N and ip : W ->• V be a chart of N such that
JN
On the other hand JdUt
J\\x\\=e
and thus converges for e \ 0 to zero. It follows that JNdr] = 0 and thus the map Hkm{M)
-> E,
[p]^
[ p JN
is a well-defined linear functional.
•
Corollary. Let M be a compact connected orientable m-dimensional manifold. Then [0] ^ 0 in H™R{M) for all nowhere-vanishing m-forms Ct on M. Proof. Let flo be a m-form defining an orientation of M. Then Cl = f • Q0 for a nowhere-vanishing smooth function / on M. We may assume without loss of generality that / > 0. Let 21 = {(Ua, <pa) I <x G A} be any locally finite, positively oriented atlas of M and {xa \ <* € A} any partition of unity subordinate to 21. Then
f n = f /.n„ = W =£[/
(^nxa/fio)
((xa/W-^-G^nno)]
and all summands on the last right hand side are non-negative, and at least one is strictly positive. Thus JM Q > 0. Since the functional HTR(M)
-+ E,
[M] H- f
M
is well-defined by the preceding proposition it follows that fl cannot be exact, i.e. [O] # 0 in H?R(M). D Remark. With the hitherto developped theory it is possible to sharpen the assertion of the last proposition to the statement that integration over M is, in the compact case, an isomorphism from H™R{M) to E. (See [AMR], pp. 552.) Bibliographical remarks. As for Section 2.7.
89
3 3.1
Symplectic geometry Symplectic
manifolds
Definitions. Let M be a (real) manifold. (1) A differential 2-form w on M is called an "almost-symplectic form" if and only if kercjp = {0} for all p in M, i.e. "w is everywhere non-degenerate". A pair (M, u>) consisting of a manifold M and an almost-symplectic form u) is called an "almost-symplectic manifold". (2) An almost-symplectic form u on M is called a "symplectic form" if and only if u> is closed, i.e. dw = 0. A pair (M,UJ) consisting of a manifold M and a symplectic form w is called a "symplectic manifold". Lemma. Let M be a manifold and u> in £2{M). equivalent:
Then the following are
(i) to is an almost-symplectic form (ii) ojp : TPM -)• (TPM)* — (T*M)P is an isomorphism for all p in M. If M is furthermore of pure dimension m then (i) and (ii) are equivalent to (Hi) wt7™/2] is an orientation form on M. Proof. The assertions follow directly from Section 1.3 and the definition of an orientation form. • Corollary. Let (M,co) be an almost-symplectic manifold of dimension m. Then M is orientable and m is even. Proof. The corollary follows immediately from the preceding lemma. • Definition. Let (M, UJ) be a symplectic manifold of dimension m — 2n. The orientation form
n :-((-l)^.I) W is called the "canonical orientation form (or Liouville form) on (M,w)". The associated orientation [^] 0 rientation *s ca-lled. the "canonical orientation on (M,w)". Proposition. Let M be a connected compact manifold of dimension m. Let furthermore D be a symplectic form on M and c = [w] in H\R{M) be the de Rham cohomology class of ui. Then the de Rham cohomology classes ck are 90
non-zero in H£R(M) for k = 1 , 2 , . . . , y . Proof. Since w is closed it defines a de Rham cohomology class c = [w]. Assume that there is a k in { 1 , 2 , . . . , y } such that ck = 0, i.e. there exists H in £ 2 f c _ 1 (M) such that d// = wfc. Closedness of u> implies that w ^ = d?j with T] := ( i A u ' * " ' * . Since w ^ is nowhere vanishing J M w ^ 7^ 0, where the integral is defined by one of the two possible orientations of M, e.g. by the canonical orientation of (M, w). Since it was shown in Section 2.8 that fMdrj = 0 for all r] in £ m _ 1 ( M ) we arrive at a contradiction. Thus there is no k in { 1 , . . . , y } such that cok is exact. • Remark. The preceding results of this section show that not all manifolds can carry a symplectic form. Examples. (1) M = R2n and u> := X^Li &xi CI = ((—1)
5
A
dxn+j- Then (M,u>) is symplectic and
• ~ T ) W " = dxi A dx2 A . . . A cfa^n-i A dxn.
The form w is sometimes called the "standard symplectic form on R2n." (2) Let £ be a two-dimensional orientable manifold and Q. an orientation form on S. Then w := fi is a symplectic form on £ (and ft = OJ is the canonical orientation form on ( S , C J ) ) . (2.1) Let E = S 2 . In the notations of Section 2.8 there is a natural orientation form Q = ( « N A ) | T S 2 (with A = dxx Adx 2 Adx 3 and N = J2j=i xj~sk~)- Then (S 2 , Q) is a symplectic manifold. (2.2) Let M = S1 x S 1 and ft = ddi A di?2, where d ^ ( g | - ) = ^, fc . Then (5 1 x 5' 1 ,0) is a symplectic manifold. (3) Let (Mi,u>i) and (M 2 ,w 2 ) be symplectic manifolds and Ai,A2 in K\{0}. Then the form Ai • (pr^wi) + A2 • (pr^u^) is a symplectic form on Mi x M 2 . (4) Let (M, w) be a symplectic manifold and U be open in M. Then (C/,w| ) is a symplectic manifold. Proposition. Let Q be a manifold, M := T*Q its cotangent bundle and •K ~ TTT ® : T*Q -¥ Q the canonical projection. Then M has a canonical differential one-form 6 := 6T ® defined by 0a„(vaq) := aq{(nt)a
for all q in Q,aq in (T*Q)q = (TqQ)* andvaq
inTaq(T*Q).
Furthermore, the two-form u> :— uiT Q := — d9 is a symplectic form on M = T*Q. 91
R e m a r k . The form 0 (respectively LJ) on T*Q is often called the "canonical one-form (respectively two-form) on the cotangent bundle T*Q". Proof of the proposition. Let
ViEF.
xn) in V C 1R™ and
n
* : V x I » 4 T*V,V(a:,») := £ > ;
( E " = i OafjL> E " = i ^ a f j Q
by
pairs
(£,77)
=
6 . • • • »fn,»/i, • • • .»/n in R and calculate:
(($ o * ) • % , „ ) & » / ) = *•(*(*,,,))((*. o **)(£,»?)) = = $ ( * ( * ,
n
r\
n
n
i.e. (S o #)*0 = £ " = 1 Vj dxj i n ^ ( V x E n ) . It follows from this local computation that 6 and OJ = —d6 are (smooth) differential forms on M — T*Q. Furthermore we have n
($ o # ) * w = ($ o V)*(-dd)
= - d ( ( $ o $)*0) = E d a ; j
A
<%
j=l
and hence w is non-degenerate in all points of the open set T*U in T*Q. Since the above local calculations are valid for all charts of Q and since the cotangent bundles over chart domains of Q form an admissible atlas of T*Q it follows that w is everywhere non-degenerate on M = T*Q. Closedness of u> is trivial since d o d = 0 and thus w is a symplectic form on T*Q. D 92
Remark. If we describe elements of V x W1 (V open in E ra ) by (q,p) = (*(gi,.. -, qn), *(Pi> • • - ,Pn)) the preceding proof yields the traditional formulas: n
($ o yye = ^pjdqj
n
and ( $ o \P)*w = ^
3=1
dgj A dpj
j=i
for the cotangent bundle forms 6 and w. Exercise. Show that the canonical cotangent bundle two-form UJT R on T*Rn is pulled-back by $ o $ = * to the standard symplectic two-form on M2n (up to renaming of the variables yj = : a;n+J- for j = 1,..., n). Remark. Given a manifold Q, a "configuration space", the symplectic manifold (T*Q, u>) with the cotangent bundle two-form ui is often called the "phase space (associated to <9)". Furthermore the dimension of Q is often referred to as the "number of degrees of freedom". Bibliographical remarks. The classics for the mathematical treatment of mechanics in the symplectic language are of course [AM] and [Arl]. We would like to add [Be], [Bry] and [GS]. 3.2
Maps and submanifolds
Definitions. Let (M,UJM) F : M -> N a smooth map.
of symplectic and (N,CJN)
manifolds
be symplectic manifolds and
(1) The map F is called "symplectic" if F*(U>N) = LJM-
(2) If F is a diffeomorphism such that F*(U>N) = WM, F is called a "symplectic diffeomorphism" (or sometimes also a "symplectomorphism"). Proposition.
Let (M,U>M)
and (N,CJN)
be pure-dimensional
symplectic
manifolds and F : M —> N a symplectic map. Then (i) TPF = (F*) p : {TPM, (WM) P ) -> (TF^N, (U>N)F{P)) is a symplectic linear map, i.e. (TpF)*({LUN)F(P)) = (w«) p for all p in M. Thus TPF is infective for all p in M, so that in particular dimR M < dimR N and the rank of F in p equals the dimension of M (for all p in M). If furthermore dimR M = dimR N = : 2n then (ii) F is a local diffeomorphism, the local inverses are also symplectic, and (Hi) F*(Slx) = QM, i-e. F preserves the canonical orientation forms and hence, a fortiori, the canonical orientations. If furthermore F is a diffeomorphism then its inverse is also symplectic. Proof. Exercise using Section 1.4. • 93
Examples. (1) Let / : Q -» Q be a diffeomorphism of a manifold Q and F := ( T / - 1 ) * : T*Q -> r * Q , defined by F ( a , ) = ( T / ( 9 ) / - 1 ) * ( a g ) for all q in Q. Then F is a vector bundle isomorphism over / and F*(dT'Q) = 9T*®. Thus we have of course F*(LJT*®) — w 7 "^, i.e., F is a symplectic diffeomorphism of T*Q. Since F is induced from a diffeomorphism of the configuration space Q, it is traditionally called a "point transformation (of T*Q)". (2) Let E be an orientable two-dimensional manifold and £2 an orientation form on E. Then a diffeomorphism of E (with the symplectic form u> := £2) is obviously symplectic if only if it is volume-preserving. (3) Let g be in GL(m, E) C Mat(mxm, E). Then Tg : E m -s- E m , Tg(x) = gx is a diffeomorphism of E m with
i.e., ((T fl )*) x can be identified with the (linear!) map Tg. Furthermore we define for A in Mat(mxm,K) a vector field XA by X^(a;) := YlTk=i a i fca; fcaf _ L' Then the flow
=
TetA(x).
Taking m = 2n and wo := ]C?=i d,XjAdxn+j it follows that the diffeomorphism Tg is symplectic if and only if g is in Sp (2n, E), In particular for n = 1 we find that
if we supply R 2n with the canonical symplectic form wo(3) Let (R 2 n , +) act on itself by vector addition, i.e., G := R2n, M := E 2 n , i? : G x M —J- M, i?(a, a;) := a; + a. Then $ is symplectic, again with respect to the canonical symplectic form on R 2 n . (4) Let ft = ijvA be the usual orientation form on the two-sphere S2 C M3 and let t? : S0(3,K) x S 2 - ) S 2 , be defined by i9(g,a;) = g • x. Then tf is a symplectic action with respect to the symplectic form w : = f l o n S 2 . Definition. Let M be a (2n)-dimensional manifold with symplectic form UJ and N a closed fc-dimensional submanifold of M. Then (TiV) z := {up G TPM \ p G N
and
wp(i>p,wp) = 0 V wp G TpAf C TpM}
is called the "skew-complement (or w-complement) of TAT in
TM\N".
Lemma. Let M be a manifold and N a closed submanifold of M. Then (i) TM\N := [J eNTnM carries a natural vector bundle structure and TN is a subbundle ofTM\N. If furthermore M carries a symplectic form u> then and dimRTpAT + dim R ((TiV) z ) p
(ii) (TN)^ is a subbundle of TM\N dimR TPM for all p in N.
=
Proof. Ad(i). Given a point p in N there is a chart
Still working in a chart as above we have now m = 2n for n in N and
we set w = (y _ 1 )*w. Clearly { g ^
, • • • ,9—-! } is a basis for
Txip{UC\N)
for all x in
(.
= {TMUnN))'=
f) j=k+l
n
ker(^(£-|)). Xj
X
Since w is symplectic, the functionals u>x( jsM ) are linearly independent so that (TN)Z holds true.
is a subbundle of TM\N
and the asserted dimensional formula •
Remark. Given M and N as in the first part of the preceding proposition, the bundle TM\N is often called the "tangent'bundle of M restricted to TV" 95
and with the canonical inclusion JN : N <-> M, j ^ (p) = p we can describe it (isomorphically) by the pull-back bundle (j^)*TM. Definition. called
A closed submanifold N of a symplectic manifold (M, u>) is
(1) "symplectic" if (TN)* n TN = Im(<70), the image of the zero-section a0 : N -+ TM\N, (2) "isotropic" if TN C
(TN)*,
(3) "coisotropic" if (TN)* C TN, (4) "Lagrangian" or a "Lagrange submanifold" if (TN)* = TN. Remark. The first condition is often written (TN)* D TN - {0}. Lemma. Let N be a closed submanifold of a symplectic manifold (M, u>) and JN : N *-»• M the canonical inclusion of N in M. Then (i) N is symplectic if and only if u>v\T N is non-degenerate for all p in N, i.e. if (JN)*LJ is a non-degenerate two-form on N, (ii) N is isotropic if and only if wp\T UN)*U
N
is the zero-form for all p in N, i.e.
= 0,
(Hi) N is Lagrangian if and only if N is isotropic and N has half the dimension of M, i.e. dimRTpM = 2 • (dimg,TpN) for all p in N. Proof. Exercise.
•
Examples. (1) Let N be a closed one-dimensional submanifold of a symplectic manifold (M,CJ). Then iV" is isotropic. (2) Let TV be a closed hypersurface of a symplectic manifold (M, u>). Then TV is coisotropic. Proposition. Let Q be a manifold and 77 a one-form on Q. Then the image of r], r](Q) = {r)(q) 6 T*Q \q € Q} is a closed submanifold of T*Q and the map r) : Q -> T*Q a diffeomorphism of Q onto rj(Q). Furthermore rj(Q) is Lagrangian in the symplectic manifold (T*Q,coT Q) if and only if rj is closed. Proof. The first part follows from the general fact that the image of a section a of a vector bundle E -^* M is always a closed submanifold such that a : M -¥ a(M) C E is a diffeomorphism with inverse T ^ ^ ) - (This fact is easily proven by using a local trivialization of E —> M.) 96
Denoting r](Q) by N and the canonical inclusion N <-> T*Q by j ^ , the equation JN — r\ o 7r| ,Qs implies that it is enough to show that r)*(uiT"Q) = 0 if and only if 77 is closed, since 7r| , Q . = TV -> Q is a diffeomorphism and clearly r}(Q) has half the dimension of T*Q. Let thus q be in Q and u in T g Q, then
fa*(*T*g)),(«) = (* T * 9 )„(,)M«)) = ^ ) ( ( T * ° *)(«)) = !?,(«), i.e. r)*(6T*Q) = 77. It follows that 77*(u/r*<2) = -dr? and thus the second part of the proposition is proven. • Corollary. Let Q be a manifold and f be a smooth function on Q. Then {df)(Q) C T*Q is Lagrangian with respect to the canonical symplectic form Proof. Obvious, since d(df) = 0 . Proposition.
Let (M,U>M)
•
and (N,U>N)
be symplectic manifolds and F :
M -* N a diffeomorphism. Then the graph I > := {{x, y) 6 MxN\y = F(x)} is a Lagrangian submanifold of the symplectic manifold (M x N, {prM)*ojM ~ (prN)*u>N) if and only if F is symplectic. Proof. Let us first observe that for any smooth map F : M -> N between two manifolds the graph Fp is a closed submanifold of M x N and F : M -> TF, x i->- (x, F(x)) is a diffeomorphism with inverse 7rj . Thus we calculate (F)*((pr M )*u>M-(pr N )*wjv) = (piMoF)*ujM-(pTNoF)*ujN
=LJM-F*CJN,
and thus Tp is Lagrangian if and only if U>M = F*U>N on M , i.e. if and only if F is symplectic.
D In order to prove that all symplectic manifolds of a fixed dimension 2n are locally diffeomorphic to the "symplectic model space" (K 2n , w0 = Y%=i dxj A dxn+j) we introduce the useful tool of "time-dependent vector fields" and the crucial computational formula which is at the base of "Moser's method". Lemma. Let M be a manifold and X a vector field on M. Then Cartan's homotopy formula Cxr) = d(ixr]) +ixdri
for all rj in
£*{M)
is equivalent to the formula jt((
= (
for all r, in
£*{M).
(Here
Inserting the right hand side of Cartan's formula for £xV yields the second formula, i.e. the two formulas are equivalent. • Definition. Let I be a connected interval in E that contains 0, M a manifold and i) : / x M -¥ AhT*M a smooth map. We define i]t(p) := r](t,p) for all (t,p) in I x M and we call rj a "time-dependent fc-form on M" if rjt is a fc-form on M for each t in I. Lemma. Let M be a manifold, X a vector field with flow
jt^tYm)
= ifP?T (d(ixv) + ixdn + ^f)
in all points p in M, t in I, where ipx is defined. (Here -ffi is of course again a time-dependent k-form on M and thus for fixed t a k-form.) Proof. The formula follows immediately from the preceding lemma and the Leibniz rule in one variable. • Definition. Let / b e a connected interval in E and X : I x M -> TM a smooth map, and let Xt(p) := X(t,p) for all (t,p) in I x M. (1) We call X a "time-dependent vector field (on M ) " if Xt is a vector field on M for each t in M. (2) Let X be a time-dependent vector field on M. A smooth curve 7 : J ->• M with J open and connected in / is called an "integral curve of the timedependent vector field X (with initial condition 7(^0) = p)" if P i s in M, to in J and 7(*) = ( 7 * ) i ( ^ | t ) = * t ( 7 ( 0 )
for all t
in
J
and 7(^0) = P -
(3) Let X be a time-dependent vector field on M , M := I x M and X(t, p) : = TtI T mlt ) + Xt(p)in (*.P)^ ® TPM- T h e v e c t o r fieId * the "suspension of (the time-dependent vector field) X".
on M
is called
Remarks. (1) Typically the interval I is either E or [0,1], or the latter "with periodic boundary conditions", i.e. X : S 1 x M -)• TM such that Xt is 98
a vector field on M. (2) Using the suspension X of a time-dependent vector field X it is not difficult to deduce existence and uniqueness of integral curves of X and of maximally defined flow maps
= (*?)'(d{iXtTH)
+ ixtdTH + ?jjf)
for all t
in E.
Proof. Exercise (possibly supported by a textbook as [AMR], [Be] or [GS]).D Theorem ("Local normal form of symplectic forms o n a manifold" or "Theorem of Darboux—Moser—Weinstein"). Let ( M , C J ) be a symplectic manifold of dimension 2n. Then for each point p in M there is an open neighborhood U = U(p) in M and a diffeomorphism ip : U —¥ ip(U) — V,V an open set in E 2 n , such that ip* ($Z?=i dxj A dxn+j) = u L Proof. Let tpi • Ui -^ i>i(Ui) = Vi C E 2 n be any chart such that ipi(p) = 0. We may assume without loss of generality that Vi is E 2 n . Let wo = ( V ^ r H ^ ) and wjfc in E be defined by w0(0) = Y^j
Denoting the inverse of the ismorphism wb : V -¥ V* on a symplectic vector space (V, ui) by w" we define a smooth time dependent vector field X for x in V2 (and i in [—e0,1 + £o]) as follows Xt(x) := {Ut{x))*(
|($?u, t ) - | ( ( * f )*<*) = (*f)*(d(tx,"t) +**,<** + ^ ) = = ($^)*(dff + w i - w o ) = 0 . It follows for t = 1 that $ := $ f : MSo (0) -»• $ f (B«0 (0)) fulfills $* W l = w0Let furthermore g in GL(2n, E) be such that (T9)* (Y^j=i dxj A <&„+.,•) = We set V := Tg o $ o ^ i : [/ -» V = V(^) C E 2 n , where £/ := t/)f (B<50 (0)) C U\ is an open neighborhood of p in M. Then ip is a chart fulfilling T/>(P) = 0 and 1
n
fp*(^2
n
dxj A dajn+jj = ipl o $* o T* ( ^
3=1
etej A ote n+ j J = Vi(wo) = u\v-
3=1
D R e m a r k s . (1) The above proof relying on the construction of the timedependent vector field Xt and the formula for ^((<J>*)*wt) goes back to [M] and is therefore also referred to as "Moser's method". Though the local normal form of symplectic forms on a manifold can be reached in a simpler way we chose this approach since it easily yields proofs for several substantial generalizations. (See e.g. [GS] and [Weil].) (2) Local coordinates (xi,..., x-^n) as in the preceding theorem, i.e. such that u) is given as ^ ? = 1 dxj A dxn+j are often called "symplectic coordinates". Writing qj = Xj,pj — xn+j for j — 1 , . . . , n the form UJ is given as X)"=i <% A dpj and we will call such coordinates ( # 1 , . . . , qn,P\, • • •,Pn) symplectic as well. (The older term for the latter version is "canonical coordinates".) 100
Bibliographical remarks. The references cited in the text of this section plus those mentioned at the end of 3.1. 3.3
Kahlerian
Definitions.
and almost
Kahlerian
manifolds
Let M be a manifold.
(1) A smooth section g of the vector bundle ®2T*M over M is called a "pseudo-Riemannian metric (on M)" if the following two conditions are fulfilled for each pin M: (i) gp is symmetric, i.e. gp(v,w) = gp(w,v) VU,UJ G TPM. (ii) gp is non-degenerate, i.e. for all 0 ^ v in TPM there is a w in TPM such that gp(v,w) ^ 0. (2) A pseudo-Riemannian metric on M is called "Riemannian" if g(v, v) > 0 V« G TPM \ {0} for all p in M. (3) A pair (M, g) consisting of a manifold and a (pseudo-)Riemannian metric is called a "(pseudo-)Riemannian manifold". Remark. A (pseudo-)Riemannian metric on a manifold is nothing else than a smoothly varying assignement of a (pseudo-)Riemannian metric to each tangent space. Examples. (1) Let M be R m and g = J2i
Furthermore, if
dZj
£b
s-i
=
dXj +
dVj
= 6j k
and
dZ;
(~dx~k ^ ( ' * ^ dx~k~ ^ ' >^Wk ^ = Using the C-linearity of the functionals dzj on TPM we find
{6j,h
'
It follows that, for k = 1 , . . . , m,
r(JL) = A jf^.J. \dxk)
dyk
,(»)...» mouu d u ""1KX uydykJ dxk D
Definition. An almost-complex structure J on a real manifold M that is canonically associated to a complex-analytic atlas 21 (as in the preceding proposition) is called a "complex structure". Remarks. (1) Though the distinction between complex structures and almost-complex structures is important, some texts on symplectic geometry are not attentive to it. (2) An almost-complex structure that fulfills a certain "integrability" condition is called an "integrable almost-complex structure". In finite dimensions 102
every integrable smooth almost-complex structure is already a complex structure by a deep theorem of Newlander and Nirenberg (see the original work [NN] or [H] for a proof). Definitions. Let (M, J) be an almost-complex manifold and g a (pseudo-) Riemannian metric on M. (1) The metric g is called "almost (pseudo-)Hermitian" if, for all p in M gp(Jpv, Jpw) = gp(v,w)ioi
allv,w in TVM.
(2) If J is a complex structure and g almost (pseudo-)Hermitian then g is called "(pseudo-) Hermitian". (3) If g is almost (pseudo-)Hermitian, the 2-form u>, denned by LJP(V,W)
= gp(Jp(v),w)
V p£ M,
Vv,w € TPM,
is called the "fundamental 2-form (on the almost (pseudo-)Hermitian manifold (M, J,
symplectic manifold is almost Kahlerian. Remark. Upon considering JU{M) as a subset of the "Frechet space" rcoo(M, End(TM)) one can use Section 1.5 to prove the following important sharpening of the preceding proposition: the topological space JU(M) is non-empty and continuously contractible to a point. (See, e.g., [McDS].) Proof of the proposition. The existence of a partition of unity subordinate to an appropriate covering of M easily shows that M carries a Riemannian metric g. Applying the theorem of Section 1.5 to V = TPM yields maps 9P :H(TPM)
^
JUp{TpM)
such that ^P(gP) = JP and (TpM,Jp,gp) is Hermitian for all p in M. Since * is real-analytic in the variable g € Ti(V) one easily poves, by going to local charts, that J is a smooth section of End (TM). Thus J is an almostcomplex structure on M. The theorem in Section 1.5 implies furthermore that the Riemannian metric gj defined by {9J)P(V,W)
= up{v, Jp{w))
is almost Hermitian on (M,J) equal to w.
Vp€M,
\/v,w£TpM
and the fundamental 2-form of (M,J,gj)
is •
Examples. (1) Let (M, 21) be a 2-dimensional real manifold with a complex-analytic atlas and g a Hermitian metric on M. Then (M, J, g) is Kahlerian since every 2 form on M is closed. (2) Let M = C 1 = R2n and g = Y^=i(dxk ® dxk + dyk ® dyk) the standard Riemannian metric on M2™. Since J ( g f - ) = " a f r t n e metric g is Hermitian and the fundamental 2-form of (M, J, g) is given as follows n
oj = ^2dxk
Adyk-
k=i
Obviously, w is closed and thus (M, J, g) is Kahlerian. (3) Given a discrete subgroup T of ( C \ + ) one easily checks that the complex structure, the metric and thus the fundamental 2-forms "descend" to the quotient O 1 /T — r\C™, since they are in fact invariant under the whole group (C n , +) acting on M = C \ Thus C / r is Kahlerian. (4) Open subsets of Kahlerian manifolds, products of Kahlerian manifolds and 104
complex submanifolds of Kahlerian manifolds are naturally equipped with induced complex structures and Hermitian metrics such that they are Kahlerian. (5) Let M = JVR be as in Example (3.2) of Section (2.2), i.e., M = i I I 0 1 z I ,w j x,y,z,w€R\
S l
3
x l
Let LJ = dy Adz + dx Adw, then w is a symplectic form on M that is invariant under the action of JVR on M given by left-multiplication. It follows that M = NZ\NR is a compact manifold with a unique 2-form such that under 7r : M ->• M we have 7T*(CJ) = u5. It follows easily that (M, UJ) is symplectic since 7r has everywhere rank four. Standard arguments from algebraic topology show that the dimension of H\R{M, E) is equal to three which shows that M cannot carry a Kahlerian metric. (See [BT] and [GriHa] for more details on the algebraic topology respectively Hodge theory needed to show the above assertions.) The manifold M was considered by Thurston to exhibit a compact symplectic manifold allowing no Kahlerian metric. (See, e.g., [Wei2].) (6) The complex projective space P n ( Q is a very important Kahlerian manifold with respect to the so-called "Fubini-Study metric" and its associated fundamental 2-form, the "Fubini-Study form" wpg. (See, e.g., [GriHa] or [J] for more details.) Bibliographical remarks. Beside the references cited in the text we would like to mention [MK] and [Wei] for the theory of Kahlerian manifolds. The reader should be aware that traditionally the class of Kahlerian manifolds is viewed as a special case of complex manifolds and not of symplectic manifolds and thus notations are rather "complex" than "real". 3.4
Hamiltonian
Remark. b
dynamical
systems
on symplectic
manifolds
Given an almost-symplectic manifold (M,w) the map
w :TM -»• T*M,u\vp)
:= ubp{vp)
for all p in M and all vp in TPM
is a vector bundle isomorphism over I d ^ , the inverse of which we denote by Definitions. (1) Let (M,w) be a symplectic manifold and H a smooth function on M. Then we define a vector field XH on M by XH
=w\dH). 105
The vector field XH is called the "Hamiltonian vector field associated to the Hamilton function H" (or "symplectic gradient of H"). (2) A triple (M, ui, H) consisting of a symplectic manifold (M, UJ) and a smooth function H on M is called a "Hamiltonian dynamical system". Remark. Since we associated to a function H on (M, to) a vector field XH, a Hamiltonian dynamical system comes equipped with the (local) flow (pXft, i.e. a local E-action on M. This explains the terminology. Remark. Since wb o w" = Idr»M the vector field XH is often denned as the unique vector field on M fulfilling uj(XH,-)=dH. This formula is clearly equivalent to the above definition and is in fact very useful in computations. Proposition. Let (M,UJ,H) be a Hamiltonian dynamical system and
~^\dpj pi \dp,
ddqj qj
dqj
dp,) '
Remarks. (1) It is understood that the function H should be read as H = H o <^_1 in the formula of the proposition. We follow the usual practice to suppress this inconvenient notation. (2) Prom the last proposition in Section 2.5 we know that (on V) the differential equation ("Hamilton's equations") for the flow of XH as in the formula of the proposition is then given by qj
dH = -—, Opj
. Pj
dH = --— dPj
. for
. n j = 1 , . . . , n.
Proof of the proposition. Let first (p : U -^ V C E 2 " be any chart on M and H = Hoip~l,u) = (ip~1)*cj. Then w is a symplectic form on V and the vector field XM = & (dH) fulfills u(XH,v)=w(Xs,
VveTmM 106
Vrr.eU.
It follows by the non-degeneracy of w and Q that (
= E L i ("fc^- + a
fib^)
with ak,pk
in f (V). Then
a\ - / a \ a# ' = W0(x5> ^ - ) = ^ ( ^ ; ) = di.
and
^ = -(^|-) = --(|-)=-fIt follows that _ y^ /9g
d
dH
d \
as claimed in the proposition.
•
Let us observe that there is another way of expressing Hamilton's equations on E 2 n (in fact, more generally, on almost Kahlerian manifolds) which brings into play the almost-complex structure: Lemma. _,
Let V be open in R 2n ,.ff o smooth function on V and J n = structure on E2™ (as in Section
J the standard almost-complex
1.5). Then Hamilton's equations are equivalent to x = - J ( V J f (a:)).
Proof.
Let x = (qi,...
,qn,Pi,
• • • ,Pn) =• {q,p)-
Then Hamilton's equations dH
q\ pi
I \
dp dH dq
are obviously equivalent to dH_
X ^
q\ _ "p I
( 0 —1 \ I dq \
I dp
\ 1 U I \ §H_ j
\
dH dq
D 107
Definition. Let (M,LJ) a symplectic manifold and X in X(M). (1) We call X a "symplectic vector field" if £ x w = 0. (2) We call X a "Hamiltonian vector field" if there exists a smooth function H on M such that X = XH • Lemma. Let (M,co) be a symplectic manifold and X in X(M). 0 if and only if (
Then C\u
=
u,
in all points where the last equation makes sense. Proof. Obviously the second formula implies the first. The equality
jt((p?ru) = c^rccx") which follows from the flow equations (compare Section 3.2) shows that £ x w = 0 implies (
for fc = 0 , l , . . . , ( d i m R A f ) / 2 .
Proof. Obvious.
•
Corollary. Let (M, UJ) be a symplectic manifold and X a symplectic manifold. Let furthermore t be in R and U open in M such that ipf is defined on U. Then for all (2k) -dimensional orientable submanifolds of U one has that CJk
JE2fc
J
(if one and then both sides of the equality is finite). Proof. Let us without loss of generality assume that cp? : E2k ->•
Assuming that dimR M = In and taking £2fc = U an open set n
(n-l) .
with finite "phase volume" fvQ = $v H ~ ^ n ! 2 — ] w n yields the result that "the phase volume is invariant under symplectic flows". Lemma. Let (M, u>) be a symplectic manifold and X in X(M). Then (i) X is symplectic if and only if the one-form i^w is closed, and 108
(ii) X is Hamiltonian if and only if ixw is exact. In particular, a Hamiltonian vector field is symplectic. Proof. Since LJ is closed, Cartan's homotopy formula implies for all X in X(M) that Cxu = d(ixw). The assertions follow now immediately. • Since a closed form on a manifold is always locally exact by Poincare's lemma the following notions are rather natural: Definition. Let (M,w) be a symplectic manifold. (1) A symplectic vector field is also called a "locally Hamiltonian vector field." (2) The set of all Hamiltonian vector fields (respectively all locally Hamiltonian vector fields) is denoted by Ham(M,w) (respectively Hamj oc (M,ui)). Lemma. Let (M,w) be a symplectic manifold and let X and Y be locally Hamiltonian vector fields. Then [X, Y] is the Hamiltonian vector field associated to the smooth function H = —io{X,Y). Proof. It is enough to show that dH{Z) = w{[X, Y), Z)
for all Z
in X(M).
Using the formula i[A,B] — [£A>*B] on £*(M) for A,B in X(M) we find: w([X, Y), Z) = X(CJ(Y, Z)) - u{Y, [Z, X}) = -Y(co(X, Z)) + u{X, [Z, Y]).
Closedness of ui implies 0=
X(LJ(Y,
Z)) -
Y(OJ(X,
Z)) + Z(u(X,Y))
- CJ([X,Y],Z)+
CJ([X,
Z},Y)
-
-LJ(\Y,Z],X).
Combining these identities yields 2OJ([X,Y],Z)
u([X,Y],Z)
= -Z(u(X,Y))+u([X,Y],Z), = -Z(OJ(X,Y)) =dH(Z).
i.e.,
n Corollary. Let (M,u>) be a symplectic manifold. Then Hami0C(M,u>) is a Lie subalgebra of (£(M),[,]) fulfilling [Hamioc(M,oj),Hamioc(M,uj)] C Ham(M,uj). Furthermore Ham(M,u>) is a Lie subalgebra of (X(M), [,]) and an ideal in Hami0C(M,u). Remark. A subspace h of a Lie algebra (fl, [,]) is called an "ideal" if [X,H] is in f) for all X in g and H in f). Proof of the corollary. Obvious from the preceding lemma. Exercise.
Let
(M,CJ)
•
be a symplectic manifold. Then the quotient vector 109
space Hamj oc (M,o;)/Ham(M,cj) is canonically isomorphic to H\R(M). Supplying the latter with the trivial commutator, i.e. all brackets are zero, the following sequence of Lie algebra morphisms is exact: {0} - • Ham(M,w) - ^ Ham l o c (M,u,) A
HJR(M)
-> {0},
where a is the natural injection and ft the projection on the above mentioned quotient followed by the canonical isomorphism. Definition. Let (M, w) be an almost-symplectic manifold and let H, Hi, H2 be in £°{M). (1) The "almost-symplectic gradient of 77" is the vector field XH = w*(d77). (2) The "Poisson bracket of Hi and H*" is the smooth function u>(Xn1, XH2) denoted by {Hi,H2}. Lemma. Let (M,u) be an almost-symplectic manifold and H\,H2 6 £°{M). Then (i) {Ht,H2} = -XHl(H2) = XH2(HI), and (ii) the map {,} : £°(M)
x £°(M)
-> £°(M),(HUH2)
is WL-bilinear and anti-symmetric,
M-
{HX,H2}
and fulfills
{Hi,H2 • Hz) = {HX,H2}
• H3 + H2 •
{Hi,H3}.
Proof. We have {HuH2}=co(XHl,XH2)
= ixH2(oj(XHl,-))
= ixH2(dHi)
=
XH2{Hi).
Since us is anti-symmetric the first assertion is thus proven. Bilinearity over E and anti-symmetry of {,} follow directly from the properties of an almost-symplectic form. It remains to show that for each Hi in £°(M) the map {Hi, } : £°(M) -»• £°(M) is a derivation: {Hi,H2H3}
= -XHl(H2
-H3) = -XHl(H2)
= {HuH2}-H3
+
• H3 - H2 • XHl (H3)
H2{Hi,H3}.
a One important motivation of the closedness condition on LJ is the following Proposition. Let (M,u>) be an almost-symplectic manifold. Then the Poisson bracket {,} fulfills the Jacobi-identity on £°(M) if and only ifui is closed. 110
Proof. For F, G and H functions on an almost-symplectic manifold one has by a simple calculation u>([XF,XGlXH)
= -{F,{G,H}}
+
{G,{F,H}}.
A lengthy but elementary calculation shows now that {dw){XHl,XH2,XH3)
= {HU{H2,H3}}
- {{HUH2},H3}
-
{H2,{HUH3}}.
Thus closedness of u implies that the Jacobi-identity holds for {, } on £°(M). On the other hand given an element ip in T* M for a p in M there exists a smooth function H on M such that
= 0 for all p
in M
and for all vi,v2,v3
i.e., dw = 0.
in TPM, •
Definition. Let (M,w) be a symplectic manifold and K : £{M) —> Ham(M, w) be defined by K{H) = —XH- The following sequence of E-vector spaces and R-linear maps is called the "fundamental sequence (on a symplectic manifold)": {0} -> ker/c -U- £{M) - ^ Ham(M,w) ->• {0}. (The map j is the injection of ker K in £(M).) L e m m a . 27ie fundamental sequence on a symplectic manifold is an exact sequence of Lie algebras and ker K is the space of locally constant functions on M. R e m a r k . A continuous function on a manifold is called locally constant if for each point of the manifold there is a neighborhood of this point such that the function is constant on this neighborhood. Obviously a continuous function is locally constant if and only if it is constant on each connected component of the manifold. A C 1 -function is thus locally constant if and only if df = 0. Proof of the lemma. Since u> is non-degenerate n(H) = —XH is zero for H in £(M) if and only if dH = 0, i.e., ker K is the space of locally constant functions on M and the Poisson bracket of such two functions is the zero function. Thus ker/c is a Lie subalgebra of (£(M), {,}). which is in fact easily seen to be an abelian ideal. Ill
Since K is R-linear and surjective it remains only to show that = [K(HI),K(H2)] for all HUH2 in £{M). Since vector fields on a (finite dimensional) manifold can be identified with derivations it is enough to prove that both act in the same way on functions. Let thus H3 be in £{M), then K({HUH2})
[«(#!),K(fT 2 )](#3) = XHl(XHa(H3)) = -({{Hs,H1},H2}
+
-
XH2(XHl(H3))
{H1,{H3,H2}}).
By the Jacobi-identity the last right-hand side equals -{H3,{HUH2}}
= -XiHliHa}(H3)
=
K({HUH2})(H3),
showing the assertion.
D
Definition. Let (M,u>,H) be a Hamiltonian dynamical system and F a smooth function on M. The function F is called a "first integral (of the motion)" or a "conserved quantity" if and only if F is constant on the integral curves of H. L e m m a . Let (M,UJ,H) be a Hamiltonian dynamical system and F in £{M). Then F is a first integral if and only if {H, F} = 0. Proof. Considering F as a 0-form on M we have
~(F(^"))
= Jt^t"TF)
= {tf"Y{CxHF)
Thus {H, F} = 0 if and only if F o y*» = F o ^ " is constant on the integral curves of XH •
= -{{
Proposition ("Noether's theorem"). Let Q be a manifold, (M,w) — (T*Q, uT*Q), and H a smooth function on M. Let furthermore
and
ipXp = (p.
Proof. Let X be the vector field on M that generates the local flow ip, i.e., X(m) — ^Jptim) for m in M. We define a smooth function F on M by setting F = 6T'Q(X). It follows that (6 = eT"Q,u = OJT'Q = -d6): dF = (do ix)0 = -(ix
o d)9 + Cx0 = ixu 112
=
J(X)
by the fact that the "lift"
[CM] P.R. Chernoff and J.E. Marsden, Properties of infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Vol. 425. Springer, Berlin-New York, 1974. [GHL] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer, Heidelberg 1990. [God] C. Godbillon, Geometric Differentielle et Mecanique Analytique, Hermann, Paris 1969. [Gol] H. Goldstein, Classical mechanics, Addison-Wesley, Reading Mass. 1980. [Gre] W. Greub, Multilinear algebra, Springer, New York 1978. [GriHa] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley, New York, 1994. [GS] V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. [H] L. Hormander, An introduction to complex analysis in several variables, North-Holland, Amsterdam-New York, 1990. [J] J. Jost, Riemannian geometry and geometric analysis, Springer, Berlin 1995 & 1998. [KL] M. Karoubi and C. Leruste, Algebraic topology via differential geometry, Cambridge University Press, Cambridge-New York, 1987. [Lan] S. Lang, Differential manifolds, Addison-Wesley, Reading, Mass. 1972: [Lau] R. Lauterbach, Klassische Mechanik und Hamiltonische Systeme, http://www.math.uni-hamburg.de/home/lauterbach/vorlesung.html [McDS] D. McDuff and D. Salamon, Introduction to symplectic topology, Oxford University Press, New York, 1995. [MK] J. Morrow and K. Kodaira, Complex manifolds, Holt, Rinehart and Winston, New York-Montreal 1971. [M] J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120, 1965, 286-294. [NN] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Annals of Mathematics (2) 65, 1957, 391-404. [Sch] F. Scheck, Mechanics. From Newton's laws to deterministic chaos, Springer, Berlin 1999. [Sp] M. Spivak, A comprehensive introduction to differential geometry. Vol. I, Publish or Perish, Berkeley 1979. [Weil] A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Mathematics, 6, 1971, 329-346. [Wei2] A. Weinstein, Lectures on symplectic manifolds, CBMS Regional Conference Series in Mathematics, 29, American Mathematical Society, Providence, R.I., 1977 & 1979. 114
[Wei] R. O. Wells, Differential analysis on complex manifolds, Springer, New York-Berlin 1980. [Wu] T. Wurzbacher, Fermionic second quantization and the geometry of the Hilbert space Grassmannian, DMV-Seminar on "Infinite-dimensional Kahler manifolds" Oberwolfach November 1995, Preprint 1998. http://www-irma.u-strasbg.fr/~wurzbach/publications.html
115
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
S P E C T R A L P R O P E R T I E S OF T H E D I R A C O P E R A T O R A N D GEOMETRICAL STRUCTURES Dedicated
to the memory
of Andre
Lichnerowicz
OUSSAMA HIJAZI Institut Elie Cartan Universite Henri Poincare, Nancy I, B.P. 239 54506 Vandceuvre-Les-Nancy Cedex, France E-mail:[email protected] These lectures aim to give an elementary exposition on basic results about the first eigenvalue of the Dirac operator, on compact Riemannian Spin manifolds with positive scalar curvature. For this, we select some key ingredients which illustrate the basic objects and some of their properties as Clifford algebras, spin groups, connections, covariant derivatives, Dirac and Twistor operators. We end by pointing out how the size of the gap around zero in the spectrum of the Dirac operator, increases when the geometrical structure is Kahler or Quaternion-Kahler. We refer to [BHMM] for an extensive and recent study of Conformal Spin Geometry.
Contents 1 Introduction
118
2 Clifford Algebras, Spin Groups and their Representations
119
2.1 2.2 2.3 2.4 2.5
Clifford Algebras Classification of Clifford Algebras The Spinor Representation The Spinor Group The Space of Complex Spinors
119 122 124 125 128
3 Connections on Vector and Principal Fibre Bundles
131
3.1 Vector and Principal Fibre Bundles 3.2 The Connection Form and the Covariant Derivative 3.3 The Curvature
131 134 136
4 Spin Structures and the Dirac Operator 4.1 4.2 4.3 4.4
The The The The
Spinor Bundle Spinorial Levi-Civita Connection Dirac Operator Schrodinger-Lichnerowicz Formula 116
138 138 139 143 148
5 Spectral Properties of the Dirac Operator 5.1 The First Eigenvalue of the Dirac Operator 5.2 Conformal Covariance of the Dirac Operator 5.3 Holonomy and Eigenvalues of the Dirac Operator References
151 151 157 163 167
117
1
Introduction
The Dirac operator (square root of the Laplacian) is a first order, elliptic differential operator which plays a central role in Mathematical Physics, Topology and Geometry. The Atiyah-Singer Index Theorem, the Lichnerowicz Theorem and the Seiberg-Witten Invariants illustrate the importance of this operator in Physics as well as in Mathematics. Unlike differential forms, spinor fields depend on the geometrical and topological data, hence the subtlety of natural operators acting on them. The aim of these lectures is to give an elementary introduction to the subject starting with the building blocks of Spin Geometry such as Clifford Algebras, Spin groups and Spinors, in order to prove some recent results concerning the spectral geometry of the Dirac operator. For convenience, we introduce basic ingredients in Riemannian Geometry as vector and principal bundles as well as the notion of connection form and the associated covariant derivative. Attached to the Dirac operator is the twistor operator, the complement of the Dirac operator, which is relevant in this setup. We explicitely define these operators in terms of projections on vector spaces. We show that there is a gap around zero in the spectrum of the Dirac operator on compact Riemannian spin manifolds and that the width of this gap depends on the geometry of the manifold. A classical reference for basic notions on Spin Geometry is the famous book of Blaine Lawson and Marie-Louise Michelsohn [LM], however in these notes we give a brief presentation of the necessary material needed to illustrate some basic results on the first eigenvalues of the Dirac operator.
118
2
Clifford Algebras, Spin Groups and their Representations
The aim of this section is to present the algebraic ingredients lying at the heart of spin geometry. First we introduce Clifford algebras, which are algebras naturally associated to symmetric bilinear forms on vector spaces. For simplicity, we restrict ourselves to complex Clifford algebras and their representations. In the Clifford algebra of the n-dimensional euclidean space, we construct the spin group Spin n , which is a 2-fold covering of the n-dimensional special orthogonal group, S 0 n . 2.1
Clifford
Algebras
Definition 2.1 Let V be a vector space over a field K of dimension n and g a non-degenerate bilinear form on V. The Clifford algebra Cl(V,g) associated to g on V is an associative algebra with unit, defined by Cl(V:g):=T(V)/l(V,g) where T(V) is the tensor algebra of V, and I(V, g) the ideal generated by all elements of the form x ® x + g(x, x)l, for x 6 V. Remarks 2.2 (1) The Clifford algebra Cl(V,g) is the algebra generated by V with the relation x • y + y • x = —2g(x,y)l, for x,y € V. (2) If ( e i , . . . , e„) is a g-orthonormal basis of V, then { e^ • . . . • eik | 1 < ii < ... < ik < n, is a basis of Cl(V,g), thus dimCl(V,g)
0< k
= 2n.
(3) There is a canonical isomorphism of vector spaces, between the exterior algebra and the Clifford algebra of (V,g) which is given by: A*V-^Cl(V,g) € i j A . . . A C{k I
r 6^ 1 • . . . • 6j f c .
This isomorphism does not depend on the choice of the basis. Examples 2.3 Let Cln denote the real Clifford algebra associated to the canonical scalar product {x,y) — Y12=i x*y* o n ^™ anc ^ ( e i> • • •» e « ) t n e canonical orthonormal basis of W1. Then - a basis of Cl\ is given by { l , e i } . Since e\ = - 1 , one has Ch ~ C . 119
- a basis of C/ 2 is given by {1, ex, e 2 , ex • e 2 }. Since e\ = e\ = (ej • e 2 ) 2 = —1, one has C72 ~ H . - a basis of C/ 3 is given by { l , e i , e 2 , e 3 , ei • e 2 , e 2 • e 3 , e 3 • eu ex • e 2 • e 3 } . One can easily check that, with the help of the following natural linear identifications with the quaternions: {1, ei -e 2 , e 2 • e 3 , e 3 -ei} ~ {l,i,j,fc} and {ei,e 2 ,e 3 ,ei • e 2 • e 3 } ~ {i,j,k,
-1}
leads to CZ3 ~ H e H . Proposition 2.4 (Universal property) Lei A be an associative algebra with unit and f:V—>A a linear map such that for all v E V f(v)2
=-g(v,v)l.
Then f uniquely extends to a K-algebra
homomorphism
f:Cl(V,g)^A. Furthermore, Cl{V,g) is the unique associative K-algebra with this property. Corollary 2.5 Let f: (V,g) —>• (V,~g) be an isometry between euclidean vector spaces. Then f uniquely extends to a K-algebra homomorphism f:Cl(V,g)^Cl(V,g). Remark 2.6 We denote by 0(V,g) the space of isometric homomorphisms of an euclidean vector space (V,g). On the Clifford algebra Cl(V,g), we have two useful endomorphisms: (1) The isometry —Id € 0(V,g) gives rise to a:
Cl(V,g) e* i
• &ih
-^Cl(V,g) i
> (.
1)
e%x • • • •
2
As a = Id, we get the decomposition Cl(V,g)=Cl°(V,g)(BClHV,g), 120
Cik
where Cr{V,g) := {
CT(V,9)
->CI(V)S)
This map is called the transposition. Another consequence of the universal property of Clifford algebras is the following: Proposition 2.7 The Clifford algebra Cln is isomorphic to the even part of Cln+i, i.e., Cln ~
Cln+l.
Proof. Let { e i , . . . ,en} and { e i , . . . , e n + i } denote the canonical basis of E n and R™+1 - with the obvious identification - and define the linear map / : K» —• Cln+l 6j
i
T e^ ' 671-1-1 •
By definition of the function / , we have /(ej) 2 = —1, thus / extends to f:Cln —7 Cln+1. Clearly, / is an injective linear map between vector spaces of the same dimension, thus the map / is an isomorphism. • In remark 2.2, we have seen that there is a canonical isomorphism of vector spaces between Cln and A*R n . The following proposition gives the relation between the multiplication in the Clifford algebra in terms of the exterior and interior products in the exterior algebra. Proposition 2.8 For all v € Rn and all
where A denotes the exterior, j the interior product and
1. If there exists ik such that j = ik then v Aip = 0 and v j
= 2. If j $. {ii,...,
e
i i ' • • • " e i f c _i
- e
i f c + i ' • • • " ei,,
— V • if
(-l)p
ip) then v J (p = 0 and vA
eip
As the equalities of the assertion are bilinear, the proposition is proved. 2.2
Classification
of Clifford
•
Algebras
Now we give some basic assertions which lead to the classification of the complex Clifford algebras. For the real Clifford algebras, we mention without proof the following proposition: Proposition 2.9 For all n G IN: Cln+S ~ Cln
(1)
a 2 m + i = C(2 m ) © C(2 m ) ~ Endc{H2m) © £ n d c ( £ 2 m ) , where C(2 m ) denotes the ring of 2mx2m over C, and T,2m — C 2 . 122
(2)
complex matrices, which is an algebra
Proof. We only give t h e proof for Q 2 m • Let (e\,..., em, em+\,..., e-zm) b e t h e canonical orthonormal basis of E 2 m a n d (zj,~Zj)j=it...,m a W i t t basis of M 2 m O R C, i.e. 1 Zj : = -(ej
and
_ 1 • Zj : = -(e,- <S> 1 + e J + m ® z).
These vectors satisfy for all j , k = 1 , . . . , m t h e equations
3C(ZJ. Zfc) = gc{zj, zfc) 9c(zj,Zk)
= 0,
— gc(zj,~z~k) = 2^'fc i
which yield Zj X Zfc + Zk -C Zj
= 0
Zj -C Zk + Zk -c Zj
= 0
Zj -C Zk + ~Zk -C Zj = — (Jjfc, 2m
since for x,y € C C Q 2 m : £ x 2/ + y -c a; = — 2gc(x, y). For simplicity we write "•" for t h e complex Clifford multiplication "-c"Let ZJ = z"i • ... -Zm a n d observe t h a t ~z~k • ZJ = 0 for 1 < k < m by t h e formulas above. Denote zir := z\x •...- Z[r, for 1 < li < ...
0
2 m
—> E n d ( S 2 m )
" = ZJp • ZKq '
• p(v) = (zLr • ZJ M- Zj p • ZK", • ZL>. • ZJ).
Obviously, p is a homomorphism of algebras. We show now t h a t p is surjective. First take v = Zj • ~2\ for 1 < j < m: p(v) (zi • ZJ) = Zj • ~z~i • zi • ZJ = Zj • ( — Zi • Ji — 1) • ZJ = —Zj • Zi • Zi • ZJ — Zj • ZJ = —Zj • ZJ
0
a n d for 2 < I <m p(y) (z( • ZJ) = Zj • ~zx • zi • ZJ = Zj • (-zi - Z i ) -ZJ
=0 123
Similarly, one computes that the image under p of a basis vector of E2 m not containing z\ is zero, whereas the image of a basis vector containing z\ is the same vector with z\ being replaced by Zj or —Zj. Denote z\ •... • zm by w. Then, for elements of the Clifford algebra of the form v
= ZJ„ • ^ • w • ~*Kn •= Zji • • • • • ZjP • w • w • z k l •. • • • z h q
the map p(v) replaces zkl • • • --Zkq by ±Zj1 •.. .-Zjp in the basis vectors of S 2 m , whereas p(v) maps all other basis vectors of S2 m to zero. Thus p is surjective, and since d i m 0 2 m — dimEnd(C?"'), it is bijective. D The above considerations yield to the classification of the complex Clifford algebras. For completeness, we also include the real Clifford algebras in the following table: n
1
2
3
4
Cln
c cec
H
Hen
C(2)
C(2) e C(2)
H(2) C(4)
on
2.3
The Spinor
5
6
7
8
C(4)
E(8)
E(8) e E(8)
M(16)
C(4) 0 C(4)
C(8)
C(8) © C(8)
C(16)
Representation
Definition 2.13 In even dimension, the complex spinor
representation
p : Q 2 m —)-End c (S 2 m ) is the isomorphism of Proposition 2.12, whereas in odd dimension, it is defined to be the projection onto the first component of the corresponding isomorphism. Definition 2.14 The Clifford multiplication is the map tn: Q 2 m ® S 2 m —> S 2 m ip® a
i—> ip • a : =
p{ip){a).
Proposition 2.15 Let ( e i , . . . , e n ) fee a g-orthonormal complex volume element wc = i^"2 ' ei • . . . • e n 124
basis of W1.
The
of Qn
satisfies
(1)
<jj%. = 1
(2)
X-LOC =
and
( - l ) n _ 1 a ; c • x for all x € Rn C Un,
where [ ] stands for the integer part. This yields the following proposition. Proposition 2.16 If n = 2m is even, the complex spinor restricted to the even part of the Clifford algebra, p°:a°2m
^
representation,
Endci^m)
admits a decomposition
where
Moreover, p°{x):Yr^m i E f \ {0}. 2.4
The Spinor
-» T^m is an isomorphism of vector spaces for each
Group
Denote by CZ* the multiplicative group of units of the real Clifford algebra Cln, i.e., the subgroup Cl*n = {
= tp'1 • (p = 1 } .
Definition 2.17 The pin group of Cln is the subgroup Pin n of Cln, denned by Pin„ = {xi •... • xk I Xj e l n ,
HZJII
= 1}
The spin group Spin n then is defined to be Spin n = P i n n n Cl°n. In other words, the spin group is the multiplicative subgroup of Cl*n generated by even products of vectors of length 1, i.e., Spin n = {x1-...-x2k\xjE Remark 2.18 125
K n , ||a;j|| = 1 }.
(1) The inverse of an element (p = Xit •... • Xi2k G Spin„ is given by i((p). (2) Denote by Cln the Lie algebra of the Lie group C7* and by spin n the Lie algebra of Spin n . Then <£l* is isomorphic to Cln, the Lie bracket being [f,il>] =
dn
y 1—• u- y-
u'1,
u G Cln. Proposition 2.20 For all x G K n , ||x|| = 1, the map Adx is an endomorphism ofW1. Furthermore, —Adx is the reflection across xx. Proof. For i £ l " , ||a;|| = 1, we have aT 1 = -x in Cln. Thus, for y G E", -Adx(y)
= x-y-x
= x- (-x • y ~2g(x,y))
=y-2g(x,y)x
a For u — xi...x2k
1
G Spin„ and y G MJ , we get Adu(y)
=u-y-u~l = u • y • tu
= xi •... • x2k • y • x2k • • • • • x i = Ad Xl
o...oAdX2k(y).
This is an even composition of reflections of E n , hence an element of SO n . By the Cartan-Dieudonne theorem, every element of SOra is a product of an even number of reflections. We thus have the proposition: Proposition 2.21 The linear map Ad\g ^n : Spinn —> S0n is surjective. 126
Ad|g j
n
is not injective, but we have:
P r o p o s i t i o n 2.22 The sequence 0 —> Z 2 — • Spinn
AH
^ 4 SOn —• 1
is a s/iori exact sequence. Furthermore, if n > 3, Spinn is the universal covering of SOn. Proof. An element u £ Spin n C C7° can be decomposed into u = a0 + e\ • a±, such that ao £ C7° and ai € Cl\, ao and ai not containing e\. Thus ao • e\ = ei • ao and e\ • a,\ • e\ = —e\ -e\-a\ = a\. Suppose now that u is in the kernel of Ad, i.e., for all y € E n Ad„(j/) = y-&u-y = y-u. For y = ei, we then get: (ao + ei • ai) • ei = ei • (ao + ei • ai) =>• ao • ei + ei • oi • ei = ei • ao - ai ==> ai = —ai ==> ai = 0.
Hence u does not contain e\. As the same procedure works for all the e / s , we get u £ {—1,1} and kerAd = {—1,1}. To prove that the covering is non trivial for n > 3, it suffices to find a continuous path in Spin n which joins —1 to the 1. One can easily see this using the path: -y(t) = (cos(-)ej + sin{-)ej)
• (-cos(-)ei +
sin(-)ej)
= cos(t) + sin(t)ej • ej.
(3)
• Proposition 2.23 The homomorphism Adt,:spmn —> son(— A 2 E n ) between the Lie algebras associated to Spinn and SOn is a vector space isomorphism. It is given by \Ad*{ei -ej)J(y)
= ifa/Ke^^y)
= 2a(e i ,y)e 3 -
2g(ehy)ei
for 1 < i, j < n. Proof. Consider the path 7(f) defined in (3). As 7(0) = 1, ^L_ 0 7(*) = &i • ej and spin n is isomorphic to TiSpin n , we can assume that e^ • ej lies in spin n , 1
= jt\t=0(Ad,{t)(y))
= | U ( 7 ( « ) • V • 7" 1 (*))
= 7 ' ( 0 ) - y • 7(0) + 7(0) • y • (7 - 1 )'(0) — €i • Bj ' y
y • Bi • €j
127
= ei-ej-y-
(-a
• y - 2g{ei, y)) • ej
= ei-ej -y + ei- {-ej • y - 2g(y, ej)) + 2g(a, y)ej =
2(eiAej)(y)
which is the stated formula and additionally proves that Ad* is surjective. Since the dimensions of both, spin n and A 2 E Tl , are equal, Ad* is an isomorphism. D We end by emphasizing that the spin group Spin„ is a compact, connected, simply connected (for n > 3) Lie group of dimension n\n~1>. 2.5
The Space of Complex
Spinors
For simplicity, we restrict to the complex case. Definition 2.24 Let p:€ln ->• End(S n ) be an irreducible representation of O n . T h e n the restriction of p to Spin„ />:Spm„ —> Aut(£„). is called the complex spinor representation and E n the space of complex spinors, dimc(£„) = 2[*]. Proposition 2.16 can now be formulated as: Theorem 2.25 If n = 2m is even, the complex spinor representation of Spin^m decomposes into p = p+ +p~. That is, the space of spinors decomposes into positive and negative spinors, £2m =
S
2 m © S2m>
where
S
2 m = § ( 1 ± <*>c) ' %2m, SO that
p±: Spirhm —• Furthermore, for x £Rn
Aut(Ztm).
\ {0}, the map
a
i—> x • a ,
±
is an isomorphism. The maps p are inequivalent, irreducible, complex representations of Spin^jn • For n — 2m + 1 odd, the spinor representation is irreducible and does not depend on the projection on the components of J5nrf(S2TO) (&End(Y,2m) chosen in (2.14). 128
P r o p o s i t i o n 2.26 ( T h e n a t u r a l H e r m i t i a n p r o d u c t ) There S n , a natural Hermitian scalar product such that {o-i,o2) =
exists
on
(x-ai,x-o2)
for every i £ l " , ||a;|| = 1, and o\, o~i 6 S n . Proof. Let Tn be the multiplicative subgroup of C7* generated by a gorthonormal basis {ex,... ,en} of W1. Using the relations (—l)2 = l , e 2 = —1 and ei • ej = —ej • e^, 1 < i, j < n, i 7^ j , we see that Tn is finite and | r n | = 2 n + 1 . Now we choose an arbitrary Hermitian product (•, •) on S„ and define for o\, a2 £ S n (CTI,
= 7fT-r 5 Z ( I / - c r i- I / - 0 '2>-
First, for ej € r n , it follows (ej • ox,ei • o2) = jp-r J~] (^ • &i •
= jf~i S ^ ^ 1 ' " " ^ = (ffi,
Then, for i £ P with ||a:|| = 1, i.e., a; = Y^i xieii with X}j£ 2 = 1, we get (x • ai,x • a2) = ^2xi(ei'
CTl ei
' '
ff2
)
+
X ^ ^ ^ ' ^ ' ai'eJ
'
ff2
)
= ^2x^(oi,a2) i
+ Y^
XiX
i (( ei • °"i > e j - ^2) + (ej • ox, ei •
= (0-1,0-2)
since, for i < j , one has (ei • ox, ej • o2) = (ei • ej • 01, e< • e^ •
CT2)
= - ( o i , e i - e ^ -<7 2 )
— — (ej • oi,ej • ei • ej • a2) = (ej • ox,ei- ej • ej • o2) = -(ej -ax,ei • a2) D
129
Corollary 2.27 For all x 6 R™ and for all o\,
=
-(<TI,X-O-2)-
Proo/. Let x be in E™ \ {0}. Then (x •
• a2) and
(X-CTI,
130
3
Connections on Vector and Principal Fibre Bundles
For convenience, we introduce the notion of vector and principal fibre bundles [KN] and their correspondence. We then define the notion of connection form and the associated notion of covariant derivative. With this, we canonically define the covariant derivative on the spin bundle in terms of the Levi-Civita connection on the tangent bundle. 3.1
Vector and Principal
Fibre
Bundles
Definition 3.1 A vector bundle of rank N over K = 1 or C is a triple (E, 7r, M) such that i) the projection •K: E —» M is a smooth map between finite dimensional smooth manifolds, ii) for all x G M, the fibre at x, Ex := 7r_1(a;), is an TV-dimensional vector space over K, iii) for every x G M, there exists an open neighbourhood U C M and a diffeomorphism, called local trivialisation, <j>:ir-1(U) -+U
x1KN
such that for all y G U,
is an isomorphism of vector spaces. Examples 3.2 1) The trivial bundle M x RN. 2) The tangent bundle TM of a smooth manifold M. Proposition 3.3 (Transition functions) Let (E, ix, M) be a vector bundle and (Ua, (pa)a€A o. cover of local trivialisations. Then the transition functions (ppai UanU0—>
GL(N,K)
=: GLN,
defined by
4>0o4>a1-(uanUp) x K s - ^ (uanUp) (x,v)
i—> (x, 131
xKN
satisfy the cocycle condition V7/3 °
be a cover of M and let
ip0a'- Ua n Up -> GLw 6e differentiable maps satisfying
/ / we define
E:=(l[[UQxK ^aeA
/ ~
where (xa,v)
~ (xp,w) :<=$> x = xa = X/3 E UaC\Up
and
w =
then E defines a rank N vector bundle over K . Definition 3.5 Given a Lie group G, a G-principal fibre bundle is a triple (P, n, M) such that i) 7r: P —> M is a smooth map between finite dimensional smooth manifolds, ii) G acts smoothly and freely on P from the right, i.e., the action PxG —> P satisfies p 9 ~ p iff g = e £ G, iii) for every point x € M, there exists an open neighbourhood U C M and a diffeomorphism, called local trivialisation,
p^(n(p),V(p)),
such that
For a cover (Ua,
4>Po0"1:(£/anf/^)xG^
(c/ttnu0)xG
(x, g) i—• ( i , ^ ( x ) 9).
and again get the cocycle conditions tp7p o ippa = y>7Q. As in Proposition 3.4, one can reconstruct the principal fibre bundle out of the transition functions. Examples 3.6 1) Every smooth manifold Mn comes naturally with the GL n -principal fibre bundle GLM -^+ M with typical fibre at x € M (GLM)X = {ordered bases of TXM} . 2) The natural fibre bundle of an oriented Riemannian manifold ( M n , g) is the SO„-principal fibre bundle SOM of oriented orthonormal bases at x € M, i.e., (SOM) x — {positive oriented orthonormal bases of TXM}. Let (P, 7r, M) be a G-principal fibre bundle. We take a finite dimensional representation p:G—>
End(S)
of G on a vector space E and define an action of G on P x £ as follows: (PxS)xG —>Px£ (p,v,g) 1—>• (pg,p{g~l)v). Dividing P x £ by the equivalence relation (p,v) ~ (p g,p{g~1)v) associated vector bundle
gives the
E := P x„ £ = (P x £ ) / ~ = (P x E ) / G . The transition functions of £ are p o <^gQ, where
On the other hand, let (E, n, M) be a vector bundle of rank N with structure group G, i.e., G C GL/v is a closed subgroup. If the transition functions ippa of a cover of local trivialisations (Ua, 4>a) can be chosen to have values in G, this defines a G-principal fibre bundle P, to which E is associated. We obtain a representation of G by composing the inclusion map of G into GL/v and the canonical representation of GL/y o n K " . 3.2
The Connection
Form and the Covariant
Derivative
Definition 3.8 Let (E, n, M) be a vector bundle. A covariant derivative is a linear map V:V{E) —>r(T*M
V(/V>) = d/ ® V- + / V^. Remark 3.9 If we take a vector field X e T(TM) and evaluate Vxip at x € M, then (Vxif>)(x) o n l y depends on the vector Xx and the values of ip in an arbitrary small neighbourhood of x. Definition 3.10 Let (P, n, M) be a G-principal fibre bundle. For any point p € P, there is a canonical injection ~ : fl —• TPP z^zp=
-g\t=d0(p exp(te)).
Its image is called the vertical space Vp and is the tangent space to the fibre .7r -1 (p), i.e., Vp — ker(7r*). Definition 3.11 A connection in a G-principal fibre bundle (P, n, M) is a distribution of n-dimensional vector spaces p >-> Hp C TPP, the horizontal spaces, such that i) TpP = Vp®Hp, and ii) it is G-invariant, i.e., Hpg = (Rg)^(Hp), The projection n induces an isomorphism
where R s : P -» P, p t-t pg. TT*\HV'-HP
-> T„.(p)Af.
Proposition 3.12 Let (P, IT, M) be a G-principal fibre bundle. The decomposition of TpP by a connection allows us to define a 1-form u on TP with values in the Lie algebra g of G u v TpP =
Vp®Hp^Vp^Q, 134
that has the following properties i) w p (z) = z, where ~z is defined in (3.10), ii) R*u> — ad(g~1)cj,
i.e.,
VX £ T(TP),
u>((Rg)*X) = adig-1)
u(X),
where ad:G —> End(g), g i-*- dag and ag:G —> G, o n gag~x. Conversely, a 1-form on TP with values in g which satisfies i) and ii) defines a connection on P by Hp := kerWp. For a connection 1-form w on a G-principal fibre bundle (P, n, M), we define a covariant derivative on every associated vector bundle E = P X n L clS follows: Take a section ip £ T(E), which is locally given by ip = [s, a], where s £ Tu{P) is a local section o n ( 7 c M and a: U -> E, a function. Since
TU^TP-^g-^
End(E),
we can define a covariant derivative on E by Vxip~[s,Xo-
+ p*({uos*){X))o}
(4)
for any X £ TU, where Xa denotes the Lie derivative of a in the direction of X. Conversely, given a K—vector bundle (E, n, M) of rank N and a covariant derivative V, we consider N linearly independent local sections of E 8 = (il>i,...,il>N):U-*GLE,
UcM
and define 1-forms u>Ra by N
V x V ' a == ^2u/3a(X)ll>0
(5)
for all X £ T(TU). ^ Now, there is a unique connection form w on the GL(K Ar )-principal fibre bundle GLE such that for any local section s £ Fu(GLE) s*u := u> = (u>pa)i
135
Remark 3.13 If the vector bundle E is endowed with a metric g and a metric connection V, i.e., Xg{fl>,ip)=g{Vx1>,
V^ver(£),
X €
T(TM),
then, the corresponding matrix of 1-forms (u>0a) is skew symmetric with respect to an arbitrary orthonormal frame s = (ipi,... ,ip^):U —> SOE. It is therefore an element of the Lie algebra SOJV of SO^r- To see this, take X e T(TM) to get 9(Vxipa,ipp)
= -sW^Vx^/s)
for the metric connection V. Thus
up*(X) = g(^2u-ya(X) V>7,#) 7
=
3.3
The
g(yxll>a,1pl3)
=
-g(f/>a,Vx1pl3)
=
-wap(X).
Curvature
For a G-principal fibre bundle (P, TT, M) with connection form u;, define the curvature 2-form ft by:
ner(A 2 (TP)® f l ) Sl(X, Y) = dw(X, Y) + M X ) , u ( Y ) ] ,
X, Y e T(TP)
As in Remark 3.13, with respect to a local section s = (r/>i, • • •, I/>N)'-U ->• SOE, U C M, let ft = s*£2, then one gets the following relation AT Qa/3 — ( d w ) a ; a + ^ ^ U)ai A W 7j g. 7=1
Definition 3.14 Let (E, IT, M) be a vector bundle with a metric connection V. Define the curvature tensor 7L: T(E) %• T(T*M ® E) 4 T(A 2 (T*M) ® £ ) , where V(a
VT/> .
(6)
Proposition 3.15 For a local section s = (tpi,...
,IPN)
G
we have
TU(SOE)
N 0=1
Proof. By defintion of V and (5), it follows V(V(V>a))=v(5^w/3a®^)
/3
^
7
'
= 5 3 ( dwi8« ~ ^ 1 <*V A UPf ) ® ^ 0 V / 7 =
5 2 ( ^W|8a + 2 2 ^ T A W^a ) ® ^ 18
^
7
'
D
Proposition 3.16 Definition 3.14 coincides with the usual one, i.e., Kx,y = [ V x , V y ] - V [ x , y ] . Proof. Take local sections X, Y e T(TM), and ip G Tu(E), then there are on G rV(T*M), i = 1 , . . . , n , and V>/3 G rV(E), /3 = 1,...,N such that
vv> = y ^ on ® y>g. Using (5) and (6), one gets V(V
- ^ai(A-) - «;([*, Y])) 1>p
i,0
-ai(X)VYil>p+ai(Y)Vx1>(i = Yl Vx(ai(Y)iP0)
- Vy(ai(X)^) -
ai([X,
Y])j>[>
i,p
= (VXVy - VyVx - V^.y])^D
137
4
Spin Structures and the Dirac Operator
Here we introduce the notion of spin structure which is needed to globally define the spinor bundle. We then extend to this bundle the algebraic properties introduced in Section 2 and compute the local expression of the spinorial covariant derivative as well as the spinorial curvature. We then end with the definition of the Dirac operator, its basic properties and give a proof of the Schrodinger-Lichnerowicz formula. 4-1
The Spinor
Bundle
Definition 4.1 Let (M™, g) be an n-dimensional Riemannian manifold. A spin structure on M is a pair (SpinM, 77), where SpinM is a Spin n -principal fibre bundle over M and r] a 2-fold covering such that the following diagram commutes: SpinM x Spin„
• SpinM
77 x A d
n
77
M
SOM x S 0 n • SOM * The maps in the rows are respectively, the actions of Spin„ and S 0 n on the principal fibre bundles SpinM and SOM. The existence of a spin structure on M is equivalent to the fact that, for the transition functions (ppa of SOM, there is a choice of lifts to transition functions of SpinM, i.e., the diagram Spin n ffia Ad MDUanU0 • SO n commutes and the
where p: Spin n -*• Aut(S n ) is the complex Spin n representation, £ „ ~ CN and N = 2^1 (see Definition 2.24). A section T/> G T(EM) is locally given by i>\v = [s,
where ? G r [/(SpinM), U C M andCT:{/ ->• £ „ . ii) The Clifford multiplication on E M is the fiberwise action given by m:
TM ® EM
—> E M
X ® •(/> = [?, a] ® [?, a] i—• [?, a • a] =: X -tp, where a • a is the Clifford multiplication on E n (see Definition 2.14), where the tangent bundle TM is seen as the associated vector bundle TM ~ SpinM x A d E n . iii) The natural Hermitian product on E M is defined by (•,•) : T(EM) x r ( E M ) —•» C°°(M,C) V'®^
i—>(il>,f),
where for all a; G M, (ip,(p)x :— (ipx,
The Spinorial
Levi-Civita
(7)
Connection
Take a simply connected open subset U C M. Then any local section s G rV(SOM) lifts to a section s G IV(SpinM), i.e., SpinM V U CM —
SOM
and we can define a connection 1-form w on SpinM as the unique connection 1-form for which the following diagram commutes 139
(8)
TUcTM The connection 1-form u is given by the Levi-Civita connection on (M, g). To get a local description of the associated covariant derivative V on E M , take an orthonormal frame s — ( e i , . . . ,e n ) e IV(SOM), U C M, and denote by: w
:=
s*u> = - Y2i<j Uij ei A e^
ft := s *fi = - J2i<j &ij ei
A e
j>
where e» A ej :=
(9)
is a basis of son. We then get: uJij(X) = -g{u{X)euej)
-
-g(Vxei,ej)
(10)
for all X G T(TM). Proposition 4.3 (local description of V and 1Z) 1) The lifted connection 1-form u satisfies S{s.{X))
:= S(A-) = —^UijiX)
ei • ed.
(11)
i<j
2) Take an orthonormal basis <j\,..., erjv of S„ ~ C {i>a)i
to get a local section
Then the spinorial covariant derivative is given locally by: 1
n
(12) «,i=i
3) Finally, if R denotes the Riemann curvature tensor of the tangent bundle, then for the spinorial curvature tensor one gets: 1 " R ei e
Kx,Yip = 7 5Z 9( x>Y > i) «i • e, • V • »,i=i
140
(13)
Proof. 1) From (8) and Proposition 2.23 follows: 5(X) = ( A d ; 1 o w o S , ) ( I ) -Ytu>ij(X)Ad:1((eiAej))
=
(see Proposition 2.23)
2) This is a direct calculation using p* — p (since the representation p is a linear map from the Clifford algebra to the vector space of endomorphisms End(S n )): Vipa = \s, p*(u> o ?*)(7a],
(see Section 3.2)
= P,P*(~2 ^2^ij e; • ej)aa]
(by (11))
i<j
= — - y j w y a • e, • ipa,
(see Definition 2.14)
i<j
= 2 Yl 5(Ve i ; ej) et • e, • ipa
(by (10))
i<3
~ 4 ] C ^(Vei, ej) e; • ej • ipa. 3) This follows directly from Proposition 3.16 and (12)
• Proposition 4.4 (Compatibility of V with "•" and (•,•)) XW,
e„) and ^»a as in (12).
1) For ip = ipa and v> = V/3; f° r
an
y vector field X, one has
141
(14) (15)
=
4 5Z
1
9iyxei,ej){ipa,erei-il)i3)
" n
i 4
which combined with the fact that {ij}a,ipp) = &ap yield (14). For arbitrary sections, we use bilinearity together with X(M,
e» • ej • (ek • tpa)
4 Yl fl(Ve»> e j) efc • e« • e j • V'o + 2 5Z #(Vefc) eJ') e i ' ^« ij
i
~'2^2l9{^ei^k)ei
-ipa.
i
We change i to j in the last term to get =
7 ] C s( Ve *> ej) efc • e ; • ej -ipa + ^2 #(Vefc> e i ) e j " ^
= efc - V i + (Vefc) • VQ,. For arbitrary Y and V), formula (15) is straightforward as in (14).
• 142
4-3
The Dirac
Operator
In the following we will often use a local orthonormal frame, denoted by s = ( e i , . . . ,e n ) e rV(SOM), U C M, which yields the relation ej • ej + ej • ei = -2<5jj
1 < i, j < n.
In the previous section, we have seen that associated to a spin structure of a Riemannian manifold (M™, g), there are three essential structures: i) The spinor bundle E M = SpinM x p E„ with the Clifford multiplication m: TM ® E M —> E M X®i>
i—•»• X • i> := p{X)ip ,
where p is t h e spinor representation. This multiplication extends t o m: A P ( T M )
i—>
^2
"tL.-v e n ' ••• ' e v
-
^>
l
where locally l
and e* = g(ej, •) is the dual basis of e^. ii) The natural hermitian product (•, •) on sections of EM. iii) The Levi-Civita connection on E M . As we have seen in (7) and Proposition 4.4, these structures satisfy the following compatibility conditions: X(xl>, if) - (Vx, if) - (V>, Vxv) = 0 , VX(Y • i>) - VXY • iP - Y • Vxi> = 0 , for all X, Y 6 F(TM), V»,
Locally, we get V:T{T,M) A
T(T*M®
EM) - A T(EM)
Lemma 4.6 TTie commutator of the Dirac operator with the action, by pointwise multiplication on the spinor bundle, of a function f: M —> C, is given by
[2>,/]V:=d/-v,
ver(SM).
Proof. A local calculation shows that [ V, f ]tf = (2?/ - /I>)V = £2=1 e* • V e ; (/V) W = E t i d/(e») ei-iP + fVi>- fTh/> = df-i(,.
a L e m m a 4.7 T/ie Dirac operator is a first order partial differential operator, which is i) elliptic and ii) formally self-adjoint with respect to (•, -)L* := JM(-, ^Vg, if M is compact, where vg denotes the volume element. Proof. i) Let x E M, £ € T£M \ {0} and / G C°°(M, K) such that (df)x = f, then the principal symbol, cr^{V) : T,XM —> S X M, is given by
atCD)^))
:=V[(f - f(x))i>](x) = {fVil> + &f-rl)-f{x)Thl>)(x) = {df)x-i>{x)
i.e., cr^{T>) is Clifford multiplication by £. To see that T> is elliptic, we have to check that, for all £ <E T*M \ {0},
f.^ = o=>-f-£-v = o-<=> -||£||V = o <^> v = o. 144
ii) To show that T> is self-adjoint, we choose normal coordinates at x € M, i.e., (V ei ej)(a;) = 0, 1 < i, j < n, and compute first n
i=l n i=l n
i=l n *
i=l
after using (14). The sum in the last term then is the divergence of a complex vector field. To see this, consider the two vector fields X i , X2 € T(TM) defined for all Y £ TM by g(X1,Y)+ig{X2,Y)
= W»Y-
Then, it follows divXi+idivX2
=J29(^ekXuek)+i^2g(VeiX2,ei) fc=i i=i n = J2 (ekg(Xuek) - g(Xu Vefcefe)) fc=i n
+ i^2(eig(X2,ei)
-
g(X2,Veiei))
i=i n
^2ek(g(Xi,ek)
+ig{X2,ek))
k=i n
fc=i Finally: (Dip, ip) = - d i v Xx - i div X2 + (tp, Dip). This equation no longer depends on the choice of the coordinates, so we can integrate over M and get f (Drl>,
[ JM
145
(
since dM — 0. D Lemma 4.8 Let n = 2m, then i) I > : r ( S ± M ) —>• r ( S : F M ) , i.e., the Dirac operator sends positive spinors to negative ones and conversely. ii) The eigenvalues ofV are symmetric with respect to the origin. Proof. i) Recall that E ^ := | ( l ± w c ) - S n (Proposition 2.25), so E+ is the subspace on which multiplication with uc is the identity, and S~ the one on which the multiplication with wc is minus the identity. We therefore get for
i>+ e r(s+M): n
n
ii) Let ip be an eigenspinor for V, i.e., Vrjj = Xxp for A € R and decompose TP = ip+ +yj~. Then X>+ + £ t y - = A^~ + \ip+, yields with i): Xty* = A^^. So the spinor ip :— V + ~ V' - i s a n eigenspinor of V associated with the eigenvalue —A, since V^> = £>(>+ -rp~) = \ip~ - A^ + = -\{4>+ -rp~) = -Xtp.
a Examples 4-9 i) Let M = W1, S I T = Kn x CN, N = 2^1, then every spinor field tp G T(SE n ) is in fact a map ip:Rn -*• CN, and the Dirac operator is given by n
V-^ei-di, i=l
which acts on differentiable maps from Rn to C ^ , where d{ = V e i . Then
^1=1
' ^ j=i
146
'
= ^2 ei • ej • didj J 3
i
i<j
i>j
i
i<j
i<j
-J2di+J2ei'
e
i ' \didi ~ didi)
= -53* i=l
ii) In the two dimensional case M = E 2 , we have Q2 = C(2) and the decomposition £2 = ^t ®^2 = C © C, which is S j = span c (ei + ie2) and Y>2 — span c (l — iei • e2). Then, each spinor field ip 6 T(EM) is given by two complex functions / , g: E 2 —» C, such that V» = / ( e i + «e2) + p(l - iex • e 2 ). The Dirac operator now becomes Vi}> = (ei •d1+e2-
d2)[(e1 + ie2)f + (1 - iei • e2)g]
= -(di + id2)f (1 - i d • e 2 ) + (di - i92)ff (ei + se2) = 2{-dzf
(1 - ei • e 2 ) + &*# (ex + ie2)),
where dj — \{d\ +id2) and dz = \{di — id2). That is 0 2dz -2dj 0 in the basis {(ei + ie2), (1 — ie\ • e 2 )} of S 2 . Hence the Dirac operator V could be considered as a generalization of the Cauchy-Riemann operator. iii) The Clifford bundle CIM. For a Riemannian manifold (Mn,g),
define the vector bundle CIM by
{CIM)X = the Clifford algebra of
(TxM,gx).
We can see this bundle as an associated vector bundle to SOM. By the universal property (2.5), one can extend p n :SO n ->-SO(E n )
to 147
pn:SOn-+
Aut(C/ n ),
so that CIM = SOM x ? n Cln. Prom (2.8) we have v • tp ~v
Atp — v Aip
under the isomorphism Cl(TxM,gx) -=•» K*(TXM). its adjoint could be locally written as n
The differential and
n
d = 2_J ej A V e i
and
8 = - V ] a j Ve; -
i=l
1=1
If we define the Dirac operator as before, we then have n
£ > : = ^ e i - V e i c±d + 6. This is the "square root" of the Laplacian, since on A* (TM) V2 ~ (d + 5)2 = dS + 8d = A . 4-4
The Schrodinger-Lichnerowicz
Formula
Definition 4.10 [Extension of (•, •) and V] i) Extend the natural Hermitian product (•,•) on F(SM) to sections of T*M
(w,»?) = 5Z( a, ( e ^' ? '( ei )) i=l
for any orthonormal basis { e i , . . . , e„} in TXM. ii) Let V* be the formal adjoint of V, i.e., V*: T(T*M
Lemma 4.11 In local normal coordinates ( e i , . . . ,e n ) at x G M, we have: n *=1
for allipGT&M). Proof. n
( V * V V > , ^ ) L 2 = ( V ^ , V ^ ) L 2 = ^ ( V e ^ , Vei¥>)L2 As in the proof of Lemma 4.7 ii), we get: n E ? = l ( V e i V ' , V e , v ) = 5 3 e i ( V e s V , V) *=1
(Ve,Ve^,
= divX x + idivX2 - ^ ( V e i VeiV,
which, by integration, gives the required condition for V* to be the formal adjoint of V. D Proposition 4.12 Let 1Z := | E i \ = i ei' ej ' ^-ei.e,-* where 1Z is the spinorial curvature. Then we get for the square of the Dirae operator:
Proof. Take normal coordinates at x € M, then
n
v
Ve +
= - 5Z « < n
n
S
ei
" ei'Ve-Ve;
n
Ve Ve
+
= - Yl - t=i
H ei • er (Ve^ V«J _ v«,- v «.) t<j
l
n
= V * V + - J Z ei-e^ •?£«....„..
D
149
Theorem 4.13 (The Schrodinger-Lichnerowicz formula) Let S be the scalar curvature of M, then V2 = V*V +
-Sldr(xM).
Proof. By Proposition 4.12, it is sufficient to show that TZ = | 5 I d r ( E M ) . Let Ric be the Ricci tensor of the Riemann tensor R and use Proposition 4.3 for 7£, to get
=
g 5Z ( 5Z R^'e* -erek\
= 7 ^ ;
2 ^ (Rijfe/ + R-jfcii + Rkiji) e-i • ej • ek
+ ^2g(Rei,ejei,ei) =
• e,
ei • ej -a + ^ g ( R e i , e i e j , e ; ) e, • e,- -ej)
« X ) ff(Rei,e3ej,e;) ej - g(R e i , e j ej,e/)ei
• et
• e(
= - ( V " -Ric(ej,ei) e^ • et - V*Ric(ei,e;) e, • e; )
where we have used e^ • ej • ek = e.j • ek • et = ek • e; • ej for i ^ j ^ k.
150
D
5
Spectral Properties of the Dirac Operator
We are now ready to study the spectral properties of the Dirac operator. For this, we start by showing that Friedrich inequality could be obtained as a corollary of the Cauchy-Schwarz inequality. We then examine the limiting case which is charaterized by the existence of a Killing spinor. In this set up, the Twistor equation and the Twistor operator appear naturally. The conformal covariance of the Dirac operator is then used to relate its first eigenvalue to that of the scalar conformal Laplacian. The last subsection is devoted to give a brief presentation, without proofs, of recent results concerning the gap in the spectrum of the Dirac operator for different holonomies. For a complete overview with proofs, we refer to [BHMM]. 5.1
The First Eigenvalue
of the Dirac
Operator
Theorem 5.1 [Lil] On an n-dimensional compact Riemannian fold (Mn,g) with positive scalar curvature S we have
spin mani-
i) ker£>= {0}. ii) IfVip — \ip for a non-trivial spinor ip 6 T(SM), then A2 > \SQ, where So := minM S. Proof. i) By the Schrodinger-Lichnerowicz formula, (see Theorem 4.13), for any non-trivial spinor field rp € T(SM), we have
/
|PV|2^= /
\V1>\2"9+ [
\s\^vg.
Thus Dip cannot be identically zero. ii) Let 2?V = M> f° r
a
non-trivial spinor ip £ T(EM). Then
/ l^l 2 " 9 -\[ SM\ = f |vvf^ >o. JM
* JM
JM
Hence, A2 - \S0 > 0. If A2 - \S0 Vyj = 0, which by i) is impossible.
= 0, then Vtp = 0 and therefore
• 151
Remark 5.2 On a compact Riemannian spin manifold, if the scalar curvature is positive, then IndAT>+ := dimkerX>+ - d i m k e r D " = 0. By the Atiyah-Singer Index theorem, this analytical index is equal to the topological index, which yields a topological obstruction for the existence of positive scalar curvature metrics. In this setup, another beautiful application of the Index theorem is given in [VW] where qualitative information on the gap between the eigenvalues of the Dirac operator is obtained. T h e o r e m 5.3 [Frl] Given a compact Riemannian spin manifold (Mn,g), then any eigenvalues A ofT>, satisfies Friedrich's inequality in — 1 where So := minj^ 5 as before. Proof. The proof we give here is due to S. Gallot [Ga2]. It is based on the Cauchy-Schwarz inequality. For an arbitrary spinor ip £ T(SM) we have n
2
\ 2
n
( £l|Ve^l) i=l
\ 2
/ n
n
'
By the Schrodinger-Lichnerowicz formula we get f -n| 2 t y | 2 v3<\ |W>| 2 ug= f |PV| 2 v JM
JM
g
JM
=•
(1 - I ) / n
|2?V|2 »
JM
S\1>\2 vg
- 4\ \
JM
g
Sty\2
>\l 4
vg.
JM
For a non-trivial eigenspinor x/> € T(EM), it follows that JM
4n-iy
4n-i
M
JM
a Remark 5.4 If r/> £ r ( £ M ) is an eigenspinor for which equality holds, that is 4n-l Then ^ satisfies the twistor equation Vxip + -X-Vrp n 152
= 0,
for all X £ TM, which specializes to the Killing equation
\>xiP + -x-i> = o,
vxer(TM),
n since ij> is an eigenspinor. Proof. In case of equality in (16), we get by Cauchy-Schwarz e» • VeiV = ej • V e j V, Vi, j , 1 < i, j < n, which is equivalent to Vi, 1 < z <
ra,
Vip = net • V e V <=> Ve,^ + -e» • I t y = 0. n D
Definition 5.5 For a Riemannian spin manifold the twistor operator is defined by V: r ( S M ) A
T(T*M ® EM) - ^ T(ker m)
(16)
where 7r is the orthogonal projection on the kernel of the Clifford multiplication m. Recall that the Dirac operator is defined by V := m o V. In fact, the twistor operator is the complementary of the Dirac operator in the following sense:
J ^ r(SM)
v
T(SM) —
r(T*M®£M)
(17) n
r(kerm)
where the composition above is the Dirac operator V and the composition below, is the twistor operator V. Remark 5.6 In fact (17) is given by the composition of the covariant derivative with the projections on the Spin„-irreducible components of T(T*M ® E M ) . This is a special case of a general situation developed in [5W]. In [Fe], it is shown that all such differential operators are conformally covariant (see next section where this fact is proved for the Dirac operator). We refer to [Brl,Br2] for further results in this direction, based on a systematic use of representation theory. Proposition 5.7 In local coordinates we have
Vip^Yl,^®
(Ve^ 153
+ -ei • Vip). n
Proof. The projection n is given by v
®VJ
i->- v ®VJ H— 2 ^ a <8> e» • m(v <8> i/>),
since one easily sees, that kerm = Qir, ir2 = -K and TV is selfadjoint.
D
1
Remark 5.8 Furthermore, for a £ (kerm)- = kerw we have (m(a),m(a)) since a = — i ^
i
—n(a,a)
e* ® e^ • m(a), which yields by Pythagoras theorem to n
Definition 5.9 A non trivial spinor ip £ T(EM) is called i) a twistor spinor if PV = 0, i.e., VXer(TM),
VxV' + -X • VVJ = 0.
(18)
ii) a generalized Killing spinor if Vip = 0 and T>ip = / ^ for some / £ C^CM), i.e., Vxip+^X-ip = 0. n for all X e
(19)
V{TM).
If ip £ T(EM) is a Killing spinor, then its associated vector field is denned by n
(The function i(ip,ejip) is real, because generally for a A;-form a we have {a-ip,ip) = ( - I ) - T 1 — (a-ip,ip)). L e m m a 5.10 For a Killing spinor ip, the associated vector field X^ is a Killing vector field, i.e., £xg = 0 - the Lie derivative of the metric in the direction of X vanishes, hence the name for such spinors. Proof. We need the following formula for arbitrary vector fields X, Y, Z (£xg)(Y, Z) := Xg(Y, Z) - g(£xY, Z) - g(Y, £XZ) = g(VxY,Z) + g(Y,VxZ) - g([X,Y],Z) = g(VYX,Z)+g(Y,VzX) 154
-
g(Y,[X,Z\)
since V is metric and torsion free. On the other hand, by definition of the associated vector field, in normal coordinates at the point x £ M, it follows VyX,/, = i ^ (Vyt/>, e j • VOej + (V1, ej • VyV) J'=l
=
_./ ^ , ( n
e i
-y-y-e,)^)
e j
since «/> is a Killing spinor. Now, for any vector fields Y and Z, we get (£Xv,ff) (Y, Z) = -if- (ty, +(V,
(Z-Y-Y-Z)-^) (Y-Z-Z-Y)-i/,)
= 0 D
Another important property of Killing spinors is the following: Lemma 5.11 The function \ip\2 corresponding to a Killing spinor rp is constant. Hence, a non-trivial Killing spinor has no zeros. Proof. For this, we prove that for any vector field X, the Lie derivative of IV'I2 in the direction of X is zero. By (14), (19) and (7), it follows X||2 = (VxV,V>)+ (, VxV) = -£((X-il>,1>)
+ W,X-il>)\
=0.
• We now examine some necessary conditions for the existence of Killing spinors. Proposition 5.12 Let tjj £ F(SM) be a generalized Killing spinor (see (19)). Then, i) f is constant and f2 = 4/n7l.1%5, and ii) (Mn,g)
is an Einstein manifold, i.e., Ric = — g.
Proof. Since ip is a Killing spinor, we have VyT/>
=-l-Y-lj) n 155
VxVyV = --X(f)Y n
-fP-i-Yn
= --X(f)Y
f
~{VxY)
.^P + L Y - X - ^ -
Vy VxV' = -~Y(f)X KX,YIP
Vxip - ~{VXY) n
-ip + Lx-Y-i;-
= --(X(f)Y
f
-{VYX)
• x/> • xP •j,
-i>+f—(Y-X-X-Y)-Tp.
- Y(f)X)
Using the first Bianchi identity and Equation 3) of Proposition 4.3, we get n 1 --Ric(J>C)-V = ^ e i - ^ x , e i V ' -
i=l
Then we compute -\Kic{X)
• i> = ^ei
-Tix,e^
i
(X(f)ei
= --YJ n
• e< • V - Ci(/) e i • X • rp)
ff2
H—j y ^ ( e » • e» • X - a • X • a) • ip. i
Using the fact that a • ej = — 1 and J2i &i- X • e^ — {n — 2)X, it follows - i R i c ( X ) • rj, = - 1 ( - n X ( / )
-rl>-df-X-rl>)
+ 4(-n-X"-(n-2)Jf)-^ nJ
n
Since for any vector X and any spinor ip, (X • ip,ip) = —(X- tp,ip), this function is purely imaginary. For X := grad/, it then follows ( - | R i c ( X ) • t/, + ^^f2X
•tf,V) = * ( / ) h / f + ^ ( d / • X • V, V)
= d/(X)|V>|2-i|d/|2h/f =:(l-l)|d/| 2 H 2 , 156
the l.h.s. being purely imaginary, thus |d/| 2 = 0, since by Lemma 5.11, |t/i|2 ^ 0. This yields X(f)1> = -df-X-tl> n
= 0,
VXer(TM),
which when combined with (20), yield
fcH/2X
Ric(X) = Thus i\^1' 5.2
f2 = f, by definition of the scalar curvature.
Conformal
Covariance
of the Dirac
•
Operator
Let (M, g) be a Riemannian spin manifold. For a conformal change of the metric g := e2ug,
u: M ->• M,
the spin structure on (Mn,g) induces a spin structure on ( M n , g ) . An isomorphism of the two SO n -principal fibre bundles is given by Gu:
SOgM
—> SOpM
( d , . . . , en) i—> ( e - " e i , . . . , e~uen). The spin structure induced on (M, g) is defined up to isomorphism by the commutative diagram Spin fl M —£=->• Spin^-M
SO s M
-£=->.
SOjM
All the arrows in the diagram are supposed to be invariant under the group action. Let p: Spin n -> Aut(E) be the spinor representation. An isomorphism of the associated spinor bundles is explicitly given by G„: S M = Spin p M x ip=[s,
p
E ^ E M = Spin^M x p E i—»^=
[G U (S),<J].
This map is an isometry with respect to the Hermitian product on the spinor bundles. Together with the corresponding isometry of the tangent bundle, 157
given by X H> X := e
U
X, we have the following relation between " • " and X~ip = X -ip
for every X G T(TM)
and ip G r ( S M ) .
Proposition 5.13 [Hn,Ba] Let (M,g) be a Riemannian spin manifold and ~g = e2ug a conformal change of the metric, then one has
for every ip G T(SM). Remark 5.14 It follows therefore, that the dimension of the space of harmonic spinors H = {ip G T(Y,M) \ Vip = 0 } is conformally invariant. In contrast to the Laplace-Beltrami operator on p-forms, the dimension of H for T> is not topologically invariant (see [Hn]). Proposition 5.15 Let (M,g) be a Riemannian spin manifold and a conformal change of the metric given by ~g = e2ug. Then 1 -^—. V*V> = VxV --X-du-rp--
r
1
X(u)ip
for all %p G r(EAf), X G T(TM). Proof. By the Koszul formula and the fact that V and V are torsion free, we have in a local orthonormal basis ( e i , . . . , e n ) uij(ek)=Uij(ek)+ei(u)8jk-ej(u)6ik,
Vi, j , k = 1,.. .n
(20)
Let (tpi,... ,XPN) be a local section of Spin 3 M, such that ipa — [s,o-a] for a basis (<TI, . . . ,(TN) of S. Then by (11) Ve^
a
= ^ 5 Z Wij(efc) ei • Cj • 1pa i,j
for a l i a = l , . . . , i V .
a
Lemma 5.16 T/ie Dirac operator associated with the metric g satisfies
V(f4>)=fV4> + e-udf-
Proof.
zw) = 5>7V*tf^ = 5>' Si(/)V» + / V e ^ Recall that e^ = e - " ei.
a
Lemma 5.17 The Dirac operators V and V are related by Vip = e'
Dtp -\
n-1
—- du • ip
Proof. i
i
= e~"^e;
7
[ V e ^ - ^ e i • grad(u) • if> -
-ei(u)tp]
T? — 1
= e-u[Vip+—r—
du -V]
•
Proof of Proposition 5.13. Let / :=e-^u. Then e-udf-i> n-1 = / e _ t t Vip + / e ~ " —r— du • ^ + e - " d / • ^
V(fiP)=fV4>
= fe~u
+
V$,
s
since d / = — ~- du.
D n
Theorem 5.18 [Hil] Let (M , g) be a compact Riemannian manifold and Dip = \ip for a non-trivial spinor field ip. Then A2>
• sup infM S eJ2u (21) 4(n-l) / In case of equality, ip is a Killing spinor. Proof. The Schrodinger-Lichnerowicz formula and IV^I 2 = - l ^ ^ l 2 + I'PV'I2 yield 0 < / „ V*? -, = j
M
(|V,/f - i|UVI2)
159
v,
for any section rp £ T(EM). For a fixed conformal change u: M -¥ E and Up :— e~v^~u tp we get
By Proposition 5.13 we get \e'uTp.
VTp = So 0
< ^
IM (A2 - i 7^1 S e 2 *) | ^ | 2 e - 2 V ?
0 < 2=i (A2 - I ^ i n f
M
5 e 2 «) J M
fefe-'Vj.
Hence A2 > I ^ - i n f ^ (Se2u). So inequality (21) holds. In the case of equality in (21) we have T>Xp = \eruTp and VTp — Q. Thus V x < £ + ^ — X - ^ = 0,
VXer(TM),
which by (5.12) applied, w.r.t the metric ~g, to Xp with / = A e - " implies that u is constant, hence ^ is a Killing spinor. • Definition 5.19 [Yamabe operator] The Yamabe operator of a manifold of dimension n > 3 is denned by (see [Be], for example): n —2 This operator is related to the Yamabe problem: Is there a metric of constant scalar curvature in a given class of conformal Riemannian metrics on a compact manifold? Corollary 5.20 Any eigenvalue A of the Dirac operator satifi.es: a) For n > 3, \ 2
A > 77 TT^I, 4(n — 1)
n > 3,
where (ii stands for the first eigenvector of the scalar conformal Laplacian L (see [Hi\]). b) For n = 2, „
27T X (M 2 )
Area(M2,g) where x(M2)
is the Euler characteristic class (see [Bal]). 160
In case of equality in a) or b) the eigenspinors associated with A are Killing spinors. Proof. a) Let u: M —> IK be a conformal change. Then the scalar curvature transforms according to Se2u = S + 2{n - 1)A« - (n - l)(n - 2)|du| 2 .
(22)
For n > 3, define the function h by h ^
= e2u .
(23)
Then Equation (22) simplifies to Sh^
= h-^-Lh.
One knows from general theory, that an eigenfunction hi to the first eigenvalue HI of L can be chosen to be positive. Moreover, it is known that Hi is positive iff, in the conformal class, there is a metric of positive scalar curvature. Thus hi > 0 and
hilLhi
= m > 0.
Now, take ui associated to hi by (23). Then
5e 2ui = Shf^
= h^Lhi = m.
It can be shown [i?i3], that if Se 2 " 1 = const., then 5 e 2 u i = sup u inf M (5e 2 u ). b) For n — 2, the transformation of the scalar curvature under conformal change of the metric, gives Se2u = S + 2Au. Now inf M (5e 2 ") < <
^ — - / Se2uug = - L — f (S + 2Au)vg 9 Area(M2,ff)7M Area(M 2 , 5 ) 7 M 1
f 2
- Area(M , 5 ) JM
,
_ 9
by Stokes and Gaufi-Bonnet theorem. 161
4TTX(M2)
Area(M 2 ,<,)'
Let t*i be a solution of n. =
2
M: Area(M
,g) JM
Then we finally get Area(M 2 ,9) JM
s
Area(M 2 ,#)
D Remark 5.21 With the help of the conformal covariance of the Dirac operator and of the Sobolev embedding theorem for pseudo-differential operators, J. Lott [Lo] showed the existence of a qualitative conformal lower bound for the eigenvalues of the Dirac operator. Moreover, for further formulation of (21) see [Hi3]. Theorem 5.22 [Hil,Li2,LiZ] a) If xjj is a Killing spinor and a £ Ctk (M) an arbitrary harmonic form of degree k ^ 0, n. Then a • ip = 0.
(24)
b) If M admits a non-trivial Killing spinor, then there exists no non-trivial parallel form a € Qk(M), k ^ 0, n. Proof. a) The Killing spinor V is an eigenvector to the smallest eigenvalue X\ of the Dirac operator T>. Using now the formulas d =^e;AVei,
(25)
8=-Y,eijVei,
(26)
By Proposition 2.8 and Yjei-a-ei
= {-l)k-1{n~2k)a,
i
we get i
i
= ((d + J)a) -ip + Ylei ~a(~~^ei ' ^ i
162
(27)
2fc, ((d + 8)a) .v» + ( - l ) f c A i ( l - — ) a - V :=A
= 0
:=V
since on compact manifolds A a = 0 ^=> d a = 5a = 0. Now |A| < |Ai| implies by Friedrich's inequality (Theorem 5.3) a • ip = 0.
b) Let a be a parallel fc-form. Then by (24), a • ip = 0. Now for any vector field X, 0 = V x ( a • ip) = ( V x a J • ip + a • V x ^ =o Since -0 is a Killing spinor, it follows that a-X
-t[) = 0.
On the other hand, by Proposition 2.8 it follows that Q
• X = (-l)k
(x • a + 2X j a ) ,
hence for any vector field X, one has (X j a) • ip = 0 ,
which by induction on k, shows that tp = 0.
• This theorem has immediate important topological consequences. For example, if there is a Killing spinor on a compact manifold, then the first de Rham cohomology is zero, Hip (M, E) = 0. Since the Kahler form Cl(X, Y) = g(X, JX) is parallel, there are no Killing spinors on compact Kahler manifolds of real dimension n > 4. 5.3
Holonomy
and Eigenvalues
of the Dirac
Operator
We have seen that on a compact Riemannian spin manifold (M™, g), any eigenvalue A of V satisfies Friedrich's inequality, A2 > j ^ - S ' o , (see Theorem 5.3). The key argument w.r.t Theorem 5.1 ii), is to replace | V ^ | 2 in the Schrodinger-Lichnerowicz formula by
I W f ^ p V f + LPVf 163
and then observe that | P ^ | 2 > 0 translates to Theorem 5.3. If A2 = \^zjS0, then for an eigenspinor ip associated with A, it follows that Vip — 0. Therefore ip is a Killing spinor, i.e., it satifies the overdetermined system Vler(TM),
V x ^ ± i x - V = 0,
(28)
where we normalized the metric so that So = n(n — 1). S. Sulanke (see [BFGK]) computed the spectrum of the the Dirac operator on the round sphere. Notice that, applying (28) with X — e^, taking its Clifford multiplication with e; and summing over i, leads to Dtp = ± f V"- It turns out that the n-sphere has a maximal number of Killing spinors. This could also be seen by using a spinorial Gauss-type formula for Sn C R n + 1 [Tr,Ba3]. For the classification of manifolds with Killing spinors one has to distinguish between even and odd dimensions. Using the twistor construction, Priedrich proved that in 4-dimensions, only the round sphere has Killing spinors (see [i
S/Nip = 0 «=4> i/)\M Note that, if (Mn,gM)
is a Killing spinor on {Mn,gM)
= (Sn,Standard), 164
•
then the warped product metric gN
is the standard metric on the euclidean space JET+1 \ { 0 } = S V x R ; . In this characterization dimensions 6 and 7 are exceptional and lead to the construction of incomplete Riemannian manifolds with holonomy G2 and Spin 7 . For example, if the dimension n is even and n ^ 6 , then the spheres are the only such manifolds. It should be pointed out that supergravity and superstring theories concern these two exceptional cases. We have seen that on compact Kahler manifolds, there are no Killing spinors. For such manifolds, K. D. Kirchberg [Kil] proved the following. For any eigenvalue A of Z>, if n = 04 (even complex dimension), then
If n = 24 (odd complex dimension), then 2>lH±^0. (30) 4 n The proof of Kirchberg's inequalities is based on the double decomposition of the covariant derivative of an eigenspinor. First, that of the spinor bundle under the action, by Clifford multiplication, of the parallel Kahler 2-form, second, as in the Riemannian case, the decomposition w.r.t the image and the kernel of Clifford multiplication. This lead to the notion of Kahlerian twistor operators. Eigenspinors which are in the kernel of the Kahlarian twistor operators are then called Kahlerian Killing spinors. Manifolds with such spinors are called limiting Kahlerian manifolds. All these manifolds have been recently geometrically described. In low dimensions, there are classification results due to Kirchberg [KiS\ in dimension 6, and to Th. Friedrich [Fr3] in dimension 4. In odd complex dimension, A. Moroianu [Mol] showed that a limiting Kahler manifold is characterized as the twistor space associated to a Quaternion-Kahler manifold of positive scalar curvature. For limiting Kahlerian manifold of even complex dimension [Li5,Mo3], it turns out that the universal cover of such a manifold is isometric to a Riemannian product M x E 2 , where M is a limiting Kahlerian manifold of odd complex dimension. The third interesting case in this setup is the family of compact spin manifolds with positive scalar curvature and holonomy Sp„ • Sp x . These manifolds are characterized by the existence of a parallel 4-form. Such manifolds are necessarily Einstein of dimension n = 0mod4. First J.-L. Milhorat [Mil] computed the spectrum of the Dirac operator on the model space, HP™. A
165
Then, in [HM1, HM2,HMZ] it is proved that for n = 8,12, any eigenvalue of the Dirac operator satisfies A
-iVT8So-
(31)
The approach used is similar to that in the Kahler situation but the argument was not sufficient to prove (31) in higher dimensions. In fact, it turned out that, in higher dimensions, some additional terms were missing in the definition of the Quaternion-Kahler twistor operator introduced in [HMZ]. Note that, in [BH] where extensions of (21) are given, it is also shown how one can make use of representation theory to systematically get optimal BochnerWeitzenbock type formulae for natural differential operators. In [KSW1,KSW2,KSW3], it is proved that (31) is true in all dimensions and that the only limiting manifold is the Quaternion-Kahler projective space. For this, the authors use representation theory to define natural twistor operators based on a double decomposition under the action of Sp n and Spj. Recently, with the help of represntation theory, J.-L. Milhorat [Mi2] gave a simple proof of (31). Summing up, we have seen that the optimal lower eigenvalue estimates depend on the holonomy of the manifold. For manifolds with holonomy SO„, U m or Spj Sp n , i.e., Riemannian, Kahler or Quaternion-Kahler manifolds, the optimal lower bounds for the eigenvalues of the square of the Dirac operator are n So n - l l >
v
'
Riemannian
"
n + 2 5o n T "
v
n n-2
~
'
Kahler, n = 2 mod 4
*
So T v
'
Kahler, n = Omod 4
~
n + 1 2 So n+8 4' *
v
'
Quaternion-Kahler
Acknowledgements These notes are based on a series of lectures given at the Summer School on Geometric Methods in Quantum Field Theory organized in Villa de Leyva, Colombia, by CIMPA and Universidad de Los Andes, July 12-30, (1999); at the "Programme Intensif Socrates" March 23-April 11 (1998), University of Nantes, France; and at the conference "Colloque en Geometrie" April 11-16 (1997), Lebanese University, Tripoli-Beirut, Lebanon. I would like to thank the organizers of these three meetings and my colleagues in Nantes for their hospitality. I also would like to thank Christoph Bohle, Bruno Colombel, Stephan Eckner and Giinter Paul Leiterer for their help in writing a first version of 166
these notes. Finally, I would like to thank Bernd Ammann, Nicolas Ginoux and Sylvie Paycha for a careful reading of these notes. References [Bal] C. Bar, Lower eigenvalue estimates for Dirac operators, Math. Ann. 293 (1992), 39-46. [Ba2] , Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509-521. [Ba3] , Metrics with Harmonic Spinors, Geometric And Functional Analysis, 6 (1996), 899-942. [Ba] H. Baum, Spin-Strukturen und Dirac-Operator en "uber pseudoRiemannsche Mannigfaltigkeiten, Teubner-Texte zur Mathematik, 4 1 , Teubner-Verlag Leipzig, (1981). [BFGK] H. Baum, T. Friedrich, R. Grunewald, I. Kath, Twistor and Killing Spinors on Riemannian Manifolds, Seminarbericht 108, HumboldtUniversitat zu Berlin, 1990. [BHMM] J.P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry, Monograph (In preparation). [Be] A.L. Besse, Einstein Manifolds, Springer, Berlin, 1987. [Brl] T. Branson, Nonlinear phenomena in the spectral theory of geometric linear differential operators, Proc. Symp. Pure Math. 59 (1996) 27-65. [Br2] , Stein-Weiss operators and ellipticity, J. Funct. Anal. 151 (1997) 334-383. [BH] T. Branson, O. Hijazi, Vanishing Theorems and Eigenvalue Estimates in Riemannian Spin Geometry, International J. Math. 8 (1997), 921934. [Fe] H. Fegan, Conformally invariant first order differential operators, Quart. J. Math. Oxford 27 (1976) 371-378. [FVl] T. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrummung, Math. Nach. 97 (1980), 117-146. [Fr2] , A remark on the first eigenvalue of the Dirac operator on 4dimensional manifolds, Math. Nachr. 102 (1981), 53-56. [Fr3] , The Classification of 4-dimensional Kdhler Manifolds with Small Eigenvalue of the Dirac Operator, Math. Ann. 295 no.3 (1993), 565-574. [Fr4] , Dirac-Operatoren in der Riemannschen Geometrie, Vieweg, Braunschweig/Wiesbaden, 1997. 167
[Gal] S. Gallot, Equations Differentielles Caracteristiques de la Sphere, Ann. Scient. Ec. Norm. Sup., 4eme serie 12 (1979), 25-267. [Ga2] Private communications. [Hil] O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162. [Hi2] , Caracterisation de la sphere par les premieres valeurs propres de I'operateur de Dirac en dimensions 3, 4, 7 e* 8, C. R. Acad. Sci. Paris, 303 (1986), 417-419. [Hi3] , Premiere valeur propre de I'operateur de Dirac et nombre de Yamabe, C. R. Acad. Sci. Paris, 313 (1991), 865-868. [Hi4] , Eigenvalues of the Dirac operator on compact K"ahler manifolds, Comm. Math. Phys. 160 (1994), 563-579. [HMl] O. Hijazi, J.-L. Milhorat, Minoration des Valeurs Propres de I'Operateur de Dirac sur les Varietes Spin Kdhler-Quaternioniennes, J. Math. Pures Appl., 74 (1995), 387-414. [HM2] , Decomposition spectrale du fibre des spineurs d'une variete spin kahler-quaternionienne, J. Geom. Phys., 15 (1995), 320-332. [HM3] , Twistor Operators and Eigenvalues of the Dirac Operator on Compact Quaternionic Spin Manifolds, Ann. Global Anal. Geom. [Hn] N. Hitchin, Harmonic Spinors,Adv. in Math., 14 (1974), 1-55. [Kil] K.-D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kahler manifolds with positive scalar curvature, Ann. Glob. Anal. Geom. 4 (1986), 291-326. [Ki2] , The first eigenvalue of the Dirac operator on Kahler manifolds, J. Geom. Phys., 7 (1990), 449-468. [Ki3] , Compact Six-dimensional Kahler Spin Manifolds of Positive Scalar Curvature with the Smallest Possible first Eigenvalue of the Dirac Operator, Math. Ann. 282 (1988), 157-176. [KS] K.-D. Kirchberg, U. Semmelmann, Complex structures and the first eigenvalue of the Dirac operator on Kahler manifolds, Geom. and Punct. Anal. 5 (1995), 604-618. [KN] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry I, II, Intersience, John Wiley, New York, 1963, 1969. [KSWl] W. Kramer, U. Semmelmann,G. Weingart, Eigenvalue Estimates for the Dirac Operator on Quaternionic Kahler Manifolds, Math. Z. 230, 4 (1999), 727-751. [KSW2] , Quaternionic Killing Spinors, Ann. Glob. Anal. Geom. 16 (1998), 63-87. [KSW3] , The First Eigenvalue of the Dirac Operator on Quaternionic 168
Kahler Manifolds, Comm. Math. Phys. 199 (1998), 327-349. [LM] H.B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton University Press, Princeton, New Jersey, 1989. [Lil] A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris, 257 (1963), 7-9. [Li2] , Killing Spinors According to O. Hijazi and Applications, Spinors in Physics and Geometry, (Trieste 1986), World Scientific Publ., Singapore (1988), 1-19. [Li3] , Spin Manifolds, Killing Spinors and the Universality of the Hijazi Inequality, Lett. Math. Phys. 3 (1987), 331-344. [Li4] , On the twistor-spinors, Lett. Math. Phys., 18 (1989), 333-345. [Li5] , La premiere valeur propre de I'operateur de Dirac pour une variete kdhlerienne et son cas limite, C. R. Acad. Sci. Paris, t. 311, Serie I (1990), 717-722. [Lo] J. Lott, Eigenvalue Bounds for the Dirac Operator, Pacific J. Math. 125 (1986), 117-126. [Mil] J.-L. Milhorat, Spectre de I'operateur de Dirac sur les espaces projectifs quaternioniens, C. R. Acad. Sci. Paris, 314 (1992), 69-72. [Mi2] , Eigenvalue of the Dirac Operator on Compact QuaternionKdhler Manifolds, Rapport de Recherche, 9 9 / 1 0 - 3 , Universite de Nantes. [Mol] A. Moroianu, La premiere valeur propre de I'operateur de Dirac sur les varietes kdhleriennes compactes, Commun. Math. Phys., 169 (1995), 373-384. [Mo2] , Operateur de Dirac et submersions riemanniennes, These, Ecole Poly technique, Palaiseau, 1996. [Mo3] , Kahler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz, Preprint 98-4 (1998), Ecole Polytechnique, Palaiseau. [SW] E. Stein, G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968) 163-196. [Tr] A. Trautman, The Dirac operator on hypersurf aces, Acta Phys. Polon. B 26 7, (1995) 1283-1310. [VW] C. Vafa, E. Witten, Eigenvalue inequalities for fermions in gauge theories, Commun. Math. Phys. 95 (1984), 257-276. [Wa] Mc. Wang, Parallel spinors and parallel forms, Ann. Global Anal. Geom., 7 (1989), 59-68.
169
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
Q U A N T U M T H E O R Y OF F E R M I O N S Y S T E M S : T O P I C S B E T W E E N PHYSICS A N D MATHEMATICS EDWIN LANGMANN Theoretical
Physics, E-mail:
KTH, S-10044 Stockholm, [email protected]
Sweden
A pedagogical introduction to quantum models of fermions is given emphasizing the interplay between physics and mathematics. The aim is to explain physical ideas and mathematical tools by looking at simple examples. T h e topics discussed include fermion systems with a finite number of degrees of freedom, examples for the quantum field theory limit and regularization, the abstract formalism of quasi-free second quantization and some of its applications like the boson-fermion correspondence, loop algebras, and the Luttinger model.
Contents 1 Introduction 2 Preliminaries 2.1 *-Algebras 2.2 Operators on Hilbert spaces 2.2.1 Finite dimensional Hilbert spaces 2.2.2 Infinite dimensional Hilbert spaces 2.2.3 Unitary representations of *-algebras 2.3 Quantum physics in a nutshell 2.4 Quantum field theory limit 2.4.1 General remarks 2.4.2 Regularizing diverging series 2.4.3 Field algebras and their representations 3 Fermion quantum mechanics 3.1 The simplest fermion system I know... 3.2 Fermion systems with a finite number of degrees of freedom 3.3 Quantum field theory limit: Heuristics 4 Quasi-free second quantization of fermions 4.1 4.2 4.3 4.4 4.5 4.6
Fermion field algebras and fermion Fock spaces Free second quantization of observables and symmetries Fermion quantum mechanics revisited Quasi-free representations Quasi-free second quantization of observables Quasi-free second quantization of symmetries 170
172 175 175 176 177 178 180 180 182 183 184 185 186 186 187 195 198 198 200 202 203 204 210
5 Loop algebras and loop groups
213
5.1 Loop algebra of complex valued maps on the circle 5.1.1 Wedge representation 5.2 Loop group of maps from the circle to U(l) 5.3 Boson-fermion correspondence 5.3.1 Bosons from fermions 5.3.2 Fermions from bosons
213 213 215 217 218 220
5.4 Bosons, fermions, and elliptic functions
221
6 The Luttinger model
223
6.1 Physical interpretation 6.2 Construction and solution of the model 7 Further developments
223 225 231
7.1 7.2 7.3 7.4
Boson-anyon correspondence and the Sutherland model Other 1+1 dimensional quantum field theory models Quasi-free second quantization and non-commutative geometry Bosons and the super-version of quasi-free second quantization
References
231 232 232 233 234
171
1
Introduction
These notes are intended as an introduction to quantum (field) theory and to (some) mathematical topics used to formulate and analyze simple fermion models relevant in physics, both for mathematics- and theoretical physics students. I should stress that this is not a systematic presentation of the subject but rather my personal (subjective!) selection of topics. My selection was guided by my strong belief that a good way to learn a complicated subject is to first look in detail at simple non-trivial examples where one can understand things explicitly.1 After that it is often much easier to understand the general structure which underlies all these examples. I also hope that these notes reflect my view that the interplay between mathematics and physics can be very fruitful for both disciplines. In my presentation I try to be mathematically precise and explicit (which I can be by restricting to rather simple examples), and I also try to explain the physical interpretation of the formalism presented. One of my goals is to give a physics- and mathematics introduction to quasi-free second quantization of fermions, a formalism which allows a mathematical precise formulation and treatment of certain quantum field theory models and, at the same time, plays a central role in the representation theory of certain infinite dimensional Lie algebras like the affine Kac-Moody algebras and the Virasoro algebra. I try to explain the physical ideas underlying this formalism, and to illustrate them in specific examples also making explicit the mathematical elegance of its general, abstract formulation. To substantiate my claim that this formalism is useful I also include short reviews of several seemingly very different research projects in which I have been involved and in which this formalism plays a central role. I have attempted to write these notes so that a student can work through them and acquire an active knowledge of the material treated. Due to limitations of space, these notes cannot be as detailed as a text book. Instead of writing ' . . . as is easily seen...' and the like as a substitute for a detailed argument (which in fact might be 'easily seen' often only by an expert) I decided to clearly mark as 'Fact(s)' all statements where beginners probably have to invest work to prove, verify, or at least fully understand them. Readers who do not find such a Fact obvious are strongly encouraged to work out a proof of it and/or look it up in the references given. I sometimes give some 'Hints' outlining a proof (not so much for those Facts which are more difficult to prove, but rather for those where I believe that the proof is important to understand their significance or illustrates a useful calculation I learned this several years ago from Harald Grosse who was my P h D supervisor then.
172
tool). I use the symbol • to mark the ending of a Fact, Theorem, Remark etc. There are several remarks marked as 'Remark (adv.)' which are either more advanced than the rest of the notes or intended for experts. I now describe in more detail the topics discussed in these notes. Fermion models are important in many different areas of physics. They are very successfully used to describe electrons in metals, protons in atomic nuclei, elementary particles in particle accelerators (like the ones at CERN in Geneva) to mention only a few examples. These and many other examples belong to the realm of quantum physics. Quantum systems with a finite number N of degrees of freedom belong to the realm of quantum mechanics, and the mathematical framework for this is very well understood since quite some time. Quantum field theory models deal with quantum systems with an infinite number of degrees of freedom, N = oo. Quantum field theory is mathematically much more challenging than quantum mechanics due to the occurrence of so-called divergences: the naive limit N —» oo often does not make sense, and one has to regularize in order to get meaningful answers. The plan of these these lecture notes is as follows. In Section 2 I collect some prerequisites. Mathematical prerequisites include some facts about *-algebras and the theory of Hilbert space operators. The latter subject is essential to fully appreciate the quantum field theory case, but I stress that I tried to write these notes so as to make them accessible also for readers not familiar with the general Hilbert space theory. Section 2 also includes a review of quantum theory intended for mathematicians, and some preliminary discussion of the significance of the quantum field theory limit TV -> oo. Fermion quantum mechanics is discussed in Section 3, mainly to introduce the physics ideas needed in the simplest context possible. As an application I mention some basic facts about Hubbard-like models which are important in solid state physics and which can be formulated in the setting of fermion quantum mechanics. I also indicate the additional difficulties arising in the quantum field theory limit. Section 4 is devoted to the general, abstract formalism of quasi-free second quantization from a mathematical point of view. In Section 5 I discuss applications in the representation theory of the loop algebras and loop groups, and I show how the mathematical results in this setting give rise to intriguing identities which are known as the boson-fermion correspondence and which play a central role in the quantum field theory application of this formalism. In Section 6 it is shown how these results can be used to construct and solve the so-called Luttinger model which is a model of interacting(!) fermions moving in one dimensional continuum space and describing a one dimensional metal (e.g. electrons in a wire). I chose this example since it is the simplest non-trivial one I know, and it nicely illustrates 173
one source for divergences in quantum field theory: they can be regarded as the price to pay for simplifying the description of a physical system by eliminating the details of its short distance structure. In this example I also show how the mathematical tools developed in the previous sections can be used to construct a quantum field theory model, and I sketch how the boson-fermion correspondence is used to completely solve this model. Finally, in Section 7 I briefly review several other developments. R e m a r k : I stress that limiting my discussion of quantum models to fermions is a severe restriction from the physics point of view: many phenomena in nature are described by bosons, e.g. gauge fields (which play a fundamental role in elementary particle physics) are bosons, and conventional quantum mechanics (as discussed in physics text books) can be regarded as quantum theory for bosons with a finite number of degrees of freedom. However, fermion models are, from a mathematical point of view, much simpler and therefore, as I believe, well suited for introducing quantum theory. Moreover, much of what I discuss here for fermions can be quite straightforwardly extended also to boson models (see Ref. [R, GL1,GL2]). R e m a r k ( a d v . ) : In the mathematical theory of infinite dimensional spaces topology plays an important role. We will encounter several such spaces in the examples discussed in these notes. Still, I will avoid to discuss topological questions (and not even define the topology in many cases) since this would be beyond the scope of these lectures.
174
2
Preliminaries
The mathematical framework of quantum mechanics is Hilbert space theory, and an useful prerequisite for doing quantum mechanics is a good knowledge of the theory of operators on Hilbert spaces. The more fundamental objects in quantum physics are actually *-algebras 2 which nevertheless, can in many cases (including all examples I discuss), be represented by operators on a Hilbert space. In quantum field theory (but not quantum mechanics) this view becomes important since there one finds truly different representations of the *-algebras of interest. In this Section I summarize some definitions and facts about *-algebras, Hilbert space theory, and quantum physics which I will use later. In particular, some of my notation used later is explained here. Of course, my presentation is too short to be in any way a substitute for systematic courses. I therefore stress that readers with no prior knowledge in Hilbert space theory should be able to understand a large part of these notes with some knowledge in linear algebra corresponding to the special case of finite dimensional Hilbert spaces CN and accepting on trust the generalizations to general Hilbert spaces which I mention. 2.1
*- Algebras
•-Algebras are an important mathematical structure in quantum physics and 1 therefore recall their definition in this section. Let A be a complex algebra (i.e. for A,B 6 A and A G C there is a definition of A + B (sum), \A (product with scalar) and AB (product) so that the usual axioms hold). Then A is a *-algebra if there is an anti linear map * : A -» A called involution so that (AB)* = B*A* and (A*)* = A for all A, B £ A. In the following we always assume that a *-algebra has a unit element which we denote as 1 (or 1^ if there is danger of confusion). A simple example of *-algebras are algebras of (complex) N x TV matrices (the natural * is given the matrix adjungation). Other important examples are algebras of operators on Hilbert spaces which will be discussed in the next section. Remark (adv.): Usually the *-algebras of interest in quantum physics come with some topology compatible with the algebraic structure, and one often requires that they are C*-algebras, see e.g. [BrR] for definition and discussion. 2
usually equipped with some suitable topology
175
Even though the fermion field algebras naturally have the structure of a C*algebra and thus one can make use of the quite powerful (but mathematically advanced) theory of C*-algebras, I will not do this for two reasons. Firstly, I believe that avoiding C*-algebra arguments makes the subject easier to learn. Secondly, if one insists on C*-algebra techniques, the treatment of quantum theory for bosons becomes very different from the fermion case, whereas the approach which I will discuss can be adapted to bosons in a straightforward manner, see e.g. [GL1,GL2,L0]. 2.2
Operators
on Hilbert
spaces
In this paragraph I collect some basic results of Hilbert space operator theory, see e.g. [RS]. Readers familiar with this subject can find a summary of my notation in the next paragraph. Readers not familiar with the terms in this paragraph can get acquainted with them in the rest of this section. For the latter readers I recommend to browse the complete section at first reading and go back to it later when needed. Notation: If 7i is a separable Hilbert space then (•, •) is the inner product in V. which we choose anti-linear in the first- and linear in the second argument. 3 We denote as C the space of all linear operators % —> Ti, and B, B\ and B2 are the bounded, trace class and Hilbert Schmidt operators on H, respectively. Moreover, Tr is the Hilbert space trace and * the Hilbert space adjoint as usual. Throughout these notes 1 will be the identity operator. We will later have several different Hilbert spaces at the same time and then sometimes write (-,•)•« instead of (•, •) if there is danger of confusion. Similarly we will sometimes write 1-n, Tr^, C(T-L) etc. For A £ C and / £ H I often write Af instead of A(f). If fn and gn are vectors H and A„ complex numbers I write A = Yln xnfn {gn, ) short for the operator defined as: Af = J2n ^nfn {gn, f)
Vfen. (As mentioned, the rest of this section is an introduction to Hilbert space theory assuming only knowledge in linear algebra). A basis (= complete orthonormal basis) in 7i is a set { e n } ^ = 1 , 1 < N < 00, of vectors in % such that (em,en) = 5m,n ('•— 1 if m = n and = 0 otherwise) for all m,n, and S „ = 1 e„ (e n , •) = 1. Fact: Af is basis independent 4 and only depends on H.O Fact: Given a basis {en}^=,1 one can represent all / £ H as / = X) n =i f^en with / „ = (e„,/).D Fact: If N < 00 3 This is the convention used by most physicists and which unfortunately is different from what is common in the mathematics literature. 4 i.e. it is the same for whatever basis one has
176
one can therefore naturally identify % with CN, and we write % = C^.D In the following I first discuss the special case % = CN for finite N. I then discuss some of the caveats which arise if one tries to generalize the results there to the infinite dimensional case N = oo.
2.2.1
Finite
dimensional
Hilbert
spaces
This section contains a summary of results from linear algebra formulated so that they can be generalized to general Hilbert spaces. The proofs of the 'Facts' should be easy and essentially only require getting used to my notation which is chosen to easily generalize to the infinite dimensional case. The simplest example (sufficient for our discussion of fermion quantum mechanics) is H — CN with (/, g)CN = ^ „ = 1 fn9n where we write CN 3 f — (/ii • • • > /w), fn S C, similarly for g, and the bar denotes complex conjugation. In this case, the distinction between the different operator algebras mentioned above is irrelevant, i.e. C = B = B\ = B2 if N < 00. Note that the set B of all linear operators on CN has a natural structure of an algebra, in particular, there is a natural definition of a product of two linear operators on CN (i.e. for A,B e B, (AB)(/)) := A(B{f)) for all fEU). Fact: The (Hilbert space) adjoint A* of A £ B is defined by (/, A*g) — (Af,g) for all / , g S 7i. (Hint: You need to check that this indeed defines A*.)D A linear operator A on % is called self-adjoint if A* = A, and it is called unitary if A A* = A* A = 1. Fact: With the * operation A ->• A*, B naturally is a *-algebra.D The standard basis in CN is defined as follows, (en)k — Sn,k for all n,k = 1 , . . . ,N. However, there are many other examples for a basis which can be of interest in applications. Fact: Given a basis, we can write all A £ B as A = Yjm,n amnem (e n , •), and the operator product thereby can be represented by matrix multiplication. Thus we get the Fact: B(CN) can be identified with the *-algebra oi N x N matrices (with (a*)mn = o ^ ) . D The importance of self-adjoint and unitary operators in quantum physics is closely related to the (generalization of the) following Fact: If A € B is self-adjoint then there a basis {en}%=1 and real numbers {A„}^ = 1 such that A = J2n=i ^n.en (en, -)-n This result implies that functions of self-adjoint operators are well-defined: f(A) := J2n=i / ( ^ n ) e n (e„, •) for any bounded C-valued function / on M. Unitary operators have a similar representation, U = ]Cre=i ^nen (e n , •), but here the A„ are phases i.e. complex numbers with absolute value equal 1. One can also easily prove the following 177
Fact *: For every self-adjoint operator A on H, U{i) := eltA exists and is unitary for all real t. Moreover, t/(0) = 1 and U(s)U{t) - U(s + t) for all s,tek, and §-tU(t)f = iAU(t)f for all / in the domain of definition of A. • One often refers to eltA as the one-parameter family of unitary operators generated by A. For all A e B and an arbitrary basis {en}^=1 in CN we can define the trace Tt(A) := X^ n = 1 ( e «' ^ e « ) which i s a linear map B -> C. One can easily prove the following Facts: This definition is basis independent, and the trace is cyclic i.e. Tr(AB) = Tr(BA) for all A,B E B.D The generalization of this to infinite dimensions has several caveats which are of central importance in quantum field theory. 2.2.2
Infinite
dimensional
Hilbert
spaces
We now consider the case N = oo. Many of the results above can be generalized to this case but this requires some care. For example, one if often interested in operators A £ C where Af is not defined for all / G rl but only for / in some subset T>{A) called the domain of A. In particular, A = B means not only that Af = Bf for all / € T>(A) but also that V(A) = V(B). Moreover, to define the sum and the operator product of two operators A,B € £ one then has to be careful about domains, and £ is not an algebra in general. Also the definition of the adjoint A* of an operator A 6 C etc. requires some care about domains. The notion of a trace also becomes more delicate (our definition of Tr(>l) above now involves an infinite sum which will not converge in general, and even if it converges for some basis, the results will no longer be independent of this basis in general). The strategy to generalize results valid for CN, N finite, to general Hilbert spaces % is to restrict them to subsets of C Fact: One subset for which all definitions and results generalize in a simple manner is the finite rank operators BQ which is defined as the set of all operators of the form A = S n = i ^n c n (fn, •) as above with M finite, Xn € C, and both {en}™=i and {/n}£Li some basis.• Unfortunately, this set BQ is rather small and many operators of interest are not in Bo • To define interesting larger sets of operators containing more operators of interest it is useful to consider dense subsets V of T-L which are such that, loosely speaking, every / £ V. can be represented as a limit of a converging sequence {fn}^Lo where all / „ £ V.5 In particular, the span of any basis {e„}^L 1 (i.e. the vector space generated by all finite linear combinations of vectors in this set) is such a dense set. Then one can define 5
for a precise definition see [RS]
178
the set B(V) as the set of all operators which have V as a common dense invariant domain i.e. for all / G V, Af and A*f exist and are vectors in T> for all A G B(V). Fact: For every dense domain V, B(T>) is a *-algebra (with product given by the operator product and the star operation by the operator adjoint).• In particular, B = B(H) are the so-called bounded operators.6 As mentioned, the definition of self-adjoint operators A G £ is now somewhat delicate but for all these operators one can prove the Spectral Theorem: For any bounded function / : R —> C, the definition of f(A) for self-adjoint finite rank operators A can be naturally extended to all self-adjoint operators A £ C such that f(A)* = J (A). In particular, Xi{A) is well-defined for all characteristic functions xi1 •, a n d Xhui2(A) = XiM) + XiM) ^ h n / 2 = 0, and Xi(A)* = Xi(A)2 = xM). Proof: see e.g. [RS\.
•
One other important result for quantum theory is that Fact * above holds true in arbitrary Hilbert spaces H (this is essentially Stone's Theorem, see e.g. [RS]). Other important subsets of C are defined such that they allow for a trace and/or that properties of the matrix trace remain true. To be specific, for 0 < p < oo, Bp is defined as the set of all A £ C for which8 ^2^=1 (e„, (AA*)p/2en) exists (i.e. is finite) for some basis {en}%=l, and one can prove the Fact that this definition is basis independentD. In particular, B\ are called the trace class operators and 62 the Hilbert Schmidt operators. The name of B\ is due to the following Fact: The Hilbert space trace Tr(A) := Yln=i ien,Aen) exists and is basis independent if and only if A € Bi.D The other subsets Bp are important due to the following Fact: If A 6 Bp and A e Bq then AB e Br with i = i + i.D In particular, the product of two Hilbert Schmidt operators is trace class. We also have the Fact: If A € B, B G Bp for some p < 00, and AB and BA are trace class then Tr(AB) = Tr(BA).0 Note also the Fact: Bp C Bq if p < q, and in particular, all trace class operators are Hilbert Schmidt.• I abuse notation here: sorry. i.e. xi(x) = 1 ( = 0) for 1 £ / ( 1 ^ / ) where / is some open subset of R One uses here the fact that for all A e C, A A* is self-adjoint and non-negative and thus (AA*)P'2 is well-defined.
7
8
179
2.2.3
Unitary
representations
of
^-algebras
In physics one is usually interested in representations n of a *-algebra A by operators on some Hilbert space H where * is represented by the Hilbert space adjoint, n(A*) = ir(A)* for all A 6 A, and TT(1A) = 1 ^ . Such representations are called unitary. Remark: Strictly speaking I should use a different symbol, e.g. f, for the *operaton in the *-algebra to distinguish it from the Hilbert space adjungation which I also denote as *. Then I could write the condition for a unitary representation as n(A^) — n(A)* for all A € A, which is somewhat clearer. This distinction between ^-operation and Hilbert space adjoint is actually important in the boson case. However, every *-algebras discussed in these notes can be indentified with the simplest of its representations -KQ which is unitary, and thus it is safe to only use one symbol *. Remark: In quantum mechanics, all unitary representations of the *-algebra of interest (see below) are unitarily equivalent, which essentially means that one can identify the *-algebra with (any of) its unitary representations. This is not the case in quantum field theory where truly different unitary representations of the relevant *-algebras exist. For pedagogical reasons I will always distinguish between the *-algebra and its unitary representation. • 2.3
Quantum
physics
in a
nutshell
In this section I summarize the general framework and the postulates in quantum theory. In a quantum physics model one usually has a separable Hilbert space H together with an *-algebra A of operators on W (with the * operation given by the Hilbert space adjoint *). (Reader not familiar with Hilbert space theory can assume H = CN, and operators on H are just complex N x N matrices.) Vectors 9 in H represent possible pure states of the system, and selfadjoint elements in A represent observables i.e. a possible measurements on the system. We denote the algebra of self-adjoint elements in A as O. Similarly, the unitary elements represent (symmetry) transformations on the system. A is called the field algebra and O the observable algebra of the model. Remark: Even though only O has a (simple) physical interpretation, it is in many cases very useful to consider the larger algebra A, for example, elements 9
to be precise: equivalent classes of non-zero vectors which differ by a phase factor
180
in O can often be constructed from simpler, non-selfadjoint elements which are not in O but in A. • In a specific model there is usually one preferred operator H in O which is called Hamiltonian and which plays, at the same time, three special roles: firstly, it represents energy measurements, secondly, it defines the time evolution of the system, and thirdly, it determines the state of the system in thermal equilibrium. Moreover, in many models there is another special element N e O called particle number operator and which corresponds to measurements of the number of particles in the system. This interpretation is reasonable only if the eigenvalues of N are integers. A unitary element in A represents then a symmetry of the model if it commutes with H and (if it exists) with N. Often one has symmetries represented by one-parameter families of unitary operators generated by some A £ O via Stone's theorem (cf. Fact *). Basic postulates in quantum physics are the following: Postulate 1: If the system is in the pure state represented by a non-zero ip G H, then the expectation value of the observable A G O (i.e. expected (average) value for the result of the corresponding measurement) is
-
(1)
In particular, < X[a,b]{A) > is the probablity for the measurement corresponding to the operator A to give a result in the interval [a, b] (—00 < a < b < 00). Postulate 2: If xp € % represents the state at time t = to, then the state at other times t > to is 1P = e - ^ C - ' o t y
(2)
provided the system is isolated in the time interval [t0,t]. Postulate 3: If the system is in thermal equilibrium and has a temperature T > 0, then the expectation value of the observable A G O (i.e. expected (average) value for the result of the corresponding measurement) is 10 =
—i
10
1
(3)
Here one implicitly assumes that e ^ ( H V-^)A is trace class which should be proven in a specific model.
181
with Z = Tr(e-^ff-'iN))
(4)
and /? = 1/T where /i is a real parameter which is determined by the requirement that the expectation value < N > of the particle number has a fixed, given value. Note that < 1 > = 1, and the normalization constant Z is usually referred to as partition function. Moreover, the constant \i is traditionally called chemical potential. Note also that < N > =
^
l 0 g Z
-
(5)
It is important to note that typcially, a necessary condition for Z to exist is that the Hamiltonian is bounded from below i.e. (ip,Hip) > EQ (ip,^) for H all ijj £ H and some EQ > - c o (this is a necessary condition for e-0( -^) to be trace class). If such an EQ is also an eigenvalue of H i.e. for some V'o, Htpo — Eotpo, then this corresponding eigenvector tp0 is called ground state of the model, and EQ is the ground state energy. Many models of interest in physics (but not all!!) have a non-degenerate ground state ipo, i.e., at zero temperature (/? ->• oo), < A > in Eq. (3) reduces to < A > in Eq. (1) with ip = ip0-
Remark: The physicist's understanding of and intuition about these postulates is mostly based on (many) specific examples of experimental setups and experimental results, together with the corresponding specific quantum models modeling and explaining them. This is the subject of quantum mechanics courses. Remark: My postulate 3 actually is usually only discussed in statistical physics i.e. systems with many degrees of freedom at finite temperature. 2.4
Quantum field theory
limit
One general aspect of quantum field theory is the occurence of divergencies, i.e. in formal computations one is lead to sums and integrals which do not converge. Over the years physicists have learnt to extract meaningful finite numbers from such expressions. The precedures used in this context are called regularization and renormalization, the latter being only necessary in more complicated quantum field theory models which are beyond the scope of these notes. Regularization, however, will be important for us, and I will illustrate its basic idea in a simple example. 182
The following section contains a few 'philosophical' remarks concerning quantum field theory. Section 2.4.2 contains the elementary examples on regularization mentioned, and in Section 2.4.3 I will outline the general strategy for constructing the quantum field theory models which I use in these lectures. 2.4-1
General
remarks
There are two different approaches to quantum field theory, one based on path integrals (a good introduction for physicists is e.g. [Ra]; for mathematicians I recommend [S]) and which today is by far dominating, and a more algebraic approach generalizing the Hilbert space approach to quantum mechanics (see e.g. [Ha]). The latter approach is difficult to use in practical computations and thus less popular today. One way to see the difficulty of quantum field theory in this latter approach is as follows. As mentioned, in quantum mechanics all Hilbert space representations of the *-algebra A of the model are essentially the same (i.e. they are unitarily equivalent), and it therefore does not matter which representation one uses. It is therefore easy to get the right setting for a given model, and the only (but in general quite demanding) task is to compute numbers which can be compared with experimental results, for example expectation values of obervables in the ground state etc. This is different in quantum field theory where one first has to construct an appropriate Hilbert space representation of the algebra A of the model which is, in general, difficult and can be done only in special cases. But then, in the end, one needs to compute also numbers. As a mathematician one would like to clearly separate these two steps, (i) construct the appropriate Hilbert space represenation of the field algebra A of the model, (ii) do computations in that setting. A 'pure' mathematician might (perhaps) not be very much interested in (ii) since he/she would regard it as a problem in 'applied' mathematics (involving expansions, approximations and numerics). However, in many examples which are important in physics, steps (i) and (ii) cannot be clearly separated. To my opinion, this is an important lesson one learns from a very successful approach to quantum field theory called perturbative renormalization (see e.g. [Co,S]). I believe that this is one fundamental reason why the algebraic approach to quantum field theory is difficult. The quantum field theory examples discussed in these notes are such that step (i) can be done in a rigorous manner, without doing (ii). These examples already show some of the complications of quantum field theory in a situtation where they can be resolved, and they also show that a Hilbert space approach (which I regard as a special case of the algebraic approach in the spirit of [Ha]) 183
to constructing quantum field theory models non-perturbatively is possible and useful, even though the examples discussed (and actually all examples where the approach I use works) are quite simple from a physics point of view. However, I believe that some generalization of this approach should be useful for understanding more complicated quantum field theory models non-perturbatively. 2.4-2
Regularizing
diverging
series
The basic idea of regularization is quite old and was used already in the mathematical theory of diverging series quite some time before quantum mechanics was discovered. For examples, the formulas 11
l + 2 + 3 + 4+--- = - i l 3 + 2 3 + 3 3 + 4 3 + • • • = —-
(6)
can be found already in a famous letter which S. Ramanujan wrote to G.H. Hardy in 1913 (see [BeR]). I will now illustrate the basic idea of regularization by explaining one elementary method to make sense of these series. I consider the series N n=l
and allow for cases where the naive limit N —> oo does not make sense, e.g. an = nk with integer k > — 1. In many such cases it is possible to extract from this limit finite numbers in a well-defined manner. A simple method for that is to introduce a regularization parameter e > 0 and consider N
Sjv(e) = £ > n e - n £ • n=l
In many cases (including an which are polynomial in n), the limit Soo(e) exists for e > 0, and we can expand Soo(e) in a series, Soo(e) = ^2kez Soo{k)ek. In this way we can determine explicity the diverging t e r m s 53fc
I stress that the sums on the l.h.s. here are regularized sums the precise meaing of which has to be defined: the naive interpretation obviously does not make sense.
184
limit can be taken. We therefore define oo
X)' a «
:=
(7)
^-(0)
n=l
where the prime here indicates that this defines a regularized sum. In this manner one can derive the following results which give a precise mathematical meaning to the Eqs. in (6): Facts: oo
-
oo
>=
1
£'» = -£• £'" l l r n=l
<8>
n=l
Hint: To prove these formulas it is convenient to introduce the function /(e) = 2 ^ i e~ne = e~ £ (l — e ~ e ) _ 1 and obtain the e-expansion of ]T)nLi nke~ne by taking derivatives of the e-expansion of f(e). O Remark: Note that my definition in Eq. (7) is equivalent to the standard regularization of this limit by analytic continuation, Soo(0) = Res(5oo(e)e _1 ,e = 0) (cf. K. Wojciechowski's lectures in the present volume). I also refer the readers to S. Paycha's lectures (also in this volume) concerning the explanation of diverging integrals. • 2.4-3
Field algebras and their
represenations
In this section I describe the strategy which I use in this notes to define and construct quantum models. First, a *-algebra A is denned by generators and certain relations amongst them. In all my examples this algebra is generated by fermion field operators obeying anticommutator relations. Even though this algebra is defined without reference to a representation, in all my examples A has a natural representation no on some Hilbert space T. A Model is given by a Hamiltonian, i.e., a self-adjoint element H E A. In case of fermion quantum field theory it is important to consider also other representations n of A, and the crucial point in a proper construction of a model is to find one such representation in which H = n(H) is a self-adjoint operator bounded from below. As I will discuss, there is a general method to find such a representation for models without interactions. This method also allows to construct certain interacting models, namely all those where there is a non-interacting model with a Hamiltonian Ho and for which the proper representation n for Ho is such that also ir{H) is a self-adjoint operator bounded from below. Thus such a model has so 'mild' interactions that it still can be constructed in the Hilbert space of the corresponding non-interacting model. 185
3 3.1
Fermion quantum mechanics The simplest
fermion
system
I
know...
... can be characterized as the *-algebra A generated by the element a obeying the following relations, a2=0,
{a,a*} = lA.
(9)
Note that this also implies (a*) 2 = 0. A simple representation TTO of this algebra is given on the Hilbert space JF = C2 with the usual inner product (•, - ^ : a
00\
= a~ = 1 i o J '
a
*
[01
= a+ = v o o
(10) and 1 = lea (= the 2 x 2 unit matrix). Note that I abuse notation here and, strictly speaking, I should write 7To(a) and 7To(a*) instead of a and a* in the previous equation. However, for simplicity in notation I identify A and no(A) for all A € A here and in the following. It is easy to see that this representation 7To is unitary, i.e., * in this representation is the matrix adjoint. 10 Note that the operator N = a* a — I j is self-adjoint and has two eigenvectors, e 0 = I
1
] and e\ = I _ J with corresponding eigenvalues 0 and
1, Ne„ = nen for n = 0,1. Moreover, the two eigenvectors provide a basis in C2. Physical interpretation: - eo is the vacuum state i.e. the state without particle. - e\ is the state with one particle. - N is the particle number operator. - a is the particle annihilation operator and a* is the particle creation operator. (Why? Noting that aei — eo and aeo — 0 resp. a*eo = e\ and a*e\ = eo and recalling the above interpretation of the states eo,i these names are kind of natural, aren't they). 186
Note that we only have states with 0 or 1 particle and thus the P a u l i P r i n c i p l e is obeyed: no more than one fermion in one state. M o d e l s : In the present simple example there is not much room to define very interesting models. One kind of Hamiltonian we can consider is H = u>a*a with real u>. Since H = wN, the eigenvectors of H are just the vectors en above: Heo = 0 and He\ = uei. Note that only for positive w the zero particle state eo is also the ground state of our model, whereas for negative w the groundstate is e\ (for u = 0 one has obviously a degenerate groundstate). It is instructive to compute the partition function Eq. (4) for the above model, Z = J2
(en,e-^H-^en)
= £
n=0,l
e-M<->»-i™)
= 1 + e-/J(—M) .
n=0,l
From this we obtain
< N > = I A i o g (Ki 0dn
+ e-««—"))=
l e 0(
ta
'-'1) + 1
which we can write as < N > = fpifjJ—(J,) where fp is the F e r m i d i s t r i b u t i o n function,
MV
(11)
= ^hi •
A n o t h e r r e p r e s e n t a t i o n of the field algebra A is given by 7r(a) = a+ = V,
vr(a*) =
(12)
Note that ip = WaU where U = I
1 n
(13)
I is an unitary operator on T: the representations no and ir
are unitarily equivalent. 3.2
Fermion systems freedom...
with a finite number
of degrees of
... can be characterized by the *-algebra AN generated by elements a •, j = 1 , . . . , N, obeying the following canonical anticommutator relations (CAR), {aj,ak}=0,
{aj,al}=Sjkl
\/j,k = l,...,N.
(14)
where 1 is the identity in AN- Note that this implies {a*j,a*k} = 0. In the following I state several 'Facts' which can be proven by (in general) simple 187
computations which I only outline as 'Hints' but otherwise leave as an exercise to the reader. Fact: The following defines a representation ir0 of the algebra AN on the Hilbert space TN = C2 ® • • •
v
v
'
^
v
'
(j - 1) times (N - j ) times a] = 03 ® • • • ® g3,
(15)
{N — j) times
and 1 = (To <S> • • • ® (To where v v ' TV times CTo
= ( o i ) 'CT3= ( o - i ) ' °"+ = ( o o ) ' CT- = ( i o ) - ( 16)
Hint: Use (&i ® • • • ® &AT) • ( c i ® • • •
for 2 x 2-matrices bj and Cj.
D
Fact: The representation n0 above is unitary i.e. * is the Hilbert space adjoint. Hint: Recall that the inner product in TN is given by (/i < 8 > - - - ® / J V , 5 I ®---®9N)TN
= (fi,9i)&
•••(fN,9N)c
2
for all fj ,gj 6 C . This implies that ( 6 i
for all 2 x 2 matrices bj.
•
Fact: The vector n := e 0 ® • • • ® e 0 € ^
(17)
iV times with e 0 = ( , ) obeys Ojfi = 0 Vj = l,...,7\T.
(18) D
12
Again I identify A and 7ro(^4) for all A e .Aw.
188
Fact: A basis in !FN is given by Fa = {al)n'---{a*N)nNQ.
for n = (nu ... ,nN),
n,e{0,l}.
(19)
Hint: Show that ^n = sneni
), ei = I n ) and s n either 1 or — 1. Use this to show that FN
— " i i , m i ' ' ' " T I N ,mjv
Vn,me{0,l}N. Check that there are enough independent vectors Fn to span TN-
(20) Q
Fact: The operators Nj := a*a5
(21)
obey N j F n = rijFn (Fn is defined in Eq. (19)), and the operator N
N := ] > > , •
(22)
NF n = j JT nj J F n Vn e {0,1} N .
(23)
therefore obeys
• Physical interpretation: In our system we have N different 1-particle states. In the state F„ we have rij fermions in the 1-particle state ' j ' , j = 1 , . . . , TV. In particular, fl is the v a c u u m s t a t e i.e. the 'empty' state without fermions. N is the particle number operator. a.j and a*j are the particle annihilation operator and the particle creation operator corresponding to the 1-particle state ' j ' . 189
Again we see the Pauli Principle at work: no more than one fermion in one 1-particle state. Fact: Let T^' be the subspace of TN spanned by the eigenvectors of N with eigenvalue n. The complex dimension of T^' is N\ (N-n)\n\
•
Moreover, TN = J** © J%> © • • • © T(NN) = 0
J#>
(24)
n=0
i.e. the particle number operator gives a natural grading for the Hilbert space on which our system is represented. • Models: In the present setting we can consider a large class of Hamitonians which are of interest for many physical applications. A very simple class of Hamiltonians is given by N
H = 22U)ja^aj
'
Ui < u)2 < • • • < U>N
(25)
equal to H = Ylj=iu,j^3P1"01*1 o u r discussion above it follows that all the vectors Fn in Eq. (19) are also eigenvectors of H, N
Vne{0,l}w.
HFB = ^2ujnjFn
(26)
Thus, if all the OJJ are positive, then fi obviously is also the groundstate of our model. However, this is not the case if some of the Wj are negative. Facts: If wi,...,u}m
< 0,
u}m+i,...,UN
> 0
(27)
then fl := al---a*mn
(28)
is the ground state of our model, and the ground state energy is m
Eb = $ > , - . 190
(29)
Moreover, a*n = 0
for j = l , . . . , m ,
ajQ = 0
for j = m + 1 , . . . , N . (30)
Hint: To see this, convince yourself that the minumum of J2j=i0Jjnj^ n G {0,1} W , is So and assumed for m = • • • = nm = 1 and nm+i = • • • = njv = 0. D Fact: The partition function Eq. (4) for the model with the Hamiltonian Eq. (25) is given by
z=
n( i + e _ / 3 ( w ^ M ) ) N
(3i)
3=1
and < N > Eq. (5) is equal to N
(32)
3=1
with the Fermi distribution function denned in Eq. (11). Hint: The proof of Eq. (31) is a simple but instructive computation which we therefore outline it here:
Z=
Y,
(Fn,e~^-^Fn) 'FN
ne^.i}" N ni,...,n^=0,l
j=l
n.j=0,l
n(l+e-«"'-")) . J'=l
D Physical i n t e r p r e t a t i o n : In the groundstate CI, all the 1-particle states with a negative energy, LJJ < 0, are 'filled' and all 1-particle states with a positive energy, LJJ > 0, are 'empty'. Moreover, for large /? (corresponding to low temperatures), < N > is close to m. The Hamiltonian Eq. (25) might look rather special. However, as I now show, it has already most of the essential features of a quite large class of 191
Hamiltonians, namely N H
= £
a
*jDi«ak •
Fact: if above is self-adjoint iff D = (Djk)fk=l i.e. Djk = Dkj for all j , k.
(33) is a self-adjoint N xN matrix D
Remark: We use the symbol '£>' since later the corresponding object will be some Dirac operator. Indeed, using the fact that every self-adjoint matrix D can be diagonalized by a unitary matrix U, D = U*uiU, where ui = diag(ui,... ,U>N), we can reduce the present case to what we discussed after Eq. (25): Using the Fact: The operators N
N
fc=l fc=l
give a unitary representation of the algebra ANO, we can write N H
= £WJ5J*S- '
(35)
and our discussion implies that all eigenstates of the model are given by Fn = ( a i ) n i - - - ( a ^ ) n " f i ,
(36)
with HFn — X^7=i WjJijFn, and these also form a basis in FN- Especially if Eq. (27) holds then fi = of • • • a*mn
(37)
is the ground state of our system. Moreover, we have the Fact: Eqs. (31)(32) also hold true in the present case.D We thus can also solve models with Hamiltonians of the form Eq. (33) quite easily by diagonalizing the 1-particle Hamiltonian D. As will be discussed in more detail later, it is natural to regard H as the second quantization of the N x N matrix D. More generally, we can define for any N x N matrix A N
dT(A) := ^ j,k=i
192
a*Ajkak
(38)
i.e. H = dT(D). One can prove several nice relations for these operators dT(A) (see the Eqs. (56) later). Here we only mention the following interesting Fact: TrrN{eidr^---eidriAk))=det(lCN
+eiM---eiAk)
(39)
for all k € N and self-adjoint N xN matrices Aj where det is the usual matrix determinant here. (The proof of this is not easy, and we will not use this result later and only mentioned it here since the generalization of this relation can be used to relate our approach to quantum field theory to the one discussed in K. Wojciechowski's lectures).• Application in solid state physics. Hubbard-like models: As we saw, models with Hamiltonians given in Eq. (33) are rather simple, as discussed. In physics one is often interested in more complicated models given by Hamiltonians N
H = H0 + V = ^2
N a D
) jkak
+
j,k=l
12
v
Jkima*a*kaeam .
(40)
j,k,£,m=l
with complex numbers Djk and Vjkim so that H is self-adjoint. The first term (i.e. the one quadratic in the cS-*>) is usually referred to as the free part of the Hamiltonian, and the second one is called the interaction term. A simple example is the one dimensional Hubbard model defined by the Hamiltonian 13 [Hu] e H
t
i a
0
u
= ~ 12 12 (°i+i,* i,o- + °i,* i+i,*) + 12ni.t"i4. i = i
(41)
j=l
where nj
a53=«W, $ = « & fari = l
/
and t and U are real parameters. The physical interpretation of this model is as follows: One has a one dimensional lattice with I sites and periodic boundary conditions. On each site j = 1 , . . . , £ one has fermions with internal spin i.e. one has two one-particle states ' f and '4-'. The first term describes hopping of the fermion, e.g. the term a^+1^a-^ corresponds to a f-fermion hopping from site j to site j + 1 (i.e. a f-fermion on site j is destroyed and a t-fermion on site j + 1 is created) etc. Moreover, the interaction term describes a strongly screened Coulomb interaction of the fermions: nJ]0- is the 13 Our notation above would suggest to write NJ]
193
operator measuring the number of cr-fermions on the site j , thus (for U > 0) the last term in Eq. (41) above describes on-site Coulomb repulsion of f- and 4,-fermions. For later reference I note the following simple Fact: For U = 0 (no interactions), the Hubbard Hamiltonian in Eq. (41) can be written as in Eq. (33) with a 2L x 21,-matrix D which has the eigenvalues E(p) = -2t cos(p) ,
p = (2n + 1) -
- _ £ _£
e
-1 (42) , n 2, 2 + l,.--,2 (I assume I even for simplicity) which all are double degenerate. (Hint: It is an instructive exercise to check that: Using (cr,j) as matrix indices (a = t , I, j = 1,.. .,£) the matrix elements of D are (.DJajV'.j' = -2t6(y(ri Ajj> where A is the I x I matrix e
/
010 0 1 0 1 0 ...0 0 1 0 1 ...0
0 0
000... 0 1 000... 1 0
0 1
A =
V - i o o . . . o i o/ which can be diagonalized by the following ansatz for the components of eigenvectors, ej — aeipj + be~ipj . ) • Similarly, the Hamiltonian for the two dimensional Hubbard model describes fermions with internal spin on a two dimensional lattice {(j, k) 6 Z2|^' = 1 , . . . ,£i, k = 1 , . . . ,£2} with ^i^2 sites, periodic boundary conditions, hopping between nearest neighbor sites and interacting via on-site Coulomb repulsion. R e m a r k : Our discussion above makes clear that conceptually, all the models defined by Hamiltonians H in Eq. (40) are rather trivial: H can be represented by a 2N x 27V-matrix, and the models can be solved simply by diagonalizing this matrix. However, from a practical point of view, these models can be very complicated, and e.g. for the two dimensional Hubbard model very little is known up to this day (even though this model has been studied very intensively in the past 10 years or so [D,cond — mat]). One reason is that one is often interested in such models on rather large lattices, but the size of matrices grows exponentially with the number of lattice sites and their numeric diagonalization very soon is beyond the abilities of the most advanced computers. Our discussion here also illustrates why models with Hamiltonians Eq. (33) 194
without interactions are so much simpler: as discussed above, if all the Vjkim are zero in Eq. (40), the model can be completely solved by diagonalizing the N x N matrix {Djk). Otherwise one has to diagonalize (or at least find the eigenstate with the minimum eigenvalue) of the 2N x 2 N -matrix H which is 'exponentially more difficult'. • 3.3
Quantum field theory limit:
Heuristics
In physics one is often interested in fermion systems with an infinite number of degrees of freedom. The simplest example showing already the main problem is the model of chiral fermions on the circle formally given by the Hamiltonian (cf. Eq. (25))
tf = 5 > < a „
(43)
ngZ
where the a „ ' obey CAR as in Eq. (14), but now for all j,k € Z. Applying naively our results in the previous section to the present model we would deduce that the ground state is (cf. Eq. (28)) 0 = VL1ai2---aL1001---O"
(44)
(which is degenerate with a^Q,, of course, but this degeneracy is not important for what we want to say here and thus will be ignored) where the quotation marks indicate that this is only a formal expression. The corresponding ground state energy should then be (cf. (29)) E0 = -"(l
+ 2 + . . . + 1001 + . . . ) "
(45)
14
which clearly is divergent. One therefore has a serious problem: the Hamiltonian of the model is not bounded from below, the partition function (cf. Eq. (31))
does not exist (the infinite product diverges) etc. As will be made clear later in these notes, the main problem is that the ground state of the model (even though formally defined in Eq. (44)) is not a vector in the Hilbert space where we have defined the model. The physical idea to remedy this problem is due to Dirac: work in a different Hilbert space where one makes sure from the start that the ground 14 After reading Section 2.4.2 you might guess Eo = -^, right? Good guess ;-) justify this is a longer story; see e.g. [Le\.
195
but to
state is present. This can be done by using the fermion field algebra as a starting point and by representing it such that a*nCl = 0
for n = - 1 , - 2 , . . . , - 1 0 0 1 , . . . ,
anCl = 0
for n = 0 , 1 , 2 , . . . .
as suggested by Eq. (30). Requiring that / f i , n \ = 1, the full Hilbert space is then generated by the states oo
II«-i)fe"(a-n)'"n 71=1
where kn,£n € {0,1} are such that £^ = 1 (fc„ + £„) < 0. The physical motivation for remedying the problem with the diverging ground state energy is as follows: all one can ever measure in a physical system are energy differences, thus we may equally well work with the Hamiltonian H = H — Eol- This enforces that fl has finite energy (namely zero). It can be achieved in a simple manner by defining normal ordering : • • • : which is linear and such that : a^an := a^an if n > 0 and : a^an := — a n a* if n < 0. One then can write oo
H = J2 n • " X := 53 n (a*nan + a _ „ _ i a l „ _ i ) ngZ
(46)
n=0
which clearly is positive definite. Similarly, oo
N = 53 n€Z
:
°n°» ~
5 3 (°n°n - O - n - l O l n - l ) n=0
(47)
is the operator measuring the particle number relative to the number of particles in the ground state. The state fi in this and similar examples is called the filled Dirac sea and is interpreted as the state in which an infinite number of negative one particle states is occupied. The fermions operators a* for n > 0 then create particles, whereas the operators an for n < 0 create anti-particles (i.e. they remove negative-energy particles from the filled Dirac sea). Thus the operator N measures the number of particles minus the number of anti-particles. Since particles and anti-particles have opposite charges it is natural to regard N as charge o p e r a t o r . R e m a r k : This example is typical: the one particle Hamiltonian for relativistic fermions always is some Dirac operator which is neither bounded from above nor below. Therefore the one particle model of relativistic fermions never is physically acceptable (there is no ground state etc.). In the next 196
Chapter we will develop a general, abstract formalism allowing to construct a physically acceptable many particle description. This formalism generalizes and gives a precise mathematical meaning to the physical ideas which I tried to explain in this section. •
197
4
Quasi-free second quantization of fermions
In this Chapter we develop the general, abstract formalism for constructing quantum models for fermions. This formalism amounts to the following: Given the one-particle description of the fermions, construct the description of the corresponding many particle system i.e. the system with an arbitrary number of identical fermions each of which is described by the given one particle formalism. In physics this is called second quantization. In the first two sections we describe the general construction of the fermion field algebra, the many particle Hilbert space, and the second quantization of observables and transformations. In Section 4.3 I show how to recover our previous results on fermion quantum mechanics from the general formalism and the assumption that one has TV < oo one particle degrees of freedom. At the end of this section I also indicate the problems one has if one tries to make the limit N —> oo and thereby motivate the necessity of the quasi-free representations which are discussed in general in Section 4.4. The material covered in the section is standard. It might be helpful for some readers to consult some other references with a different emphasis, e.g. [A,CaR,KR,Mi,0,Wu]. 4-1
Fermion
field
algebras and fermion
Fock spaces
Let % be a separable Hilbert space. The fermion field algebra A = A(Jt) over H is defined as the *-algebra generated by elements a*(f) and a(f) = a*(f)* such that / -> a*(f) is linear, and the following CAR hold, {a(/),a(s)} = 0,
{a(f),a*(g)}
= (f,g)nl
Vf,g£H.
(48)
Note that this also implies {a*(f), a*(g)} = 0. Remark ( a d v . ) : Usually one defines A(H) as C*-algebra by also defining a norm via ||a^*H/)ll — (/>/)-« ( s e e e-S- [BrR]). As mentioned, we will not use C*-algebra arguments in the following. • There is a simple representation no of this algebra A on a, Hilbert space F = F(H) which is completely determined by the following conditions (as will be explained below): (i) the representation TTQ is unitary, (ii) there is a vector fl in T such that
(ft, n> = 198
I
(49)
and 15 a(/)fi = 0 V / G H .
(50)
The Hilbert space T^H) is called the fermion Fock space over "%, and TTQ is the so-called Fock-Cook representation of the fermion field algebra A{H). I now sketch how one can reconstruct T from the conditions above: The a(f) and a*(f) are operators on T, thus all vectors 16 / i A - - - A / „ := a * ( / i ) - - - o * ( / n ) n
(51)
with fj £ H, n an arbitrary non-zero integer, are in Jr. Then one can show that these vectors span J7, and the Hilbert space structure of T is determined by the relations above. In particular, one can prove the Fact: The above conditions imply ((fi,9i)n
(/i A • • • A / „ , 51 A • • • A gm) = <5„,m det \(fn,9l)-H
•••
:
(fi,9n)n\
• •. •••
:
. (52)
(fn,9n)nJ
Hint: Your can prove Eq. (52) by induction. One important step is the following computation (where one uses repeatedly a(fi)a(gj)* = (/i, gj)H 1 — a(gj)*a(fi) to move a ( / i ) to the right until one can use a(/i)fi = 0 and thus get rid of it) (0,,a{fn) • • • a(fi)a*(gi) n
• ••a*{gm)Q,) =
a
(/i.9\)n ( » Un) • • • a(f2)a(g2)* •••a(gm)*Q,} - (n, a ( / n ) • • • a(f2)a(g1)*a(fl)a(g2y • • • a*(gm)fl) = m
• •• = E t - 1 ) ' " 1
(53) H
/ i A - ^ A ' - A /
n
(n > 1) .
15 As before, I should write 7ro(a(/)) in the following equation, but for simplicity in notation I identify A and no(A) for all A G A in the following. 16 You can regard the following equation as definition of a useful and suggestive notation.
199
D Remark (adv.): I am a bit sloppy here, but if you know more about Hilbert space theory you can fill in the missing details (see e.g. [BrR]): denoting the vector space spanned by the elements A A • • • A / „ Eq. (51) as AH, one can take Eq. (52) as a definition of a sesquilinear form on AH and then prove that it has the properties of an inner product making AW. into a pre-Hilbert space (see e.g. [i?5]). Then T is recovered as the norm completion of AH. • Remark (adv.): My discussion of the representation TTO here is similar to the one in most physics texts. A more elegant argument is as follows (see e.g. [CaR]): first construct the Hilbert space T, and especially the vectors fi and / l A • • • A fn, without reference to the fermion field algebra A and such that Eqs. (52) and (49) provide the definition of the inner product in J-. One then takes Eqs. (53) and (50) as a definition and proves that this defines a unitary representation -KQ of A on T. • 4-2
Free second quantization
of observables
and
symmetries
Let i b e a bounded operator on the one particle Hilbert space H- Then aT(A)a
= 0
(54)
and n
dT{A)hA---A}n
:= ^ / i A - A ^ A - A / „
(55)
define an operator on T (since the vectors H and /1 A • • • A / „ span H). By simple computations one can prove the Facts: [dT(A),dT{A)] = dT([A,B]) dT{A)* = dT(A*)
(56)
for all A,B G B. In particular, if A is self-adjoint then dT(A) is also selfadjoint. • Let U be an invertible, bounded operator on W. Then T(U)ft = 0
(57)
and T(t/)/i A • • • A /„ := Ufi A • • • A Ufn 200
(58)
define an operator on T, and similarly as above one proves the Facts: r ( l / ) r ( V ) = T(UV)
nii)-1 = r{u-1) T{U)* = T{U*)
(59)
for all invertible U,V £ B. In particular, if U is unitary on 7i then T{U) is unitary on T. Moreover, T(eiA) = e idr < A)
(60)
and r(C/)dT(yl)r(C/- 1 ) =dT{UAU-1) for all A,U eB,U
(61)
invertible. •
We also observe the Facts: We have [dT(A),a*(f)}
= a*(AJ)
(62)
and r{U)a%f)T(U)-1=a*(Uf)
(63)
for all / £ ~H, A, U € B, U invertible. Moreover, Eqs. (54) and (62) completely determine dT(A), and Eqs. (57) and (63) completely determine ]?([/).• As will be demonstrated in specific examples later, for self-adjoint A it is natural to regard dT(A) as the many particle observable associated with the one particle observable A. Similarly, for unitary U, T(U) is regarded as the many particle (symmetry) transformation associated with the one particle (symmetry) transformation U. dT and T are therefore often called the second quantization map for observables and (symmetry) transformations, respectively. Remark: Note the Fact: The algebra B{W.) is naturally a Lie algebra with involution * and the Lie bracket given by the commutator (with a topology given by the operator norm).D Since A —> dT(A) obviously is linear, Eq. (56) shows that dT provides a * preserving representation of this Lie algebra B{%) on the fermion Fock space over H. Similarly, one has the Fact: The set 1 of invertible operators in B is a Lie group with involution * • , and Eqs. (59) show that r is a *-preserving representation of this Lie group. Moreover, Eq. (60) establishes a nice and natural relation between the former Lie algebra and the latter Lie group which motivates the notation dT vs. I1. For self-adjoint 201
operators A, Eq. (60) also gives a nice relation of Stone's theorem on % and on T{U). D R e m a r k ( a d v . ) : For infinite dimensional Hilbert spaces % the full story is a little more complicated since the dT(A) are unbounded operators on T even if the operators A are bounded on H- However, it is quite safe to regard this as technicality which one can ignore since it is easy to construct a common, dense invariant domain for all operators dT(A) and to establish Eq. (56) on this domain. Moreover, in physics application, one is often interested in dT(A) for unbounded operators A on H. It is quite straightforward to extend the construction dT to this situation. In particular, if £ is some algebra of (in general) unbounded operators on H with common, dense invariant domain T> then Eq. (55) for fj 6 V makes sense and defines dT(A) for all A € C on some common, dense invariant domain in T such that Eq. (56) etc. hold on this latter domain (for details see [GL2]). In particular, Eq. (60) holds true also for unbounded self-adjoint operators A on %. •
4-3
Fermion
quantum
mechanics
revisited
We now show how to recover our results in Section 3.2 from the general formalism above. Facts: Fermion quantum mechanics with N one-particle degrees of freedom corresponds to the special case H = CN, i.e. AN = A(CN) and FN — T(
(64)
The eigenvectors {ij}1JL1 corresponding to negative eigenvalues ojj of D obviously span a subspace H- of H = CN. One can interpret H- as the negative energy subspace corresponding to the operator D. We now observe the Fact: The ground state of the model only depends on the negative energy subspace H- of CN, i.e. for any basis {Ej}™=1 in %_, the vector ft
=a*(E1)---a*(Em)Q,
is equal to fi in Eq. (64) up to a phase. Hint: Write Ej = X^fcLi Vjk&k where V is some unitary mx m matrix. Then m
n'=
2
Vlkl---Vmkma*(ekl)---a*(ekm)n
= det(V)Cl
fci,...,fcm=i
where in the last step one uses that the a*(ekj) anticommute with each other. D This last fact will be an important motivation for the construction in the next section. 4-4
Quasi-free
representations
Let H- be some closed subspace of 7i such that the operator P_ projecting onto U obeys PJ! = P* = P _ . We write P+ = 1 - P_ and V.+ = P+U. Note that this gives a splitting of H in two orthogonal subspaces, H = H+ © H-, i.e. every f £ H can be written as / = /+ + /_ with f± = P±f G H±, and (f-,f+)n = 0 (the latter follows from P_P+ = P+P_ = 0). As discussed, in the examples we have in mind %- is the negative energy subspace corresponding to some Dirac operator, but the following construction is completely general. We now define a non-trivial representation of the fermion field algebra A on the fermion Fock space T corresponding to 'filling up the Dirac sea' i.e. the states in the subspace H-, as discussed in Sections 3.3 and 4.3 above. For that we need an antilinear operator J :% —> H commuting with P_ and obeying J 2 = 1 and (f,g)n = {g,f)H for all f,g £ H. (In the following the explicit form of J will play no role, but for the sake of completeness we note the following Fact: Let {e n }„ e z be any basis in V. such that {e Tl }°__ 00 is a basis in H-, i.e. P_ = Yl°-oo e™ (e™> ')• Then
203
defines an operator J with the desired properties.•) We then define r{f)
:=a*(P+f)+a(JP-f),
i>(f) := a{P+f) + a*(JP-f)
Mf € 7(65)
where a,(*\f) are the fermion field operators in the Fock-Cook representation as defined above. We observe the following important Fact: For every projection operator P _ , the operators in Eq. (65) define a unitary representation np_ of the fermion field algebra A on the fermion Fock space F, ^)(/)=TP.(O
W
(/)),
(66)
which is such that
(67)
Hint: One needs to check that / ->• tp*{f) is linear, ^ ( / ) = ip*(f)*, and that the CAR in Eq. (48) are obeyed for the operators ip^*\f)- This can be proved by simple computations using the corresponding properties of the operators «(*>(/). Eq. (67) follows trivially from Eq. (50). • R e m a r k : One can prove that the representations np_, P- projection operators on 'H, are irreducible (see e.g. [BrR]). O R e m a r k : As mentioned, in a specific model with a 1-particle Hamiltonian (Dirac operator) D, a natural choice is P_ = 9{—D) (defined via the spectral theorem) with 6(x) = 1 for x > 0 and 9{x) = 0 for x < 0. Then the state O above represents the filled Dirac sea of the model, and one can prove that it is indeed a ground state of the many particle Hamiltonian corresponding to D. 4-5
Quasi-free
second quantization
of
observables
We now discuss the second quantization of 1-particle operators in the quasifree representation ivp_ defined above. In case of the Fock-Cook representation, the second quantization of 1particle observables and transformations was always possible. In a quasifree representation this is no longer the case: loosely speaking, the second quantization of a 1-particle operator exists only if it does not mix the subspaces 'H+ and V.- above 'too much'. To be more specific: the decomposition 204
7i = %+ © H- of our one particle Hilbert space provides a natural decomposition of all operators A on H as follows,
where A++ = P+AP+ maps U+ to %+, A+_ = P+AP- maps %_ to %+, etc. As we will explain in more detail below, the second quantization of a (bounded 17 ) operator A exists if and only if A-+ and A^— are HilbertSchmidt operators (which loosely speaking means that these operators are 'not too big'). This is the celebrated H i l b e r t - S c h m i d t condition which we will encounter in several places. R e m a r k : The notation introduced in Eq. (68) is very useful in calculations. Note that in this notation,
and the multiplication of operators is simply by the rules of 2 x 2 matrix multiplication. For A a bounded operator on H, one would like to construct a self-adjoint operator dtp_ (A) = dt(A) on T such that [dt(A),r(f)]=r(Af)
(70)
for all / e H, and (n,dr(A)fl\
=0.
(71)
It turns our that if this operator exists, the relations in Eqs. (70)-(71) determine it uniquely, and a necessary and sufficient condition for existence is the above mentioned Hilbert-Schmidt condition. Thus the set of all operators allowing such a second quantization is 5i := {AeB\A-+,A+_eB2}
.
(72)
We now summarize the important properties of the quantization map dT in a T h e o r e m : Let A e B. Then a unique operator dt(A) on T obeying the relations (70)-(71) exists if and only if A £ gi. Moreover, A -)• dt(A) is linear, and the following relations hold true, df(A)*=df{A*) 17
(73)
This condition can be relaxed but then one has to cope with some additional technicalities.
205
and [at(A),at(B)]
= at([A, B}) + S(A, B)lr
(74)
where S{A, B) = Trn(A^+B+„
- B-+A+.)
,
(75)
and dt(A)n
= 0<& A-+ = 0 .
(76)
R e m a r k : Note that Eqs. (74)-(75) show that the Hilbert-Schmidt condition is not just a technicality: the term S(A, B) has to be finite (otherwise one has a serious problem with the relation in Eq. (74)) and for that it is crucial that the operator ( A _ + B + _ — JB_ + 4+_) has a finite trace — this is (essentially) just the Hilbert-Schmidt condition. • Outline of Proof: Instead of giving a full mathematical proof of this Theorem (which can be found e.g. in Ref. [CaR]) I present a more down-to-earth construction of dT(A) for a restricted set of operators A which will (hopefully) give a more intuitive understanding of aT, and especially of how the Schwinger term in Eq. (74) appears. For that I use a basis {e n }„ e z in H such that {en}n=-oo *s a basis in ~H-, i.e. P_ = S _ o o e " (e"> ')«• ^ e c a n then write bounded operator A on B as A = Y2m nezamn^m (en, -)-H w n e r e amn = (em,Aen)H. For simplicity (i.e. to avoid convergence problems in the following) we assume that A is such that only a finite number of the amn are non-zero. I will refer to such operators as finite rank operators. I then define Q(A) :=
^2amnil)*{em)ip(en),
and by a simple computation (using the CAR) one can check the Fact:
iQ(A),r(f)} = r(Af) for all / € H.O This shows that Q{A) is quite close to aT(A) but exactly so due to the following Fact:
We thus see that the operator dt(A)
:= Q ( A ) - T r ( A _ _ ) l ^
obeys both relations Eq. (70) and (71). A simple computations (using again the CAR) proves the Fact: [Q{A),Q(B)] = Q([A,B]) (similarly as for the operator dT(A) is the Fock-Cook representation).• With that we obtain \df(A),dt(B)} = [Q(A),Q(B)] = Q([A,B}), which implies Eq. (74) with S(A, B) = Tr([A, B]__) = Tr(A_ + B+_ + A _ _ £ _ _ - 5 - + A + _ -
B—A-).
The second and the forth term on the r.h.s. in this equation cancel due to cyclicity of the trace, and we obtain Eq. (75). It is also easy to check that Eq. (73) holds true. We now have proven the Theorem for a (small) subset of g\. As will be explained below, if we were only interested in this subset of operators, the Schwinger term S would be irrelevant. However, the construction of dT{A) can be extended to all A € g\ so that all the relations derived here remain true, and then 5 becomes important. In particular, there are operators A E gi such that neither Tr(yl__) nor Q(A) exist (we will give a simple example for this below). Note that the argument above does not apply to such operators since Tr([^4,B] ) will not exist in general, even though S(A,B) in Eq. (75) is well-defined (i.e. [A ,B ] is not trace class, thus the equation Tr([^4 ,B ]) = 0) used above does not make sense.) As mentioned already above, Eq. (75) indicates that the Hilbert-Schmidt condition is necessary, so the non-trivial step in the proof of the Theorem is to show that it is also sufficient for dT(A) to exist. The proof of this (see e.g. [CaR]) is beyond the scope of the present notes. Remark: Note that if dim(H) is finite, the Hilbert Schmidt condition is empty: all operators can be second quantized. The same is true if P_ = 0 which obviously corresponds to the Fock-Cook representation: dTo = dT. • Remark (adv.): Again, the operators dT(A) all are unbounded, but one can construct a common, dense invariant domain in T so that all the relations above hold on this domain [CaR]. Moreover, the construction of dT(A) can also be extended to certain Lie algebras of unbounded operators A on 7i obeying the Hilbert-Schmidt condition [GL2], and this extension actually is required in several important applications. • 207
Remark: Observe the Fact: gi is a Lie algebra with the Lie bracket given by the commutator and involution * . • The Theorem above claims that dt is a ^-preserving so-called projective representation of this Lie *-algebra on T: it is not a representation due to the appearance of the term S(A, B)lj? which is called Schwinger term by physicists. From a mathematical point of view, S is a 2-cocycle of the Lie algebra g\: The relations dT(.4),df (B)\ = - [df(.B),df(,4)] and \dt(A), \dT(B),dt(C)l| valid for commutators) imply the Fact:
+cycl. = 0 (= Jacobi identity
S(A,B)+S(B,A)=0 S(A,[B,C})
+ S(B,[C,A])
+ S(C,[A,B})
=0
D
(77)
which are called 2-cocycle relations. Finally, the relation in Eq. (76) is called highest weight condition. In fact, gi is a prominent example of an infinite dimensional Lie algebra in which many of the well-known infinite dimensional Lie algebras can be naturally embedded, for example the loop algebras. One reason for the importance of the Theorem above is that it immediately gives interesting (and in fact very important) representations of infinite dimensional Lie algebras like the affine Kac-Moody algebras, the Virasoro-algebra 18 etc. We will come back to this in the next Chapter. • Remark: There is one important conceptual point in the extension of dT from finite rank operators to g\ which I would like to stress: The normal ordering prescription I used to define dV(A) was just one amongst many possible, and another possible prescription would lead to a different second quantization map dt'(A)
= dt(A) - b(A)lf
(78)
with 6 some linear function gx -> C so that b(A*) — b(A) for all A £ gi. For example, we could have a model with a ground state O' G T different from fi, and then we would rather normal order with respect to fi' i.e. use dT' above with b(A) - /tt',df(A)tt'\.
Fact: dT' obeys relations as in Eqs. (73) and
(74) with S replaced by S" where S'(A,B)
= S(A,B)
+ b([A,B}).
(79)
Moreover, S is a 2-cocycle if and only if S' is.D It is thus natural to regard the two quantization maps dT and dT' as equivalent, and therefore the two 18 This actually requires the extension of the Theorem to unbounded operators mentioned above.
208
2-cocycles S and S" related as in Eq. (79) are also equivalent 2-cocycles. Thus the Schwinger term we discuss here is interesting only because of the following Fact: The 2-cocycle S in Eq. (75) is non-trivial: there is no linear function b so that S'(A,B) Eq. (79) vanishes for all A, B £ gx. (Hint: Convince yourself that this is proven if one can find two commuting operators A± in g\ such that S(A+,A-) is different from zero. Show then that A± — ^2n£Z en±i (en> -)-H (notation as above) are two such operators.)• It should be obvious now why the Schwinger term S is irrelevant for finite rank operators. This also explains why Schwinger terms are irrelevant in fermion quantum mechanics and becomes important only in quantum field theory. D Remark: What we discussed in the previous two remarks is a special case of Lie algebra cohomology (see e.g. [ChD] for a concise introduction). • Remark: In the physics literature one often writes dt(A)
= ^2amn:
V*(e m )V(e n ) :,
(80)
where : • • • : is the normal ordering symbol mentioned already in Section (3.3) and which (more generally) is defined as the linear map such that
: ••(«.» W = = { ^ ( X T ' ZZZX °
<«»
equivalent with : A : = A — (0, ACl) Ijr (i.e. subtraction of the vacuum expectation value). As already mentioned, the Theorem above shows that : A : can be well-defined even if A and (0, Aft) do not exist separately. A simple and generic example for this is for the operator An > 0 for n > 0
(82)
and Am < \ n for m < n. In this case, Tr(D—) = ^ n < o ^n does not exist in general (the sum does not converge). However, £>_ + = D^ = 0 (i.e. both are trivially Hilbert Schmidt), and our theorem above 19 therefore implies that dT(D) is well-defined. Moreover, dt(D) = J2 n€Z
X
n : V*(e„Men) = = E
^WMtM
+ J^ |A„|V(e„)V*(e„)
n>0
n<0
which shows explicitly that this operator is non-negative and dt(D)fl 19
To be precise: its extension to unbounded operators; see [GZ>2].
209
= 0. •
R e m a r k : The example in the last remark is actually very important from a physics point of view and leads us back to my discussion in Section 3.3: We can now see explicitly that the quasi-free representation can be chosen such that it obeys an essential physical requirement on any reasonable quantum model, namely the existence of a ground state. As discussed, in a specific model, the 1-particle Hamiltonian is given by self-adjoint D on % and dt(D) represents the corresponding many particle observable Hamiltonian. Generically, D is some Dirac operator on a compact spin manifold, and such an operator can always be represented as in Eq. (82) with some complete orthonormal basis {e n } n gz and An —>• —oo for n —> —oo. Such an operator does not have a ground state, and dT(D) in the Fock-Cook representation does not have a ground-state, either. However, our discussion in the previous remark makes explicit that in the quasi-free representation np_ with F_ = 0(—D) = ^ n < 0 en (en, - ) w , the Hamiltonian dT(D) has a ground-state, namely il. I assumed here that D has a pure point spectrum for simplicity, but the result holds actually true for all self-adjoint operators D. • 4-6
Quasi-free
second quantization
of
symmetries
I now discuss the Lie group version of the results in the previous section. Much of the structure there is already contained in the Lie algebra, (which can be regarded as the infinitesimal version of the Lie group), but (as I will only indicate) there are a few features reflecting some interesting topology of the Lie group which is lost when going to the Lie algebra. For unitary operators U on 7i, we are interested in a unitary operator t(U) on T such that t(UMf)f(Uy=*P(Uf)
V/eW.
(83)
Using the results in the previous section it is easy to obtain such an operator T{U) if U = elA with a self-adjoint A € 51: In this case, standard Hilbert space arguments imply the Fact: 20 f{eiA)
= eidt^
.
(84)
(Hint: Existence of the r.h.s. of this follows from Stones Theorem since dT(A) is a self-adjoint operator on T. Moreover, the identity
20
T h e results also holds for a large class of unbounded, self-adjoint operators A obeying the Hilbert-Schmidt condition [GL2].
210
can be proven by expanding the l.h.s. in a Taylor series (in ( 6 l ) using repeatedly Eq. (70) which shows equality with the Taylor series of the r.h.s.)D However, there are interesting operators U for which T(U) exists but which are not of the form elA, A € g\. I will not go into this too much and only state the Theorem: Let U £ B be unitary. Then a unitary operator t(U) obeying Eq. (83) exists if and only if U is in the set d
:= {U unitary on U \U-+,U+-
e B2 } •
(85)
Instead of a Proof: I believe that this result is quite plausible by now, and for a proof I refer to Ref. [R] which also contains an explicit (and to my opinion very instructive) construction of the operators T(U) for all U € G\. To understand the significance of this result it is useful to convince oneself by direct computations of the following Facts: If A e
•
Remark: Observe the Facts: The set G\ is a group closed under *, and T(U) is uniquely determined by Eq. (83) up to a phase. In particular, if T(U) and f ( F ) exist also t(U*) and t(UV) exist, and t{U*) = t(U)* .
(86)
Moreover, t(U)t(Y)
= a(U, V)f(UV)
(87)
with a a phase valued function on G\ x G\ satisfying a(U, V)a(UV, W) = a(U, VW)a(V,
a(U*,V*)=WM)
W)
(88)
for consistency. Hint: To prove the group property of G\, use that A e Bi and B € B implies AB,BA 6 6 2 ' To prove uniqueness up to a phase, show that if t(U) and f'(U) both obey Eq. (83), then A = t(U)*t'(U) commutes with all ip*(f) and ip(f), f € %. Moreover, show that such operators A necessarily are 211
proportional to the identity. A similar argument can be used to prove Eq. (87). ^ • To give an explicit formula for a one needs to fix the phase of f ([/). A convenient choice is /fi,f(£/)fi\
real and positive
(89)
which, however, does not work for all U € G\ but only in some neighborhood G° C G\ of the identity [R]. An explicit formula for the resulting 2-cocycle a : G\ x G\ -> U(l) was derived in [LI]. We also note that t(eiA) is equal to lb A eidT{A) o n jy u p t o s o m e phase e ( \ and an explicit formula for this phase (which has interesting physics applications) was also derived in Ref. [LI]. R e m a r k : As already mentioned, Eq. (83) determines the operator only up to a phase, i.e. up to to changes t(U) -> f'(17) = $(U)T(U)
(90)
where /3 a phase valued function on G\ such that 0(U*) = 0(U). It is easy to see that such a transformation changes a in Eq. (87) &{U,V) -» a'(U,V) = £ ( t / , F ) f f i f f i l .
(91)
Similarly as in the Lie algebra case, two 2-cocycles a and &' related as in Eq. (91) are called equivalent, and the 2-cocycle a appearing in Eq. (87) is interesting only because it is non-trivial i.e. not equivalent to one which is identical to 1. Moreover, what I discussed here and in the previous remark is a special case of Lie group cohomology [ChD].
212
5
Loop algebras and loop groups
The set of maps from the circle to some Lie algebra of matrices are infinite dimensional Lie algebras with a beautiful representation theory which is treated in several excellent text books, see e.g. [K, Mi, PS). Historically, the developments of this theory went in parallel with interesting activities in theoretical physics, namely string theory and progress in the understanding of certain two dimensional quantum field theory models, and there was a fruitful interplay between these two subjects (see e.g. [GO, CaR] for discussion and reference). In this section I discuss a few simple example from these developments. In the following, x G [—ir, IT] is a coordinate on a circle of length 2TT. For V = C, K, U ( l ) , . . . , we denote as C(V) the set of smooth (i.e. C°°) maps S1 —> V, and we denote such maps as loops. 5.1
Loop algebra of complex valued maps on the
circle
1
The set £(C) of smooth maps S —>• C has a natural Lie algebra structure i.e. the linear structure and the Lie bracket can all be defined pointwise (e.g. the Lie bracket is [ai,a 2 ](a;) := [ai(x),a2(x)] = 0 for all a,- 6 £(C) etc.). This Lie algebra has an interesting central extension £(€)' which is the simplest examples of an affine Kac-Moody algebra and which I now define. As a set, £(C)' = £(C) ©C, i.e. elements in this set are pairs (a, a) with a 6 £(C) and complex numbers a. Fact: This set has a natural Lie algebra structure: the linear structure is obvious, and [(a1,a1),{a2,a2)}
:= ([a1,a2],S(a1,a2))
= (0,5(ai,a2))
(92)
with
s(a
^=^Ldx
(^ a 2 ( x ) - a i W ^ J •
(93)
defines a Lie bracket.• Remark: What is interesting here is that S defined in Eq. (93) is a nontrivial 2-cocylce of the Lie algebra £(C) (see my discussion on 2-cocycles in Section 4.4).D 5.1.1
Wedge
representation
I now describe how the mathematical results in Section (4) can be used to construct interesting representations of the Lie algebra £(C)'. We take as one 213
particle Hilbert space % = L2{SX) with the basis en(x) = ^ L e i ( n + § ) * ,
n
GZ
(94)
V27T
and define P_ := £ n < o e n (e„, •). Remark: Note that P_ = 8(—D) where D is a self-adjoint extension of —idx which is the (chiral) Dirac operator on the circle S1. We will come back to this further below. The loop algebra £(C) has a natural representation r on H. Fact: For every a G £(C), (r(a)f)(x) = a(x)f(x) for all / G H, defines a bounded operator r(a) on %. Moreover, r([ai,a 2 ]) = [r(a 1 ),r(a 2 )]-Q In the following I will simplify notation and write a for r{a) (this is a common abuse of notation). It will also be convenient for us to decompose such maps a into positive, negative and zero Fourier components, a(x) = a+ (x) + a" (x) + a a±
&
(P)eipX> " = ^«(0)
(*) = ^ E
05)
±p>0
where I use the following convention for Fourier transformation of loops, d(p) = /
dxa(x)e-ipx
p GZ.
J — 7T
A central result in the theory of loop groups is the following Corollary of Theorem 1 in Section 4.4: T h e o r e m : The map (u, a) —>• dT(u) + aljr provides a highest weight representation of £(C)' on the Fermion Fock space ^ 7 (L 2 (5 1 )), i.e. [df(oi),df(a 2 )] = 5 ( a i , a 2 ) l ^
(96)
with S in Eq. (93), and d f ( a - ) f i = dt (a + )*fl = 0
(97)
for a l i a , a, G £(C). Proof: One needs to show that all a G £(C) define operators in
( ^ T ^ ^ )
'
To prove these results it is convenient to use Fourier transformation. crucial step is to show that Tr((a1)T(a2)±) =
£
The
0 ( T ( m + §))0(±(n + §))<*!(m - n)a2(n - m) =
= $3^(±P)Hai(-p)a2(p) P6Z
which follows by straightforward (and instructive) computations.
•
Remark: All results discussed here can be extended in a straightforward manner to the loop algebras C(glN) of maps from the circle to the complex NxN matrices gl w . In this case, the Lie bracket is [a\, a2](x) — a\(x)a2(x) — a.2{x)a.i{x) and is non-zero in general, and the 2-cocycle is given by S(0l,£K,
=J L / _ >
t r
„(^«
! W
-«
l W
^)
(98,
where tr jv is the usual trace oi NxN matrices (i.e. sum of diagonal elements). It is an instructive exercise to extend our discussion above in all detail to all these loop algebras. Note that one needs to show that S obeys the 2-cocycle relation in Eq. (77). D 5.2
Loop group of maps from the circle to U(l)
I now discuss the Lie group £(U(1)) of smooth U(l)-valued maps on S1 which has £(E) C £(C) as Lie algebra. Much of the structure of £(U(1)) is a simple consequence of the structure of £(C), but there are some interesting additional features reflecting non-trivial topology appearing only on the Lie group level. Fact: Point-wise multiplication of loops, {
u{x) = wx + a(x)
(99)
where w = [u(n) — u(—n)]/2ir is an integer called the winding number, and a G C(M) is a periodic function.• The winding numbers w ^ 0 correspond to the structure of £(U(1)) which is not contained in £(C) (more precisely: w labels the connected components of £(U(1))). It is known that £(U(1)) has an interesting central extension £(U(1))' which I now define: as a set, £(U(1))' equals £(U(1)) x U(l) i.e. its elements 215
are pairs (eiu,j) where 7 is a phase and eiu € £(U(1)). Moreover, multiplication in £(U(1))' is denned as follows, ( e i u i , 7 i ) • (e iU2 ,72) := {eiui • eiu^,es^^/2lll2)
(100)
with S(ui, u 2 ) = — [ui (?r) U2
1 r
+
{-IT)
d
J
f M*)
- ui
,,
x
{-IT) U2 {IT)]
+
, ,du2{x)\
Mx) Ui{x)
^nJ_J {-^r
-
^r)
•
(101)
Note that this equals the 2-cocycle in Eq. (93) for loops with winding number zero, but for other loops there is a non-trivial boundary contribution. It is therefore not completely obvious that Eq. (100) defines a proper multiplication. However, one can convince oneself of the following important Fact: The product defined in Eq. (100) makes £(U(1))' into a group. {Hint: One needs to check that 2-cycle relations similar to the ones in Eq. (88) hold true.) • We now sketch how the wedge representation of £(C)' discussed above can be extended to £(U(1))'. We first note the simple Fact: {r{
= df{a±),
R~WQRW
= Q + wl
(102)
for all integer w where Q ~ dt{l).
(103)
Hint: Check that tpi = X)nez en + X (en> ")> which is just the unitary operator discussed after the Theorem 2 in Section 4. Eq. (102) can be checked by using R%jj*{en)R* = ip*{en+i) and the notation introduced in Section (4), for example, Q = J2 {fp*{en)^{en) - ^(e_ n _i)V*(e_„_i)) n>0
etc.
• 216
From that we obtain the T h e o r e m : The map (eiu,j) -> 7f(e i u ) with f<(eiu)
+
: = eiaQ/2ffWeiaQ/2eidf(a
+a~)
(104)
provides a unitary representation of the Lie algebra £(U(1))' on the fermion Fock space ^ ( L 2 ^ 1 ) ) , i.e. f (eiui)f
(eiU2) = e s(«i,« 2 )/2f (ei(ui+u2))
( 1 0 5)
with 5 given by Eq. (101). Moreover, (n,f(eiu)n\
= 5wfie~sia
'a+)/2
(106)
where oo
_p 5(a",a+) = X;^j5fi(-P)fi(p) p=l
is positive definite. Proof: Using
• e a i e a 2 = e [ a i> a 2]/ 2 e a i + a 2 n irQ
(107)
for oj = idT(aj) together with Eq. (96), and R e Rr and integer n, one obtains Eq. (105) with
n
inr ir
= e- e Q
for real
5(ui, U2) = (wi5:2 - aiu;2) + 5 ( a i , a 2 ) identical with S in Eq. (101). A similar computation implies Eq. (106). 5.3
Boson-fermion
•
correspondence
The wedge representations of loop groups and algebras discussed above provide powerful mathematical tools to construct, analyze, and in some cases even solve non-trivial 1+1 dimensional quantum field theory models. In this section I discuss some results which play an important role in these developments. I first recall that the quasi-free representation used to construct the wedge representations of £(C)' and £(U(1))' has a simple physical interpretation: it describes a quantum field theory of chiral fermions in the circle 5 1 with the 1particle Hamiltonian D = —idx which is the chiral Dirac operator. Formally, this model can be defined as follows, H=
I
dxil>*(x)(-idx)ip(x)
J — 7T
217
(108)
with (formal) CAR {i>(x),ip*(y)} = S(x — y) etc., and anti-periodic boundary conditions, ip(—Tr) = —ip(ir). The quasi-free representation we discuss provides a rigorous construction of this model. I now show that this quantum theory model of fermions is in fact equivalent to a quantum field theory model of bosons, i.e., quantum field operators with commutator (rather than anti-commutator) relations. This is actually a simple consequence of Eq. (96) but an important result for quantum field theory applications. 5.3.1
Bosons
from
fermions
We consider the representations of £(C)' and £(U(1))' on the fermion Fock space T = T(l?{Sx)') discussed above. Defining ep(x)=e-ipx,
p(p) = df (c,,),
peZ,
(109)
it is easy to see that the Theorem in Section 5-1.1 implies the Facts: p(— p) = P{P)\ \P(P),PW)]=PS-P,P>,
(no)
for all p,p' G Z, and p{p)n = Q V p > 0 . D
(111)
Note that /3(0) is identical with Q defined in Eq. (103). Due to the relations in Eq. (110) physicists interpret the p(p) as b o s o n field o p e r a t o r s . R e m a r k : Formally one can write p(p) = f
dxe~ipx
: xp*{x)xp{x) : ,
(112)
J — 7T
and since p(x) =: ip*(x)tp(x) : is what physicists call a fermion c u r r e n t , the result above means that the Fourier modes of the fermion currents are bosons. The following seemingly technical result has then several important consequences: L e m m a : The vectors
Bv({np}peN) := ( JJ ££=*£= W n , oo
np€N0,
v € Z,
^n P=I
218
p
(113)
provide a basis in the fermion Fock space T. Comments: I will not try to prove this result (see e.g. [CaR] and references therein) and only give a few comments. This lemma means that the loop operators r(<^), p € £(U(1)), span T, or, put differently, that any operator on T can be expressed also in terms of the operators p(p) and R. This Lemma also provides a convenient tool to prove that two operators on T are equal: one only needs to prove that the two operators are the same on all vectors given in Eq. (113). I also note that the vectors defined by Eq. (113) are convenient in practical computations and an alternative to the fermion basis in the Fock space T: F ( { m + , m - } £ o ) = f n ^ ( e n r V ( e - n - i ) m » J ft, oo
m±e{0,l},£(m++m-)
(114)
ra=0
I will give examples for this further below.
•
An important result which can be proven using this Lemma is that the fermion Hamiltonian oo
H = £ ( n + I ) (V*(e„Me n ) - V*(e_ 1 _ n )^(e_ 1 _„))
(115)
71=1
can be expressed in a simple manner in terms of the boson operators. Fact: oo
P=I
(Hint: Check that the operators on the r.h.s. of Eqs. (115) and (116) both obey [H, p(p)} = -pp(p),
and HRun
= \v2R"Sl
Vp, v € Z ,
which implies that they are equal on all vectors in Eq. (113).D) This striking result is usually referred to as Kronig identity, and it is a special case of the so-called Sugawara construction (see e.g. [GO]). 219
5.3.2
Fermions
from
bosons
My discussion above implies that it should be possible to express the fermion field operators in terms of the boson operators p(p) and R. I now sketch how to obtain explicit formulas for that. The idea is to construct loops elu,J-e S £(U(1)) so that uVte(x) approximates the step function interpolating between — TT and TT and jumping at the point y € S1. The role of s > 0 is to provide a regularization (smoothing), i.e., uyio(x) = sgn(:r — y)/ir is the sharp (i.e. non-continuous) step function and Uy>e(x) is smooth for e > 0. Fact: A convenient choice for the uVtS is uy,e{x) = (x-y)
+ a+e(x) + a~e(x)
(117)
with 00
a ± (x) = ± i l o g ( l *'
e ±<(*-v)-e)
1 = ± V __ e ±
(H8)
(Hint: Check that dxuyt£(x)/(2ir) equals ^r E n 6 Z e i n ( x ~ 2 / ) " e | n | w h i c h i s obviously an approximate (and periodized) ^-function, and uy<e(y) = 0.)D Note that the winding number of the uy<€ equals 1. By a straightforward computation one can check the Fact: S(uy,e,uy
= insgn(y -y';e
+ e')
(119)
where sgn(j/ — y';e + e') = uyiie(y)/ir is an approximate sign function.• We thus obtain from Eq. (105) the Fact: t(eiUy'c)t(eiuv'
e
') = e™ss"(v-y';e+e')f(e"V.<=')f(eiUv-c)
l
,D
E
i.e., the operators r(e "» ) in the limit e \. 0 formally behave like fermions! This latter limit, however, is quite delicate: one can prove that c e r(e t U ! / £ ) for 1
ce =
V2TT(1 -
(120) e-a«)
converges to ip*(y) as e I 0. To be more precise: r (/) = lim r
dy f(y) c £ f ( e ^ )
(121)
for all / in a dense subset of L 2 (5 1 ). Remark (adv.): This latter equation holds in the sense of strong convergence on a dense domain, for all / which have a Fourier transform with compact 220
support. Note that one can regard ipe(y) = c e f(e™' je ) as approximative fermion field operators: For e > 0 they are proportional to unitary operators and thus 'nice' operators on J-', and in the limit they converge to operator valued distributions. For more details and a proof close in spirit of my presentation I refer to Ref. [CaH\. 5.4
Bosons,
fermions,
and elliptic
functions
As an application I show in this section how the boson fermion correspondence can be used to prove identities for elliptic functions. I discuss the simplest example I know. I compute Z(z) = Trr(e-pfi+izQ),
z€C
(122)
for the operators given in Eqs. (115) and (103). Note that Z(z) has a simple physical interpretation: it is the the partition function of the free fermion model formally defined in Eq. (108). I have thrown in the variable z since it does not make the computation more difficult but the result becomes more interesting. We will obtain an identity by computing Z(z) first using the fermion basis in Eq. (114) and then by using the boson basis in Eq. (113). Due to the boson-fermion correspondence, all these vectors are eigenstates of # and Q. Using [H,ip*{en)} = (n + ±)ip* (en) and [Q,ip*(en)] = ip*(en) etc. it is not difficult to prove the Fact: Every vector in Eq. (114) is a common eigenvectors an( of H and<3, and the corresponding eigenvalues are Y^=i(n~\){mn+mn) i m m X)^Li( n ~" n)> respectively.!!] Recalling that the vectors in (114) provide a basis in T, we can therefore compute Z(z) as follows, Z(z)=
V
e-0T,^1(n-^)(mi+m-)+izj:^=1(mi-m-)
_
(/3(n-i)+iz)m- \ _ 71=1
\m+=0,l
/
°°
\m-=0,l
1
1
= Y[(l + e ^ ( n - 2 ) + i z ) ( l + e -/3(«-2)- i z ) . Using the notation q = e-W 221
(123)
we can write this result as oo
Z{z) = J J (1 + g 2 "- 1 cos(z) + g 4 "- 2 ) .
(124)
n=l
Using [ # , p(—p)] = pp(-p) it is not difficult to prove the Fact: Every vector in Eq. (113) is a common eigenvectors of H and Q, and the corresponding eigenvalues are | i / 2 + X ^ i J W p and v, respectively.D Using that the vectors in (113) provide a basis in f, we can therefore compute Z(z) also as follows, oo
Z(Z) = Y1
£
e-/3E~1Pn1,->
2
+^
=
i>£Zm,n2,...=0 oo
2
oo
oo
.
=£e-^+- n E «-*"*=z^ ^ n Tir^) • iygZ
p=l n„=0
ygZ
p=l
v
'
Recalling the definition in Eq. (123) we can write this result as
z{z}=
^/""w^-y
(125)
Equating Z(z) in Eqs. (124) and (125) and replacing z by 2irv we obtain oo
1 3 0
oo
] T g^e * " = J J ( 1 - q ) X I C1 + < ? 2n ~ 1 coB(27rt;) + g 4 n ~ 2 ) i/gZ
2p
p=l
(126)
n=l
for all v € C, which is a well-known identity in the theory of elliptic functions: the l.h.s. in this Eq. is the definition of the theta function 93(v), and the r.h.s. is the product representation of this very function (see e.g. [EMOT], pp. 355 and 357).
222
6
The Luttinger model
In this section we discuss quantum field theory models for interacting fermions in one dimensional space. To be specific I concentrate on the Luttinger model [ML], a simple model for a one dimensional metal. My purpose is to illustrate how the mathematical results summarized above can be used to construct and solve 1+1 dimensional quantum field theory models. 6.1
Physical
interpretation
We start with a physical motivation for this model. We consider spin-less fermions in a one dimensional metal (wire) which is a lattice with lattice constant a and £ sites and which can be characterized by a so-called band relation E(p) = E(—p) describing the energy as a function of the (pseudo-) momentum p. For example, if we would consider the Hubbard Hamiltonian in Eq. (41) for U — 0, we would take the band relation in Eq. (42). We are interested in the continuum limit where one keeps the length L = al of the system fixed while the lattice constant a is sent to zero. A physical motivation for this is as follows: the lattice model (finite a) has obviously more structure (i.e. is more complicated) since any physical quantity in the model will depend on a. Physicists are mainly interested in properties of systems which do not depend on this short-distance structure i.e. properties which are (essentially) independent of a. Instead of doing computations at finite a and then taking the limit a —> 0, it seems easier to directly do computations in a model where the short-distance structure (i.e. dependence on a) has been eliminated from the start: such a model has less structure and therefore should be easier to treat. I will outline how such a model can be found in our example. As we will see, this model is indeed simpler, but one has to pay a price: in contrast to the finite-a model, it is a quantum field theory model with all the difficulties discussed (and solved) in previous sections. A formal argument leading to the continuum model is as follows. We consider a metal, i.e. there is a finite density of fermions. We therefore assume that the energy eigenstates are all filled for energies E(p) less then the chemical potential \x. The so-called Fermi surface separating filled from empty states consists in our example of the two points p = ±pp where E(p) — fj, vanishes. Physical intuition suggests that in the continuum limit only states close to the Fermi surface are relevant, whereas states far away from the Fermion surface (i.e. where \E(p) — fj,\ is large) should not have much influence on the physical properties of the system and only provide some inert background. To obtain a non-trivial continuum limit, we label these states as p = ±pF + ak where k 223
is ^7 times a half-integer (cf. Eq. (42)), and then we Taylor expand the band relation about the Fermi surface, E(±pF
+ak)-fi
= ±vFk + ...
(127)
where vp = aE'(pp), for example, vp = 2tasin(pp) in case of the band in Eq. (41). We thus obtain two branches corresponding to the excitation close to the two points p = ±PF of the Fermion surface. The constant vp characterizes the slope of the band at the Fermi surface and is usually called Fermi velocity in the physics literature. If vp has a non-zero limit as a —• 0 (which for the band in Eq. (42) is achieved by tuning t = t(a) and fj, = fj,(a) carefully) it is plausible that we can neglect the higher order terms ' . . . ' in the Taylor expansion above. In the continuum limit we then obtain E±(k) = ±vpk with k e AQ where A* = jfc=^(n+!)|nezj
(128)
the Fourier space for anti-periodic functions on a circle of length L. R e m a r k : The upshot of my discussion above is: I replaced the energies for states 'far away from the Fermi surface' by something simpler, and justified this by arguing that the precise properties of these states should not matter much. Note that the energy band E± (k) above is precisely the energy spectrum of the 1-particle Hamiltonian D = -idxvF
°z=(\ ? ) ^0-1,
(129)
(with anti-periodic boundary conditions) acting on the 1-particle Hilbert space U = L2(S\)®£2
(130)
where 5 [ is the circle of length L, and x G [-L/2, L/2] is a coordinate on S^. Formally, the second quantization of this Hamiltonian is ,L/2
H0=vF
dx {r+{x)(-idx)rl>+{x)
- p_(x){-idx)il>-(x))
(131)
J-L/2
where {ip±(x),ip±(y)} = S(x - y) etc. The Hamiltonian of the Luttinger model is obtained by adding an interaction terms H' to H0, H = H0 + H' 224
(132)
with pL/2
L/2
/ -L/2
dx /
./-L/2
dy p+(x)V(x - y)p-(y),
p±{x) = V 4 ( * ) l M * )
(133)
describing a 4-point interaction of the fermions. The real-valued function v{r) describing the spatial dependence of the interaction is arbitrary up to some (minor) restrictions which we will discuss and specify further below. The mathematical methods discussed in these notes obviously allow a construction and solution of the model formally defined by the free Hamiltonian HQ. We now argue that this framework allows also to construct the interacting model with the Hamiltonian H. Remark: It is interesting to note that HQ for VF = 1 is identical with the Hamiltonian of free relativistic Dirac fermions in 1+1 dimensions. Therefore HQ is also the free part of the Hamiltonian of 1+1 dimensional relativistic interacting fermion models, for example, in the Schwinger model [Ma] which is electrodynamics with massless fermions. This and other models can therefore be treated by similar methods as the ones discussed below. Remark: Above we discussed an example for the so-called continuum limit motivated by the physicists' expectation that many properties of a system should not depend of its short-distance structure. One also expects that many properties should not dependent on the size of a system (as long as it is sufficiently large that surface effects play no role) and therefore often performs the so-called thermodynamic limit where the system size L is sent to oo. This latter limit is also very interesting (and non-trivial) in many models but its discussion is beyond the scope of the present notes. I only mention that the divergences associated with a continuum- and thermodynamic limit are often called ultra violet- and infra red divergences in the physics literature. 6.2
Construction
and solution
of the
model
Note that the Hamiltonian H0 in Eq. (131) is the sum of two terms, H0 = vp(H£ + HQ), where H£ is precisely the chiral Hamiltonian in Eq. (108) which we used to construct the wedge representation of the loop spaces £(C) and £(U(1)) in Section 5 (up to a rescaling x -+ 2irx/L etc.). Moreover, HQ is the same as H^ up to some sign change. It is thus quite obvious that the results which we derived in Section 5 for HQ should also hold for HQ (up to some minor changes). As we now show, these results are essential for constructing and solving the Luttinger model. 225
To simplify notation we set vF = 1
(134)
from now on (this corresponds to assuming vp > 0 and measuring energies in units oc Vf). We can construct the model in the quasi-free representation defined by the 1-particle Hilbert space in Eq. (130) and the projection P_ = 0{—D) with D given in Eq. (129) (anti-periodic boundary conditions). Denoting as i>±{k) = rp(ef) with e±(x) = ^e-ik*E±
and E+ = ( j ) and E~ = ( J )
we can completely characterize this representation by the following equations, ij)±(±k)n = tpl(?pk)n = 0 Vfc>0
(135)
and I tpa(k),•>!)*, (k1) > = 8aai ^8k,k' etc. In this representation we can give a precise meaning to the free part of the Luttinger Hamiltonian i.e. HQ = dT(D). In a physicist's notation,
ft 0--T2 ? r H
53 k : (r+(k)^+(k) - j>L(k)ij,-(k)) :
(136)
fceAj
where : • • • : is normal ordering with respect to the free Dirac sea ft. To give a precise meaning to the interaction term of the Luttinger Hamiltonian we construct p±{p) = d T ( e p | ( l ±
(137)
the Fourier space for periodic functions on S],. The arguments explained in Section 5 can be easily adapted to prove existence of these operators together with the following relations, [p±(p),P±(P')]=±P^6p,-1>l [PHP)JT(P')]=0,
(138)
[ffo./S*^)] =±PP±(P) for all p, p' € A* and p+(p)n = p-(-p)fl 226
= 0 Vp>0.
(139)
Formally, p±(x) = : -ip±(x)ip±(x) :, suggesting to try to make sense of the interaction term in Eq. (133) using Fourier transformation, + fl'4E^ L A W"(-!')
(140)
where r£/2
wp
- / dr V(r)e~%pr = W-p = Wp . «r J-L/2
(141)
As I will discuss in more detail in a remark below, if (and only if) the potential function V(r) is such that |WP|<1
Vp
(142)
Y^PWP<0C p>0
and
does H = Ho + H' define a self-adjoint operator on T which is bounded from below, i.e., is the quasi-free representation appropriate for the free model Ho also appropriate for the Luttinger model. The relation which allows a solution of the Luttinger model is the Kronig identity in Eq. (116) which adapted to ^ o becomes ^o = ^
( \{Q\ + Q2-) + £ (p+(-p)p+(p) V p>0
- p-(jp)p-(-p))\
. (143) /
As discussed, this relation makes explicit that the free fermion Hamiltonian is identical with a free boson Hamiltonian. The simple but crucial observation now is that the interaction term H' is also quadratic in the boson fields, and therefore the Luttinger Hamiltonian H is identical with a free boson Hamiltonian. Hamiltonian of this latter kind can be diagonalized explicitly, and all what is needed for that is some simple facts about the harmonic oscillator problem which one learns in an introductory quantum mechanics courses. In the following we outline this diagonalization. We write
H
= T^2hp
(144)
p>0
and introduce the notation p+ (p) for p > 0
«&>) = < V V , ^ for i vp ^< 0n p (p) VM
227
(145)
so that hp = p[c*(p)c(p) + c *(-p)c(-p)] - Wp[c*(p)c*(-p) +
c(~p)c(-p)] Vp>0,
and h0 = \ (Ql + Q2.) + WoQ+Q- .
(146)
Note that [c(p),c*(p')] = —6py
etc.
(147)
for all p,p' ^ 0, which are precisely the relations of the creation- and annihilation operators in a harmonic oscillator problem. Moreover, we see that we can diagonalize the Luttinger Hamiltonian by diagonalizing the hp separately, and each hp is a harmonic oscillator-type Hamiltonian. For that we introduce new operators C(±p) = cosh(A p )c(±p) + sinh(Ap)c*(Tp)
Vp ^ 0
(148)
obeying the same relations as the operators c(p), and we determine the real parameters Ap so that hp can be written in the following form, hp = OJP (C*(p)C(p) + C*(-p)C(-p))
- ifcl
(149)
with some constants uip, r)p to be determined. By straightforward computation one finds that hp has the desired form if we choose tanh(2A p ) = Wp
(150)
which has a solution Ap only if the first condition in Eq. (142) is fulfilled, and in this case, tfe=p(l->/l-Wp2).
cvp=p^l-W*,
(151)
With that we can now find all eigenstates of the Luttinger Hamiltonian: the ground-state O is determined by the following conditions, C(p)O = 0 V p ^ O ,
Q ± H = 0,
(152)
and the ground state energy is E
° = ~T^p{1 ~ \A~ I ^ ) •
(153)
p>0
Note that the ground-state energy E0 in Eq. (153) is finite if and only if J2p>oPWp is finite, and we see that the second condition in Eq. (142) is 228
necessary for the construction to make sense (otherwise there cannot be a ground-state ft in T). Once Cl is constructed it is easy to also find all other eigenstates of the Luttinger model: they are given by J ] C*(p)""Ru++ BTsCl,
npeNo,v±eZ
PT±O
where R± = f (elx'27r/L2(1±CT3>), and the corresponding energy eigenvalues are Eo +
T\
^U+ + ^ V
+ w v v
°+-
+ Y^wp(np+
n
-p)) J
P>O showing that also |Wo| < 0 is required for H to have a ground-state. From that we can compute the partition function etc., and we have completely solved the model. I finally mention that physicists are also interested in the so-called Green's functions in quantum field theory models which are (in general) distributions allowing to predict results of measurements. In case of the Luttinger model, these Green's functions are
(n,^(t,x)n) where * ^ ( t , x ) stands for an arbitrary product of finitely many fermion field operators r/>± (t, x) and V>± (t, x), and the t dependence of these latter operators is given by by A(t) = eiHtAe~lHt. We note that there are formulas similar to Eq. (121) allowing to express the fermion field operators ip±{x) in terms of the boson fields p±(p), and with the results summarized above one can compute all Green's functions of the Luttinger model explicitly (see [HSU] for more details). Remark: The construction and solution for the Luttinger models described here was for zero temperature. A similar construction and solution of the Luttinger model at finite temperature was given in [CaHa]. Remark: Another prominent quantum field theory model is the the so-called massless Thirring model which is obtained from the Luttinger model in the limits where space becomes infinite, L —> oo, and the interaction becomes local, i.e. Wp = g
independent of p, 229
|g| < 1.
(154)
These limits can be quite easily performed in the Green's functions. However, they are highly non-trivial and interesting on the operator level and provide an example for multiplicative regularization in a fully solvable model (see e.g. [GLR] for more details).
230
7
Further developments
In these notes I have tried to give a pedagogical introduction to the theory of quasi-free second quantization of fermions, a formalism which has many applications and which, to my opinion, provides a fine example of a fruitful interplay between mathematics (functional analysis, Lie group theory, differential geometry, theory of special functions...) and theoretical physics (quantum field theory models in elementary particle physics and solid state physics, string theory, integrable models...). I therefore hope that these notes can be useful for mathematicians who want to learn some physics and for theoretical physicists interested in the mathematics discussed here. As I will describe in this final section, this formalism has been playing also a central role in my research. Again, my selection of topics here is personal, and I restrict my discussion to projects in which I have been directly involved. 7.1
Boson-anyon
correspondence
and the Sutherland
model
In 1+1 dimensions it is possible to have quantum fields ip(x) which neither are bosons nor fermions but which obey relations
Vx,y6S\x^y
(155)
with a > 0 a so-called statistics parameter. For a an odd integer these fields are fermions (i.e. they anti-commute), for a an even integer they are bosons (i.e. they commute), and since a can have any value such fields often are called anyons. Anyons have received quite some attention in theoretical physics during recent years. For example, they are known to be relevant in the theory of the fractional quantum Hall effect. We recently gave a construction of a quantum field theory model of anyons based on the representation theory of the loop group £(U(1)) and generalizing the boson-fermion correspondence discussed in Section (5.3) to a boson-anyon correspondence [CaL]. Moreover, we showed that this model of anyons naturally provides a second quantization of the so-called Calogero-Sutherland model which is defined by the Hamiltonian
S^ + Jfe,,-'«*-*>/*>
<156)
and describes a system of N interacting particles moving on the circle. Using this anyon model we were able to derive in a simple manner an algorithm allowing to solve the CS model for arbitrary particle number N and coupling a [CaL], and we showed that this algorithm allows to recover the full solution 231
of the CS model originally found by different methods by Sutherland about 30 years ago. Recently I generalized this construction to the elliptic CS model which is the generalization of the Hamiltonian in Eq. (156) where sin~ 2 ((-)/2) is replaced by the Weierstrass elliptic function p ( ) with periods 2ir and i/3, /? > 0 (the CS model corresponds to the limiting case /? —¥ oo). The idea underlying this construction is to construct the anyon model at finite temperature 1/0. In this way I obtained an algorithm for solving the elliptic CS model [L4]. This can be regarded as a continuation of the theme illustrated in Section 5.4: use relations of operators on the fermion Fock-space JT(L 2 (5 1 )) to derive non-trivial identities involving elliptic function. 7.2
Other 1+1 dimensional
quantum field theory
models
Other interacting quantum field theory models which can be constructed and solved by methods similar to the ones we described in our discussion of the Luttinger model include the Schwinger model [Ma], i.e. 1+1 dimensional quantum electrodynamics with massless fermions, and the LuttingerSchwinger model, i.e. the gauged Luttinger model [GLR]. A similar construction of massless Q C D i + i , i.e. the non-Abelian version of the Schwinger model, was given in [LS]. 7.3
Quasi-free geometry
second quantization
and
non-commutative
There is an interesting relation of quasi-free second quantization of fermions and Connes' non-commutative geometry [Con]. The cocycles in Eq. (98) and (75) offer the simplest non-trivial example of this. This can be seen by introducing some simple notation: using integration by parts we can write Eq. (98) as follows, 5(ai,a2) = - — : / ZlTl
txN(aida2)
(157)
Jsi
where the r.h.s. is proportional to the integral of the de Rham 1-form a\da2 ai^-dx over the manifold S 1 . We can also write Eq. (75) as follows, S{A,B)
= -±Tic{A[F,B])
=
(158)
where F = P+ - P- and Trc(A) = | T r ( ^ + FAF) = Tr(A++ + A—) is the so-called conditional trace. (This can be checked easily by using F2 = 1 and the notation introduced in Eq. (68).) According to Connes, dA := i[F, A] can be regarded as a natural definition of an exterior differentiation 232
on algebras of Hilbert space operators. Thus we can interpret the equation S(ai,a2) = S(ai,a2) naturally as follows: S is the natural generalization of 5 from de Rham form on the circle to an algebra of operators on a Hilbert space according to the following rules, 0-forms 1-forms
a € Map{S1, gl w ) ->•
/ ^-Trc(-). (159) Js1 n This is a simple example of Connes' quantized calculus [Con] and the starting point of a quite long and nice story in which the following topics play a central role: current algebras and anomalies in higher dimensions (e.g. the analog of the 2-cocycles above to 3+1 dimensions etc.), quantum gauge theories, noncommutative geometry, I have described this story in more detail in previous reviews [L3, L2] and will not repeat it here. As an appetizer, I only state the following identity which is a generalization of the correspondence in Eq. (159) to de Rham forms on higher dimensional manifolds, 21 integration
i™Trc(o;o [F, ai] • • • [F, a„]) = cn
t r w ( a 0 d a i • • • dan)
n
Vaj e Map(R , gl w ) , n = 1 , 3 , 5 . . .
(159a)
where d is the usual exterior differentiation, F is the sign of the chiral Dirac operator on R™, and
are normalization constants [LI]. 7.4
Bosons and the super-version quantization
of quasi-free
second
In these notes I have restricted myself to fermions for simplicity. However, many results discussed can be adapted to bosons. Formally the boson version can be obtained by replacing anti-commutators {•,•} by commutators [•,•]. For example, the boson quantum mechanics with N degrees of freedom can 21 for simplicity I write the formula only for odd dimensions n, but there is a similar one for n even [LI].
233
be characterized by the following relations, [a,-, ofc] = 0,
[aj, c£\ = Sjk 1 Vj, k = 1 , . . . , N
(160)
with the usual representation on the Hilbert space L2(RN). This is usual quantum mechanics but, as mentioned, mathematically more difficult than the fermion version since the a,j can only be represented by unbounded operators on an infinite dimensional Hilbert space. Still, there is a natural boson version of the formalism discussed in Section (4.4), and there is even a superversion which is a generalization corresponding to a model where one has bosons and fermions and also allows for transformations mixing bosons and fermions [GL1,GL2]. Surprisingly, most results obtained in this formalism have a natural interpretation as a Z 2-graded generalization of the fermion results discussed in Section (4.4): all algebras etc. in the formalism can be given a natural Z2-grading, and all one has to do is to replace (anti)-commutators by graded commutators. As an application, this formalism naturally yields representations of the Lie super-algebra generalizations of the affine Kac Moody algebra and the Virasoro algebra which play an important role in string theory. Acknowledgments I would like to thank Sylvie Paycha and Tilmann Wurzbacher for carefully reading the manuscript and suggesting several improvements. This summer school in Villa de Leyva was a wonderful experience for me, and it is my pleasure to thank the organizers for the invitation and the excellent hospitality. References [A] Araki H.: Bogoliubov transformations and Fock representations of canonical anticommutator relations, in: Operator algebras and mathematical physics, Iowa City (1985), Contemp. Math. 62, American Mathematical Society (1987) [BeR] Berndt B.C. and Rankin R.A.: Ramanujan. Letters and Commentary, History of Mathematics Vol. 9, American Mathematical Society (1995) [BrR] Bratteli O., and Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics. Vols. 1 and 2. Berlin: Springer (1981) [CaHa] Carey C.A. and Hannabuss K.C.: Temperature states on the loop groups, theta functions and the Luttinger model, J. Func. Anal. 75, 128 (1987) 234
[CaH] Carey A.L. and Hurst C.A.: A note on the boson-fermion correspondence and infinite dimensional groups. Commun. Math. Phys. 98, 435 (1985) [CaL] Carey A.L. and Langmann E: Loop groups, anyons, and the CalogeroSutherland model, Comm. Math. Phys. 201, 1 (1999) [CaR] Carey A.L. and Ruijsenaars S.N.M.: On fermion gauge groups, current algebras and Kac-Moody algebras, Acta Appl. Mat. 10, 1 (1987) [ChD] Choquet-Bruhat Y. and DeWitt-Morette C : Analysis, Manifolds and Physics. Part II: 92 Applications. Amsterdam: North Holland (1989) [Co] J. Collins: Renormalization. Cambridge Monographs on Mathematical Physics, Cambridge University Press (1984) [Con] A. Connes, Noncommutative Geometry, Academic Press (1994) [cond-mat] On the archive h t t p : / / x x x . s i s s a . i t / f i n d / c o n d - m a t one can find about 900 papers since 1992 discussing the Hubbard model [EMOT] Erdelyi A., Magnus W., Oberhettinger F., and Tricomi F.G., Higher Transcendental Functions. Vol. II, New York-Toronto-London: McGrawHill Book Company, Inc. (1953) [D] for review see e.g. Dagotto E.: Correlated Electrons in High Temperature Superconductors, Rev. Mod. Phys. 66, 763 (1994) [F] Frenkel I.B.: Two constructions of affine Lie algebra representations and boson-fermions correspondence in quantum field theory, J. Fund. Anal. 44, 259 (1981) [G] Gilkey, P.B.: Invariance Theory, the Heat Equation, and the AtiyahSinger Index Theorem, Publish or Perish: Dilmington (1995) [GL1] Grosse H. and Langmann E.: A Superversion of quasifree second quantization. 1. Charged particles, J. Math. Phys. 33, 1032 (1992). [GL2] Grosse H. and Langmann E.: On current superalgebras and superSchwinger terms, Lett. Math. Phys. 21, 69 (1991). [GLR] Grosse H., Langmann E. and Raschhofer E.: On the LuttingerSchwinger model, Ann. Phys. (N.Y.) 253, 310 (1997) [GO] Goddard P. and Olive D.: Kac-Moody and Virasoro algebras in relation to quantum physics, Int. J. Mod. Phys A l , 303 (1986) [Ha] Haag R.: Local Quantum Physics: Fields, Particles, Algebras, Berlin: Springer (1992) [HSU] Heidenreich R., Seiler R., and Uhlenbrock D.A.: The Luttinger model, J. Statist. Phys. 22, 27 (1980) [Ho] Hormander L.: The Analysis of Linear Partial Differential Operators III. Grundl. math. Wiss. 274, Berlin: Springer (1985) [Hu] Hubbard J.: Electron correlations in narrow energy bands, Proc. R. Soc. London, Ser. A 276, 238 (1963) 235
[IZ] Itzykson C. and Zuber J.-B.: Quantum Field Theory, New York: McGraw-Hill (1985) [J] Jackiw R.: Topological investigations of quantized gauge theories, in: Relativity, Groups and Topology II, Les Houghes 1983, DeWitt B.S. and Stora R. (eds.), North Holland: Amsterdam (1984) [K] Kac V.: Vertex Algebras for Beginners, Univ. Lecture Series Vol. 10, American Mathematical Society (1991) [KR] Kac V.G. and Raina A.K.: Highest Weight Representations of Infinite Dimensional Lie Algebras, World Scientific: Singapore (1987) [L0] Langmann E.: Cocycles for boson and fermion Bogoliubov transformations, J. Math. Phys. 35, 96 (1994) [LI] Langmann E.: Noncommutative integration calculus, J. Math. Phys. 36, 3822 (1995) [L2] Langmann E.: Traces of noncommutative geometry in quantum field theories, Contribution to conference 'The Standard Model of Elementary Particle Physics — Mathematical and Geometric Aspects', Hesselberg, March 1999 (to appear) [L3] Langmann E.: Quantum gauge theories and noncommutative geometry, Proc. of German-Polish Symposium New Ideas in the Theory of Fundamental Interactions (Zakopane, Poland, September 1995); Acta Phys. Pol. B 27, 2477 (1996); hep-th/9608003 [L4] Langmann E.: Solution algorithm for the elliptic Calogero-Sutherland model, math-ph/0007036 [LC] Langmann E and Carey A.L.: Loop groups, Luttinger model, anyons, and Sutherland systems, Proceedings of International Workshop Mathematical Physics in Kiev, Ukraine (May 1997), Ukrainian Jour. Phys. 6-7 vol. 43, 817 (1998) [LS] Langmann E. and Semenoff G.W.: QCD(l-fl) with massless quarks and gauge covariant Sugawara construction, Phys. Lett. B 341, 195 (1994) [Le] Lepowsky J.: Vertex operator algebras and the zeta function, See: math.qa/9909178 [Lu] Lundberg L.-E.: Quasi-free "second quantization", Comm. Math. Phys. 50, 103 (1976) [Ma] Manton N.S.: The Schwinger model and its axial anomaly, Ann. Phys. (N.Y.) 159, 220 (1985); see also Schwinger J., Phys. Rev. 128, 2425 (1962) [Mi] Mickelsson J.: Current Algebras and Groups, Plenum Monographs in Nonlinear Physics, Plenum Press (1989) [ML] Mattis D.C. and Lieb E.H.: Exact solution of a many-fermion system and its associated boson field, J. Math. Phys. 6, 304 (1965); see also 236
Luttinger J. M., J. Math. Phys. 4, 1154 (1963) [O] Ottesen J.TV. Infinite Dimensional Groups and Algebras in Quantum Physics, Lect. Notes in Physics 27, Berlin: Springer (1995) [P] Polchinski J.: String theory, Vols. I and II, Cambridge University Press (1998) [PS] Pressley A., and Segal G.: Loop Groups, Oxford Math. Monographs, Oxford (1986) [Ra] Ramond P.: Field Theory: A Modern Primer, Frontiers in Physics Vol. 74, Redwood city, USA: Addison-Wesley (1989) [R] Ruijsenaars S.N.M.: On Bogoliubov transformations for systems of relativistic charged particles, J. Math. Phys. 18, 517 (1977) [RS] Reed R., and Simon B.: Methods of Modern Mathematical Physics I and II, Academic Press, New York (1968 and 1975) [S] Salmhofer M.: Renormalization: An Introduction. Berlin: Springer (1999) [Se] Segal G.B.: Unitary representations of some infinite dimensional groups, Commun. Math. Phys. 80, 301 (1981) [We] Weinberg S.: The Quantum Theory of Fields. Vols. I and II., Cambridge: Cambridge University Press (1996) [Wo] Wodzicki M.: Noncommutative Residue, in: K-theory, arithmetic and geometry, Yu. I. Manin (ed.), Lecture notes in Mathematics 1289, Springer: Berlin (1985) [Wu] Wurzbacher T.: Fermionic second quantization and the geometry of the Hilbert space Grassmannian, DMV Seminar on "Infinitedimensional Kahler manifolds," Oberwolfach November 1995, Preprint 1998, to appear in the Birkhauser series on DMV-Seminars. See: h t t p : / / w w w - i r m a . u - s t r a s b g . f r / wurzbach
237
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
HEAT EQUATION A N D SPECTRAL GEOMETRY. I N T R O D U C T I O N FOR B E G I N N E R S . KRZYSZTOF P. WOJCIECHOWSKI Department of Mathematics IUPUI (Indiana/Purdue) Indianapolis IN 46202-3216, U.S.A. E-mail:kwojciechowski@math. iupui.edu These notes contain an introduction into the theory of ^-determinants of 1Dirac operators. We discuss here the simplest possible case of the operator on S . We study in details the necessary facts from the theory of Heat Equations concentrating more specifically on the small time asymptotics of the trace of the kernel of the Heat Operator.
Contents Introduction
239
1 (^-determinant
240
2 (^-function and 77-function of a Dirac operator and Heat Kernel
244
3 Heat kernel and ^-function on S
1
247
4 Duhamel's Principle and (^-function of the operators A /
253
5 Heat kernel of the operator T>\
260
6 77-invariant, the phase of the ^-determinant
263
7 Modulus of detcVa
271
8 Heat Equation and Index
276
9 Spectral Flow ^-invariant and ^-invariant
279
10 Index and spectral Flow
281
11 A Simple Example
286
12 Bibliographical Remarks
290
References
291 238
Introduction In the beginning of 19991 was invited to spend the majority of July in Villa de Leyva, a beautiful small town located in Colombia. I was supposed to deliver a set of lectures on spectral geometry of elliptic boundary problems for Dirac operators in the Summer School on Geometric Methods in Quantum Field Theory organized by CIMPA and Universidad de Los Andes. However, once in Villa de Leyva I found out that the majority of the audience were graduate students at different stages of development. Therefore, I faced the difficult task of presenting some introductory material on this beautiful subject to beginners, who did not know global analysis and some of whom were not particularly advanced in functional analysis. I decided to present a rigorous introduction into theory of the ^-determinant for the simplest possible Dirac operator, which is the operator T>f = —i-^ + f(x) on the circle. In fact, I tried to keep the presentation open to advanced undergraduate students. The only two prerequisites, beyond the advanced undergraduate level, are the notion of the meromorphic extension to the complex plane of a function h(s), holomorphic on the half-plane Re s > r for certain real number r , and some fundamental results on the trace of an operator acting on a separable Hilbert space. I usually tried to organize lectures as follows: first, I outlined a theory for a general Dirac operator on a closed manifold, then I proved the results discussed in the first part in the case of the operator T>f .
239
1
^-determinant
Many fundamental calculations of Quantum Field Theory reduce to the computation of the determinant of some Dirac operator. In fact, most physicists will tell you that, at the one loop level, any such theory reduces to a theory of determinants. Therefore, the notion of the determinant of the Dirac operator plays a central role in many recent studies in the area of Quantum Field Theory, and in the Spectral Geometry and Topology as well. We discuss here one of the possible schemes of making precise the notion of the determinant. We use ^-function regularization. The starting point is a well-known formula, which holds in the case of a linear operator T : C " —» C n , which we also assume to be positive. We denote the eigenvalues of T by 0 < «i < ... < an and then the following equality holds n
detT=J[ak
,
fc=i
which may be further rewritten using an auxiliary complex variable s in the following way det T = e s £ = i l n
ak
= e~&^*=* °DU=<> .
(i)
Hence, if we introduce ^-function of the operator T as n fc=i
the equality (1) can be rewritten as d e t T = e _ * ( C ; r W ) l «= 0 .
(2)
The important fact here is that formula (2) holds in the infinitedimensional case as well. More precisely, if we assume that P : C°°(M; S) —• C°°(M; S) is a positive elliptic differential operator of order d > 0 (i.e. Dirac Laplacian) acting on sections of vector bundle S over a closed manifold M of dimension m , then (P(S)
= Tr P~s 240
(3)
is a well-defined holomorphic function of complex variable s , for Re s > y , which has a unique meromorphic extension to the whole complex plane with simple poles at s = Sk = y - k . However, s = 0 is not a pole and formula (2) makes sense in this context. The proofs of all necessary analytical results can be found for instance in Chapter 1 of [G]. We introduce the ^-determinant of the operator P by the formula detcP = e - ^ K ' W ) ! ' - 0 .
(4)
The goal of these simple notes is to show how things work out in the case of the operator Vf . So far we have discussed only the case of a positive operator. Dirac operator is an elliptic operator of first order and therefore has a spectrum which includes infinitely many positive and negative eigenvalues. Let us assume now that we replace the operator P by a Dirac operator V : C°°(M;S) ->• C°°(M;S) , which has spectrum consisting of positive eigenvalues, which we denote by {Afc}fc£N and negative eigenvalues {— ^ J K N Now, the C-function of V looks as follows
cx>(*) = £ A r + £(-i)-vr k
k
2 k
which can be written as
&>(*)-(-!)
+
g
(5)
2
where r)-p(s) = Y^k ^k~S ~ !CfcM^s is the ^-function of the operator V introduced by Atiyah, Patodi and Singer (see [APS1]). Once again it is holomorphic for Re(s) large and has a meromorphic extension to the whole complex plane with only simple poles. There is no pole at s = 0 and therefore we can study the derivative of Cx>(s) at s = 0. We have
C©(0) - — 2 — + ^ { ( - 1 ) 241
}|.=o
2
•
The ambiguity in defining ( - l ) _ s = e±ns (i.e. a choice of spectral cut) now leads to an ambiguity in the phase of the ^-determinant. We choose the sign in such a way that the definition of the C-determinat of the Dirac operator V is as follows det!:V = e'i-^(o)-vr>(o)).e-c-^
^
(6)
where the first factor on the right side of (6) is the phase of the determinant and the second factor is the absolute value of the determinant of the Dirac operator. In these lectures we study the case of the operator T>a = — i-^ + a : C^iS1) ->• C^iS1) . We show that all ingredients in formula (6) are welldefined and that in fact (^-determinant is the true algebraic determinant. At the end of this introduction let us explain why it is enough to study operators Va , when it seems that we should be investigating operators of the form r,
• d
\
where f(x) denotes a smooth real-valued function on S1 . Let us consider two such operators
-idi
+
Mx)
and let us introduce functions 9i(x) = / fi(s)ds Jo We have now the following result
.
Proposition 1.1 Assume that 5I(2TT) =
then the operators V^
52(27r)
,
and Vf2 are unitary equivalent 242
(7)
Proof. We define operator U acting on functions on S1 via formula (Us)(x) = e i ( 9 l ( l ) - 9 2 ( a ; ) ) s ( x ) . The operator U is a unitary operator on L 2 (5 X ) , and the straightforward computation shows
hence T>jl and Vf2 are unitary equivalent, which among other things implies that they have the same spectrum. •
Corollary 1.2 The operator —i-^ + f{x) is unitary equivalent to the operator —i-^+a , where
Corollary 1.3 The spectrum of the operator —ij^ + f(x) a}fc€z, where a is given by the formula (8).
is equal to {k +
This last corollary follows from the fact that we know the spectrum of the operator T>a — — i~^ + a . It has eigenvalues k + a corresponding to eigenfunctions <j>k{x) = -4=e lfcx .
243
2
^-function and fj-function of a Dirac operator and Heat Kernel.
We study the C-determinant of the operator Va using the Heat Operator determined by T>\ . We start with a discussion of the situation in the case of a general Dirac operator. Later, we prove all the results for the operator T>a on S1 . Let V : C°°(M;S) -»• C°°(M;S) denote a Dirac operator acting on sections of a bundle of Clifford modules over a closed manifold M . We want to solve the Heat Propagation problem for the operator T>2 , which means that having / 0 £ C°°(M; S) , we want to solve the problem: {^l+V2)f{t,x)=Q fort>0 with /(0,a:) = /o(a:) (9) at The problem (9) has a unique solution for each smooth initial data fo{x) (see for instance [R,Tl,T2]). The usual way to get the solution is to construct a family of operators e~tv2 = E(t) : C°°(M;S) -+ C°°(M;S) such that E(0) = Id and for each /o and t > 0 we have (E(t)f0)(x)
= f(t,x)
.
It is not difficult to see that E(t) satisfies the semigroup property i.e. for each s , t > 0 w e have E(t + s) = E(t)E(s) . Moreover, operator E(t) has a smooth kernel, which means that there exists a smooth function e(t;x,y) , where for each x,y £ M e(t;x,y) is a linear map from Sx to Sy such that (e~tv2f0)(x)
= f(t,x)
= f
e(t;x,y)f0(y)dy
.
(10)
JM
Assuming that we know the spectral decomposition of the operator T> we have a nice abstract formula which gives the kernel of the Heat Operator. Let us denote by A^ an eigenvalue of V , which corresponds to the eigensection (frk- Then we have +oo
e(t;x,y)
= Yle~tXl(t>k(x)®(f>k(y)
•
—oo
In other words, we have equality + 0O
(E(t)s)(x)=Yl(/
< s{y);
— oo JM
244
(11)
where < •; • > denote an inner product on the fibre Sy . We refer to [G] and [Tl], [T2] for more details. In the following we concentrate on a very easy special case. Although representation (11) looks nice, it is not of very much use if we want an explicit formula for the kernel e(t;x,y) . We start with the differential operator, for which we know the exact formula for the kernel and then study how the perturbation of the operator affects the Heat Kernel. This is what we do in the next Section in order to study C-function of the operator VQ = —-£? on the circle. Here we justify the introduction of the Heat Operator. We use this operator in order to study ^-function and 77-function of the Dirac operators. Proposition 2.1 The following equalities hold for a Dirac operator V
Cp2(s) =
W) / ° ° t S ~ l T r e~iT>2(it
for Re{s)
> d^r~'
(12)
and
*><*> = rviW [°°tS-lTr
Ve W2dt
f°r
~
Re
^ >i
± i
T^ •
2
where in the discussion of £z> (s) we assume that T> is invertible. sumption is not necessary in the case of T]x> (S) •
This as-
Proof. We prove second equality in (12). The proof of the first one is completely analogous. We have /*oo
/
/>oo
+00
t^1TrVe-tv2dt
t^\ke-tx*dt
= Y/
J
0
-00 +00
°
-oo
= X>(A 2 fc )-^ / (t\l)^e-^d(t\i) = ^jTsign -00
=
J
\k-\Xk\~s- / J
°
r^e~rdr
= T(^-—
=
)-7]T,(S)
.
l
We discuss the suitable domain of s for which (12) is valid only for the operator Vf. a 245
R e m a r k 2.2 If we assume that V has eigenvalue 0 , i.e. there exists a nontrivial solution of the equation Vs = 0 , then of course the first formula in (12) does not hold as the integral on the right side is divergent. To cure this problem we proceed as follows: the operator V is an elliptic operator on a closed manifold hence ker T>, kernel ofT>, the space of all solutions of Vs = 0, is finite dimensional and consists of only smooth sections. This is obvious for the operator Vf and we refer to [G], or [Tl], [T2] for the proof of this fact for general Dirac operator. We consider the integral in (12) on the orthogonal complement of this space, but to stay consistent with definition (3) we have to add the dimension of ker V . More precisely, if we denote by l i p the orthogonal projection onto ker V , then the first formula in (12) is replaced by <Ws) = - ! - r't'^Trie-'V-ILvW+dimkerV r ( s ) J0
246
for Re(s) > A T " M . 2 (13)
3
Heat kernel and £-function on S1.
We now discuss the ^-function of the operator T>\ = A = — -^ on 5 1 . We will use formula (12) so that we need information about the kernel of the operator e~tA . It is not difficult to check that the e^i (t; x, y) kernel of the corresponding operator on R 1 is given by the formula ,
N
1
eRi (t; x, y) = -j==e
(*-»)2
«
Let me repeat again that this means that the function f(t,x) formula
f{t,x) = (e-^-^f0)(t,x)
= f -±=e-^
given by the
f0(y)dy
solves the problem (9) with the initial data fo(x) . Now we define kernel on S 1 using the formula es*(t;x,y)
= ^
e R i (i; a;, ?/+ 27rn) .
(14)
nGZ
In (14) we use the representation of S1 as R/27rZ . Now as an exercise the reader may check the following fact (see for instance [R])
Proposition 3.1 The kernel of the operator e esl(t;x,y).
tA
on S1
is equal to
Proposition 3.1 has the following extremely important consequence
Theorem 3.2 Let us assume that 0 < t < 1 , then there exist positive constants c\, ci such that the following equality holds \eSi(t:x,y)-eni(t;x,y)\
^
R e m a r k 3.3 (1) The statement of the Theorem 3.2 is usually written as 1
2
esi(t;x,y)
=—==e **~~ + 0 ( e ~ ^ ) for t small . (15) v47rf (2) In the following we really need (15) for (x,y) close to the diagonal hence the distance between x and y is indeed given by \x — y\ .
Proof. We have
eAt;x,y) = -7==e « +-7z=rL e 715^0
and we estimate the sum as follows
Ee
ix-y-lnn)-*
4
= 2_,( e
<
00
(x-j|-2»n)2
V—V
4<
2
1
(i-«+2Jn)-i ,
+ 1
2.£(e-^r = 2.-L.-Jr x %-i
e*t
P 4t
J-
~~T?5 ~?£
e
"' 2
e"*t
1
(™)2
r-^
) < 2- ^
e
2 „ ,2 < —5- = 2 - e ~ ^
est
•"
<
.
Now (15) follows from the elementary estimate 1 D
Let / , g : (0, oo) —>• R be smooth functions, then we write if for any natural number m we have Iim
/Wz2W=0.
In particular we have just shown that for each x,y € S1 248
/ ~ g if and only
1 esi(t;x,y)
_te
7==e
Vint The next Corollary is the first result, which ties the spectral geometry that we are studying with Number Theory. It is also at that point that we introduce the trace of the heat operator.
Corollary 3.4
(16)
E ^ ~ ^~ 1/2 - \
Proof. We study the trace of the Heat Operator e " t A on S 1 . The operator A has a complete set of eigenvalues with corresponding eigensections given by the standard basis of L2 (S1) . Namely, for any integer k , k2 is an eigenvalue with corresponding eigenfunction
i A
<^) = E
fc€Z
e
= ! + 2- E
fceZ
e
-in''
n6N
where
(f;9) = /
f(x)g(x)dx
is standard L2 product on S1 . On the other hand the trace of an operator with a smooth kernel is also given by the integral of this kernel over the diagonal Tr e - '
= /
e s i (t; x, x)dx
Therefore, we have 249
.
00
£e-t«2
I =
( T r e-tA
1 f _ 1) = i ( / eSI(t;x,x)dx - 1) =
= i ( - L / dr-l) + 0(c"*) , -i V47Tt i s 1
which finally gives (16).
•
Now we investigate CA(S) the ^-function of the operator A . Let us observe that this analysis is immediately related to Number Theory. Namely, if we introduce Riemann ^-function CO
CTC(s) = X > -
s
,
n=l
then we have the equality CA(S) = $ > 2 ) " S = 2< w (2a) + 1 .
(17)
fcgz
This is the reason that we can formulate the main result of this Section in terms of Cn(s) • We have the following Theorem
Theorem 3.5 Function CK(S) is a holomorphic function of s for Re(s) > 1. It has a meromorphic extension to the complex plane. Point s — 1 is the only pole of £-fc (s) . It is a simple pole and we have Ress=1Cn(s)
= 1 , <w(0) = - | and fr(-2l) = 0 for I = 1,2,3,... . (18)
Theorem 3.5 follows from the corresponding result for the ([-function of the operator A = T% on S1 . We use representation (13) . We also need two properties of the function T(s) . First, let us recall that in the neighborhood of s = 0 function T{s) has the following form
r(s)=1-
+
,+
s f
(s)=1
s
+ s
" + s2f^ s
250
,
(19)
where 7 denotes Euler's constant. We also use the identity «r(s) = r ( s + l) ,
(20)
in order to extend T(s) from the holomorphic function on Re(s) > 0 to a meromorphic functionon on the whole complex plane. Points Sk = —k k = 0,1,2,.. are the only poles and Ress=^kT(s)
(-l)k = ^ -
,
(21)
as follows from (19) and (20). Let us also observe that now it is easy to show that function pjjy is a holomorphic function of s on the whole complex plane and the only zeros of ^h) a r e points Sk = —k . Now, we are ready to analyze the function CA(S) •
P r o p o s i t i o n 3.6 Let function h(s) be given by the formula h
1 f°° ^ = T v T / tS~lTr{e-tT>2 r ( s ) Ji
- Uv)dt
.
(22)
Then h(s) is a holomorphic function of s on the whole complex plane. Proof. We estimate Tr(e~tv Tr(e~tv2
- lip) for 1 < t as follows
- lip) = 2 £ fceN
e~h#•<.-&
<
i£N
2_^e -*K < 2-e 2 • 2_, e 2 < c-e 2 , fceN fceN and the result follows from elementary complex ananlysis. 2-e
2
e~tk* = 2 £
•
It is the integral on the interval 0 < t < 1 , which determines singularities of CA(S) • We have
251
-ml!*'"irf'+'w=fci~mI!•"*+m™ • where p(s) is yet another function holomorphic on the whole complex plane and we use the fact that Tr II(X>) = dim ker V = 1 . We have proved the following Theorem Theorem 3.7 There exists a function g(s) holomorphic on the whole complex plane such that CA(«) has the following representation
<-M = ^ 7 ^ T - r 5 ) 4 + I + F5)*)-
< 23 »
We combine this result with T ( - ) = v ^ and ——=0
fork
= 0,1,2..
,
in order to obtain the next result Corollary 3.8 The only pole of CA(S) is located at s = \ . It is a simple pole and the reesidue of T(s) at s = 1 is equal to 1 . Moreover £A (0) = 0 and Ui-k) = 1 for k= 1,2,3,.. . Theorem 3.5 follows immediately from Corollary 3.8 since now we can use (23) to represent CTC(S) as *
I X
1/^
lS\
-\
\FK
1
C*(-) = -2(U(-2) -1) - r ^ - j T T -
252
1
m
1
1
,S.
• - + 2f(f)-5(2} •
.„,.
(24)
4
Duhamel's Principle and ^-function of the operators A / .
In this section we study the ^-function of the perturbations of the operator A = A 0 = VQ . We discuss in order of increasing technical difficulty the operators AQ = A + a , A / = A + f(x) , where f(x) is a smooth function on S1 , and the operator V2. — — ^ — 2ia-^ + a2 . The easiest example here is of course operator A a . It is natural to expect the equality e -*A„
_ e-tA-ta
_
e-tae-tA
to hold, especially because the bounded operator (Bf)(x) with the operator A and the equality e-t(A+B)
_
= af(x)
commutes
e-tAe-tB
holds for commuting matrices A , B and for bounded commuting operators in a Hilbert space. The operator A is an unbounded operator on L 2 ( 5 : ) , which creates technical problems. We discuss the standard way of getting the Heat Operator for the perturbation of the given (Dirac) operator. We introduce now Duhamel's Principle . We offer a formal formulation without precise assumptions, and later on we will show that everything can be made rigorous in the case of a Dirac operators on 5 1 .
Duhamel's Principle Let A and A + B be operators acting on a separable Hilbert space such that the operators e~tA and e~^A+B^ exist. Then e-t(A+B)
=
e-tA
_ f* e-s(A+B)Be-(t-s)Ads
(2g)
JQ
Proof. The Heat Operator is equal to the identity at t = 0 , therefore e-HA+B)
_ e-tA
=
= [ e~
fl
dL,e-s(A+B)e-(t-s)A)ds
Jo
d
=
s
+ B) + A)e-«-')Ada. 253
•
Now we apply formula (25) to e~
=
e-tA.
• f e~sABe-i-t-s'>Ads+ Jo
f ds Jo Jo
and
_
['dre-r(-A+B'>Be-(-s~r'>ABe-(-t~s'>A
l A+B The discussion above shows a way of constructing the operator \ e~ ^ t B We express operator e- (^+ ) as an infinite series in terms of the operators e~~tA and B . We introduce some notation before getting more specific. Let B(t) and C(t) denote two 1-parameter families of bounded operators in a Hilbert space. We introduce (B *C){t) convolution of B(t) and C{t) as the operator
(B*C)(t)= f Jo
B(s)C(t-s)ds
We take B(t) = e~tA
, C(t) = Be'tA
and Cn(t) = (C *C * .. *C)(t)
,
where we convoluted C(t) n—times in the last formula. Then the following formal equation gives the Heat Operator of A + B e-t(A+B)
= e-tA
+
£ ( _ i ) » ( B * Cn)(t) .
(26)
n=l
At this point it is a formal expression, which becomes the rigorous identity whenever we are able to show that the series on the right side of (26) is absolutely convergent. Actually, in our case, we will show even more. We use (26) in order to construct the kernel of the operator e~tA' . We take A = A and B = f(x) then enj(t; x, y) kernel of the operator (B * Cn)(t) is equal to enJ(t;x,y)=
/ dsi / duy / Jo Js1 Jo
ds2 / du2 Js1
eA{sn;x,un)f(un)e&(sn-i;un,un-{)
/ Jo f(ui)eA(t
254
dsn I dun Js1 - si;m,y)
(27)
.
We use this representation in order to prove the absolute and uniform convergence of the series which represents kernel of the operator e~tAf .
Theorem 4.1 There exists to such that the series 00
e A (*;i,l/) + 5](-l) B e„,/(t;a! > y)
(28)
n=l
converges uniformly on S1 for 0 < t < to .
Remark 4.2 1. Although we prove Theorem 4.1 for small to the choice of to is unsignificant. We can easily prove the following variant of the Theorem: For any to there exists a constant Mto such that for any 0 < t < to and any x,y e S1 oo
= \'£t(-l)nen(t;x,y)\<Mt0
\e±f(t;x,y)-e±(t;x,y)\
.
(29)
?l=l
We leave the estimates leading to (29) to the willing reader. 2. We do not present the optimal result. In fact the estimates in the proof were written down half an hour before the lecture. This is not a very important matter. We want to show to the uninitiated reader an unpolished way of proving that the existence of the Heat Operator for the operator A / follows from the perturbation of the operator e~tA . We will use extra information on the kernel egi (t; x, y) later on. Proof. There exist positive constants ci,C2 such that for any x,y £ S1 and for any 0 < t < 1 we have \eA(t;x,y)\ We start with the term
< -j= , |/(a;)| < c2 .
eij(t;x,y) 255
\ei,f(t;x,y)\
4TT JQ
= \
JSI
ds
Jo
Js1
du eA(s;x,u)f(u)eA(t-
y/s(t - s)
,
2
ft
ClC2
J0
y/sJT^)
f* ds
V27o 7^
s;u,y)\
1
~
<
Jo
J^I
„ ,
= 2ClC2
•
The second inequality is even easier. Let K > 0 denote a constant, then we have pt
/•
ft
| / ds I 1 du Kf(u)eA(t 7o JS
- s\u,y)\
Now, we can deal with
\en,f(t;x,y)\
Js1
e&(sn;x,un)f(un)eA(sn-i rt
rt
r
ds 2 /
rsi
p
/•
du2-.. I
Js1
/-si
(2ciC2)(47rciC2) / cfsi / dui / Jo Js1 Jo ^/Sn-2-\f(Un-2)eA(sn-3
Js1
dun
r
ds„_i /
dun-i
Js1
Jo
/ ( « l ) e A ( * ~ « 1 ! " l , 2/)| <
/•
rSn-3
ds2 / du 2 ... / Js1 Jo
~ Sn-2] « n - 2 , « n - 3 )
256
dsn /
i-Sn-2
ds2 /
Jo
/ Jo
/(ui)eA(i - si;uuy)\ <
|/(un_i)eA(s„_2 -Sn-i;«n-l,Wn-2)
/•*
J
/ —z=z=ATrciC2Ky/i ./o V* - s
du2
Js1
Jo
dui I
Js1
Jo
d«i /
- sn;un,u„_i)
2c\c2 I ds\ I ds\ I Jo
2TTCIC2K-
enj(t;x,y)
= \ / dsi / Jo
<
r
dsn-2
I dun_2 Js1
/ ( « l ) e A ( * - «1! « 1 , 3/)| <
.
(2ciC2)(47rcic2) / dsi / 1 dui I ds2 I 1 du2---^/sn-z Jo Js Jo Js |/(un_2)eA(sn-3 -s„_2;un-2,'"n-3)
(2ciC2)(47rcic2)2 / dsi / Jo
\f(un-3)eA(sn-4
dm
Js1
I Jo
f{ui)eA{t
/
dsn-3
I dun-2 Js1
- Si; u i , y ) | <
/
Js1
Jo
- sn_3;un-3,it„-4)
dsn-2
f(ui)eA(t
dun-3sn-3
- si;ui,y)\
< ( 2 c ? c 2 ) ( 4 7 r c 1 c 2 ) n - 1 - ^ = (2c21c2)(ATrc1c2Vi)n-1
<
.
Now, finally we can estimate the whole series. oo
oo
| ^ ( - l ) n e n ( < ; a ; , j / ) | < ^(2c?c 2 )(47rc 1 c 2 V / t)"- 1 = 71=1
71=1
= (2c?c 2 )(47rcic 2 v'i)--
T
.
1 — 47TCiC 2 V*
We have proved that the series (28) is uniformly convergent for 0 < t < 1 (47TC1C2)2
r-i '
The terms in the series are expressed in the succesive powers of y/i , which leads to the following fact Corollary 4.3 There exist a family {u„(a;)} ne N of smooth functions on S1 such that for any natural number N we have N
eAf(t;x,x)-^2ts^1un(x) n=0
257
=0(tN) .
(30)
The last result is not really the best we can get. We know (see for instance [G],[T2]) that actually we have an expansion in powers of t . This result is actually quite difficult to establish in the case of a general Dirac operator. In the case of S1 we can use the fact that, for any 0 < t and any x,y € S1 eA(t;x,y) > 0 (see (14)). We also use the fact that identity e -sA e -(*- s )A _ e~tA reads as follows on the level of the kernel of the operators / e^,uMt-s;u,„)du = e^,y) Js*
.
(31)
T h e o r e m 4.4 There exists a family {vn(x)}n^T
eAf(t;x,x)
= 0(tN)
-^r-hnix)
.
(32)
n=0
Proof. We show that each next term in the series has one more power of t . Let us explain this on the level of the first term \
ds du eA(s;x,u)f(u)eA(t-s;u,y)\ Jo Js1
C2- / ds dueA{s\x,u)eA{t—s;u,y) Jo Js1
<
— c-i- \ eclL{t\x,y)ds = Jo
C2t-eA{t\x,y)
The estimate on the n — th term of the expansion follows in the same way. • Corollary 4.5 For any smooth function f(x) on S1 we have CA/(0) = 0 .
Proof. The result follows from the corresponding result for A = A 0 , as we have just proved that there exists a positive constant c such that the following estimate holds 258
\eAf(t;x,y)-e&.(t;x,y)\
,
1
for any x,y £ S and any 0 < t < 1 . Let TLA, denote the orthogonal projection onto the kernel of the operator A / . Then we have CA,(s) = ^ r r /
-!— / ts'xTr{e'tAf T(s) J0
i s - 1 T r ( e - t A * - UAf)dt + dim ker Af =
- UAf)dt + dim ker A , + ^^-Hs) 1 (s)
,
where h(s) is a function holomorphic on the whole complex plane. Trace of I I A / is equal to dim ker Af and therefore we have CA,(S) =
_L £ ,.-i Tr e-t±,dt + {dim ker A/).(1 _ _L £ ,-!*) + J_. Ms) = W) I! ts~1{Tt + 0{Vi))dt + {dim ker AfHl ~ ft)] + W)Ms) = ts loiVi)dt+{dim ker A/H1
~ m}+mm
W T Y T ^ + m I! '
•
We take l i m ^ o and obtain 0 .
• As an exercise reader may use Duhamel's Principle to check that eA+a(t;x,y)
=e-taeA(t;x,y)
259
.
(33)
5
Heat kernel of the operator T>\.
In this section we use Duhamel's Principle to construct the kernel of the operator e~tv« on S1 . Our goal is to prove the analogue of Theorem 4.4 in this new situation. We have V\a = - A - 2ia— + a2 = - A + 2aV0 + a2 . dx We know that there is no problem with a2 . We simply have: e-A+2aT>o+a 2 (*! * ' V) ~
e
~'a
e
- A + 2 a P 0 (*5 ^ V) •
Now we have to face the perturbation of order 1, which is not a bounded operator on L 2 (5 1 ) . Still we can study the kernel of e~A+2aT'a the way we did it in a previous Section. We have to show the absolute convergence of the series
eA(t;x,y)
+ ^2en^Va(t;x,y)
,
(34)
n=l
where en,va(t;x,y)
= Jo
dsx l dm / Js1 Jo
ds2 / du2 Js1
eA(s n ;a;,u n )2aX>oeA(s n _i;u„,u n _ 1 )
/ Jo
dsn / dun Js1
2aT>oeA{t - si;ui,y)
.
Let us first figure out the straightforward estimates on e n ,o 0 (t; x, y) . We have \ei,va(t;x,y)\=2a-\
n
ds
eA(s;x,u)(—)eA{t
l f% ds f 4TT y 0 js(t - s) Js>
2a— f
dS
-
<*-»)2 | « - s r | 2(t-a)
f
|M U —y| V\
~
°47T J0 y/sQ - s) Jsi 2(t - s) 260
s;u,y)\ <
<-»)» ^
(u-y)-'
2 /•* ds f°° 4TT y 0 ^ ( t - s ) J0 1 /•' ds 2a— / . 2TT y 0 ^ ( t -
z _£ e •" 2(t-s)
=
, „ 1 2 f' 1 < 2a/ ds = 2 a 2-Kt J0 IT
n
s
More generally we have 1 <~ «\en,Va(t;x,y)\<-(2a)n—3
1
'1 ^-i, ^-(-7=)"-^— ,
(35)
and as in Section 4, we have proved the uniform and absolute convergence of the series which formally gives the kernel of the operator e - A + 2aP <> . hence, we have just constructed this kernel. Again we expect that terms with odd n should disappear. This follows from the general theory (see for instance [G] Chapter 1, or [T2]). However, in our case we can offer a simple argument which shows that Tr e~tV" expands in powers of t rather than \ft .
Theorem 5.1 There exists a sequence of real numbers {rfc}£L0 such that for any natural number N we have N
Tr e-iV-
- £
tk~hk
= 0(tN)
.
(36)
fc=0
R e m a r k 5.2 Working a little harder we would be able to prove that there exist a sequence of smooth functions {/fc(z)} such that N
eV2(t;x,x)
-Y,tk~hk(x)
=
0{tN)
fc=0
This "Local Variant" of Theorem 5.1 for the general Dirac operator is proved in [G],[T2]. 261
Proof. We use the fact that operators A and T>0 = ~ij^ commute. Now we write a series which gives the operator e-*(A+2ai>o) e-t(A+2aV0)
= e _tA +
(-^^(j)
,
where En(t) =
(2a)" f dsi P ds2... fSnl Jo Jo Jo (2a)n(V0)ne-tA-
dsne-s"AV0e-^-i~^AVo...Voe^t-s^A
f dsx ['* ds2... f^'1 dsn = Jo Jo Jo
=
n (2a) ~(V0)n.ec - t A n -
Theorem 5.1 follows from the fact that X>o = — *^r has a symmetric spectrum which implies the equality TrVln+1e~tv2
=0 ,
for any natural number n . D Corollary 5.3 For any real a CP2(0)=0.
(37)
Proof. We have just proved w
Tr e-
° - Tr e~tvZ < 4 ,
for 0 < t < 1 .
•
262
6
^-invariant, The Phase of the (^-determinant.
In this section we study the 77-invariant of the operator Va . The main result is
Theorem 6.1 The function Re{s) > - 2 .
r)va(s)
is a holomorphic function
of s for
Once again we prove this result by studying Heat Kernels. We recall the formula
We now decompose Tr Vae~w« 0 Tr Vae-W*
and use the fact that Tr V0e~tv°
is equal to
= Tr X\,e -rt> 2 - Tr X>0e~tP° =
Tr(Va-V0)e-tv'-TrV0(e-w'-e-t'Do)=a-Tre-tv'-TrVo(e-m'-e-tv') We know the expansion of the first summand on the right side a - T r e - t p ' - a J ^ + ci-v/< = 0 ( ^ ) .
(38)
We remember that e-«>«
=
2 e-ta e-t(A-2a730)
_ e~ta2 / e - i ( A - 2 a © 0 ) _ e~tA\
2 e~ta £-tA
+
and now we have to study TrD0(e-^-e-fD»)= ta2
(39) t
2aVo)
tA
Tr e- V0{e- ^-
- e~ )+Tr 263
a
( e -*° - Id)V0e~
tA
.
The second term on the right side is again equal to 0 and finally we only have to show that Tr e~t0?T>0(e~t^A-2a'D^ - e~tA) has the correct asymptotic. It was already observed in Section 5 that e-t(A-2aV0)
_ e-tA _ > p (2a*)"-„„, n tA £^-' ^--D n! oe-
•
(40)
n=l
Hence we do have Tr Voie-^
-ta? T l r ep
ST^ (2at)nvn+l 2_, n ! Vo n=l
- e~tvi)
e
- t -^ ( AA- 2 a T > 0 ) Tre-taVoe
=
_
-tA _ T -ta2 V"" (2at) ' jytkr~tA -ire ^ {2k - l)\ °° e v k=l
'
Now let us use an extra symmetry we have in this formula TrVlke-tA
= (-l)k(±)k(Tre-™2)
= (-l)Hj/(^
+ 0(e-i))
. (41)
We have proved the following result P r o p o s i t i o n 6.2 There exist a sequence of constants {bk} such that TrVae-tv*~^z+b1Vi+...
.
(42)
However, we are not out of trouble if we take into consideration the situation in the neighborhood of s = 0 . This is due to the fact that we have to study 1 P°° lim — - r r - / t^Tr . - 0 r ( 3 ± i ) J0 264
Vae-tv°dt
and in this situation factor ^h\ is replaced by
*+!, ; hence, the singularity
which comes from the heat kernel is not cancelled by singularity of T(s) . Now, when we study more precisely formulas (38) and (41) , we come to the conclusion that actually we have bo = 0 ,
(43)
in (42), but as in these notes the focus is on different methods related to Heat Kernel we offer yet another argument. This type of argument is used quite often in different contexts in Spectral Geometry. We define a function n{a)
= Ress=0riT>a{Q)
•
This is a smooth function of the parameter a , and we have
Theorem 6.3
f-0
(«,
We need the next Lemma in the proof of Theorem 6.3. Lemma 6.4
A(e-t^)
=
_|V^(XVD0 +
VaVa)e-^v°ds
(45)
Remark 6.5 Of course (45) is the formula which holds for the smooth, 1parameter family of Dirac operators over a closed manifold. In the particular case of the operator Va = —i£ + a on S1 , we have t>a = ^ = 1 and (45) becomes
A ( e -^) = _2 y":Vae-«%ds = -2Wa e 265
«*
Proof. In the Lemma 6.4 we study the variation of the Heat Kernel under the smooth change of the Dirac operator d
-{e-tv*} da
= lim I.( e -">2+r_c-">2) = lim I . / ' ±(e-^l+re-(t-s)vl)ds r-+o r r—^0 T Jo (IS lim I . jf' e - ^ ( 2 3 2 - l ^ J e ^ - ^ d s
lim f r-*° Jo rlim
^°Jo
f
=
=
e-^l^z3±LeHts)vlds+
r
Ue-<*+r
- e~^)(Vl
- Vl+r)e^-^ds
+ 0 = - /*e- s T , «(V a V a + Jo
- f e-^Vle-^-'^ds Jo
=
Vai)a)e-^-s^ds
a Proof. Now we are ready to prove Theorem 6.3. We have I
/-00
H(a) = lims — - T J T / r
"->°
(
t^Tr
Vae-tv°dt
,
(46)
2 ) JO
and we study the variation of the integral on the right side of (46)
£ { |
t^TrVae-w*dt} /»0O
rOO
/ Jo
t^rTrVa{e-t'Dl}dt=
/•OO
/ Jo
=J
I Jo
Jo
+
/>0O
t^-TrVae-w«dt+
/-OO
t^TrVae-tv°dt-2
t^TrVa2ee-tv°dt
t^TrVa.e-Wdt e
Jo
=
/*00
r ^ T r tPaXfrr"'-= / Jo 266
t^TrVae-tTlldt+
t^±(TrVae-w°)dt=
2/ Jo
t^TrVae-w*dt
dt
+
Jo
2 l i m ( i ^ T r X> a e- £D »)|J - 2 /
^ ( t ^ J T r P„« .e-^-d*
The limit l i m ( t ^ T r X> a e- c7, °)|| is equal to 0 for the invertible operator Va and s > 0 and we obtain a crucial formula f ^ T r P a e - t 7 , » d i } = -s /
-H /
t ^ T r Vae-tv°dt
.
(47)
Once again the formula simplifies in the case of Va on S1 as Va = 1 . Now we finally have information about the variation of the residue dll da
d ,. aos-+o das->o
, , '
av ,UaK
,. s d r f°° - 1 «->o Tt^f1) da J0 ^or(^)
—Llims2/ V7T «->0
t^Tre-tT,*dt
„,
.7,2 , ,
=0
J0
D Let us observe that in fact we also obtain the formula for the variation of the ^-invariant i.e. the number Tfpa (0) . 77po(0) - lim — = ^ - [^ t^Tr s->o r ( ^ ± i ) J0
e'^dt
-—=lims-
+ 0(ti))dt=
V*M0
t^u/-+nVi
io
Vt
= - ~ lim s- f t^Tr y/ir s^o J0
= lims/
V* a-^° 267
Jo
e~tv^dt =
t ^ <-dt
V*
=
/•I
- lim s- /
1
i2
I
d* = — l i m s - — = - 2 .
Theorem 6.3 implies that r)va (s) is a holomorphic function of s in the neighborhood of s = 0 , because we know that 71(0) = 0 , and by Theorem 6.3 , this extends to any real number a . In fact, we have proved more, because equality (42) implies that there exists constant c > 0 such that for any 0 < t < 1 we have \Tr Vae-^']
< cy/i ,
(48)
which gives us this representation TrVae-w^=b1Vi
+ b2ti+0(ti)
.
Now, we can discuss the structure of the 77-function of T>a I
-I
rOO
- — ^
/
l i1 t*i TrVae- >-dt
L
J. \ — ) Jo
—irjT f 1
(
2
/»00
t^1 (hVi + b2t% +0{t*))dt
= —mr
t^X(blVt
=
Jo
(2)
+ b2t§+0(t§))dt
+ h(s) =
) JO
m ftidt
+
r(^±i
wik-\ f1ti+1 + hl(s) + h(S) ,
where h(s) is a holomorphic function on the whole complex plane and hi(s) is holomorphic for Re(s) > - 6 . This shows that there exists h2(s) a function holomorphic for Re(s) > —6 , such that _2_
_&2_
2
+ + h {s) wAs) = rjsq-j^2 * • , ~+ 2 r r^ITI ( ^ +
2
In particular Theorem 6.1 is proved. 268
(49)
Corollary 6.6 rtDM
-TjTr Vae-W°dt
=- r l
.
(50)
Proof. Theorem 6.1 implies that we can apply the second formula in (12) for any s with Re(s) > —2 . •
Corollary 6.7 rIVa(0) = -~limy/i-TrVae-e'Dl y/TT
,
(51)
e-+0
which again in the case of operator Va on S1 gives ?)x>a (0) = —2 . Proof. We differentiate equation (50) d_ -VVa(0) dai
= -£-(4= r da y/n J0
4= r St~ATr V7r Jo dt
^rTr y/t
Vae-tv*dt)
Vae-tv*)dt
9
= 4 = f ° ^Tr yf-K J0 s/t
l
-=• l i m ^ - I Y Vae-iV")\I V7T E->0
1
V^^dt-Y
/*oo -i
- -== / V^ Jo
2
Vae-tv*dt+
= -L71 J" ±Tr V " Jo Vt
~T?r Vae~tv'dt
=
Vt 2
y=-\im y/e-Tr Vae~eV"
.
In the case of Va on Sl we have
- ~
lim v^-Tr T>ae-eT>" = ~ ~ lim ^.{.fl
+ 0(y/i))
= -2 .
n 269
Theorem 6.8 i7x>.(0) = l - 2 a .
(52)
Proof. We have just proved that WVaity = -2a
mod Z .
More precisely the continuous part rjr>a (0) of r)-pa (0) (as a function of o) is equal to ret
pa
\ r)Vr(Q)dr = - 2 - / dr = - 2 a . Jo Jo We have a formula m>As) = — + ^r,sign(k a fc#o
+ a)\k + a[
which gives lims-jo T]va (s) — 1+ fjva (0) = 1 — 2o .
D
It follows now from Theorem 6.8 and Corollary 5.3 that we know the phase of the ^-determinant of the operator Va Corollary 6.9 The phase of the det(Da is equal to e J 2 L(2a_1) . In the next Section we deal with the modulus of the (^-determinant.
270
7
Modulus of detcpa
There are some reasons beyond the scope of these notes (see for instance [ScW2],[ScW3], which tell us that we should pick the operator P i = -i-£ + \ and assume that its (^-determinant is equal to 1 and then obtain the value of the ^-determinant for other Va by studying the variation of the determinant with respect to a. Therefore we discuss now the variation of the modulus with respect to the parameter a. We know that absolute value of the £determinant is equal to - half of ^{0) — ^ C P J ( S ) U = O • The function 2 Cv (s) is holomorphic in the neighborhood of s = 0 and we can take the derivative with respect to s
lim 4-ij^
r ^ T r
e-^dt)
K
v
s^ods r(s)j0 l i myT(s)ds ( - i - ^KJ- ( r s^o 0
e~1!D-dt)+
= l i m ( - ^ [°° t^Tr 2
s->o r(s) j0 t'^Tr
e~tv*dt)) = s->o l i mv ( r"(£s )42 JT 0
Remark 7.1 The fact that lims^o{f^j£Uo°°
t lTr
e
"
e'tV-dt))
t^Tr
"^d*) •
disappears is s l
due to an absence of the singularity of the function IC(s) = /0°° t ~ Tr e~tV"dt at s = 0 and it is characteristic for dimension 1 , or more generally for odd dimensional manifold M . In the even-dimensional case this part may produce additonal contributions. We know that T'(s) = — Jr + h(s) , where h(s) is a holomorphic function in the neighborhood of s = 0 , hence we have arrived at the equation
THFTT / ds^T(s) J0
* S _ 1 2> e-tv*dt)\s=0
= lim / s^oJo
ts~lTr
L
-Tr
e~tv"dt e-tv°dt
= .
(53)
We use Lemma 6.4 to study the variation of the right side of (53) and obtain 271
f Tr e~w'dt ^
/ da Jo
= -2- / Jo
Tr VaVae~tVl«dt
=
/•oo
T r Vae-tv*dt
-2- /
= -2-»^.(l) .
(54)
We work on the expression J 0 Tr Vae tv<>dt in order to figure out the exact value of the variation . We may assume that Va is invertible, hence /•oo
/ Jo
/*oo
TrVae-tv'dt=
/»oo
TrV-lVle-iV*dt
= -
Jo
7o
lim(Tr V7le-tTZ)\l
i
~(Tr «*
= + lira Tr P " ^
e—>0
-
V^e'^dt
=
^
e—>0
We have just proved
Proposition 7.2 da
(55)
£->o
"
We need a formula for the operator T>~x in order to get the right side of (55) . The point is that we have an explicit formula for ka(x,y) the kernel of the operator V~l
1-e2
ka(x,y)
•nr tor x < y
:= < . +\_e-i*i« for x > y
Now, we have
s«*<°> - - 2 - l i m
/
da;/ 272
dy
ka(x,y)eV2(t;y,x)
(56)
-2-lim z^JS1
dx
dy ka(x,y)et)2(t;y,x)
.
J\x-y\<6
The last equality follows from, earlier established, fact that the Heat Kernel e(t;x,y) is exponentially decaying when the distance between x and y is bounded away from 0 and time is going to 0 . In other words lim ka(x,y)eT>2(t;y,x)
s—yO
=0
for
\x - y\ > 6 .
"
Therefore, we have a sequence of equalities
—2-lim dx dr{ka(x;x+r)e-o2(£;x+r,x)+ka(x;x—r)ex>2(e;x—r,x)} e -*°JSi Jr<5 - 2 - lim I
dx f
dre-ira{~-
^-r- +
\—}^=e-£
=
=
sin 2ira ,. f , _, 1 r2 sin 2na hm / dre %Ta . e~^ = -2n-— 1 - cos 2na e-vo Jr
Proposition 7.3 d ^i ,ns c sin 27ra —Cp 2 (0) = - 2 T T — . da I — cos 2-na The variation of the modulus of the determinant is given respectively by IT
sin 2ira 1 — cos 2ira 273
.
,
, 57
Theorem 7.4 The following equality holds for any 0 < b < 1
=sin
5g
S^f " •
( )
2
Proof. We have 1 ., ,„. - ,_ 1 ., ,-.. f sin 2-na , 1 f sin 2-na „ , -«&?(0) ( 9^! ° = *• Ji / .1 —/ vcos da =-• 2 /i - 1 — cos —Z7ra 2nda = i a 0 27ra 2 <• 2 A ^ 2 2 1 f1-™3 -• / 2 J2
2irb
du 1 , 1 - cos 2-Kb 1 , . 2 , , — = —-m = —•In sin 7ro = In sin irb . u 2 2 2
• Corollary 7.5 de*c£>a = e^^-^-sin
na-det^Vi
,
(59)
/or any 0 < a < 1 . It is quite natural to assume now that det(T>\ — 1 and take ,if-(2a-l).
sin na
as the (^-determinant of the operator Va . This is even more sound if you notice that e ^ ( 2 a _ 1 ) - s m ira =
,
which actually plays the role of the determinant. We can generalize the whole story as follows: let us observe that the operator Va is from the spectral point of view, equivalent to the operator — i-^ on the interval [0,2n] subject to the boundary condition /(2TT)
= e 2 ™/(0) , 274
(60)
This means that we consider the operator X>„ = — i-^ on the subspace of C°°([0, 2ir\) , which consists of only functions which satisfies condition (60). This becomes a nice, self-adjoint operator on L2([0,27r]) with a spectrum equal to the spectrum of the operator Va on 5 1 . The last can be easily seen by the straightforward solution of the eigenvalue problem -i-^
(61)
which gives
where
X2TT
= 2ira + 2nk ,
for any integer k . More generally we can assume that — i-^ acts on C n valued functions f(x) and that the boundary conditions is given by a unitary matrix T : C n ->• C n . We denote the resulting operator by ( - * ^ ) T • Then using the fact that we can diagonalize the operator we can apply our method to prove det^-i
— ) T = det
Idc
"^~T
y
(62)
where once again we use the "relative £-determinant", i.e. we assumed that detc(-i—)_Id
= 1 .
Actually, we can compute the value of det^{—ij^)-id using Elementary Number Theory and it appears to be equal to 2n . We refer to paper [BSW] for details. It is much more amazing that most of this generalizes to the general situation in which we deal with elliptic boundary problems for the Dirac operatos (see [Scl],[5c2],[5cWl],[5cW2],[5cW3],[?]).
275
8
Heat Equation and Index
So far, we stayed on the elementary level and a smart undergraduate student having [J?S1],[.RS2] in sight, should be able to go through this text alone without help from more advanced colleague. In the last part of my lecture I move to a more complicated set-up. I discuss the index of a single Dirac operator and its counterpart in the self-adjoint case, the spectral flow of the 1parameter family of Dirac operators. This gives me an opportunity to discuss a new application of Duhamel's Principle, which is a technical tool used in these lectures and arguably the main hero of the story. Still, I think that a smart undergrad can deal with this presentation by himself/herself looking some things up in the first Chapter of [G]. Let A : C°°(M;E) ->• C°°{M\F) denote an elliptic differential operator acting on sections of smooth vector bundles E and F over a smooth, closed manifold M . We assume that we have given Riemannian structure on M and Hermitian structures on E and F . The spaces kernel of A and cokernel of A are defined as ker A = {s£Coo(M;E);As
= 0} ,
and coker A = ker A* = {s € {s € C°°(M; F); A*s = 0} . They are finite dimensional. The number index A = dim ker A — dim coker A is a homotopy invariant of the operator A , which means that it does not change under smooth deformation of the operator A . The model here is the Dirac operator acting on sections of S , a bundle of Clifford modules over a closed manifold. This is a self-adjoint operator hence ker V = coker V , and the index is trivial in this case. However, we know that in the case dim M is an even number then spinor bundle splits S — S+ © S~ onto the subbundle of spinors of positive and negative chirality. This gives a decomposition 276
V
=
[ V+
0 )
:C
°°(M'>5+)
® C°°{M; S~) -> C°°{M; S+) © C°°(M; S~) , (63)
where the operators T>± : ^^(MjS 1 1 1 ) -> C 0 0 (M;5 = F ) now, in general, have nontrivial indices. Let us observe that V~ = (D+)*
hence index T>~ = —index T>+ .
There is a formula which expresses the index in terms of topological invariants determined by manifold M , bundles E and F , and coefficients of the operator V . This formula is a content of the famous Atiyah-Singer Index Theorem proved by Atiyah and Singer in early 60th and we refer reader to [G, ?] for formulation, proof, and extensive bibliography related to this result. One can use the Heat Equation Formula in order to prove the AtiyahSinger Index Theorem. Let us consider the operator A from the beginning of the Section. The starting point of the proof is the following formula due to Bott, which holds for any t > 0 : index A = Tr e~tA'A
- Tr e~tAA'
.
(64)
Proof. The operators A* A and AA* are self-adjoint elliptic operators, hence, they have non-negative spectrum. Formula (64) follows from two facts. First, it is easy to observe that the 0-eigenspace of A* A is equal to the kernel of the operator A and the 0-eigenspace of A* is the cokernel of A . Now let A denote a positive eigenvalue of the operator A* A corresponding to the eigensection
We use this information to compute the right side of (64). Let £^(^4*^4) and E\(AA*) denote the eigenspaces of A*A and AA* corresponding to A We have just seen that those are isomorphic spaces and in particular dim EX(A*A) — dim E\(AA*) . Now we compute 277
Tre-tA'A-Tre-tAA' e~txdim
Y,
EX(A*A)
=
-
XEspec(A" A)
dim ker A-dim
e~txdim
^
EX{AA*)
=
\€spec(AA*)
coker ^ 4 * + ^ e~tx(dim
EX{A*A) -dim EX{AA*)) = index A
XjtO
• The Index Theorem follows from the analysis of the limit lim(Tr e~tA*A -Tr
t-K)
e~tAA')
.
Using the construction of the Heat Kernel we can show that at least for the Dirac operators we have a "local" density a(x) such that index A = lim(TY e~tA'A
- Tr e~tAA')
= [ a(x)dx
,
(65)
where "local" means that a(x) is constructed (in a complicated way) from the coefficients of the operator A at the point a; .
278
9
Spectral Flow ^-invariant and ^-invariant
The index of a self-adjoint elliptic operator is equal to 0 . However, there is an invariant which corresponds to the index in the self-adjoint case. Let us consider {B u }o<« C°°(M;E) . The space ker Bu = coker Bu has finite dimension for any u , which implies that only finitely many eigenvalues change sign when u travels from 0 to 1 .
Definition 9.1 The number of the eigenvalues which change sign from — to + minus the number of eigenvalues which change sign from + to — , when u varies from 0 to 1 is called spectral flow of the family {Bu} and is denoted by
sf{Bu} .
Spectral flow was introduced in paper [AP52] in order to study a noncontinous part of the ^-invariant. To explain the situation let us think about the family { 2 ? a } _ i < 0 < i . We have the equality Vi =e~ixV
ieix
,
~2
2
hence P i is unitary equivalent to the operator V__i and Vvi(0)
= Vv_i(O)
,
equal to 0 . On the other hand we have computed the variation of the ninvariant (see Corollary 6.7)
•n-ba = - 2 -
Hence, at least formally we expect r?pi(0)=r/p_i(0)-2 . 279
However, this is compensated by the appearence of the spectral flow of the family {Va}_±
dim ker V + r)v(0) -z •
(66)
Invariant fp is a well-defined function on a space of Dirac operators. Once again this function has a well-defined continous part £x> and integer value component, which this time is equal to the Spectral Flow. To be more precise we have the following result
Theorem 9.2 Let then
{^UJCKUO
denote a smooth family of Dirac operators,
£z?i - £x>0 = Cz>i - £c>o + sf{Vu}0
,
(67)
and
da-^du
&-&o=J0
•
Instead of offering the proof of the Theorem for a general case we allow the reader to figure out this result in our simple scenario on S 1 .
280
10
Index and spectral Flow
The Theorem which we prove in this Section is due to Atiyah, Patodi and Singer (see [APS2]). The novelty here is the proof, which is due to the author of these notes (see [Wl]). In the unpublished paper I studied a slight generalization of the famous Witten's Holonomy Theorem and the proof of the equality of the Index and the Spectral Flow offered in [Wl] was a simple variant of a general method used to prove Holonomy Theorem. Let for any 0 < u < 1 Bu : C°°(M; S) ->• C°°(M; S) denote a compatible Dirac operator acting on sections of a bundle of Clifford modules over closed, odd-dimensional manifold M . We assume that {Bu} is a smooth family and we also make a technical assumption which makes the pasting of the operators easy Bu = BQ for 0 < u < s and Bu = Bx for 1 — e < u < 1 . Moreover, we assume that there exists a unitary bundle automorphism g : S -4- S , such that Bx = gBg-1
.
(68)
The equality (68) lets us think about {Bu} as the family over S 1 , as Bi is now unitary equivalent to the operator BQ . On the other hand, we can construct a corresponding object, an elliptic operator i o n a manifold S1 x M. We introduce Sg a smooth vector bundle over S1 x M as follows: we take bundle I x S over I x M and now we identify (l,V,s)~(0,y;g(s)) ,
(69)
to obtain a bundle over S1 xM , where S1 = R / z • Let u denote a coordinate on S 1 , then we define operator A : C°°(5 1 x M; Sg) -» C 0 0 ^ 1 x M ; Sg) by the formula A = du + Bu . Though it is not essential, here is the point when we use the assumption that family {Bu} is constant near 0 and 1 in order to have a well-defined operator 281
A . To see that A is indeed an operator acting on sections of a twisted bundle let us take section s £ C°°{S1 x M; Sg) . We have «(!,
for any y € M ,
and now we apply operator A As(l,y)
= (du + B1)s(l,y)
(du + gBg-1)g(y)s(Q,y)
=
= g{du + Bo)s(0,y)
= gAs(0,y)
,
which shows that As is again a section of bundle Sg
T h e o r e m 10.1 index A = sf{Bu}
.
(70)
We use (64) in order to prove Theorem 10.1. One would expect that we need to construct exact Heat Kernels for the operators A* A and AA*. However, the information we need is not hidden deep in those kernels. We show that it is enough to know an appriopriate approximations of those kernels. To get rid of the unnecessary terms we use the Adiabatic Argument . We introduce an auxiliary parameter e and define the operator Ae as A£ = edu + Bu . This procedure corresponds to replacing the original 5 1 of the length 1 by the circle of the length |- , as basically we replace normal coordinate u by v = j . Index of the operator is unchanged under the smooth deformation hence we have index A = index Ae = Tr e~tA'Ac 282
- Tr e~tAcA'
,
(71)
for any positive number e . We use Duhamel's Principle to approximate the operators e~tA^Ae and e~tAcA^ . This time; however, we do not start with the solution of the Heat Equation. Instead of that we use an operator close enough to the Heat Operator to be a good starting point for the whole procedure described earlier in these notes. Such an operator is usually called a Parametrix in the literature, though this may lead to a misunderstanding as this term is also used in some other parts of the theory of partial differential equations. We define Q£(t) as the operator which serves as the parametrix for both e~tA'Ae and e~tAcA' as follows Qe(t) = e-tB2ee2td*
.
The operator Qe{t) is a composition of the operator e~tB at each u G S1 we have the operator
, or more precisely
QE{t\u) = e*2ta?>e-tB* . The reason that it is not the Heat Operator define by A*eA£ (or AeA*e) is the fact that operators du and Bu do not commute. We have A*£Ae = -e26l
+ Bl-
and AeA* = -e2d2u + Bl+ eBu
eBu
.
(72)
Now we use Duhamel's Principle e-tAtAe
= Qe{t)
Qe(t)+
[ e-sA'At{-A*eAe Jo
+
r d{e_sA.AeQ Jo ds
/•t
Qe{t) + J
J
ds j
S
dre-rA'^{-AlAe
_
+ ^-)Qe{t-s)ds as
=
=
,
Qe(sK-A*eA,
+—)Q£(t
+ ^)Qe{s-r){-AtA£ 283
- s)ds+
+ ~)Qe{t-s)
.
This equality may be used in order to construct e~tA'Ae . Calculations are longer and more complicated then before, but they are not really more difficult. The new element here is due to the fact that the operator — e2d\ in A*A£ does not commute with e~tB» in Qe{t) and we have t c w r i t e the first term in the expansion of e-tAeA^ w i t h respect to Qe(t) as follows
f Qe(s)(-A*eAe
+ -^)Qe(t-s)da
= e-J
Qe(s)BuQe(t
e2 f Qe{s)[e-^B«;dl)ee2(t-s^artial»ds Jo
- s)ds+
.
Now, the careful reader observe that all other terms in the series which gives e~iA'Ac are of order at least e2 with respect to the parameter e . Similarly we see that e-tAcA
* - Qe{t) + e- [ Qe{s)BuQ£(t
- s)ds = 0(e2)
Jo
We reached the main technical result of this Section
Theorem 10.2 The index of the operator A is given by the follwing formula which holds for any t > 0 Tr Bue~tB»du
index A = index Ae = —= I
.
V7T y s i
Proof. We have index A = lim(Tr e~tA*A' £->(T
2- Elim e f Tr{Q£{s)BuQe{t _f0 Jo
- Tr
e~tAcA')
- s))ds = 2- £_>0 lim eTr Bu f QE(t)ds = Jo 284
(73)
2- lim(et)Tr BuQe{t)
= 2- l i m ( e t ) - - 7 = - /
Tr Bue~tB"du
,
where in the last line we do have trace on S 1 x M (or more precisely of the operator acting on L 2 (5 X x M;Sg)) on the left side and the trace on M on the right side. The Theorem is proved. • Now we use Corollary 6.7 (see (51)) and Theorem 10.2 to show the equality
index A = -^= /
Tr Bue~tB»du
=
VTi" Js*
l i m ^ l f Tr Bue-tB-du=-]t-+o ^TT 7 s i
f nB (0)du . 2 ysi
(74)
We use Theorem 9.2 to make the final step in the proof of the Theorem 10.1 index A = — - /1 r]Bn(0)du = — / 1 £,Budu = sf{Bu} 2 Js Js
285
.
11
A Simple Example
We refer the reader to Section 26 of [BW] for the extended analysis of the two-dimensional case of the Index Theorem. Now you can read this Section without looking anywhere else in this book, which is devoted to the study of Elliptic Boundary Value Problems. However, for your convenience I decided to extract simplest possible example of the equality of the index and the spectral flow. We already know that spectral flow of the family {—i-^ + u}o
,
(75)
for each couple of integers k, I £ Z . It is not difficult to see that both operators d and d = d have trivial index. You can use method presented below, in the non-trivial case, to prove that. We also have a constant family of the operators in the normal direction {Bu = i-^} , which produces no spectral flow. To change that we replace smooth functions by sections of non-trivial line bundle over T 2 . We replace condition (75) by f(u + k,x + 2irl) = eikxf{u,x)
,
(76)
All smooth f(u,x) satisfying (76) are sections of the bundle Sg over T2 , with g(x) = elx (see (69)). The corresponding operator dg : C°°(T2;Sg) ->• C°°(T2;Sg) is equal to dg(u,x)
= du + idx + u ,
as follows from the following identifications 286
(77)
{du +idx+u+
+ l)(eixf(u,x))
l ) / ( « + l,a:) = (du +idx+u
eixduf{u,x)+eixdxf{u,x)-eixf(u,x)+(u+l)f{u,x)
=
= eix(du+idx+u)f{u,x)
.
We calculate the index of the operator dg . We start with the kernel. We have to find all smooth solutions of the operator (du + idx + u) on R 2 , which satisfies pewriodicity condition (76). Any smooth / satisfying (76) is of the form
f(U,x) = Y,fk(uykx
.
fc€Z
where coefficients fk{u) fulfill condition fk(u) = fk+1(u
+ l) .
(78)
Now we see that (du + idx+u
+ l)fk(x)eikx
= (f'k(x) + (u-
k)fk(x))eikx
,
hence fk{u) = cke-^-V2
.
This implies that solution of the equation dgf = 0 has the form
fcez Condition (78) implies that ck = ck+i for any k £ Z and we proved
Proposition 11.1 Kernel of the operator dg is the one-dimensional subspace ofC°°{T2;Sg) spanned by £ f c 6 Z e~h(n-k)2eikx 287
We can compute kernel of the operator dg in the same way. This time the solution is B +i(«-fc)
2
ikx
fcez
The coefficients of this series explode as k —>• ±00 , hence it is a formal solution only, and we have her dg = coker dg = {0} . We have just proved the main result of this Section.
T h e o r e m 11.2 index dg = sf{i-f~ + u}o
(79)
Students who already took differential geometry should notice that number 1 is nothing else but integral JT2 ch(Sg) , where ch(Sg) denotes Chern Character of Sg (see [G]). More generally, let us introduce a unitary transformation — e0ikx gk(x) =
which determines bundle Sk over T2 . The corresponding operator has the form dk -du
+ idx + ku ,
and we have formula A
C
index dk = sf{i— + ku}o
(80)
It is more difficult to show that we can replace gk by any map h : S1 ->• U{\) of degree k and the index of the resulting operator dh = du + idx +uh~1(x) is still equal to k — deg h .
289
— (x) , ax
12
Bibliographical Remarks
At the end let me discuss a little history of a subject. My comments are far away from being complete and the reader is again referred to different places for more accurate information. The Atiyah-Singer Index Theorem is a breakthrough result, which shows how to use analysis of elliptic operators to obtain a topological information and also how the topology and geometry determines the analysis on a manifold. The first proof using Cobordism Approach was published in [AS] (see [P] for the detailed exposition). Soon after the first proof of the Index Theorem had been published, Bott observed that we can use Heat Kernel, or equivalently C-function of an elliptic operator to obtain an index formula. The C-function entered global analysis a long time ago, but the serious systematic study of the relation between analysis and geometry based on C-function and Heat Kernel started with the famous work of Seeley [Sel]. Singer and McKean used Heat Kernel to study geometrical information contained in the Laplace operator (see [MS]). The proof of the Index Theorem based on the Heat Equation method was found by Patodi and independently by Gilkey. We refer to [G] for bibliographical remarks on this subject. The original Heat Equation proof was greatly simplified in 80th. The detail of the modern approach are presented for instance in [T2]. The C-determinant of the positive elliptic operator was introduced by Ray and Singer in [RaS]. This notion of the determinant was first used by physicists in the middle of 70th (see [H]) and has become an established mathematical tool of Quantum Field Theory since then. The ^-function was introduced by Atiyah, Patodi and Singer in [.APSl]. Singer observed the relation between ^-invariant and the phase of the determinant and he first used (6) as the definition of the C-determinant of the Dirac operator (see [Si]). The corresponding theory of the C-determinant of the elliptic boundary problems for the Dirac operators was studied by author and his collaborators (see [Scl],[Sc2], [ScWl],[ScW2],[ScW%[W2]). We also refer to [BW] for the presentation of the theory of elliptic boundary problems for the Dirac operators and to [M] for a different approach to the analysis of the Dirac operators on a manifold with boundary.
290
References [APS1] Atiyah, M.F., Patodi, V.K. and Singer, I.M.: 1975, 'Spectral asymmetry and Riemannian geometry. I', Math. Proc. Cambridge Phil. Soc. 77, 43-69. [APS2] Atiyah, M.F., Patodi, V.K. and Singer, I.M.: 1976, 'Spectral asymmetry and Riemannian geometry. I', Math. Proc. Cambridge Phil. Soc. bf 79, 71-99. [AS] Atiyah, M.F. and Singer, I.M.: 1963, Bull. Amer. Math. Soc. 69, 422-433. [BSW] Boofi-Bavnbek, B., Scott, S. G. and Wojciechowski, K. P.: 1998, 'The ^-determinant and C-determinant on the Grassmannian in dimension one', Letters in Math. Phys. 45, 353-362. [BW] Boofi-Bavnbek, B. and Wojciechowski, K.P.: 1993, Elliptic Boundary Problems for Dirac Operators, Birkhauser, Boston. [G] Gilkey, P. B.: 1995, Invariance Theory, Heat Equation and the AtiyahSinger Index Theorem, CRC Press, Boca Raton. [H] Hawking, S. W.: 1977, 'Zeta function regularization of path integrals in curved spacetime', Comm. Math. Phys. 55, 133-148. [MS] McKean, H. H. P. and Singer I. M.: 1967, 'Curvature and the eigenvalues of the Laplacian', J. Diff. Geom. 1, 43-69. [M] Melrose, R.: 1993, The Atiyah-Patodi-Singer Index Theorem, A. K. Peters, Boston. [P] Palais, R., ed.: 1963, Seminar on the Atiyah-Singer Index Theorem, Princeton. University Press, Princeton. [RaS] Ray, D. and Singer, I.M.: 1971, 'i?-torsion and the Laplacian on Riemannian manifolds', Adv. Math. 7, 145-210. [RSI] Reed, M. and Simon, B.: 1972, Methods of Modern Mathematical Physics 1: Functional Analysis, Academic Press, New York. [RS2] Reed, M. and Simon, B.: 1975, Methods of Modern Mathematical Physics 2: Fourier Analysis, Self-Adjointness, Academic Press, New York. [R] Rosenberg, S.: 1997, The Laplacian on a Riemannian Manifold, Cambridge University Press, Cambridge. [Scl] Scott, S.G.: 1995, 'Determinants of Dirac boundary value problems over odd-dimensional manifolds', Comm. Math. Phys. 173, 43-76. [Sc2] Scott, S.G.: 1998, 'Splitting the curvature of the determinant line bundle', Proc. Am. Math. Soc, to appear. [ScWl] Scott, S.G. and Wojciechowski, K.P.: 1997, 'Determinants, Grassmannians and elliptic boundary value problems for the Dirac operator', 291
Letters in Math. Phys. 40, 135-145. [ScW2] Scott, S.G. and Wojciechowski, K.P.: 1999, 'C-determinant and the Quillen determinant on the Grassmannian of elliptic self-adjoint boundary conditions', C. R. Acad. Sci., Serie I, 328, 139-144. [ScW3] Scott, S.G. and Wojciechowski, K.P.: 1999, 'The C-determinant and Quillen determinant for a Dirac operator on a manifold with boundary', GAFA, to appear. [Sel] Seeley, R. T.: 1967, 'Complex powers of an elliptic operator'. AMS Proc. Symp. Pure Math. X. AMS Providence, 288-307. [Se2] Seeley, R. T.: 1969, 'Topics in pseudodifferential operators'. In: CIME Conference on Pseudo-Differential operators (Stresa 1968), pp. 167-305. Cremonese 1969. [Si] Singer, I.M.: 1985, 'Families of Dirac operators with applications to physics', Asterisque, hors serie\ 323-340. [Tl] Taylor, M. E.: 1996, Partial Differential Equations. Basic Theory., Springer, New York. [T2] Taylor, M. E.: 1996, Partial Differential Equations. Qualitative Studies of Linear Equations., Springer, New York. [Wl] Wojciechowski,K. P: 1993, 'Witten's Holonomy Theorem in the NonLoop Case', Preprint. [W2] Wojciechowski, K.P.: 1999, 'The C-determinant and the additivity of the ^-invariant on the smooth, self-adjoint Grassmannian', Comm. Math. Phys. 201, 423-444.
292
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
R E N O R M A L I Z E D T R A C E S AS A G E O M E T R I C TOOL SYLVIE P A Y C H A Departement de mathematiques, Complexe des Cezeaux Universite Blaise Pascal, 63 177 Aubiere Cedex, France E-mail:pay [email protected] ^-function regularization techniques are commonly used by physicists in Quantum Field Theory - e.g. to compute partition functions using ^-function determinants and by mathematicians in geometry and topology - e.g. to define invariants such as the 7) invariant which plays a fundamental part in the Atiyah-Patodi-Singer index or the analytic torsion involving again ^-function regularized determinants. Using ^-function regularization techniques we define renormalized traces of (possibly non trace-class) classical pseudo-differential operators ; these renormalized traces serve as a tool to generalize some well-known finite dimensional geometric concepts involving traces to a class of infinite dimensional vector bundles and manifolds. We express the obstructions preventing a straightforward extension in terms of Wodzicki residues of classical pseudo-differential operators and describe some cases for which these obstructions vanish.
Contents Introduction
295
1 Traces in riemannian and complex geometry 1.1 Traces on vector bundles 1.2 Variation of traces 1.3 Geometric concepts involving traces
299 299 300 301
2 Renormalized limits
308
2.1 Finite parts 2.2 Mellin transform 2.3 Renormalized limits 3 Pseudo-differential operators
308 314 316 318
3.1 Pseudo-differential operators on an open set of Md 3.2 Pseudo-differential operators acting on sections of vector bundles 4 Traces on operator algebras
318
4.1 Some remarkable ideals of the algebras of bounded operators 4.2 Renormalizable procedures 4.3 Traces on algebras of classical pseudo-differential operators 293
322 327 327 330 332
5 Algebraic obstructions
335
5.1 A fundamental property 5.2 Obstructions in terms of the Wodzicki residue 5.3 Further obstructions of algebraic type 6 Geometric obstructions 6.1 6.2 6.3 6.4 6.5
335 341 344 346
Weighted vector spaces Families of weighted vector spaces Weighted traces on weighted vector bundles Some geometry in infinite dimensions Conclusion
References
346 348 350 352 357 358
294
Introduction Traces play an important role in finite dimensional differential geometry. Given a (finite dimensional) Riemannian manifold (M, g), traces arise in different concepts such as (see e.g. [GHL], [J], [BGV], [CDD], [N]...) (i) the divergence of a one form given by minus the trace of the covariant derivative 6a := —tr(Va), where V is the Levi-Civita connection, a a one form on the manifold M, (ii) the Laplacian on functions given by minus the trace of a Hessian A / := -tr(Hessf) = —tr(Wf), where / is a C 2 function, V the LeviCivita connection, (iii) the Ricci curvature given by the trace of an operator involving the curvature, Ricc(U, V) := tr(Q(U, •), V), where fJ is the curvature of the LeviCivita connection, U, V two vector fields on the manifold. In complex geometry, they arise in a fundamental way when building characteristic classes using invariant polynomials (see e.g. [BT], [BGV], [KN], [N]....) For instance the j-th characteristic class on a complex vector bundle is defined using the 2j form ir($V) in terms of any connection V on the bundle, fi denoting the corresponding curvature tensor. These lead in turn to the Chern character tr(e~n) and its de Rham cohomology class. Our goal here is to try to understand how some of these concepts may be extended to an infinite dimensional framework, namely a class of Frechet, resp. Hilbert manifolds and vector bundles [L]. Among the manifolds we shall consider are loop groups (see e.g. [PS]) which are manifolds of maps on the circle with values in a Lie group. They offer an interesting but yet tractable example of infinite dimensional manifold and some of their geometric properties have been studied extensively in the literature (see e.g. [DL], [F], [Pr], [PS], [SW]). A first step towards the generalization we are aiming at is to extend the notion of trace on the algebra of matrices to a class of operators acting on a class of infinite dimensional spaces, namely to the algebra of classical pseudodifferential operators [Sh], [T] on some given closed manifold. Since we do not want to restrict ourselves to trace-class operators for which there is a welldefined notion of trace, we build these "traces" from the finite part [H], [Sc] of otherwise divergent expressions. When the operators happen to be traceclass, they coincide with the ordinary trace. Here we use the terminology "trace" abusively since there will be obstructions to the tracial property tr[A, B] = 0 which can be expressed in terms 295
of a Wodzicki residue [K], [MN], [0], [W]. To compute this obstruction we use methods developped (independently) by Kontsevich, Vishik [KV1, 2] and Okikiolu [O]. Typically such algebraic obstructions arise as cocycles [Mi], [MN] in the Hochschild cohomology on the algebra of classical P.D.O.s. These extended "traces" are defined with the help of some extra data, an elliptic operator and their dependence on the choice of such an operator can be measured (once again) in terms of a Wodzicki residue. This gives rise to the notion of weighted vector space ( r ( M , E),Q), namely a Frechet or Hilbert space T(M, E) of smooth or Sobolev type sections of some (Riemannian/ hermitian) vector bundle E based on a closed Riemannian manifold M, combined with a positive self-adjoint elliptic operator Q acting on T(M, E) (or possibly only on a dense subspace of that space). A first rather trivial example is the space C°°{S1,lRn) of smooth maps on the unit circle S1 with values in Mn, in which case the manifold M is the unit circle and the vector bundle E is the trivial bundle S1 x Mn. A natural weight would be the Laplace-Beltrami operator [R], [G], [BGV] on S 1 . A slightly more sophisticated example is that of a space Hs{M,E),s > 0 (in fact s should be larger than dl™M if we want continuous sections) of Sobolev type sections of a finite rank vector bundle E based on a closed Riemannian manifold M. If E is equipped with a connection V, then there is a natural weight given by the associated generalized Laplacian A := —ir(VV). Given a weight Q on T{M, E), we can build weighted traces on the algebra CL(M,E) of classicalP.D.O.s acting on C°°(M,E). This carries out to families of vectors spaces {T(Mb,Eb),b € B} in the above class parametrized by a manifold B, if equipped with a family of operators {Qb,b € B} which are locally invertible admissible (this notion is recalled in section 3) elliptic operator. In that setting we define a notion of weighted trace {trQb ,b 6 B} as a bundle map on the bundle based on B with fibres given by algebras {CL(Mb,Eb),b G B} of classicalP.D.O.s. This construction makes sense provided we require that the transition maps of the infinite rank bundle £ with fibre T{Mb,Eb) be elliptic operators. With this at hand, we can introduce the concepts of weighted Laplacians, of weighted Ricci curvature, weighted first Chern form... defined -in a similar way to the finite dimensional case- as (minus) weighted traces of Hessians, weighted traces of operators involving the curvature... We can also extend characteristic forms tr(Q?) to this infinite dimensional setting naively replacing the trace by a weighte'd trace. Classical results of Riemannian geometry such as Weitzenbock type formulae 296
or classical constructions in the complex setting such as the Chern-Weil construction do not go through to this infinite dimensional setting in a straight forward manner since obstructions of geometric type can arise which again are described by a Wodzicki residue (see [ABP],[Pa], [PR]). However, as we shall show in section 6, these classical results can be generalized up to some extra Wodzicki residue type term involving the commutator [V, tr®] where V is the connection and trQ the weighted trace; in the finite dimensional case, the connection "goes through" the trace so that such a commutator does not arise. In particular, when defining the weighted first Chern form on a weighted complex vector bundle equipped with a connection as the weighted trace of the curvature, unlike in the finite dimensional case, one does not expect it to be closed in general. Only in very special cases, as in the case of loop groups and given a good choice of weight (this corresponds to the two step trace of [F]), can one check that the obstructions vanish [CDMP], [Ma], [Pa]. These obstructions being rather cumbersome, we might want to circumvent them introducing some compensating terms. This can indeed be done when the weight arises as the square of a family of Dirac type operators using tools such as trace forms familiar to non commutative geometers but it would be out of the scope of this article to describe such a procedure in detail (see [PR], [Pa] for further investigations along these lines). We hope that these lecture notes - through the question we raise of how to extend classical geometric concepts to the infinite dimensional setting - beyond the relevence of this problematic, will offer the reader a different outlook on some classical and therefore maybe well-known to him/her basic concepts of Riemannian and complex geometry. Along the way, our approach to this problem brings in as well as some geometric tools mentioned above, various algebraic tools (such as algebraic cocycles in section 5), analytical tools (such as finite parts and pseudo-functions in section 2) and functional analytical tools (such as renormalization procedures in section 4), which we feel are interesting for their own sake. There are yet many other ways of approaching the question of how to extend classical geometric concepts to the infinite dimensional setting, one of which is to focus on measures rather than on traces. A natural thing to do from this point of view is to replace the Riemannian volume measure by Gaussian type measures in infinite dimensions, this leading - in the case of path manifolds - to some fascinating constructions on Wiener space. This second approach raises the question of how to construct a Laplacian which is self-adjoint, a problem which in our context does" not make sense since there are no measures around and hence no notion of self-adjointness. However, the two approaches do have 297
some fundamental common feature. Namely elliptic operators arise in both constructions; here, the Laplacian that underlies the construction of Brownian motion has been replaced by the weight Q.
298
1 1.1
Traces in riemannian and complex geometry Traces on vector
bundles
In this section we recall briefly some basic concepts of Riemannian and complex geometry involving traces. Our presentation, which might in some places seem a little unusual to the reader, is organized in view of generalizing the concepts to the infinite dimensional setting in section 6. A trace on an algebra A over a field K (in practice K = JR or K — <S) is a /^-linear map tr : A —> C with the tracial property: tr([a, b}) := tr(ab -ba) = 0 Va, b € A.
(1.1)
A well-known example is the algebra A = M(K,n) of n x n square matrices with coefficients in K and tr(a) := Y17=i a " ^ a = (a»i) ^ AAs a consequence of the tracial property, given an invertible element c of A, we have tr(a) = tr{c~1ac) Va S A. Let now n : E —¥ X be a (smooth) finite rank jftf-vector bundle based on a (smooth) finite dimensional manifold X. A vector bundle morphism of •K : E -¥ X over the identity of X is a fibre preserving map that is linear on each fibre; i.e a map u : E —> E such that TTU = IT and such that for any x € X, ux € Hom(Ex) where Ex denotes the fibre above x. Let Hom(E) denote the homomorphism bundle of E with fibre Hom(E)x — Hom(Ex). Given a local chart ([/,
-»• 4>(U) x Kd
(x,vx) - • (
-+(x,tr(ux))
(1.2)
defines a bundle morphism which we shall refer to as the trace on the bundle Hom(E).
299
1.2
Variation
of traces
+
Let ut,t € M — {0} be a smooth one parameter family of matrices in M(K, d), then the map t ->• tr(ut) is also smooth and we have j^tr(ut) = tr(-^ut). More generally, if n : E —> X is a trivial vector bundle over X, and if x —> u x is a smooth section of Hom(E), then dtr(u) = tr(du).
(1.3)
Let V be a connection on a (finite rank) vector bundle n : E —>• X. It induces a connection V f f o m = V* ® 1 + 1
tr{d{*lu))
using (1.3) lt
(1.4) t
=
tr{VHomu).
Hence the connection V commutes with the trace, namely [V, tr]u := dtr(u) - tr([V, u]) = 0 Vu 6 C°°(Hom{E)) 300
(1.5)
where for a bundle F, C°°(F) denotes the space of smooth sections of F. We have used two ingredients to prove this elementary result, a purely algebraic property, namely the tracial property of the trace c.f. (1.1) combined with an analytic property, namely the fact that the trace commutes with the ordinary differentiation c.f. (1.3). 1.3
Geometric
concepts
involving
traces
We refer the reader to e.g. [BGV], [GHL], [J], [N] for the concepts in Riemannian geometry needed here. We also refer the reader to the lecture by O.Hijazi in this volume. We shall focus on some basic concepts involving traces. Let (X, g) be an n dimensional smooth Riemannian manifold equipped with the Levi-Civita connection V. • T h e divergence: Given a one form a on X, the tensor V a is a section of T*X
(1.6)
Exercise: Show that n
n
8(a) = -^2eida(ei)
+ ^a(Ve;ei)
where (e;) is a local orthonormal frame. The operator 8 coincides with the formal adjoint d* of the exterior differential d : C°°(X) ->• C™{T*X) for the scalar product induced by the metric i.e {df, a) = (/, d*a) where for two smooth functions f,fonX the scalar product is given by (/, / ) := Jx dvol(x)f(x)f(x) and where for two one-forms a, a on X the scalar product is defined by (a,a) = / dvol(x)g^(x)ai(x)aj(x)
— V* /
.-_, Jx
Jx
301
dvol(x)a(ei(x))a(ei(x))
for any local orthonormal chart (ei(x)) and where (gV) is the inverse matrix of (gij). Here and thereafter, we shall often make use of the Einstein summation convention omitting to write the summation symbol which is subintended on repeated indices. • The Laplacian: Given a smooth function / , and taking a = df yields the Laplacian A / := 5df = d*df = -tr(Wdf)
(1.7)
It therefore corresponds to minus the trace of the Hessian Hessf
:= Vdf of
/ •
Let us recall that for a fc-covariant tensor T we have: fc (Vt/TXfi, • • •, Vfc) = Vu(T(Vu • • •, Vjt)) - £ ) T O . • • •, Vi-uVuVi,
• • •, Vk)
t=i
(1.8) for any vector fields U, Vj., • • •, 14. In particular for any two vector fields U, V and any smooth function / we have (V 2 /)(C7 ® V) ~ (Vi/V/XV) = VuVvf
-
Vvvvf-
Hence we can write: n
n
- A / = Y, V2/(e< ® e4) = ^ Vei V e ,/ - Vv e , e ,/. i=l
(1.9)
i=l
For two smooth functions / i and /2 on X we have: A ( / i / 2 ) = ( A / i ) / 2 + / i A / 2 - 2tr(dfx ® d/ 2 ). The notion of Laplacian extends to vector bundles as follows. Let E be a vector bundle based on a Riemannian manifold equipped with a connection V s . The Levi-Civita connection V on X combined with thes connection V E yields a connection on V T **® E on the tensor product T*X ® E so that we obtain an operator V T * X ®- B V B : C°°(X,E) ->• C°°(X,T*X ® T*X ® £ ) . In other words v r * x ® B V £ ; € C°°(X,T*X ® T*X ® Hom{E)). Identifying TX with T*X via the musical maps, given a section a of E we can therefore view V B ® T * X V E (T as a map from T X to TX ® i? and we can define: ABa
-tr(VT'x®EVEa)
:=
= -^(VT*x®BVBa)(ei®ei) = - £ ( V i ™ * V j=l
i=l
for any local orthonormal frame (ei)ilt...tn 302
of T X .
e
» + Vfeieia
Remark: The sign convention sometimes vary from one text book to the other; the one we choose here leads to a positive operator (exercise). • The Ricci and scalar curvature: Prom the curvature Q associated to the Levi-Civita connection V, one can build the curvature tensor R € C°°(T*X®4) defined by R(u,v,w,t) = (Sl(u,v)w,t) where (•, •) is the scalar product induced by the Riemannian metric. For fixed vector fields u, v, R(u, •,«,•) defines a symmetric bilinear form on each tangent space and the Ricci curvature is the symmetric 2-tensor 71
Ricc(u,v)
:= (tr2,iR){u,v)
= 2_J.R(u,ej,i>,e;).
(1-10)
The scalar curvature is the trace of this two tensor: Scal = tr(Ricc).
(1.11)
The Bochner-Weitzenbock formula relates the Laplacian on functions to the Ricci curvature on a Riemannian manifold: A Bochner-Weitzenbock formula: ([A,V]f)(V)
= Ricc(V,Vf)
Vf eC°°{X,<S),W
tC^iTX)
(1.12)
Proof. Let us first recall that for any C 2 function / on a manifold X equipped with a connection V without torsion, for any point x £ X, the operator Hessf(x) is a bounded symmetric (with respect to the scalar product induced by the Riemannian structure) operator acting on the tangent space TXX at point x. This follows from the fact that V2f(U
®V-V®U)
= WuWvf
- V v „ v / - Vy Vuf +
Vvvuf
= \Vu,Vv]f -Vv„v-Vvuf = [Vf/,Vi/]/-V[C/,v]/ = 0 where we have used the fact that [U, V] = VVV torsion. Let V € TXX and let (e*, i = 1, • • •, dimX) point x.
— VyU because V has no
be a local orthonormal frame at
([-A,V]/)(V) = ( - A V / ) ( V ) + V v A / = ^2 V 3 / ( e i ® et <8> V) i
303
Vvtr{Hessf)
= J2
v3
/ f e ® v ® e*) - M W V 2 / )
i
where we have used the fact that
Hessf
is symmetric
= J2 ( v 3 / ( e * ® y ® e 0 - V3f(V ® e i ® <*)) = ^V2V/((eiAV)®ei) = E(Ve.AvV/)ei = 5Z(n(ei,V)V/,ei) = -iJ*cc(V,V/).
•
First Chern form on a vector bundle: Let £ be a smooth complex vector bundle of rank d based on a Riemannian manifold X equipped with a connection V s . The curvature tlB lies in C°°(T*X <8> T*X
:=tr(flE(U,V))
V U,VeC°°(TX)
(1.14)
where the trace is taken on the local matrix representation of the curvature tensor. This definition is of course independent of the choice of local chart. Equivalently, if E is hermitian we can write: n
rfl(U,V)
'£l(nE(U,V)ei,ei)
= i=i
where (e») is a local orthonormal frame, i.e. a local section of E
0(E).
E
It follows from the Bianchi identity [\7 , fl ] = 0 and from (1.4) that the first Chern form is closed. Indeed we have: drE
=dtr{nE) = tr([VE,nE}) = 0.
Thus we can define a de Rham cohomology class: E
Cl(E):=[r
}GH2(M)
called the first Chern class. It has the following properties: 304
(1.15)
(i) It does not depend on the choice of the connection and is a topological characteristic of the bundle (ii) Naturality. Given a bundle morphism / , Cl(rE)
= rc1(E)
where f*E is the pull-back of the bundle E by / . (hi) c1(E) = c1(E')
iiE~E'.
Let us now generalize this construction to other invariants involving the curvature for which we shall give a proof of the independence of the choice of the connection for the corresponding cohomology classes, thus proving (i) as a particular case. • Chern-Weil invariants: It also follows from the above considerations that any polynomial P of degree k on gln(K) gives rise to a well-defined form tr(P(ftB)) for we have (using again the cyclicity of the trace) tr(P(¥flB))
= tr (C (&fLB) C r 1 ) = t r ( P ( * » n B ) )
with C := \&_1 o $ G GLn{K). Let us consider the case of a monomial P(M) :— Mk,k G IN, since the case of a general polynomial then easily follows taking linear combinations. Since tr{P{C~1MC)) = tr(P(M)) for all C € GLn(K), setting C := etB,B G gln(K) and differentiating w.r.to t we find it
'52tr(P(A1,---,Ai-1,[B,A&An.u--,Ak))=0 \/B G gln(K),Aj
€ gln(K),j
= 1, • • •, k.
(1.16)
Here P(Ai, • • •, Ak) := A\ • • • Ak is the polarized fc-form corresponding to P. Let E be a, vector bundle of rank d equipped with a connection V s . Let fiE be the corresponding curvature and let P be a polynomial on the algebra of d x d matrices. Then the following classical result holds: dtr(P(nB))
=0
(1.17)
and the de Rham class of P(ClE) is independent of the choice of the connection Proof. As before, we can* restrict ourselves to a monomial P(A) :— Ak. Let cti, • • •, ak be 7?om(£:)-valued forms on X and let V s = d + 6B be the local 305
representation of a connection V s in some given local trivialization (£/, $, <j>). Applying (1.16) to B := 6E, Ai = $*ai we can write: dtr (Pi&ot!,
fc • • •, *»a fc )) = ^2 tr(P(¥au-
• •,<*(*»<*<) , • • •,
¥ak))
i=l
fc
= 5 3 t r ( P ( # » a i , • • -,<*(*<<*), • • •
Mak))
i=i fc
+ 5]tr(P($»a1,-",[flE>$»ai],---,*»afc)) i=l fc
= I ] M ^ ( « i , •' •. [ V E , a i ] , • • •, ak)).
(*)
i=i
Applying (*) to a* = fi£ Vi =,l---,k and using the Bianchi identity [ V s , ftB] = 0 yields that dtr{P{QE)) = 0 and hence that tr(P(QE)) is closed. Let us now check the independence of the corresponding de Rham class of the choice of the connection. Let V s and V B be two connections and let E - \>E, V f := tVE + (1 - t)VE = VE + trj. It is easy to check n := V that fif := VEVE = QE + t[WE,rj] +t2r) At] and from this to deduce that •§-tVlE = [Vf ,rj\. The following holds: |*r(P(fif))
=^*r(P(fif,-",nf))
i=i
= 2M^(«f.-",[vf)T/],nf,--.,nf)) j=i
where
\Vf,rj\
is at place j
= X)(tr(p(nf l ..-,[vf > i7],nf,---,nf)) + X;'r(^(«f,-",[vf > nf],--- > i7 > -",nf,-..,fif)) where
[Vf,rj], [Vf, 77] are at place j 306
= £dir(mV-,r?,nf,-..,nf)) where
77
is at place j .
To establish the last equality we applied (*) (replacing V s by V f ) to on = Of Vi ^ j and a, = r\. Hence ir(P(Of)) - ir(P(fif)) = ^ Jo S?"=i dtr{P(Vlf, • • •, 77, Of, • • •, ilf))dt is an exact form, which ends the proof. Using these polynomial invariants, we can also define the Chern character [a]
E
Ch(V ) = tr^)
:= £ t ^ t r ((ft')*) fc=0
(where n is the dimension of the underlying manifold) which is again a closed form, the cohomology class of which is independent of the choice of the connection. We recover the first Chern form introduced above as minus the term of degree 2 of the Chern character of the bundle: r f = tr(nE)
= tr((VE)2)
= - [tr{e-n*)][2]
= -Ch(VB)[2].
(1.18)
We refer the reader to [BGV],[CDD], [KN], [N] ...for a more detailed account of these concepts.
307
2
Renormalized limits
In this section, preparing for the notion of renormalized traces to be described in section 3, we recall the notion of finite part of an otherwise diverging function. In contrast to other sections, this one might seem rather technical and the reader might like to skip it at first reading. I however strongly encourage her/him to come back to it later since renormalized limits are an essential tool in various approaches to Quantum Field Theories. The main reference used in this section is [Sc] but other more recent references on related issues are e.g. [EF], [EK] and some mentioned therein. I also refer the reader to the lectures by K. Wojciechowski and E. Langmann in this volume where similar renormalization techniques are used. Let me finally address my thanks again to Marc Arnaudon at this point since it was some informal discussions with him that served as the initial motivation to write up this section. 2.1
Finite
parts
In this section we consider a (complex valued) measurable function g on ]0,1[ such that g e L}oc(]0,1]). Hence g € Lx(\e, 1[) for any e > 0 but g might not belong to L1(]0,1[). We make the following Assumption: There is a finite subset J of IN, complex numbers b, aj, and Xj indexed by J and there is a function h £ L1(]0,1[) such that: g(t)
= Y^ aJrXj
+
b r l
+ h(t)
(2-1)
jeJ
with Re(\j)
> 1, Xj # 1 Vj <E J.
• Finite part of an integral: The notion of finite part was first introduced by Hadamard (see [H], [Sc] for the history of the concept) and then further investigated by Schwartz [Sc]. Under assumption (2.1), for any e > 0 we can write:
r1
a
/ g{t)dt =Y^ J Je j€J
r
i
L-Aj+1
*~
Xi+1
l1 J£
j€J
~bloge+
JfcJ
= I(e) + F(e) 308
r1 h
h(t)dt
where I{e) := - E j e J ^+i^Xi+1
- ^loge and F(e) := £ . £ J ^
+
J h(t)dt. When £ tends to zero, ^(e) converges to a finite expression:
f.p. £ g(t)dt := ]T —^— + f1 ft(*)d* = lim ^
,(«)* + £
z ^ " ^
1
+ Mogej
(2.2)
called the finite part of the integral J0 g(t)dt whereas 1(e) diverges. Notice that whenever g e i 1 (]0,1[), then f.p. f0 g(t)dt = JQ g(t)dt. We leave to the reader the easy exercise to prove that f.p. JQ : L]OCQO,1]) —» C defines a linear map. • Behaviour under a change of variable: The finite part of the integral of a function satisfying assumption (2.1) is not in general invariant under a change of variable in the integral. Let t —>• u(t) be a smooth map from [0,1] to [u(0) = 0, u(l)] which is a diffeomorphism on ]0,1[. Let u —> t(u) denote its inverse; we want to check that j : « i - > g(t(u))t'(u) satisfies condition (2.1). k )
Let fc0 be the first integer for which # » ) (0) ^ 0, then t(e) = Y,k=k ^
o(e«) = ^ 3 ^ ( 1 + E £ i * e
k
k0\
J€J
v M ^ . ^ + °(cK-k°))
-d
(°) +
'
fco!#+fc°)(0) + o(u K - fe °) (fc + *o)!t(*»)(0)
k=0
^»*-'''£ OL/rffW-^* (fc + fco)! <( fc o)(0)
(*b - 1)!
\ fc=o f( fc °)(Q)^o fK-ka
1 E ^ ^ L , , :^^^-) (fc + fco)! *( :°)(0) fc=i
+ +
+
fc
k0b fe t( o)(0)
,—ko
^ °
+ h(u)
f(*o)(Q)u*°-1 / ^ °
^ fc
fc
fc0!
f(fc+fc°)(Q)
" (* + *<>)! <(fc°)(0)
v-Av + o(u -
fc0! *(fc+fc°)(Q) + o(u K - f c °) (fc + fco)! t(k°)(0) 309
K fc0
+
)
W
[£ri
+0{u
(fc +fco)!t(*o)(o)
\
for some /i € -^QO, 1[). ^From this (rather complicated !) expression, one can see that the term u~h°'Xi gets multiplied by terms of the form ul,i G IV this giving rise to terms of the type u~k°'x'+l,i £ JN which are compatible with condition (2.1) provided —ko • X3•, — i = — 1 or ko • Re{Xj) — % > 1. This single out the following cases: Case fc0 = 1 :
i)Xj G IV, ii)Re(Xj) g IV, iii)Re{Xj) G IV
Case fc0 7^ 1 :
i)Xj G Q,
ii)Re{X3) $ Q,
iii)Re(Xj)
and ImXj j£ 0.
G Q and ImXj ^ 0.
In cases i) and ii) the map g satisfies condition (2.1) but not in case iii) which we shall therefore leave aside. It follows from (2.1) that Mi) / g(t(u))t'(u)du Je
Mi) = 2J / jeJ Je
Oj(t(u))" A ^'(u)du
Mi)
Mi)
t'(u)t(u)-1du+
+b
h{t{u))t'{u)du
= £ -ZTTi rXi+%) + *P°g*]Jw + f Kt)dt. we have: u(l)
/
g(t{u))t'{u)du
= h{e) + Fl{e)
where -Aj+l
-Aj+l
*~*°
fco!
^
(
0
)
^
,
and where
j G J
-v • -
Jt(e)
Let us first assume that t'(0) ^ 0 so that &o = 1. 310
-
^
(i) If Re(Xj) £ IN for any j e J and 6 = 0 then ii(e) leads to a divergent expression when e ->• 0 and we have: f.p. ( V * ^ g(t(u))t'(u)du
= lim I /
j := J2 r j ^ y + ^
M*)*
3(i(u))t'(u)du+
£ ^rtr £ - A ' + 1 C(o))~A'+1 (i+«(C*' +1 ) = /.p. / /(t)d* Jo
(2.3)
where we have set a(e) := ^2kS2
H^'WOV
~*~ ° ( £ i f _ 1 ) -
(ii) If -Re(Aj) ^ IV for any j G J and 6 / 0 then the expression h(e) gives rise to an extra finite term in the limit e —> 0:
^
fl(t(u))t'(u)d«
= lim ( / " ( 1 )
ff(t(«))f
J := J ] - ^ - y + Jo h(t)dt - 61og(t'(0)) (u)du + £
— ^ — e - ^ + 1 (t'(0))-Ai+1 •
•(l + a ( £ ) ) - ^ + 1 + 6 1 o g £ j
(2.4)
Comparing (2.4) with (2.2) we see that f.p. ( £
' g(t(u))t'(u)du
j - /.p. (J'
g(t)dt\
= -61og(*'(0)).
(iii) If there exists jo £ IV such that AJ0 6 IV, say Aj0 = n0 e IV then the divergent term in e~no+1 multiplied with the term in £ n o _ 1 arising in the expansion (1 + a{e))-n°+1 = 1 + (1 - nQ)a(e) + ( " a " 1 1 " n a(£) 2 + • • • gives rise to an extra finite term in the limit e —)• 0, thus adding an extra term to the finite part. Let us now consider the case when t'(0) = 0 i.e fco > 1. We also distinguish three cases. 311
(i) If Re(Xj) £ 0 and we have:
f-P- if
g{t{u))t\u)du\ = f-P-f f(t)dt.
(ii) If Re(Xj) ^ Q for any j € J and 6 ^ 0 then the expression Ji(e) gives rise to an extra finite term in the limit e —> 0 and we have: f-P- (f
1
g(t(u))t'(u)du\
-f.p. f1
f(t)dt = -61og(£<&°>(0))+61og(fc0!).
(iii) If for some jo G J, Xj0 € Q, then for any fc0 such that (—A,0 + l)ko £ Zfi, a compensation arises between the divergent term g(_Ai+1)fco a n ( i a term in the expansion of of (1 + a(e))~Xi+1 thus giving rise to an extra finite term. Hence only if Re(Xj) £ Q for any j £ J and 6 = 0 (none of the powers in the asymptotic expansion (2.1) are rationals) is the finite part invariant under a change of variable: f.p. ( /
g{t{u))t'{u)du)
\Ju(0)
=f.p. )
[
g(t)dt.
JO
Our conclusion therefore differs from [Sc] where it was asserted that the invariance under a change of variable held true for all g of type (2.1). • A meromorphic extension: For z 6 € with Re(z) > 0 and large enough, the function G(z) := JQ g(t)tzdt is well defined. Indeed, let Re(z) > sup(Re(Xj)) - 1 or Re(z) > 0 if J is void, then:
G{z) = [ g(t)tzdt Jo = V a j / tz~x'dt + b f tz~1dt+ f h{t)tzdt •eJ
Jo
Jo
Jo
" E r ^ T T + j+jfAW" is well defined. It extends-to a meromorphic function on the upper half complex plane Re(z) > 0 with a finite number of poles {0, Xj— 1, j G J } . Let us focus on the pole at zero. 312
The case 6 = 0: since A_, ^ 1, the function z ->• G(z) extends continuously to 0 and we have: G
+ f h^dt = f-P- I 9(t)dt
(°) = E ~ T V T
(2.5.a)
~xi + l Jo Jo In other words, the finite part arises as an analytic extension of an ordinary integral. jeJ
The case 6 ^ 0 : we first need to extract a (simple) pole at zero before taking the limit at z = 0: lim (G{Z) - -)
= Y) ~^-7+
f
Ht)dt = f.p. f
g(t)dt.
(2.5.6)
• P s e u d o - f u n c t i o n s : We now assume that 6 = 0 and that none of the Xj,j G J are integers. Then for any n 6 If, in a similar way as in the case n = 0, we can define G n :=
( )
E
1TXT
+
/ h^edt
= f-P- I 9(t)tndt.
(2.6)
j^n-Xj + 1 J0 J0 Let <j> be a smooth function on ]0,1[. The Taylor-Lagrange formula at order K yields: K
<j>{t)
= E ^r1^ + k=0
t
-wrl!{1-
u K (K+1) du
)* ^
so that choosing K = [supj€j(Re(Xj))], the integral J0 g(t)~j- JQ(l — u)K(p(K+1)(tu)du is well-defined. Moreover the map t -> g(t)(j>(t) satisfies condition (2.1) and hence it follows from the linearity of the finite part of an integral that:
f.p. j * g(t)4>(t)dt = J2 ^r-f-p-
( ^ g{t)tkdt^
l!9{t)t~^r ((( 1 - u )V + 1 ) w<*<) • i,From the above f.p. f0 g(t)tkdt is tegral so that the the map
(2.7)
discussion, we know that provided Re(Xj) ^ Q and 6 = 0, independent of the choice of parametrization inside the inintegral symbol on the l.h.s. of (2.7) makes sense. Moreover fQ g(t)cp(t)dt is linear and continuous on C°°([0,1]) for the 313
topology of uniform convergence of all derivatives. Furthermore for some large enough integer K there is a constant C > 0 such that \f.P. [ 0(iMt)dt|
Mellin
transform
[BGV]
Recall that T(z) = /0°° e~tt~zdt is a meromorphic function with poles at the negative integers. Moreover T(z)~l = z + jz2 + 0{z3) where 7 is the Euler constant 7 = limn_*.oo(logn - Y^kH £)• Unlike in the previous section where we considered a finite interval, here we will be working on ]0, +co[. We shall therefore need some condition on the functions at infinity in order to have well-defined integrals. Let us consider functions / 6 C°°(]0,oo[) such that: (i) 31 > 0, 3C > 0, such that \f(t)\ < Ce~xt for large enough t (ii) There are positive integers N,q ^ 0 and real numbers a, aj,/3j,jj j g J V such that 00
00
f (6)^0^2^^^+
00
E
/ ^ ^ l o g £ + 5 > ^ - (2-8)
j=o,'-"-N zzc
3=0
with
i=°
Note that when a is an integer, there is a redundancy in this expression since constant terms can arise as 70 or as aa+N- By the symbol ~ we mean that for any J £ N, 3Kj := [a] + qJ + N € IV such that Kj
._ _N 1
/(e) = J2 a / ^ 3=0
Kj
+
Kj
_ _N
S
Pje'^logp
+ E 7j£j + o(eJ).
j=o,>-°--"ez
i=°
For such functions the Mellin transform (see e.g. [BGV]) M(f)(z)
1 r°° := = p r / mt'^dt 1 z \ ) Jo 314
(2.9)
is well defined for Re(z) large enough. It is useful to notice that for A > 0 we have: -1
\~z = M(t -> e~tx){z)
fOO
tz-\ -tx dt.
= Tq-^ L\Z)
J0
Let us show that M(f) extends to a meromorphic function with poles in { N+a~3; j g .ffV}. Provided Re(z) is chosen large enough we can write: pl
/•CO
z 1
T(z)M(f)(z) = j f(t)t - dt + J Kj
f(t)tz~ldt
Kj
„i
ri
log t dt J
/*1
J
tj+z~1dt
+ Y"7j / ,-_n 3=0
q
/>O0
E 3-Q.-N a,i
Z^
(2.10)
Jl J
a
V^ sr^ ,
/(i)t* - 1 d<
+
JO J
a
*-
o(tJ)tz~1dt
+
•'O
Kj
'
fl
Pi (j-a-N
^, vV^ ^ _ TJ ,
N2
Z ^ j _|_
+ z
CO
R(z) + / /
f{t)tz-ldt
(using an integration by parts to compute the integral involving the logarithmic terms) where z —> R{z) is holomorphic for Re(z) > ~Kf+l. The map z —> f± f(t)tz~1dt is analytic in z since f(t) decreases exponentially at infinity. Hence the map z -¥ T(z)M(f)(z) is meromorphic with poles in the set {N+g~3,j G IV}. Since T _ 1 is analytic, the map z ->• M(f)(z) is also meromorphic with poles in the set { N + a ~i ; j g iV}. For our purposes, it is useful to know how the Mellin transform behaves at zero. It only has a simple pole at zero and: M(f)(z) + 0oz~l = aa+N + 70 - 7 / W + O(z) = constant term — 7 • (coefficient of the logarithmic term) This follows from (2.10) combined with the fact that T(z)-1 where 7 is the Euler constant. 315
(2-11)
= z + jz2 + 0(z3)
2.3
Renormalized
limits
[BGV]
• Renormalizable functions: We shall call a smooth (complex valued) function / on C°°(]0,oo[) renormalizable whenever it satisfies conditions (i) and (ii) of (2.8). Since the poles of the function M(f) lie in { +^~}, j € IV}, if a is not an integer, then 0 is not a pole and we can define M(f)(0). However for a renormalizable function / of the type (2.8) with a € 2Z, one expects a pole in zero and we need to "extract divergences" before taking the limit. On the grounds of (2.11), for f j g J R w e define the fx-renormalized limit of / at 0: LIMj^0f(t)
:= aa+N
= constant term
+ jo - fi/3a+N — // • (coefficient of the logarithmic term)
(2.12)
Prom (2.11) it follows that: lim(M(/)(z) + p0z~l)
= aa+N + 70 - lPa+N = LIM^0f(t).
(2.13)
Notice that if a is not an integer, then po = 0 and LIM^0f{t)
= limz^M(f)(z)
=
M(f)(0).
• A class of renormalizable functions: Let g be of the form (2.8) with Pj = 0 for all j € N. We set Xj := n + ^ - J , Then /(e) := /e°° g(t)dt satisfies assumptions (2.8) and /(e) ~ 0 A) - S A ^ I -\?+i£~Xj+1 ~ aq(a+N) l°g£ f° r some constant A0 = E A ^ I =%f+i+JO [9^ ~~ ^ # 1 ajt~x'dtj + f™ g(t)dt. Since g G Ll(\e, +oo[) for any e > 0, we can extend the notion of finite part defined in (2.2) for the integral J0 g(t)dt on ]0,1[ in a straightforward way to the integral J"0 g(t)dt on ]0, oo[: /*oo
f.p.
/ Jo
rco
/*1
g(t)dt
:= f.p. / g(t)dt + / Jo Ji
= E -irzi
g(t)dt
+ / ^(t) - £ ajt-^dt + / g(t)dt
and we have Ao = f.p. J0°° g(t)dt. As before f.p. /0°° g(t)dt is invariant under a change of variable if Re(Xj) ^
LIM^0f(e)
= f.p. / Jo 316
g(t)dt +
tia{a+N)r
In particular, taking /z — 7 and using (2.13) yields: /•OO
lim (M(f)(z) z ^°
- a ( o + J v ) 9 z - 1 ) - f.p. I Jo
g(t)dt +
ja(a+N)q.
Exercise: Let g(t) := -s-^—, A > 0 and fe(t) := - / e ° ° s-j—dt. Check that LIM"*(f) = logX = - C A ( ° ) w h e r e w e h a v e s e t CA = M(t -> e~tx). Let us summarize where we now stand. Proposition: For g satisfying (2.8) (i) and (ii) with a £ Z then: (i) the finite part f.p. (J"0 g(t)dt) is invariant under a change of variable, (ii) 4> —> JQ g(t)4>(t)dt defines a distribution of finite order, (Hi) z —• M(f)(z)
has no poles.
Proof. This is a direct consequence of the above discussion since a $. Z, => 0j = 0 and Re(Xj) = -J+°+N <£ Q Vj € JV.
317
3
Pseudo-differential operators
This section gives a brief presentation of the basic tools in our framework, namely classical pseudo-differential operators and particularly elliptic ones, their logarithms and their complex powers. Classical references are [GJ, [LMJ, [St], [Sh]. We also refer to [D] for a detailed presentation of the tools briefly described in this section. An ordinary partial differential operator of order m G IV is a polynomial expression
\a\<-m
where Pa G C°°(IRd) and D% := H ) w § ^ - • •• + ! £ . Here |a| = a x + • • • ad. It can also be written: Pu(x)=
[
e te -«
where
jRd
= ^ pa(s)f |«|< m
and where • denotes the canonical scalar product in M . Allowing more general expressions for a(x,£) leads to the notion of pseudo-differential operator. 3.1
Pseudo-differential
operators
on an open set of Md
• T h e symbol set: Let U be an open subset of Md. Given a G M, let us denote by Sa(U) the set of real valued smooth functions
u x md -> m with compact support in x and satisfying the following property. Given any two multiindices 7 = (71, • • • .7d) and 5 = (81, • • • ,64) in Nd , there exists a constant Clt$ such that \DZDl
< C 7 , 4 (1 + m\)a-lSl
Vz G U,V£ G JRd
(3.1)
where we use the symbol || • || to denote the norm on M . An element of Sa(U) is called a symbol of order a. Notice that the above condition is a requirement on the behaviour when ||£|| goes to infinity of the expression (l + U\\)\s\-"\D2Dl
valued symbols but also to matrix valued symbols U x Rd -> (x,£)
Mrur2(M) -KT(X,£),
the components of which are symbols of order a as defined above. Exercise: Let A := - E t = i •&?• S h o w t h a t CTA(Z,0 := ||£|| 2 on Md is a symbol of order 2 and that 07 1+A )-*(a;,£) := (1 4- ||£|| 2 )~' : is a symbol of order -2k with k£ IN. The principal part of the symbol a € Sa(U) follows:
(or leading symbol) is defined as
lim —75—• t—>+oo t
(3-2)
A smoothing symbol is a symbol in S-°°(U)
:= p | S a (l7 )
and the relation
a-d^a-aZS-^iU) defines an equivalence relation on S(U) := |J
Sa(U).
For a real valued function a on U x iRd with the property that there exist aj £ Smj(U),j € JN with (mj)j&jN a decreasing sequence in .ZT verifying the following condition: J
VN G JV,3JW
e JV,
CT^O-X^^^^)
eS-N(U)
v
J>JW
we shall write: oo
j'=o
A symbol of order a is called classical if there exist aa-j such that: oo
j=0
319
€ Sa~j(U),
j € iV
which are positively homogeneous, i.e
vtem+,vtemd.
(3.4)
Let * € C°°{Md) such that <3> vanishes for for ||£|| < \ and #(f) = 1 for Hfll > | . Then 0-Q_j(a:,£) - *(0<7 a -j(a;,f) belongs to 5 _ 0 0 (C/). Hence for a symbol to be classical, it is sufficient to require that CO
j=o a
with crQ_j(a;,^) € S ~i(U) and positively homogeneous in f. Following Kontsevich and Vishik [KVl], we shall say that a classical symbol lies in the odd-class if it has integer order and the positively homogeneous components cra^j are moreover homogeneous i.e: 0a-j{x,t£)
= ta-j
VteM
V£ e Md.
(3.5)
Polynomial symbols provide examples of symbols in the odd class. A symbol a in Sa(U) is called elliptic if its principal part
and
$(aa - J) € S-°°(U).
(3.6)
All the above mentioned notions extend in an obvious manner to complex or matrix valued symbols. • From symbols t o pseudo-differential operators: Following the prescription described in the introduction for partial differential operators, to a symbol a e Sa(U) we can associate the pseudo-differential operator (P.D.O.) with symbol a defined by: A : C~(17) —• C°°(]Rd) u —> (x ->• Au(x)) where Au(x)=
e*-x
[ d
JlR xU
eii<x-^a(x,Ou(y)dyd^.
f d
JR xU
320
(3.7)
Here C£°(E7) denotes the space of complex valued smooth functions with compact support in U. The leading symbol of A is given by the leading part of its symbol. If the symbol is classical, we shall call the corresponding P.D.O. classical and the set of classical P.D.O.s of order a is denoted by CLa(U). The Laplacian A := — J2i=i ~§£? 1S a partial differential operator of order 2 with symbol
:= p |
CLa(U).
Such an operator sends any function in a Sobolev class Hs to a smooth function and can be represented by a smooth kernel, i.e K G CL-°°(U)
&3k£
such that
C°°(U x 17),
Ku(x)=
Ju
k{x,y)u(y)dy
Vu G C~((7),
Vz G U.
Two P.D.O.s P and Q are called equivalent (and we note P ~ Q) if their difference P — Q is a smoothing operator; one often works with P.D.O.s "up to equivalence". A partial differential operator is local i.e. (u — 0 => Au = 0). But a pseudodifferential operator, because of the smearing produced by the Fourier transform, is not local. However, it has some pseudo-locality properties [G], [LM]: (i) For any e > 0, there is an e-local P.D.O. P £ to which P is equivalent i.e P ~ Pe and such that the support of PE is contained in the e neighborhood {a; G lRd,d(x, U) < e} of U, (ii) If u is smooth on an open set U then Pu is also smooth on this open subset. an Given two partial differential operators P = YLa<Jaix)^x d P = ^2g^p(x)D^, a direct computation shows that the symbol of the product PP reads
a(PP) =
J^D^D^. a
A similar type of relation holds "up to equivalence" for products of pseudodifferential operators, replacing the symbol "=" by the symbol "~":
321
(3-8)
The various classes of symbols introduced previously induce corresponding classes of pseudo-differential operators. An odd-class classical P.D.O. is a classical P.D.O. with integer order with symbol in the odd class, an elliptic operator is a P.D.O. with elliptic symbol. Ordinary partial differential operators with integer order provide examples of (classical) P.D.O.s in the odd class. Prom the characterization of ellipticity of the symbols of pseudo-differential operators by the invertibility of their leading symbol, it follows that a pseudodifferential operator A is elliptic whenever its leading symbol O-L(A){X,£) (which is defined globally) is invertible for any (x,t; ^ 0) € T*M. From the characterization of the ellipticity of the symbols by "invertibility up to a smoothing operator" described above, it follows that the ellipticity of a P.D.O. P can also be characterized by the existence of a P.D.O. P such that both $(PP — I) and <&(PP — I) are infinitely smoothing. We shall refer to this by saying that its localized version is invertible modulo a smoothing operator. For a partial differential operator, there is no need for a plateau function $ and it is elliptic if and only if it is invertible modulo a smoothing operator. From the product rule for symbols of ordinary pseudo-differential operators P and P, it follows that o-L{PP)=aL{P)o-L{P).
(3.9)
This combined with the characterization of ellipticity by the invertibility of the leading symbol shows that that the product of two elliptic pseudo-differential operators is elliptic. 3.2
Pseudo-differential bundles
operators
acting on sections
of
vector
The notion of pseudo-differential operator can be carried out to operators acting on sections of vector bundles. Let ni : E\ -» M, ir2 : E2 -» M be two smooth vector bundles of rank n and r2 respectively over a smooth compact and boundaryless (i.e closed) Riemannian manifold M of dimension d. A linear operator A : C^iM^Ex)
-)•
C°°(M,E2)
acting from the space C0? (M, Ei) of smooth sections of a vector bundle E\ to the space C°°(M, E2) of smooth sections of a vector bundle E2 is called a pseudo-differential operator of order a if it is locally given by a (r 2 x ri) matrix 322
of pseudo-differential operators of order a. Let us make this last requirement more precise. Given a neighborhood U of any point m £ M and any two smooth functions X, x with compact support in U, there are local trivializations:
where r» is the rank of Ei and where (17, <j>) is a local chart on M (so that (U) is an open subset of Std), such that the induced linear map Cc°° (<£(!/), € r i ) -»• C°°(0(£/),
->pr2o$2(x,4x)$rlu
is a P.D.O. of order a. In other words, its symbol which takes values in (r 2 x n ) matrices has components in Sa(
power of the cotangent bundle T*M. We shall denote this section called the leading symbol of the P.D.O. by OL{A). The ellipticity of A is characterized by the invertibility of its leading symbol ai {A). •Elliptic (admissible) pseudo-differential operators: Let E —> M be a finite rank vector bundle based on a closed Riemannian manifold M. We shall denote by Ell*(M,E) the set of invertible classical elliptic operators, by Ell*rd>0(M,E) the subset of invertible classical elliptic operators that have strictly positive order. If E is hermitian, we denote by Ell*0^d>Q{M, E) the subset of operators in EU*ord>0(M, E) with positive leading symbol. Using the fact that (TL(A)*<7L(A) =
(3.10)
where T ^ = {A = reie,r > R},T2,g = {A = Re^,6 >4>> -0}, T3ig = {A = r e i(0-27r) r > ^ i a n d r$ = p M ,j Y2$ y p 3 e H e r e A z = exp(ziog\) w here logX = log\X\ + id on Y\j and logX = log\X\ + i(6 — 2n) on r3>g. This definition is independent of the choice of R but depends on the choice of 6 and yields for any z £
It can be extended to all complex powers of A by the semi-group property Az = Az~kAk for k large enough. When M is Riemannian, E is hermitian and A is essentially self-adjoint, then Azg is independent of the choice of 6 and coincides with the complex powers defined using spectral representation. For 9 = — 7r, we shall drop the explicit mention of the angle 6 writing simply Az. • Logarithms of elliptic operators: For an admissible invertible elliptic operator of order 0, the spectrum is bounded (because the operator is bounded) and does not cross 0 so that it is contained in D(0,R)/D(0,e) for some e > 0 small enough and some R > 0 large enough, where D(0, R) denotes the disc centered in 0 with radius R. Let Lg be a spectral cut and define the logarithm of A by: logg A ••=TT[ 27r
loge^A-Xy'dX
(3.11)
Jru.s.B
where TRtSzg = U»=i ri,fl,e,0 and Tx,R,e,e = {A = reie,e
dzA<
(3.12) z=0
This defines a (non classical) P.D.O. operator of zero order and hence a bounded operator from HS(M,E) to HS~£{M,E) for any e > 0 and any s E 1R. For 9 = —TV, we shall drop the explicit mention of the angle 9 writing simply log A. In local coordinates (a;,£) on T*M, the symbol of logs A reads: viog,,A{%, 0 = ard{A)log\(\Id
+ a classical P.D.O. symbol of order 0.
Although the logarithm of an admissible invertible elliptic classical pseudodifferential operator with spectral cut Lg is not itself a classical pseudodifferential operator, for two operators A E Ell™$™0(M, E), B E EHZrd>o(M,E) admitting spectral cuts Lg and L$: ^-l^ECL\M,E) ordA ordB 325
(3.13) '
v
where CL°(M,E) denotes the algebra of zero order classical pseudodifferential operators acting on sections of E. As a consequence, given Q € Ell*+(M,E) with strictly positive order, we have: logg(A) = —~j^}°gQ +
a
classical P.D.O. of order 0.
326
(3.14)
4
Traces on operator algebras
In this section, we consider a particular class of bounded operators, namely trace-class operators which will play an important role in the sequel. They do not form a closed ideal for the topology of bounded operators so that there is no reason for the trace of a family of trace-class operators to converge when the operators converge in the operator norm. However, when restricting ourselves to another class of operators, classical pseudo-differential operators, we can circumvent this problem extracting a finite part of the otherwise diverging limit of traces of certain families of pseudo-differential operators. Let H be a separable Hilbert space equipped with an inner product (•,). Let B(H) denote the *-algebra of bounded linear operators on H. We briefly recall (without proofs) some basic facts about trace-class and Hilbert-Schmidt operators. We refer the reader to [RS]. 4-1
Some remarkable operators
ideals of the algebra of
bounded
• Compact operators: An operator A € B(H) is called compact if it sends the unit ball (and hence any ball) of H into a set with compact closure in H. Alternatively, A is compact if given any bounded sequence (un) in H, one can extract a convergent subsequence (Au^n)) of its image (Aun). Finite rank operators, i.e operators with finite dimensional range are compact operators. In fact they form a dense subset of the closed subset of compact operators for the operator norm convergence in B(H). We shall denote by K,{H) the set of compact operators which form a two-sided *-ideal in the algebra B(H), i.e the sum of two compact operators is compact, the adjoint of a compact operator is compact and the product of a compact operator with a bounded operator is compact. Smoothing operators defined in the previous section provide examples of compact operators on H = L2(M, E) the space of L 2 -sections of a hermitian vector bundle E based on a closed Riemannian manifold M. Here the L 2 -hermitian product is built using the Riemannian volume form on the base manifold and the hermitian product on the fibre: (cr,p):= / (a{x),p{x))xdvol{x)
(4.0)
JM
where (•, -)x is the hermitian product on the fibre above x, dvol[x) the volume measure on M at point x and p, a are any two smooth sections of E. 327
Let us recall that the non-zero spectrum of a compact operator coincides wth the set of its non-zero eigenvalues and that when the operator is self-adjoint, its spectrum is either reduced to zero, or finite, or a real sequence converging to 0 (see e.g [G]). •Trace-class operators A compact linear operator A on H is called trace-class if there is a complete orthonormal system (C.O.N.S) (e„,n € IN) of H such that J2™=1(\A\en,en) < oo where \A\ := \/A*A (the square root is defined using the spectral representation of ,4M,see e.g. [RS]). If this condition is fulfilled for one C.O.N.S. it is also fulfilled for any other C.O.N.S. and the expression: oo
tr(A)~^2{Aen,en)
(4.1)
n=l
is well-defined and independent of the choice of C.O.N.S (en,n 6 IV). It is called the trace of A. The set of trace-class operators on H build a two sided ideal ZX(H) in B(H) i.e. M^+MH)
C MH),
MH)B{H) C liiH),
B(H)MH) C Ii(ff).
Moreover it is stable under the *-operation and tr(A*) = tr(A). Exercise: Check that (A 4- l)~z is trace-class in L2(Sl,M) where A := — 4^.
for, Re(z) > 1
at-*
Hint: Use a Fourier expansion to investigate the behaviour of the eigenvalues ( A n , n € JV) of (A + 1)" 1 . In a similar way, one can check that e~tA is also trace-class for t > 0. It follows immediately that e~tA lies in all Schatten classes, i.e in every XP{H) — {A € IC(H),tr(\A\P) < oo},p e JN - {0} where \A\P is defined via the spectral representation of A* A. •Hilbert-Schmidt operators: A compact operator is called Hilbert-Schmidt whenever there is a C.O.N.S (e„,n G N) such that Yl^Lii^n, Aen) < oo. In other words a bounded operator A is Hilbert-Schmidt whenever A* A is trace-class. The Hilbert-Schmidt norm of such an operator is defined by \\A\\2HS := tr{A*A).
(4.2)
Hilbert-Schmidt operators on H also build a two-sided ideal T2(H) of B(H). The adjoint of a Hilbert-Schmidt operator is also Hilbert-Schmidt so that like li(H), the set X2{H) is a *-ideal in B(H). For A € Xi{H) we have ||^4|| ffS = ||A*||//s- The ideal of trace-class operators is contained in the ideal 328
of Hilbert-Schmidt operators li(H)
C 12(H).
The product of two Hilbert-Schmidt operators is a trace-class operator (exercise) and we have: tr(AB) = tr(BA)
VA,B € 12(H).
(4.3)
Furthermore the Hilbert-Schmidt norm derives from a hermitian product on
MH): (A, B) := tr{B*A)
VA, B € 12(H).
(4.4)
It will be useful for our purposes to keep in mind that for any Q £ EU*o+d>0(M,E) the smoothing operator e~tQ lies in 12(H). Exercise: Let Q € Ell+*d>0(M, E) of order q. (i) Show that the operator Q
z
is Hilbert-Schmidt for Re(z) >
dimM 2q
•
Hint: Use the asymptotic behaviour A„ ~ Cn*>™M of the eigenvalues (A n ,n € W) of Q (which has purely discrete spectrum, see e.g. [G]). (ii) Show that for any A G CL(M,E) and any t > 0 the operator Ae~tQ is trace-class (it is in fact infinitely smoothing) and that tr(Ae~t(^) = tr(e'tQA) = tr(e-iQAe-?Q). •HS-pseudo-traces: Given K £ 12(H), from the properties of HilbertSchmidt operators we recalled above, it follows that for any A 6 B(H) the operator K*AK lies in li(H) (exercise) and hence that: B(H) -+ € A -> trK(A)
:= tr(K*AK)
(4.5)
is a well-defined linear operator which we will refer to as a HS-pseudo-trace ("HS" standing for Hilbert-Schmidt). Moreover from the above properties it also follows that (exercise): trK(A)
= tr(AKK*)
K
= tr(KK*A). K
In general tr does not define a trace, namely tr have (exercise): trK([A,B])
= tr([K*K,A]B)
(4.6) K
(AB) ^ tr (BA)
= tr([B,K*K]A).
and we (4.7)
If H is finite dimensional, we can choose K = Id in which case this obstruction vanishes and we are back, with the usual trace. • Fredholm operators: A Fredholm operator A : H -> H1 where Hi is 329
another separable Hilbert space, is a bounded operator A £ B(H,H\) such that there exist operators B £ B(H, Hi) and C £ B(Hi,H) with the property that both BA — I and AC — I be compact. In other words, A is invertible up to a compact operator. Any operator of the type A — I + K where K is compact, is a Predholm operator. A Predholm operator can be characterized by the fact that it is bounded, both the kernel KerA of A and cokernel KerA* of A* are finite dimensional and both the the range R(A) of A and the range R(A*) of A* are closed. Let us assume for a moment that H C Hi and that the inclusion is compact. If a Predholm operator A : H -t Hi is self-adjoint, then A : H n Ker(A)x ->• Hi fl Ker(A)1- is invertible and one can define its inverse A-1 : Hi —> H extending it by zero on the kernel. Then A~x : Hi —> Hi is compact as product of a bounded and a compact operator given by the inclusion H c Hi. It therefore has purely discrete spectrum and hence so has its inverse A. Thus a self-adjoint Predholm operator A : H —> Hi with H C Hi has purely discrete spectrum when the inclusion is compact. Let us now take H = HS(M, E), s > 0, Hi = L2(M, E) where n : E -» M is a finite rank vector bundle based on a closed Riemannian manifold as before. A self-adjoint elliptic operator A £ Ell(M,E) of order s induces a bounded operator A : HS(M,E) ->• L2(M,E). Being "invertible up to a smoothing operator" (see above discussion) and a smoothing operator being compact, it is Predholm. The manifold M being compact, by Sobolev embedding theorems, we know that the inclusion HS{M, E) -> L2(M, E) is compact for s > ^f^. It follows that the Predholm operator A : HS(M,E) ->• L2(M,E) has purely discrete spectrum. Since its inverse (extended to zero outside the kernel) is compact and has spectrum tending to zero, if A is positive, its eigenvalues tend to +00. A more detailed investigation shows that the asymptotic behaviour of a partial differential elliptic operator of order s is given by Xn ~ Cnd^M [G]. 4-2
Renormalization
procedures
A renormalization procedure (R.P. for short) on a Hilbert space H is a one parameter family {K\,\ G A} where A is an open subset of M+ := {A e + M, A > 0} or of
Vw G H
If A =
s.t.
K\ £ 12{H) if Re(X) > A0.
In general tr\A):=tr(K*xAKx)
(4.8)
does not converge when A —¥ 0 and we shall need to renormalize the limit taking -whenever possible- the finite part of this divergent limit. Because of the obstruction (4.7) arising for fixed A, one expects to see an obstruction of similar type in the renormalized limit, an obstruction we will be investigating in section 5. •Heat-operator renormalization procedures: The spaces we shall be considering are spaces of sections of some hermitian finite rank vector bundle ? r : £ - > M o n a c l o s e d Riemannianmanifold. Typically, H = L2(M, E) where the L 2 -scalar product is denned by (4.0). To simplify, let us first consider the case when E = M x <Sd is a trivial vector bundle and let us consider the operator defined componentwise by the Laplace Beltrami operator A on M. For A £ A := St+, the operators Kx := e"~sA,A g A are infinitely smoothing and hence Hilbert-Schmidt. It is easy to check that it defines an R.P. on H. This generalizes to a general vector bundle E based on M replacing A by any operator Q 6 Ell*o*d>0{M, E) (see section 3) acting on smooth sections of E and considering the family K® :=e^rQ,\G
A
(4.9a)
which also defines an R.P. on I? (M, E). • ^-function renormalization procedure: Let us come back to the trivial case E = M x
This generalizes to a vector bundle n : E -»• M replacing A by any element of Ell™j™0{M,E) and considering the family K?:=Q-i,\eh
(4.9.6)
(such complex powers were defined in section 3) which also defines an R.P. on L2{M,E). • Renormalizable families of traces: Let 7r : E —> M be a finite rank vector bundle on a closed Riemannian manifold M of dimension TV. Given an element A G CL(M, E) or order a and Q G Ell+*(M, E) of order q > 0 then the map t -> tr(Ae~tQ) defines a renormalizable functions ([GS], [Le]) and OO
tQ
s
CO
a ti lIL
tr{Ae- ) ~ jTl i ^
+
OO
1
J2 j=0,'~a~N
3=0
Pjt^T
log* + J ^ t *
€2Z
(4.10)
i=°
for some complex coefficicents aj,(3j,-yj,j G IN. When Q G Ell**d>Q(M, E), from the fact that X~z = M(t -»• e~tx)(z) it follows tr(AQ~z) = M{fA,o){z) with fA,Q(t) := tr(e~t(^) and hence, using (2.11) that + Poz~l = a 0 + 7o - iPo + O(z).
M(fA,Q)(z) 4-3
Traces on algebras of pseudo-differential
(4.11)
operators
Let as before A G CL(M,E), Q G Ell*$™0(M,E). It follows from the discussion in section 2 that the map z ->• M{fA,o){z) has at most a simple pole at zero and with the notations of (2.12) we define a linear map: trQ>» : CL(M,E)
-»• C°°(M,
:= LIM^0tr(AQ~z).
(4.12)
and the Wodzicki residue res(A) := ord(A)Resz=0tr(AQ'z)
(4.13)
where Resz=Zo stands for the complex residue at the point zQ. When n = 7 the Euler constant, then we shall simply write trQ(A) and refer to this expression as the Q-weighted trace of A. We have: trQ(A)
=LIMz^0{tr(AQ-z)) := lim (tr(AQ-z) a.->0 \_
= lim (tr(AQ~z) z-yo \ 332
-
z
-resz=0tr(AQ-z)
—res{A\ z • ordQ
In general the weighted trace depends on the spectral cut chosen to define the complex power Q~z; if Q G Ell*+d>0(M, E) we take L* := {reix,r > 0} as common spectral cut. Whenever the order a of A is such that a < —dimM then A is trace-class and tr®(A) = tr(A) coincides with the usual trace of A so that the Q dependence disapears altogether. As we shall see shortly, tr® does not define a trace on the algebra of classical pseudo-differential operators. In fact the Wodzicki residue is (up to a multiplicative constant) the only trace on the algebra of classical P.D.O.s [W], see also [K], [Le]. • T h e Kontsevich-Vishik t r a c e [KV], [Le]: The linear functional tr® restricts to the canonical trace introduced by Kontsevich and Vishik trace [KV] which we denote by TR on classical pseudo-differential operators with non-integer order. If the order a of A is not an integer then it follows from the results of section 2.2 that zero is not a pole of M(/^Q) so that we can take the (ordinary) limit trQ(A) :— tr(AQ~z)z-oThe linear map tr® has the tracial property on non-integer order operators: P r o p o s i t i o n [Le]: Given any Q G EU*o+d>0{M', E) (i) TR{A) := tr®(A) is independent of the choice of Q. (ii) TR([A, B}) = 0 for A, B € CL(M, E) of orders a and b such that a + b is not an integer. Idea of proof. It is based on the following formula:
tQL%Ae-tc*+RN(A,Q,t)
e-^A = £ 3=0
J
'
where for N large enough i?#(i) = 0(t) and where V Q . 4 = A, V^Q1 A := [Q,%A]. a d (i) Given a one parameter family {Qu,u e M+} e EU* and l 0'r d "^(M, E), ord>0\ d applying the above U U Y C formula 1 U 1 U 1 U 1 C X to I U fL A = -£^Qu, the Duhamel formula (see e.g [BGV]) yields:
A-triAe-^)
= - j \
Ae-V-^'f-^-Q^e-^dh
333
As can easily be seen from definition (4.12), we can assume without restriction that Q has integer order since tr®(A) = tr®' (A) for any s > 0. Thus A having non-integer order, so has AVJQ ^^- non-integer order and hence the asymptotic expansion of tr(AS/Q^Le~iQ")dt around t = 0 has neither terms of type i _ J _ 1 nor terms of type i - ^ - 1 log£. Thus the constant term as well as the logarithmic term vanish in the asymptotic expansion of -^(tr(Ae~tQ") around t = 0 from which it follows that z 1 ±(tr{AQ- )) = j ^ y /0°° t*- fttr{Ae-tQ) has no pole and has limit zero when z —>• 0. (ii) Let A,B be two P.D.O.s then tr{[A, B]e-tQ)
= tr(ABe~tQ) °° (—1V'
and the rest of the proof goes as above.
334
-
tr(Ae~tQB)
5
Algebraic obstructions
The weighted trace tr® defined above is not a trace and this fact leads to obstructions. In order to investigate these obstructions, it is useful to establish some properties of the Kontsevich and Vishik trace defined prviously on classical pseudo-differential operators of non-integer order. 5.1
A fundamental
property
Kontsevich and Vishik introduced a functional A —> TR(A) on classical pseudo-differential operators with non-integer order for which they proved a property which plays a fundamental part for our needs. Let us first describe the general lines of the construction of TR in a heuristic way. For a (classical) pseudo-differential operator A 6 CL(M, E) (see notations of section 3) locally given by: Au(x):=f
a(x,Qu{Od£
(5.1)
where a(x, £) is the (locally defined) total symbol, we would like to define "TR(A):=
[
[
trxa{x,Odtdvo\(x)"
(5.2)
- here trx denotes the trace on the fibre Ex above x of the vector bundle E which in general does not make sense since a(x, £) typically has components of degree > —dimM. However, it does make sense for an operator of order < —dimM and yields the ordinary trace. We should therefore find a way of picking up only the finite term in such an expression. We shall come across similar problems to the ones described in section 2, namely that of defining a finite part invariant under a change of variable, which means here defining a covariant finite part. To go from the approach we are about to describe back to the zeta-function approach to traces described in section 4 (see e.g. (4.12)), one way is to find an asymptotic expansion of tr(A(Q — XI)"1) as |A| -» oo and then using a contour integral from which one deduces the asymptotics of tr{Ae~tQ) when t —¥ 0 from which follow in turn the asymptotics of tr(AQ~z) around z = 0 [GS]. Let us first recall some basic facts about homogeneous distributions. •Homogeneous distributions ( [H] chap. 3.2): An element u of L}oc(Mn - {0}) - which in particular is a distribution on Mn - {0} - is called 335
homogeneous of degree z when u{tx) - tzu(x)
Vt>0,x/0.
(5.3)
n
If Re(z) < —n then for a smooth function \I> on JR which is zero in £?(0, | ) and equal to 1 on JRn-B{0, §) we have * u e ^(M71). Indeed writing x - r£, with ||£|| = 1 we have \<S(x)u(x)\ = rRe^|tf(£)"(£)! so that for any R > 0 the integral : f
JB(O,R)
f
JO
droit
JM\\=I
converges. However problems might arise for negative integer values of z because of divergences coming into play. Take z = —n in (5.3). Then in a similar way to what was done in section 2, we have: /
^{x)u{x)dx
= /
JB(0,R)
$(x)u(x)dx
+
JB(0,1)
=
Jl
^(x)u(x)dx
u(£)r -1 dr
/ JU\\=1
+ logR
JB(0,1)
u(£)d£, JU\\ = l
which diverges when R —• oo. Following methods presented in section 2, we can define the finite part f.p.(
/
(x)u{x)dx) := lirnR^+0O
K
JjR"
'
(/
<2{x)u(x) - logR /
\JB(0,R)
\
u($)d£ )
JM\\ = 1
J
$(x)u( i{x)dx
JB{O,I)
but this is not invariant under a change of variable anymore (we refer the reader to the discussion in section 2). Take z = —n + k, k > 0 for some k £ IV — {0}. A similar computation yields: /
^(x)u(x)dx
JB(0,R)
= /
%(x)u(x)dx +
«'B(0,1)
/ Jl
u(£)r~1+kdrd£
•'||?||=l
= /
9(x)u(x)dx)
:= / i m R ^ + 0 0 ( / 336
k J\m\=i
fc
*(rc)u(a:) - ifl f c f
•/||«||=i u(0# )
= /
V(x)u(x)dx - \ f
JB(O,I)
u(Odf •
K J\\S\\=I
• T h e Kontsevich-Vishik canonical density: Let us consider a classical P.D.O. A of order a with symbol a = J2j=o ^aa-j + a(w) where aa-j G Sa-i(Mn), a{N) G S " * - ^ 1 ^ ™ ) an d w h e r e a s before * is a smooth n function on lR which is zero in B(0, | ) and equal to 1 on Rn — B(0, | ) . Since O(jv)(a;,0 = Oflf| a - J V ) we have / B ( 0 > f l ) O ( A T ) ( 0 ^ = / K » a(iv)(z,£)dC + 0 ( i ? o + n _ N ) . On the other hand, splitting the integrals /
tyaa-j(x,£)d£=
JB(0,R)
tyaa-j(x,Z)dZ+
JB{p,l)
aa-j(x,£)dt,
J B(0,R)-B(0,1)
we use the fact that homogeneous of order a — j to express the last integral. If a is an integer then there is an integer jo such that a — j + n = 0 and we have for N > j 0 : Y,
[
9aa-j(x,t)dt = itf
j=oJB(0,R)/B(0,l)
f
j=0Jl
N
i
+ logi? /
rO-^-iaa-jfatWdr
J\£\=1
r
a_ n (a;,£)d£ + constant term.
•/|ei=i
Finally this yields the existence of an asymptotic expansion in R —»• +00 and of a constant C(a(x, •)) dependent of R such that N
r
1
+ logR [ J
r
a-n{x,t)d£
+ C(a(x,-)).
(5.4)
\t\=i
Because of the logarithmic term in R, one does not expects the finite part f.p.{[
a(x,Z)dt)
= LIMR^OO
J Rn
[
a(x,0d£
(5.5)
JB(0,R)
-defined taking the limit when R —> 00 of the r.h.s of (5.4) after substracting out divergences- to be invariant under a change of variable of Mn. However if the order a is not an integer, there is no logarithmic divergence and f.p. (/ f i „ a(x, £)d£) is independent of the local representation a(x, £) of A: 337
Proposition [KV] (see also [Le] in Prop. 5.2): Provided A £ has non-integer order: TR(A)~
[ trEJf.p.([ a(x,0d0)dvol(x) \ Jm." / and coincides with the previous definition of section 4-3.
CL(M,E)
(5.6)
JM
• The Wodzicki residue: When A has integer order, there is a pole at zero arising in tr(AQ~z) which by the Mellin transform corresponded to a logarithmic divergence in the asymptotic expansion of tr(Ae~£Q) . This in turn gives rise to a logarithmic divergence in fB,Q R. a(x,£)d£ when R —> oo where a was the symbol of A. We have: Proposition ElCd>0(M,E),
[W] (see also [Le]):
For any A
res(A) := ord(A)Resz=0tr(AQ~z) = — ord(A) (coefficient of logf in the asymptotic expansion of tr(Ae~t®) trx(a,-n(x,£))d£ JM
e
CL(M,E),
Q €
as t —¥ 0)
I dvol(x)
\JU\\=I
— coefficient of log R as R —> oo in the asymptotic expansion of I
trxa(x,(;)d£
(5-7)
JB(1,R)
(where trx denotes as before the trace on the fibre Ex above x) defines a trace on the algebra CL(M,E) i.e. we have res(AB)
= res(BA)
VA,B €
CL{M,E).
It follows from the definition that the Wodzicki trace vanishes on trace-class operators. This is the reason why we shall not be using it as a tool to extend the finite dimensional trace, and why we consider weighted traces. However it plays an important role since it is - up to a multiplicative constant factor the only trace on the algebra of classical P.D.O.s [W]. The notion of Wodzicki residue also generalizes to a class of non classical operators (which includes the classical ones) as described in [Le]. The third indentity in (5.*7) also shows (exercise) that the residue vanishes when the underlying manifold is odd dimensional and the operator in the odd class (a concept introduced in section 3). 338
• S m o o t h and holomorphic families of P.D.O.s. The algebra CL(M, E) inherits a Prechet structure equipped with a topology induced by the natural Prechet topology on symbols given by the following family of semi-norms indexed by multiindices 7 6 Nd, 8 £ ]Nd and i £ {1,---,JV}, fee JV: lkll7,*,*
:=
max su
i( Pyevi,S€iR*M\\=^DlDtak(y,^]i)
where crfc is the homogeneous component of order k, { V i } ^ ! , . . . ^ is a finite open cover (with Vi compact) of M associated to a partition of unity {Vi,€i}i=i,---,N on M subordinated to some finite open covering {£/i}i=i,...,w. This topology combined with natural semi-norms on smooth kernels of compact operators, induces a Prechet structure on the space of classical pseudodifferential operators CL(M, E) via the identification of operators with their symbol up to a smoothing operator (see e.g [KV1] section 3 and [D] for further details). This topology leads to the following definitions. Given a domain D c G (resp. an open subset U C Mn), we shall call a family of PDOs A(z) € CL(M,E) of order a(z) with z € D holomorphic (resp. A(x) of order a(x) with x S U smooth) if: (i) z -> a(z) e € is holomorphic (resp. x -¥ a(x) is smooth) (ii) the homogeneous components z -> a a (z)-j (resp. x ~> aa(x)-j ) of the symbol of A(z) (resp. .A(:r)) are holomorphic (resp. smooth) in z (resp. in x) as functions on the cotangent unit sphere S*M restricted to some local chart, (iii) the kernel KA(Z)(U,V) (resp. KA(x)(u,v) ) of the family A(z) (resp. A(x)) is holomorphic (resp. smooth) for u, v in disjoint local charts, (iv) there is a local chart V such that on V x V and for all N 6 JN there exists an integer M^ such that the map (u,v) -»• KA{z)(u,v)
-
Z / ^ G K I D I K i r ^ X ^ ^ ) (a, j i j ^ 339
exp{i(u-v,t))d£
(resp. the same expression with z replaced by x) is a CMN smooth (local) kernel on this neighborhood of the diagonal which is holomorphic in z (smooth in x) together with its partial derivatives in u,v up to order MM- Here $ is a cut-off function already introduced previously (see below formula (5.3)). •Fundamental property [KV] (see also [Le] Prop.5.5): Let A(z) e CL(M, E) be a holomorphic family of operators on a domain D C
=
±-res(A(z0)) a (zo)
(5.8)
when a(zo) € {—n, —n + 1, • • •}. Proof. We shall only prove the case a(z) = z, leaving the general case as an exercise for the reader. Let n denote the dimension of M and let a(z)(x,£) denote the symbol of A(z). We have: N
a(z) =
J2*(0az-j(z)+g{N)(z) j=o
where g(N)(z)(x,0 = 0(lf I -N ) » S llfII ->• °°- Notice that for Re{z) < N-n the map £ -> g(N)(z)(x,£) n e s i n ^(M71). As before and with the same notations, for Re(z) negative enough we can write: /
r N a 2 _ j ( 0 =*»"»*_•«, J2 /
JBt"
*{t)az-j(z){x,£)dZ
+
[~^JB(0,R)/B(0,1)
+ [ 9(N)(z)(0d£. Hence the map z ->• JRn $(£)az-.j(£) extends to a meromorphic map on the complex plane with simple poles at —dimM, —dimM + 1, • • • and residue at ke 2Z given by resz=ktr(A(z)) = - / s „ _ a a-dimM(k)(x,Qd€ = -res(A(k)). 340
5.2
Obstructions
in terms
of the Wodzicki
residue
We saw at the end of section 4 that the Q-weighted trace tr® is tracial on the set of pseudo-differential operators with non-integer order (where it is denoted by TR) and that it is independent of Q. These properties fail to hold on the whole algebra of pseudo-differential operators and we shall express the corresponding obstructions in terms of a Wodzicki residue. • T h e weighted Radul cocycle (see also the lecture by E.Langmann in this volume): We first need the notion of cocycle in the cohomology of Lie algebras (see e.g. [Mi]). Let L be a Lie algebra, then a cochain of degree n (or n-cochain) is an antisymmetric multilinear map: C : L x • • • x L —>•
• • •,£», • • •,Xj • • •,xn+1)
i<3
has the property 62 — 0. An n-cochain c is called a cocycle if 6c = 0 and cocycles of the form 6c are called coboundaries. Two n-cocycles c, c' are equivalent whenever their difference is a coboundary i.e if there is an (n — l)-cochain c" such that c — c' = 6c". A two-cocycle c on L gives rise to a central extension of L namely L := L © C with bracket [(a, a), (b,/3)] := ([a,b],c(a,b)). The fact that c is a cocycle is equivalent to the Jacobi identity for the bracket on L. Proposition-Definition : Let A,B Then cQR{A,B) := tr^([A,B})
£ CL(M,E),
= lim tr([A,B]Q'z)
=
Q £
Ell™j™0(M,E).
1
—res([logQ,A]B) (5.9)
is called the Q -weighted Radul cocycle. This was proved independently in [MN] and [CDMP]. Proof. Let us first note that for z with large enough real part, we have tr([A,B]Q-z) = tr([Q-z,A]B) and let us set C(z) := [Q~Z,A)B which is a P.D.O. of order a(z) = a-z-q + b,a = ord{A),b = ord(B),c = ord{C). z -> 341
C{z) is a holomorphic family and we have C(0) = 0. Hence resz=oTR(C(z)) = —res(C(0)) = 0 and z -> TR(C(z)) is holomorphic at zero. Here TR denotes the canonical trace as in section 4. Let us compute its value in zero. We have: trQ([A, B]) = lira TR(C(z))
Resz=0TR{-C^
=
z-J-0
z
TR([Q-',A]B) = Resz=0~z = Resz=0TR([Q 1
a'(0)
Z
*,A]B)
res([\ogQ,A]B)
= --res([logQ,A]B) where we have used the fact that D(z) := [
by
(5.8)
~ ,A]B is also a holomorphic
family at zero of order a(z) = a + b — qz and that ( " ~ z ~ I J ^
= — logQ.
' U=o
The Q-weighted Radul cocycle c^ carries this name because it generalizes the ordinary Radul cocycle [Ra],[CDFW],[Mi] which we recover for M — S1 and Q = \D\, D being the Dirac operator on S1 introduced at the beginning of section 3. Two different weights <5i,Q2 lead to two different weighted Radul cocycles c^ , i = 1,2 and we have:
We recall from the end of section 3 that the difference P := £ ^
-
J2JL2L
2
is a classical P.D.O. so that we can write c# — c^ = 5res{P •). Thus two weighted Radul cocycles are equivalent since their difference is a coboundary. • T h e Q-dependence of trQ: Proposition [CDMP]:Let {Qt,t £ I C M} be a smooth one parameter family of elements of Ell^y0(M,E) with constant order q. Let {At e CL(M,E),t & I C M} be a smooth one parameter family of elements of CL(M, E). Then for any t0 e / , -£
trQ'(At)
= tr^o{±
At)„lres{Ad
342
iogQt).
(5.io)
Proof. We shall only prove ± trQ<(At0) = \imJtr{AtoQ^) dt\t=t0 z->odi| 1=to
= --res{Atoj q
dt\t=tQ
\ogQt)
since (5.10) then easily follows. Let us first notice that although logQt does not in general belong to (log Qt - logQ t o ) does since CL(M,E), the operators £. _ logQt = dt\t it is expressed in terms of a difference of logarithms of same order. The family C(z) := Ato dt\ Qtoz) is a holomorphic QTZ = A£u__to(Q t=t0 family in CL(M,E) and hence by the fundamental property (5.8), the map z -»• TR{C{z)) = TR{Atof QJZ) = ft, TR(AtoQ^) is meromorphic \t
tQ
|t
tQ
with simple poles. In fact the map z -> TR(C(z)) is holomorphic at z = 0 since using again the fundamental property we can write (here a(z) = a — zq): Res\z=0tr{C{z))
= -res(C(0))
Since C(0) = 0, the family z ->• z~lC{z) dt\t=t0
trQ<(At0)
dt\t=t0 =
=
1 -res
is holomorphic at zero and we have:
TR{AtaQTz) U=o
Resz-oTR
because
= 0.
C(z)
z —> TR(C(z))
has no pole at
z =0
Z
(QT ~QT: M0 ± dt\t=t0 y z
z=0/ J
d logQi dt\t=tl
res At0 q Notice that unless Qt or •£, _ Qt commutes with Ato, in general, one has =
res(Ato^f
log Q t ) + res {A^Q^1^
Qt). d*|i-.0"
This generalizes to (smooth) families of operators {At € CL(M,E),b G B} and {Qb G Ell+*(M, E),beB} parametrized by a manifold B [CDMP], [PR]: dtrQ(A)
= trQ(dA)
ordQ 343
res(Allog<3).
(5.11)
5.3
Further
obstructions
of algebraic
type
Let us briefly mention other obstructions of algebraic type that can occur. • Multiplicative anomalies: For A 6 Ell*o+d>0(M, E) we can define the C determinant detc{A) := exp{-(,'A(fy) where CA(Z) •= M(t -»• tr(e~tA)){z) (see (2.9)). In general it does not have the multiplicative property i.e. given another B € EU*o+d>0(M, E) we have detc(AB) ^ detc(A)detc{B). This phenomenum is referred to as the multiplicative anomaly [KV1,2], [O] which is also investigated in [D] from a different point of view. The anomaly is an algebraic obstruction which can be expressed in terms of a residue of the type res P ( C^^ ~ 3*$J)) where P £ CL(M,E) is of order zero (cf. [KV1.2]). • Schwinger functional: Let D £ EU*ord>Q(M', E) be a self-adjoint operator. We assume it is invertible for simplicity but this construction can be extended to the non-invertible case. Adopting notations which are frequently used in the context of second quantization, let e(D) :=• D\D\~~X (or s(D) := (D + PD)(\D\ + PD)~l if D is not invertible where PD denotes the orthogonal projection onto its kernel) denote the sign of D which defines a classical P.D.O. of order 0. The operator Q :— \D\ is a positive self-adjoint operator in Ell^Td>Q{M, E) with strictly positive order. We define the Schwinger functional: CL(M,E) (A,B)
->
(5.12)
The terminology " Schwinger functional" is motivated by the fact (as we shall see shortly) that it coincides with the usual Schwinger cocycle on some subalgebra of CL(M, E). A straight forward computation yields: c°(A,B)=trM([A,e(D)}B)=trM(A[e(D),B}).
Proposition [CDMP]: Given a subalgebra A of CL(M,E) the following conditions are equivalent (all the cocycles are Lie algebra cocycles): c§ is a 2 -cocycle on A O- c§ is an antisymmetric bilinear form on A O- (A, B) -> c$R(A, B) := i r | D | ([e(D)A, B]) defines a 2 -cocycle on <& Cj-R is an antisymmetric bilinear form on A «*• res (e(D)[A, [log \D\,B]]) = 0 V.4, B e A. 344
A
Provided one of these conditions is fufilled, the following relation holds:
c?R-c°=6tr? where trj? := tr^D^(e(D)-) so that the cocycles c^R and d? are cohomologous. The fourth equivalence shows that the obstruction to the cocycle property of the various functionals involved arises as a Wodzicki residue res(s(D)[A, [log|D|,B]]). Here again the obstruction expressed in terms of a residue is purely infinite dimensional. If res(e(D)-) is tracial on A i.e if res(e(D){A, B]) = 0 VA, B e A, the obstruction vanishes. The algebra CL?es(M,E)
:= | A e CL(M,E),ordA
< 0,ord[e(D),A]
= g?es(L2(M, E)) n CL(M, E),
< _ ^ M | (5.13)
where g?es(L2(M, E)) := {A 6 Hom{L2(M, E)), [e(D), A]
is Hilbert-Schmidt} , (5.14) satisfies the equivalent assumptions listed above and the restriction of the Schwinger functional c^ to this subalgebra is therefore a cocycle. There it coincides with the usual Schwinger cocycle [CFNW], [Mi] and is cohomologous to the twisted Radul cocycle c^R [CDMP].
345
6
Geometric obstructions
The purpose of this section is to give some geometric applications of the apparatus of weighted traces developped in the previous sections. The notion of trace underlies many constructions in finite dimensions, some of which were briefly described in the first section. Many of them extend to the infinite dimensional context naively replacing the ordinary trace by a weighted trace. However the algebraic obstructions combined with some geometric ones we are about to describe, impel a naive generalization of some of their properties; typically additional Wodzicki residue terms arise. This prevents us for example from extending the Chern- Weil construction to build caracteristic classes; only in very special cases (we shall briefly comment on the loop group case) do we obtain closed forms naively replacing the trace in the usual invariant ploynomials by weighted traces. In the "half weighted" graded case (see [PR]), the "half-weight" being a family of Dirac type operators (the square of which yields a weight), one sees from previous works on families of Dirac operators ([BFJ, [DF], see also [BGVJ) that a better approach to invariant polynomials on such infinite rank complex vector bundles might arise from modifying the initial connection using the "half-weight" into a family of Bismut connections [B[. In finite dimensions the choice of the connection is irelevent at the level of characteristic classes, here it is the very choice of the connection which makes all the difference! Although because of lack of space, we shall not be able to report on these possible extensions, we hope to give the reader an idea of how weighted traces could be used in the infinite dimensional geometric context and thereby also the limits of their possible use. As a result we allow ourselves a "looser" tone in this section than that used in the previous ones and will only brush over notions without giving technical constructions. We refer the reader to [Pa] for further developments and speculations along these lines. 6.1
Weighted
vector
spaces
We shall call a weighted space a couple (T(M,E),Q) where T(M, E) denotes a Frechet or resp. a separable Hilbert space of smooth or resp. Sobolev type sections of some finite rank vector bundle E based on a closed Riemannian manifold M and where Q 6 Ell*°%™0(M,E) is called a weight. Generalized Laplacians provide weighted structures on spaces of type T(M, E). By generalized Laplacian acting on sections of E, we understand a second order differential operator with leading symbol OL(X, £) := ||£|| 2 - Given a connection V s on E, the Levi-Civita connection on M induces a connection 346
® £; and the operator given by a -> -tr{VT*M®BVEa) is a generalized Laplacian. Notice that when M is reduced to a point, then the vector bundle E reduces to a vector space V and (T(M, E),Q) coincides with (V, Q) where now V is a vector space of dimension the rank of the vector bundle and Q a matrix with spectral cut. Hence Q can be viewed as a generalized metric, a metric being given by a positive definite matrix. In fact in our infinite dimensional setting, it is useful to think of Q as a kind of "quantized metric": a positive definite matrix has been replaced by an admissible elliptic operator. V
T * M ® S o n T*M
From the data (T(M, E),Q),Q of order q > 0, in addition to the L 2 -hermitian product (•, •) mentioned in (4.0), one can build for any s 6 M+ an Hs hermitian product: (
To a weighted space (C°°(M,E),Q), we can associate a weighted trace tr® on the algebra CL(M,E) of classical pseudo-differential operators acting on smooth sections of E as defined in the previous section. Exercise: Show that whether one takes this weighted trace using the Hsscalar product defined in (6.1) or the L 2 -scalar product does not make any difference. In the following we shall use the L 2 -scalar product. Remark: When E = M x V is a trivial vector bundle, V being a finite dimensional vector space with scalar product (•, -)y, we can view CL(M, E) as the tensor product CL{M,G) ® Hom(V). When the weight Q is scalar, meaning that Q := Q0
:= -trQ'^(Hessf(x))
V/i G M,Vx 6 HS(M, E) 347
(6.2)
Let us point out at this stage that there are many other ways of defining Laplacians in infinite dimensions. Here we have picked out the finite part of the asymptotic expansion tr(AQ~z) when z —»• 0. But one could have instead picked up the residue term using the Wodzicki residue to build Aresf = tr g -res(Hessf). Or we could have taken —limt^0 ' "f/-?6) which leads to a Levy-type Laplacian [ABP], similar to Levy Laplacians arising in white noise analysis. The latter Laplacians are of purely infinite dimensional nature because they vanish on the trace-class part of Hessf(x). The weighted Laplacian however coincides with the usual one whenever Hessf(x) is tracial, which is the reason why we shall keep to it. In particular, if M reduces to a point, only the weighted Laplacian is non trivial. Example: Let (C°°(M, E), Q) be a weighted Hilbert space. Let for 0 < t < s, f : HS(M,E)
-> C be the C 2 function defined by f{x)
= (|M|?)
for
S
x 6 H (M,E) where || • \\f is the norm associated to (•, -)f (see (6.1)). Then D2f(x)(u,v) = (Q^u,Q^v)f. Hence Hessf{x) = Q 2 ^ e CL(M,E) and we can define A®,flf i n
_
s <
= —trQ'^(Q
5 ) which coincides with
—tr{Qt~s)
_2dimM_
With the same notations as above, let f,g € C2(HS(M,E),(D) such that for any x G M, the operators Hessf(x) and Hessg(x) both differ from a trace-class operator by an element of CL(M, E). Then tr(Df(x) <8> Dg(x)) := Y,Denf{x)®Deng{x) is well defined for any C.O.N.S. ( e n ) n e J V of HS(M,E) and A° , M (/S)(s) =
A.Q'"(f)(x)g(x) ~2tr(Df(x) ® Dg{x)) + f(x)AQ'flg(x)
V/u G R.
(6.3)
This easily follows from the property of the Hessian Hess(fg) = (Hessf)g + 2Df ig> Dg + f(Hessg). It shows that AQ behaves like a second order differential operator. 6.2
Families
of weighted
vector
spaces
Instead of a fixed weighted space we now consider families of weighted spaces which leads us to the notion of weighted vector bundle. • A class of vector bundles: We first need to restrict ourselves to a class of vector bundles, namely vector bundles n : £ —• X based on a manifold X with fibres modelled on a Prechet, resp. Hilbert space T(M, E) of smooth, 348
resp. Sobolev type sections of a fixed finite rank vector bundle IT : E —• M based on a closed Riemannian manifold M and such that there is a local trivialization of the bundle with transition maps given by elliptic classical pseudo-differential operators. (We will also allow P.D.O.s which are locally given by multiplication operators by a Sobolev type function). For a vector bundle £ in that class with fibres modelled on some T(M, E), it makes sense to call a linear operator A acting (possibly only densely) fibrewise on £ a (classical) P.D.O. whenever, given a local trivialization (U,
(6.4)
The transition maps Qoty-1 being classical P.D.O.s by assumption, whenever ^A is a classical P.D.O. so is $ ' A Notice that the order of the operator is also unchanged under a change of local trivialization so that it makes sense to talk about the order of an element of CL(£). • Weighted vector bundles and weighted manifolds: Recall that a Riemannian structure on a finite rank vector bundle n : E —> X corresponds to a reduction of the frame bundle GL(E) to an orthonormal frame bundle 0(E). A metric g on a finite rank vector bundle E can be seen as a section of gx, x S X of Hom(E, E*) which is locally a positive definite matrix. Similarly we can equip a bundle £ modelled on some T(M, E) where E is a Riemannian finite rank bundle, with some data Q given by a section of CL(£) which is locally an invertible elliptic operator of positive order and with positive leading symbol; i.e, there is a local trivialization (U, <j>, $) of £ where (U, <j>) is a local chart on the base manifold, such that &Q g Ell^d>0(M, E). Provided £ is Riemannian, we can reduce the structure group Ell*(M,E) of the bundle £ (notice that it coincides with GL(V) when E = {*} x V) to the subgroup Oll{M,E) := {A e Ell* (M, E), A* A = A A* = Id} (notice that it coincides with 0(V) when E = {*} x V). In that case, the condition § ' Q e Ell*^d>0(M, E) does not depend on the choice of the local trivialization (C7,4>, $) since, given another trivialization (V, ip, VP) with U fl V ^ 0, we have &Q = C^QC'1 for some C € Oll{M,E) so that 349
aL{^Q) = aL{C)ai{^Q)aL{C)~1 is also positive and hence * ' Q also lies in EU*0+d>0(M, E). (Notice that the order is in fact independent of the choice of local trivialization.) This confirms the fact that Q can be seen as a generalization of a metric. However, if we do not want to restrict ourselves to Riemannian bundles, we need to weaken the requirements on the data Q; a weight Q on a vector bundle £ modelled on some T{M, E) is a section of CL(£) which is locally an admissible (see section 3) invertible elliptic operator; i.e, there is a local trivialization (U,
Weighted
traces on weighted
vector
bundles
Let {£, Q) be a weighted vector bundle. For a section A of CL{£) and given a local trivialization (£/,
M€ M
(6.6)
where $'(6)Ab, resp. &(b)Qb denotes the operator Ab, resp. Qb in the local trivialization (U, <£,$). This definition is independent of the choice of 350
local trivialization. Indeed, let C be an invertible elliptic operator and let Q G EU*o°$™0(M,E), A G CL{M,E). Then provided Re{z) is chosen large enough so that CAQ~% and Q~iC~x are Hilbert Schmidt and using the fact that tr(KiK2) = tr^^Ki) for Hilbert Schmidt operators K\,K?., with the notations of section 4 we have: fr.CQC-'./.^c.-i)
=
LIM^0tr
(cAC'1
(CQC"1)^)
= LIM^0tr
(CAC~lC
{Q)~z C T 1 )
= LIM^tr
(CA {Q)~z
= LIM^tr
(CA (Q)"* (Q)~* C " 1 )
= LIM^0tr
((Q)-* C - ^ A (Q)~f )
C'1)
= LJM*L+0tr((Q)-*A(Q)-*)
= LJMJJLf0*r(A(Q)-*(Q)-*) = LIM^0tr =
(AQ-Z)
trQ'»{A),
where we have also used the fact that (CQC~X) = C {Q)~z C~x which follows from the very definition of complex powers (see formula (3.10)). Applying this to C := $(6) o
(yt(b)Ah)
l C~ )
tr*tWQ'"H**(b)Ab).
• Variation of the weighted trace: We saw in section 5, formula (5.11) that given a smooth family {Ab,b G B} of P.D.O.s and a smooth family {Qb,b G B} of invertible positive elliptic operators with constant order q, B being a smooth (possibly infinite dimensional) manifold and for any (j, G 1R we have: dtrQ^(A) = trQ^{dA) - ^res(Ad logQ). We want to generalize this to sections of CL(£) where (£, Q) is a weighted vector bundle. The weighted traces tr Q, '*,/i G M extend to CL(£) valued forms: trQ^
: Q(X) ® CL(£) -> Cl(X) a® A 351
-*trQ>,l(A)a.
In a similar way, we can extend the notion of Wodzicki residue to CL{£)valued forms. Proposition [CDMP], [PR]: Let (£,Q) be a weighted vector bundle based on B and let a be a smooth CL{£)-valued form on B. Let us assume £ is equipped with a connection V f such that [V £ ,a] is also a CL(£)-valued form on B. Then [Vf ,tr°'"](a) := dtrQ^(a)-trQ''1{[V£,a\)
= (-l)l a l + 1 -res(a[V £ ,logQ])
V^ 6 M
(6.7) where a is the degree of the form a. Proof: We only prove the result for a bundle morphism A i.e. a CL{£) valued zero form A on B. The proof easily generalizes to forms of higher degree keeping track of the signs. We write V £ = d+6£ in a local trivialization (U, <j>, $) (see notations of section 1). Since A -> [V f , A] defines a bundle morphism of CL(£), for A G
-res(AdlogQ)
= trQ'lt(\ye,A])
- trQ^{{e£,A\)
= trQ'i*([V£,A})
- -res(A[6£,logQ})
= trQ'»([V£,A})
-
-
-res(Ad\ogQ) -
-res(AdlogQ)
-res(A[V£,logQ]).
We shall refer to the bracket: [V f , trQ^} : (1(B) ® CL{£) -> B x R 1 a -»• — r e s ( a [ V , log <2]) as a geometric obstruction of the weighted trace tr®'1*. Notice that it is independent of n G M. 6.4
Some geometry
in infinite
dimensions
Going back to section 1, let us see how properties such as the BochnerWeitzenbock formula (1.12) or the Chern-Weil construction briefly described in section 1 could be naively extended using weighted traces. 352
Let (X, Q) be a Hilbert weighted manifold equipped with a Riemannian metric and the corresponding Levi-Civita connection V. Then the tangent bundle TX is also equipped with the weight Q. • A weighted Ricci curvature: Let Cl be the curvature of V. If the map (U, V) -> Q(U, -)V defines a section of TX
(6.8)
• A generalized Bochner Weitzenbock formula [ABP]: We use the notation of formula (6.2) for the Q-weighted Laplacian.
[AQ'», V]f(V)
= RiccQ>»(V,Vf)
for any a vector field Hessf, (Vy(-,;V) sections of CL(TX). residue vanishes and
+ -^-res(Hessf[V,logQ\)
VM 6 R
(6.9) V o n J , for any C 2 function / such that the two tensors , VyHessf on X can be seen (via Riesz theorem) as Notice that if X is finite dimensional, then the Wodzicki we recover (1.12).
Proof. The proof of (6.9) closely follows that of (1.12) (see [ABP]) up to the fact that exchanging the covariant differentiation with the weighted trace (which goes without saying in the finite dimensional setting) requires more care here; it gives rise to a residue term by formula (6.7). Let V € TXX and let (ei, i € IV) be a local orthonormal frame at point x of eigenvectors of Qx. ( [ - A « , V]f){V) = Lim^0
= (-A«V/)(V) + VVA«'"/
J2 V 3 / ( e " £ Q e i ® e~eQei ®V)-
VvtrQ
(Hessf)
i
J2 V 3 / ( e " £ 0 e i ® V ® e~eQei) - i r Q ( V v V 2 / ) +
= Lim£^0
i
+ -^^res(Hessf[W, where we have applied (6.7) to = LIM^0
3
J2 {V f(e-
eQ ei
log Q]) a = Hessf
®V® e-'^ei)
i
+
oraQ
^—res(Hessf[V,logQ]) 353
- V 3 / ( V ® e~eQei ® e ^ e * ) ) +
= LIM^0
V V 2 V / ( ( e - £ « e i A V) ® e ^ e ; ) + - L - r e S ( t f e S S / [ V , l o g Q ] ) '-r' oraU i
= LIM^Qy2(n(e-^ei,V)Vf,e-sQei) i -r'
+
-^rres(Hessf[V,logQ}) oraU
i
= -RiccQ^(V,
V / ) + ^ - r e s ( F e s s / [ V , log Q]).
• Obstructions to a (weighted) Chern-Weil theory Let us go back to a general weighted vector bundle (£,Q) based on X and equipped with a connection V £ . When a in formula (6.7) is the curvature tensor H £ of V£ seen as a CL(£) valued two form, then by the Bianchi identity we have [V £ , ft£] = 0 but this does not imply anymore that dtrQ(fl£) = 0 (as it did in the finite dimensional case, see (1.15)) since it follows from (6.7) that dtrQ(Sl£) = -±res(Q£[V£,log<2]). Hence in general, trQ(Q£) does not define a characteristic class because of the above obstructions unless Q and V £ are chosen in such a way that res(fi £ [V £ ,log<9]) = 0. One can generalize this observation to traces of higher order polynomials in the curvature tensor. Let us see how the Chern-Weil construction we briefly recalled in section 1 could be extended to our infinite dimensional framework with the help of weighted traces. Since we have replaced the algebra of matrices by the algebra CL(M, E) and the trace on the algebra of matrices by weighted traces trQ where Q € Ell™$™0(M, E), it is natural to introduce the following definitions. Given a weighted vector bundle (£,Q) based on some manifold X where the model space of £ is a space of sections T{M, E), M being the base manifold of E, and given a connection V f on £, to a polynomial P on CL(M, E) we can associate a form trQ,IM(P(il£)) on X where /J, e JR. It is a well defined expression since we can write in a similar way to the finite dimensional case (see section 1) and with similar notations: tr**^(P(¥n£))
= i r c * ' « 0 ~ 1 - " ( C P ( * ' , n e ) C r 1 ) =
using the covariance property trCQC (CAC*1) = trQ(A). However obstructions come into the way at various stages, and to start with when trying to generalize (1.16). As in section 1 we take P(X) = Xk and consider its canonical form P(Xi, • • •, Xk) = Xi---XkDifferentiating w.r.to t the covariance 354
property just mentioned applied to C := etB and using (5.9) we find: k
Y, tr^{Pk(Au-
• • ,Ai-U [B,Aj),Ai+1,
• • • Ak)) =
= -^res(Pk(Al,---,Ak)[A,\ogQ\) (6.10) ordQ for any B 6 CL(M,E),Aj € CL(M,E),j = 1, • • • ,k. Furthermore we come across geometric obstructions of the type (6.7) and we have (compare with (1.17)): dtr^(P(Q£)) =
^--re S (P(ft £ )[V £ ,logQ]). (6.11) ordQ Proof. Let us try to mimick the proof of the corresponding finite dimensional result as shown in section 1. As we did there, we first restrict ourselves to a monomial P(A) := Ak and compute dtr{P{Sl£)). Let Vs = d + 6£ be the local representation of the connection in some given local trivialization (U,9,4>). Applying (6.10) to B := e£(
= trQ'»(dP(¥n£,-
• •, ¥n£))
^—res(P(fl£)dlogQ) ordQ
k
= V trQ'ViPi&tf,
•••, d(¥n£),
•••, ¥n£))
i=l
where
d(&Q£)
— res(P(n£)d\og ordQ
Q)
is at place i
k
= ^ tr(P(^n£, • • •, d(^n£), •••, &si£)) + Y2 tr(P(&n£, • - •, [e£, $»o £ ], • • •, ¥n£)) j=i
where 1
ordQ
di&tf)
and [6£,¥n£]
res(P(n£)[6£,logQ])
are at place i l
ordQ
—res{P{9.£)dlogQ)
k
= J2 trQ^(P(V£, where 1
ordQ
• • •, [V£, n% • • •, n£)) ordQ
?£ fcc,£-\ £ [V , fi ] is at place i
-res(P(n£)[V£,\ogQ)) 355
J—res(P(n£)[V£,logQ})
where we have used Bianchi identity [V£, fl£] = 0 in the last line. Given another connection V £ on £, one can check (we leave this as an exercise for the reader, see [Pa]) that ir Q '"(P(ft £ )) - trQ'»(P(n£)) is an exact form whenever the residue res(P(fl£)[V£,log Q]) vanishes for any connection V £ on £. Thus, whenever res(P(fi £ )[V £ ,log<3]) vanishes for any connection Vs on £ (this is the case in finite dimensions of course), then trQ'^(P(Cl£)) defines a characteristic class given by its cohomology class in the de Rham cohomology which is independent of the choice of connection. Under such an assumption, these weighted Chern- Weil invariants indeed define topological invariants as they do in the finite dimensional case. • Cases for which the obstructions vanish: Finite rank bundles. As expected, when £ reduces to a finite rank vector bundle, tr® coincides with the ordinary trace and this residue which is a purely infinite dimensional obstruction vanishes. Hence we recover the finite dimensional Chern-Weil invariants. Weighted loop groups. Let G be a Lie group, M a compact manifold and let us take B := C^iS1^) and £ := TB where the index 0 means the loops are based at a fixed point. As before we choose a weight given by a left invariant family of Laplacians Q1 := L^AL^1 where A is the scalar Laplacian on C°°(S1, Lie(G)) and L 7 left multiplication. Following [F] let us consider a left invariant JET 5 metric built up from the Killing form on the Lie algebra on C°° (S1, G) and the corresponding Levi-Civita connection V s . There is a natural almost complex structure J on H 2 ( 5 1 , G) which commutes with the connection and H 2 (S 1 , G) is a Hilbert Kahler manifold (see [F], [Pr]). The curvature applied to smooth vector fields is a classical P.D.O. of order - 1 and hence has divergent trace. However, its trace along the Lie algebra is a P.D.O. of order —2 which is therefore trace-class. For smooth vector fields U, V, W, one can show that res (p*{U,V)[Vfo,logQ]\ = 0 [Ma] so that trQ(tti) defines a closed form and gives rise to the first Chern class of the weighted complex bundle (£,Q). Since the curvature is a P.D.O. of order — 1, its A;-th power is of order — k and therefore of trace-class in dimension 1 for k > 2. Higher Chern classes are thus well-defined as ordinary traces, or equivalently the weighted traces trQ((fl£) ) coincide with ordinary traces for k > 2 and hence give rise to higher Chern classes. 356
6.5
Conclusion
It is rather frustrating to come across so many obstacles arising as Wodzicki residues preventing a straight forward generalization to the infinite dimensional context of some basic geometric concepts and their properties involving traces. These obstructions are due to the additional data given by the weight, so one might ask whether it is possible to avoid using this extra data. My guess is that it is not since the weight, even if not always explicit in the various approaches (at least the ones that I know of) to the geometry of infinite dimensional (continuous, non discrete) manifolds or vector bundles, it always seems to be hidden in some form or other. Just to quote two striking examples, when using Brownian motion to tackle path spaces, or "conditioned traces" as in [DL] and [F] to handle the geometry of loop groups, there is a weight hidden behind the construction in both cases, the Laplacian in the first case as a basic ingredient to construct Brownian motion, and the "conditioning" of the traces in the second case which can be interpreted as a choice of a weighting procedure for the otherwise ill-defined traces. So there seems little chance of getting rid of these weights. Then a natural question arises, namely whether we can compensate the Wodzicki residues due to these weights in the traces. In certain cases, noncommutative geometry provides a way to counterbalance these obstructions, at least when weights arise as squares of some family of Dirac type operator as in the case of infinite rank bundles built up from families of Dirac operators as in the context of the family index theorem. It would be out of the scope of this article to describe such mechanisms here and we refer the reader to [PR], [Pa] for further developments along these lines.
357
References [ABP] M. Arnaudon, Y. Belopolskaya, S. Paycha, Renormalized Laplacians on a class of Hilbert manifolds and a Bochner- Weitzenbock type formula for current groups to appear in Infinite dimensional Analysis, Quantum Probability and Related topics (2000) [B] J.-M. Bismut, "Localization formulae, super connections and the index theorem for families," Commun. Math. Phys. 103 (1986), 127-166. [BF] J.-M. Bismut et D.S. Freed, The Analysis of elliptic families I, Comm. Math. Phys. 106(1986) p.159-176. [BGV] N. Berline, E. Getzler, M. Vergne , Dirac operators and Heat Kernels, Springer-Verlag, 1997 (2nd. ed.). [BT] R. Bott, L. Tu, Differential forms in algebraic topology, Graduate Texts in Math., Springer Verlag 1982. [C] B.Y. Chen, Geometry of submanifolds, Pure and Applied Mathematics, Dekker (1973). [CDD] Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, manifolds and physics, North Holland Publishing Company, Revised Edition 1982. [CDMP] A. Cardona, C. Ducourtioux, J.P. Magnot, S. Paycha, Weighted traces on algebras of pseudo-differential operators and geometry on loop groups, Preprint 2000 [Co] A. Connes, Non commutative geometry, Academic Press, New York, San Francisco, London 1994. [CFNW] M. Cederwall, G. Ferretti, B. Nilsson, A. Westerberg, Schwinger terms and Cohomology of Pseudodifferential Operators, Comm. Math. Phys. 175 , 203-220 (1996). [D] C. Ducourtioux, Renormalized traces and determinants, Ph.D. thesis in preparation [DF] X. Dai, D.S. Freed, rj invariants and determinant bundles, J.Math. Phys. 35 , 1994, p. 5155-5194. [DL] B. Driver, T. Lohrenz, Logarithmic Sobolev Inequalities for Pinned Loop groups, Journal Funct. Anal. 140 (1996). [EF] R. Estrada, S.A. Fulling, How singular functions define distributions, Preprint June 2000. [EK] R. Estrada, R.P. Kanwal, Regularization, Pseudofunction, and Hadamard Finite Part, Journ. Math. Anal. Appl. 141, p.94-207 (1989). [F] D.S. Freed, The geometry of loop groups, Journal Diff. Geometry 28, p.223-276 (1988) [G] P. Gilkey, Invariance Theory,the Heat Equation and the Atiyah-Singer 358
Index Theorem, Publish or Perish (1984) [GHL] S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer Verlag (1990) (2nd. ed.). [GS] G. Grubb, R. Seeley, Zeta and Eta functions for Atiyah Patodi Singer theorems, Journal Geom. Anal. 6 (1996), p. 31-77. [H] L. Hormander, The analysis of linear partial differential operators, I. Grundlagen der mathematischen Wiss. 256, Springer Verlag (1983). [J] J. Jost, Riemannian geometry and Geometric Analysis, Springer Verlag (1998). [Ka] M. Karoubi, K-theory (An introduction), Springer Verlag 1978. [K] Ch. Kassel, Le residu non commutatif [d'apres Wodzicki], Seminaire Bourbaki 708, Asterisque 1989, p. 199-229. [KN] S. Kobayashi, S. Nomizu, Foundations of differential geometry I, I I , J.Wiley and sons (1963). [KVl] M. Kontsevich, S. Vishik, Determinants of elliptic pseudodifferential operators, Preprint of the Max Planck Inst, for Mathematics in Bonn n.94-30 (1994). [KV2] M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators in Functional Analysis on the Eve of the 21st Century Vol. I (ed. S. Gindikin, J. Lepowski, R.L. Wilson), Progress in Mathematics (1994). [Ku] N.H. Kuiper, The homotopy type of the unitary group of Hilbert spaces, Topology 3 (1965). [L] S. Lang, Differential and Riemannian manifolds, Springer Verlag 1995. [LM] B. Lawson, M.L. Michelsohn, Spin Geometry, Princeton Mathematical Series, 1989. [Le] M. Lesch, On the noncommutative residue for pseudo-differential operators with log-polyhomogeneous symbols, Annals of Global Analysis and Geometry 17, p. 151-187 (1999). [Ma] J-P. Magnot, Ph.D thesis in preparation. [M] J. Marsden, Applications of Global Analysis in MathematicalPhysics, Publish or Perish, 1974 [Mi] J. Mickelsson, Current algebras and groups, New York, Plenum Press, 1989. J. Mickelsson, Second Quantization, anomalies and group extensions, Lecture notes of a course delivered at the "Colloque sur les Methodes Geometriques en Physique", C.I.R.M, Luminy, June 1997. J. Mickelsson, Wodzicki residue and anomalies on current algebras in "Integrable models and strings" ed.A.Alekseev and al., Lecture Notes in Physics 436, Springer 1994. [MN] R. Melrose and V. Nistor, Homology of pseudo-differential operators I. 359
manifolds with boundary, Preprint: funct-an/9606005, Oct. 98. [N] M. Nakahara, Geometry, topology and physics, Graduate student series in physics, Adam Hilger (1990). [0] K. Okikiolu, The multiplicative anomaly for determinants of elliptic operators, Duke Mathematical Journal, 79, n.3 p.723-750 (1995). [Pa] S. Paycha, Renormalized traces as a looking glass into infinite dimensional geometry, Preprint 2000. [PR] S. Paycha, S. Rosenberg, Curvature on Determinant bundles and first Chern forms, Preprint 2000. [Pr] A. Pressley, The energy flow of the loop space of a compact Lie group, Journal of the London Math. Soc. 26 (1982). [PS] A. Pressley, G. Segal, Loop groups, Oxford Univ. Press 1988. [Ql] D. Quillen, The determinant of Cauchy-Riemann operators over a Riemann surface, Funktsional Anal, i Prlozhen. 19 (1985), n.l, p. 37-41. [Q2] D. Quillen, Superconnections and the Chern character, Topology 24 (1985), p.89-95. [Ra] A.O. Radul, Lie algebras of differential operators, their central extensions, and W-algebras, Funct. Anal. Appl. 25, p.25-39 (1991) [RS] M. Reed, B. Simon,Methods of modern mathematical physics, Vol I-IV, Academic Press (1975). [R] S. Rosenberg, The Laplacian on a Riemannian manifold, Cambridge Univ. Press, 1997. [Sc] L. Schwartz, Theorie des Distributions, Hermann 1966, New edition 1978. [Se] R. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math.10 (1967), p. 288-307. [Sh] M.A. Shubin, Pseudo-differential operators and spectral theory, Springer Verlag (1980). [SW] M. Spera, T. Wurzbacher, Differential geometry of Grassmannian embedding of based loop groups, to appear in Journal Diff. Geom. and its Applications (2000). [T] M.E. Taylor, Pseudo-differential operators, Princeton University Press (1981 ) [W] M. Wodzicki, Local invariants of spectral asymmetry, Inv. Math. 75. P .143-178 (1984).
360
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
C O N C E P T S I N G A U G E T H E O R Y L E A D I N G TO ELECTRIC-MAGNETIC DUALITY TSOU SHEUNG TSUN Mathematical Institute, Oxford University 24-29 St. Giles', Oxford 0X1 3LB United Kingdom. E-mail: tsou @ maths, ox. ac.uk Gauge theory, which is the basis of all particle physics, is itself based on a few fundamental concepts, the consequences of which are often as beautiful as they are deep. In this short lecture course I shall try to give an introduction to these concepts, both from the physical and mathematical points of view. Then I shall show how these considerations lead to a nonabelian generalization of the wellknown electric-magnetic duality in electromagnetism. I shall end by sketching some of the many consequences in quantum field theory that this duality engenders in particle physics. These are notes from a lecture course given in the Summer School on Geometric Methods in Quantum Field Theory, Villa de Leyva, Colombia, July 1999, as well as a series of graduate lectures given in Oxford in Trinity Term of 1999 and 2000.
Contents Introduction
363
1 Gauge invariance, potentials,
fields
1.1 Notations and conventions 1.2 Gauge invariance, potentials, fields: abelian case 1.3 Gauge invariance, potentials, fields: nonabelian case 2 Yang-Mills theory in action—the Standard Model of particle physics 2.1 2.2 2.3 2.4
Particle classification The strong interaction The electroweak interaction The Standard Model
3 Principal bundles, connections, curvatures 3.1 3.2 3.3 3.4
Principal bundles Connections and curvatures Bundle reductions Holonomy and loop space variables 361
364 364 365 367 371 371 372 373 375 378 378 380 383 384
Gauge group and charges 4.1 4.2 4.3 4.4
387
Locally isomorphic semi-simple Lie groups Specification of the gauge group Charges and monopoles Examples
387 388 391 395
Action principle and symmetry breaking
398
5.1 5.2 5.3 5.4
Maxwell equations and minimal coupling Yang-Mills equation Wu-Yang criterion Symmetry breaking
Electric-magnetic duality 6.1 6.2 6.3 6.4 6.5 6.6
398 399 400 403 406
Abelian theory The star operation in Yang-Mills theory Generalized electric-magnetic duality for Yang-Mills theory 't Hooft's theorem and its consequences Magnetic monopoles from symmetry breaking Seiberg-Witten duality
References
406 408 412 413 415 416 420
362
Introduction In this lecture course I shall be mostly concerned about some basic concepts of gauge theory, mainly to answer the question: why are they introduced in physics? Almost all of these concepts appear already at the classical level, so that we shall deal principally with the motion of a quantum charged particle in a classical (gauge) field. In so doing we shall find that certain fundamental questions will arise which are usually not addressed by physicists, because they are usually overwhelmed by many seemingly more pressing questions. As a by-product, I shall also try to clarify a few notations to make it easier for mathematicians to read physics literature. Since Maxwell's theory of electromagnetism, that is, abelian gauge theory, is the best understood gauge theory, I shall take it as our reference point. So most definitions will start with the abelian case, and most results will be checked against it. However, we shall be extremely careful in not assuming abelian results for the nonabelian case. For most of the basic concepts, I shall be following my book. Apart from other textbooks both in mathematics and physics which I have found useful, I have included in the bibliography only a few papers. For those who are interested to pursue further, the references in the cited material will easily lead them to other relevant articles.
363
1
Gauge invariance, potentials, fields
In the first lecture, I shall use very simple mathematics only, so as to put the emphasis on the physical reasons behind these concepts. In subsequent lectures, however, I shall not hesitate to use more sophisticated mathematics, because it helps both in further understanding and further developments. In this way, I hope to show you that it is the physical situations which force us to use various sophisticated mathematical tools, and not the other way round, by which I mean looking for or inventing physical theories to apply the mathematical tools we have on hand. 1.1
Notations
and
conventions
I shall use the following groups of terms synonymously: • Maxwell theory, theory of electromagnetism, abelian theory; • Yang-Mills theory, nonabelian (gauge) theory (however, nonabelian may take a truly mathematical meaning); • Spacetime, Minkowski space. I shall use the following notations unless otherwise stated: • X = Minkowski space with signature H • n,v,... • i,j,...
— spacetime indices = 0,1,2,3 = spatial indices or group indices
• repeated indices are summed • G = gauge group = compact, connected Lie group (usually U(n),SU(n),0(n)) I shall make the following convenient assumptions: functions, manifolds, etc. are as well behaved as necessary; typically functions are continuous or smooth, manifolds are C°°. I shall use the units conventional in particles physics, in which h = 1, c = 1, the former being the reduced Planck's constant and the latter the speed of light. Caveat In a fully quantized field theory, particles and fields are synonymous. Where there is no confusion, I shall use the two terms interchangeably. 364
1.2
Gauge invariance,
potentials,
fields:
abelian
case
Consider an electrically charged particle in an electromagnetic field. The wavefunction of this particle is a complex-valued function ip(x) of X (spacetime). The phase of ip(x) is not a measurable quantity, since only |V>(a:)|2 can be measured and has the meaning of the probability of finding the particle at x. Hence one is allowed to redefine the phase of IJJ(X) by an arbitrary (continuous) rotation independently at every spacetime point without altering the physics. We say then that this theory possesses gauge invariance or gauge symmetry. In view of this arbitrariness, how can we compare the phases at neighbouring points in spacetime? In other words, how can we 'parallelly propagate' the phase? We can, if we are given a vector-valued function A^(x), called the gauge potential. Then for the phases at x and a nearby point x + Ax to be 'parallel' we stipulate that the phase difference be eAll(x)Axfl, where e is a proportionality constant which will later be identified with the charge of the particle. This concept of 'paralleUism' has to be consistent with gauge invariance. In other words, if we effect a rotation of eA(x) on the phase of ip(x),
V>(aO •->• e t e A <*ty(aO then the phase rotation at x + Ax^ will be e(A(x)+d^A(x)Ax'i), so that for consistency the gauge potential must transform as A^ix)
^ A^x)
+ d^A.
Next, by iterating the parallel phase transport along a given path T, we can obtain a finite phase difference: e I
A^x)
dx^.
v This depends on the path in general. Now the phase difference (with V =
r 2 - rx) e
A^dx" - e f
A^dx",
at the same point P (see Figure 1) is a physically measurable quantity (as observed in interference phenomena), depends on the potential, and is in 365
Q
Figure 1. Phase transport along a closed loop.
general nonzero. By Stokes' theorem e<j)(T) = - e ff
F„l/(x)dx^dxl/,
where F^(x) = d^A^x) — dtlAl/{x). The quantity -eFfll/(x)dxlidxu is the infinitesimal phase change on going round the infinitesimal parallelogram ABCD, Figure 2: C
D x + dxv
x + da;'1 + dxu
x + dx^ A
- ^
B
Figure 2. Infinitesimal transport
It follows that the tensor F^ is a gauge invariant quantity, as can be checked directly. Now this change in phase, apart from the factor e representing the charge of the particle in question, acts on all wavefunctions in a universal way and is hence a physical property of the spacetime under consideration. It represents therefore a physical field, called the electromagnetic field tensor. 366
Historically, in classical physics, it was the components of this field tensor, the electric and magnetic fields, which were introduced first: '
p _ *>*" ~
0 Ei E2 E3 \ — E\ 0 —B3 B 2 -E2 B3 0 -B1 • \-E3 -B2 Bi 0 /
The potential was introduced as a mathematical convenience only, since in classical physics it is not necessary to specify the potential, only the field. Bohm-Aharonov
experiment
To demonstrate that in order to describe fully quantum mechanics in electromagnetism one needs the potential, not just the field, Bohm and Aharonov devised the following experiment, successfully performed by Chambers. Electrons are made to travel along two different paths Ti and T2, as
Magnetic Core
Source p
Diffraction Patterns
—0
Screen Figure 3. Schematic Representation of the Bohm-Aharonov Experiment.
illustrated in Figure 3, enclosing a magnetic core. Outside this magnetic core, the field vanishes but the potential is nonzero. A diffraction pattern, made by the interference of the two beams, is observed on the screen beyond the magnetic core, and this diffraction pattern varies as the magnetic field and the paths are varied, according to the phase difference we calculated above. This is one of the most important experiments of modern physics, and its positive outcome is widely regarded as the strongest single evidence supporting the basic tenets of electromagnetism as a gauge theory. 1.3
Gauge invariance,
potentials,
fields:
nonabelian
case
Yang-Mills theory, as originally proposed by Yang and Mills in 1954, is a generalization of electromagnetism in which the complex wavefunction 367
tp(x) of a charged particle is replaced by a wavefunction with 2 components V> = {ipl(x), i = 1,2}. By a change of 'phase' we now mean a change in the orientation in internal space of tp under the transformation: t/j i->
Sip,
where, to preserve probabilities, S is to be a unitary matrix. In this way, gauge invariance can now be interpreted as the requirement that physics be unchanged under arbitrary SU(2) transformations on ip independently (but continuously) at different spactime points. We note that it is also possible to have the more general U(2) transformations. We can now consider parallel transport of this 'nonabelian phase' in a way similar to the abelian case. Introducing the gauge potential A^{x) as a matrix-valued spactime vector-valued function, we again stipulate that at two neighbouring points x and x + Ax, the 'phases' are parallel if they differ by gAfi(x)Ax'Jj, where the porportionality contant g again represents a 'charge': ip(x + Ax) —
exp(igAIJ,(x)Ax'J')ip(x).
Note that now A^ (x) € su(2), because it represents an infinitesimal change in the phase of ip. Under a gauge transformation S(x):ip'(x) = S(x)ip(x), so that ip'(x + Ax) - S{x + Ax)ip(x + Ax). But ip'(x + Ax) = exp(igA'>1(x)Ax'i)
tp'(x),
where A'^ is the gauge transform of A^. Equating, we get, for all ip(x), S(x + Ax) exp(igAIJ,{x)Ax'J') ip(x) = expiigA'^Ax1*)
S(x) ip{x).
Expanding to leading order in Aa; and dropping ip(x), A'^x)
= S(x) A^x)
S-^x)
S-X(x).
- ( i ) d^Six)
When S(x) is infinitesimal, in the sense S(x) = exp(igA(x))
~ 1 + igA(x)
then we get A'pix) = A„{x) + d„A{x) + ig[A(x),
A^x)],
which reduces to our previous formula in the abelian case. 368
Next, we consider transport over a finite path T as before. Remembering that our primary concern is how the wavefunction ip changes, we see that what matters is the limit of the product of the matrices S as the step size tends to zero, and not the sum of the 'phases' A. (Remember we are being unsophisticated!) For matrices, in general eA+B ^ eA eB, so that the answer we are after is not expig / Afj,(x) but the product of S as we said. However, for sentimental reasons about the abelian case, this product is usually written
$(T) = Pexpigf
A^x),
where the letter P denotes path-ordering. This Yang called the Dirac phase factor, it is also known as the Wilson loop. For a closed infinitesimal path in the form of a rectangle ABCD, Figure 2, we can evaluate the change in phase as before (where repeated indices on the right are not summed): + dx^ + dxv) = exp(igA1/(x + dxfl)dxl/) • exp{igAll(x)dx'1) i>ADc(x + dx*1 + dxu) = exp(igAfl(x + dx")dx^) • exp{igAv(x)dxu) ^ABC{X
ip(x) ip(x).
Hence the phase difference at A, on expanding to leading order, = {g(d„Au - dvA„) + ig*{AvA„. - A^)}
dx* dx".
As before, we can define the field tensor F/il/(x) by equating this to —gFllv(x)dxti
dxv,
whence FpV{x) = dvAil(x)
-
dpAv{x)+ig\Ap(x),Av{x)\.
By similar considerations as before, we see that under a gauge transformation S(x), FM„ transforms as Flv{x) =
S{x)FIMV{x)S-1{x).
We see that F^ is no longer gauge invariant, as in the abelian case, but gauge covariant. 369
Important Remarks 1. In the above description of classical electromagnetism the gauge group symmetry (that is, the phase freedom) plays no role (or only a trivial one) if there are no charges, because FM„, the physical fields, are gauge invariant. However, when we consider the dynamics in the form of the Maxwell equations, then this symmetry is the symmetry of those equations, and is therefore an important ingredient even in the classical field theory without charges. This remark does not apply to Yang-Mills theory, since there the field F^ being covariant and not invariant the gauge symmetry should be taken into account already at the outset. 2. We note an extremely important difference betwen the abelian and the nonabelian case: the finite phase difference is no longer related to the surface integral of the field tensor in the nonabelian case. First, the Dirac phase factor is not given by a line integral along the closed path T. In fact, the line integral has no significance at all in the nonabelian case. Moreover, the surface integral of F^ does not make sense, since there is no way to order the matrices F^ on a surface! 3. Because it is only covariant and not invariant the field tensor here is no longer a physically measurable quantity, not even in classical physics. 4. Yang has shown that, both in abelian and nonabelian theories, what describes the physics exactly is the Dirac phase factor $ ( r ) in the sense that the same physical situations correspond to identical $ ( r ) and different physical situations to different $ ( r ) . The obvious question then arises: why do we not use $ ( r ) as variables to describe Yang-Mills theory and forget about the gauge dependent potential ^4M(a;)? The answer is: the space of of closed loops is infinite dimensional, which means that $ ( r ) is more difficult to handle, and there are vastly too many $ ( r ) variables. Roughly speaking, we would be using functions of infinitely many variables as opposed to 4 functions of 4 variables! We shall have a bit more to say on this later.
370
2
Yang-Mills theory in action—the Standard Model of particle physics
Before we proceed any further and in order to exhibit the pivotal role that Yang-Mills theory plays in modern physics, I have to tell you some facts about particles. This lecture is strictly for mathematicians only! 2.1
Particle
classification
Particles are classified by their interactions among themselves and with the fields in spacetime. Apart from gravity, which we shall neglect totally in these lectures, there are two kinds of interactions: strong and electroweak. Both are gauge theories, but with a significant difference. We shall take them in turn presently. Fundamental particles are also distinguished by their spin. Particles with integral spins (in suitable units) are called bosons, and those with half-(odd)integral spins are called fermions. Known bosons all have spin 1, and are sometimes called vector bosons. There are some spin 0 bosons, called scalars, which are postulated to exist but have not yet actually been detected. All known fermions have spin | . Fermions which take part only in electroweak interactions are called leptons. There are many (all unstable) particles called resonances which are composites of those above. (The only possibly stable composite is the proton.) They are mainly divided into mesons (such as pions) or baryons (such as protons, neutrons), but we shall not study them here. There are more than 150 such so far discovered. For our purposes, the following are the fundamental particles: Vector bosons (also known as gauge bosons): 7; W+, W~,Z°; g (photon; massive vector bosons; gluons) Leptons: e, ve\ /i,t"M; T,vr (electron, electron neutrino; muon, muon neutrino; tauon, tau neutrino) Quarks: u,d; c,s; t,b (up, down; charm, strange; top, bottom) In a full quantum theory, these particles all have corresponding antiparticles. The notation is: the antiparticle of x is denoted x. Remarks Fact 1. All known fundamental bosons have spin 1. 371
Fact 2. All known fundamental fermions have spin | . Fancy 1. (—> Fact 3.) Theory postulates existence of certain scalars called Higgs particles. Fancy 2. Supersymmetry needs spins 0, | , 1, §, 2. Terminology changes as our understanding goes further. 2.2
The strong
interaction
The strong interaction gives rise to nuclear forces and is governed by an SU(3) gauge theory. This gauge symmetry is fancifully called "colour", and so the corresponding quantum field theory is usually called quantum chromodynamics or QCD. The gauge potential A^ takes value in the Lie algebra su(3), and has hence 8 components. Interpreting these as particles (when the field is fully quantized), they form the 8-dimensional adjoint representation of SU(3). They are called the gluons. They are vector bosons and are massless. They do not have electric charges. The massive particles are in the 3-dimensional fundamental representation of SU(3). These are called quarks. They are spin | fermions, and have charges of | or | , in units of the electron charge. There are 6 known species of quarks: u,d; c,s; t,b; each of which has 3 components corresponding to SU(3). These components are said to have different "colours", e.g. red, green and blue. These have no relation to the genuine colour that we see. Just as the phase in electromagnetism can be arbitrarily rotated at each spacetime point, and so cannot be measured, so is this colour symmetry. We say in both cases that the gauge symmetry is exact. Where QCD differs from QED is that particles with nontrivial colour charges, i.e. in any representation of SU(3) other than the trivial one, cannot exist in the free state. They cannot therefore be directly observed, but indirectly their existence is fairly well established experimentally. This peculiar property goes under the name of confinement and is special to nonabelian theories. To prove confinement is one of the most important aims of theoretical physics at present. Experimentally this is true so far, since only singlets of SU(3) have ever been observed. This is the probable origin of the name "colour", as in the hidden colours of white light. Quarks and antiquarks combine to produce observable particles as resonances. We can see which combinations are possible by looking at the tensor product of the fundamental and conjugate fundamental representations and 372
picking out the singlet. For example: 3®3 = l©--3 ® 3
The electroweak
interaction
The electroweak interaction gives rise to both electromagnetic phenomena and radioactivity, and is governed by a gauge theory with a gauge group usually denoted as SU(2) x 17(1). (We shall give more details about the specification of the exact gauge group in Lecture 4.) The SU(2) part is often referred to as (weak) isospin, and the U(l) part as (weak) hypercharge. In electroweak (or Weinberg-Salam) theory an important novel ingredient is introduced in Yang-Mills theory, that is, (spontaneous) symmetry breaking. In addition to the gauge bosons (vector bosons) and the massive fermions as in QCD, we introduce some scalar (i.e. spin 0) particles called Higgs fields <j>. They are in a 2-dimensional representation so that they are in fact gauge spinors:
-(?)• The lowest energy state (called vacuum) occurs when <j> jt 0, so that a physical system in such a vacuum state corresponding to o will no longer be invariant under the whole of SU{2) x U(l), only under a U(l) subgroup which leaves the spinor (f>0 invariant. This is the situation of symmetry breaking: although the theory has SU(2) x 1/(1) invariance, the actual physical system has a smaller invariance. This 17(1) subgroup is generated by a linear combination of T3 and Y, where Ti,T2,T3 are the generators of su(2) and Y is the generator of weak hypercharge (7(1). (We shall go into more details later.) Reminder. The notation for Tj is exactly the same as for ordinary spin, where T3 is represented by the diagonal matrix
373
and the commutation relation is [Tj,Tj-] = eij^TkThe residual gauge group U(l) is identified with the U{\) gauge group of electromagnetism, as observed in the physical world. The mechanism of symmetry breaking is often visualized as a phase change. In the early universe when the temperature was high, the whole SU(2) x U(l) symmetry was exact. As the temperature decreased to a critical value, symmetry breaking occurred and the electromagnetic gauge symmetry "froze out" to produce the phase we are in today. As a result of the symmetry breaking, 3 of the 4 gauge bosons combine with some components of the Higgs doublet
VL
+ V'H-
Since ip has spin | , it is a spacetime spinor and ipL and ipR are eigenstates of the chirality operator j 5 . Then the representation assignments are:
SU(2) doublets L
eR
\r
/
HR
L
\
•
/l
TR
SU{2) singlets
Notice that neutrinos are supposed to have only left-handed components in this assignment. In view of the recent Super Kamiokande results on neutrino oscillations, it may be necessary to revise this assignment and suppose the neutrinos have also right-handed components (and a nonzero mass).
374
Table 1. Particle content of the Standard Model
Gauge symmetry Gauge bosons Force (gluons) Strong SU(S) (QCD) Electroweak SU(2) x 17(1) r,w+,w-,zu (Weinberg — Salam)
2.4
The Standard
Matter (quarks) leptons (Higgs)
Model
The particle spectrum is summarized in Table 1. Note that the particles in brackets are not (or have not been) directly observed, but they are part of the gauge theory. The standard model is an amalgamation, a knitting together, of the above two theories in such a way that all of known particle physics, up to the present day, is encompassed. It is a Yang-Mills theory with a gauge group which is usually written as SU(3) x SU(2) x [/(l)—we shall examine this in more detail in Lecture 4. The particle content can be schematically represented as (QCD + Weinberg-Salam) x 3. The multiplication by 3 is necessary to model another aspect of the particle spectrum known as generation. Take the charged leptons as an example. There are 3 of them: e, yu, r. Except for their very different masses: mr-.m^rrie^
3000: 200:1
they behave in extremely similar fashions. The 3 neutrinos ve,u,j., vT also have similar interactions. The quarks also come in 3 generations: u,c,t with charge I (called the up-type quarks) and d, s, b with charge — | (called the down-type quarks). But the standard model is not just putting the 2 theories together, because although the leptons do not transform under SU(3) (since they have no strong interaction), the quarks are in nontrivial representations of weak isospin SU(2). In fact, we can set up Table 2 for the 3 generation, and Table 3 for the lightest generation (and similar ones for the other two generations). In Table 3 the electric charge satisfies: Q = I3 + \Y. Other normalizations exist in the literature. 375
Table 2. The 3 generations of leptons
Quarks
Leptons
{d)L'UR'dR
(?),•«"
(Cs)L'CR'SR
( ; * ) , -
(l)^n,bR
(7).-
Table 3. Charges of leptons
SU{Z) 3 Ul 3 di UR 3 3 dR 1 VeL 1 ez, 1 eR
SU(2); 2- I ^
2
4 |
2
-±
i; o
h t / ( l ) y tf(l)em 1 a 3 3 1 i
2
1;0 2 I > 2 2- - ± *' 1; 0 2 Z
376
32 3
43 3 3
-1 -1 -2
I 3
0 -1 -1
The standard model is a very good representation of the state of our present knowledge, but most physicists doubt that it constitutes a full final theory. G o o d points 1. Purports to explain all of particle physics. 2. Survives all precision tests so far. Open questions 1. Why 3 copies (generations)? 2. Where do the Higgs fields come from? 3. Why are quarks confined? 4. There are over 20 parameters which are not explained by the theory.
377
3
Principal bundles, connections, curvatures
In this lecture we change tack altogether and make contact with differential geometry. Most mathematicians working on Yang-Mills theory work on vector bundles, but following Yang I tend to think in terms of principal bundles. The two ways are of course equivalent. In order to simplify definitions and so on and avoid all unnecessary troubles, I shall make the following general assumption: 'Things are as nice as possible.' For example, manifolds and maps are smooth, Lie groups are compact connected, equivalence classes can be confused with their representatives. Also, no formal proofs will be given. 3.1
Principal
bundles
Definition. A principal coordinate bundle V is a collection of the following: 1. a manifold P called the total space, 2. a manifold X called the base space, 3. a projection w.P—tX,
with
TT~1(X),X
£ X, called the fibre above x,
4- a Lie group G acting on itself by left translation, called the structure group, 5. an open cover {Ua}a&A °f Xy 6. Va 6 A, a diffeomorphism called coordinate function
Va; e Ua,g G G,
(b) if we define Vx £ Ua
<-
71 M/
X Figure 4. Sketch of a principal bundle.
(c) the map (ppa:UanUp
-+G
is smooth—it is called the transition or patching function. A sketch (Figure 4) may be helpful. Definition. Two principal coordinate bundles V and V are said to be strictly equivalent if they have the same total space P, the same base space X, the same projection •K, the same structure group G, and their coordinate functions {(f>a), {
£Uar\U'p
is left multiplication by an element of G. Definition. A principal bundle is an equivalence class of coordinate principal bundles under strict equivalence. 379
Definition. A trivial principal bundle is one in which P^X
xG.
This is the case in most applications to physics. N o t e . We shall often call P the principal bundle. Dictionary 1 base space structure group principal bundle principal coordinate bundle 3.2
Connections
and
spacetime gauge group gauge theory gauge theory in a particular gauge
curvatures
First we recall a few definitions. 1. Give a map / : X -> X' we can define its differential /* at x £ X, as a linear map TXX -> Tf(x)X' as follows. Given a tangent vector U at x, choose any curve x(t) in X such that x(0) — x and U is the tangent to x(t) at x. Then the image /*£/ is the tangent vector to the image curve in X' at f(x). It can be shown that the definition is independent of the curve x(t). Similarly, given any 1-form w' on X', we can define a 1-form f*u' on X by (f*cj')V = w ' ( / . n for any vector field F o n l . 2. Denote by La the left multiplication by an element a £ G. Let g be the Lie algebra of G. A vector field A on G is said to be left invariant if (La)*A — A, Va € G. Recall then that g can be characterized as the set of left invariant vector fields on G. 3. Suppose a group G acts on the right on a manifold P. Then for each A € g, the action induces a vector field a (A) on P as follows. At each u £ P, consider the action of the 1-parameter subgroup expt.,4, whose orbit is a curve in P passing through u at t — 0. The tangent to the curve at u is the required vector. We call a(A) the fundamental vector field corresponding to A € 0380
Now we come to the definition of a connection on P. Consider the action of G on P given by, Va e G, u £ P, Ra{u) = (f>a{x,((f>~^{u))a), where x = n(u) £ Ua. Note that this action moves points along the same fibre, and is indeed a right action since Raia2(u) = Ra2(Rai(u)), u £ P. We also write: Ra(u) = ua. Definition. Given a principal bundle P as above, a connection 1-form w on P is a g-valued 1-form on P satisfying 1. w(«r(A)) = A,VA£
g,
2. oj((Ra)*V) = ad(a - 1 )w(V), Va £ G,V vector fields V on P, where the adjoint action ad(a _ 1 ) of a-1 on A £ g is often written as a~1Aa. We wish now to show how to define, using a set of local sections {Ua}, local 1-forms {u)a} which are equivalent to the given connection UJ. Choose local sections ua:Ua —• P such that cf>a(x,e) = ua(x),
x
£UaC\U0.
Then up{x) = ua{x)
-»• TU(X)P.
We have dup(V) = dua{V)
(V) ua(j>0a(x)(4>0a(x))-1d(t>pa(V) u0(x)(<j)0a(x))~1d(l>0a(V).
So acting with w on both sides we get
up{V) = uj{dua(V)(j)0a{x)) = ad(^(x))ua(V)
+ (^{u^x^paix))-1
d(j>pa{V)
u{ue{x){(j>0a{x))-1d(j>!3a{V).
+
Now i^paix))"1 d
— -Ofj,j S 381
In fact, since <j>pa goes from patch a to patch /?, and S goes from unprimed to primed patch, we should re-write this last formula by using W = S~l:
which, apart from the physical dimensional the first formula. See [O] for reference. In particular, we see that u> which is a collection of 1-forms oja on X. In the special trivial, we need only one such 1-form on X.
factor -, is exactly the same as
1-form on P can be replaced by a case when the principal bundle is This is the usual case in physics.
A connection w on P defines a decomposition of the tangent space at each point u £ P into a vertical and a horizontal subspace: J-u
=
Vu ©
flu,
where Vu consists of all those tangent vectors which are tangent to the fibre through u, and 7iu those tangent vectors annihilated by u>. Definition. p-form r\ by
We can now define the exterior covariant derivative Dr] of a DV(VU...,
Vp+1) =
(dri)(hVlt...,hVp+1),
where hVi denotes the horizontal component Definition.
ofV.
The curvature 2-form il of the connection u> is defined by Cl =
DOJ.
Theorem. (The structure equation of Cartan)
du(X,Y) = -%[u{X),uj(Y)} + n(X,Y),
X,YeTu(P),
u£P.
Theorem. (Bianchi identity)
Definition.
We say that a connection OJ is flat if Q, = 0.
Definition. One can also define local curvature forms: Qa =
Doja.
It can then be shown that the following patching condition holds: Up = a d ( < ^ ) f t „ . 382
Dictionary 2 connection curvature
<—> gauge potential <—> gauge field
Translation. (The structure equation of Cartan) FpV = dvA^ - d^Av + ig [A^, Av]. v
'
"curl" (This formula appeared in Lecture 1.) Translation. (Bianchi identity) DpFvp + DVFPI1 + DpF^
= 0,
where D^F^p = dilFvp - iglA^ 3.3
Bundle
reductions
Let P be a principal bundle with structure group G and let H c G be a subgroup. We say that P is reducible to H if there exists an open cover of X such that all the transition functions
<—> symmetry breaking <—l Higgs fields
Examples. To be given later. 383
3.4
Holonomy
and loop space
variables
Consider a principal bundle P A X. Let £(s), s = 0 —> 27r, be a piecewise differentiable curve in X. Then a horizontal lift of £(s), denoted by £*(s), is a curve in P such that TT(£*(S)) = £(s), and all its tangent vectors are horizontal. Through any point u € P such that ir(u) = £(0) there is a unique horizontal lift of £(s) which starts at u. Suppose now we are given a curve £(s) in X starting from xo and ending in X\. Let uo be an arbitrary point in 7r -1 (:ro), a n d consider the horizontal lift £*(s) of £(s) through t*o- Let ui be its end-point, so that we have ir(ui) = x\. Thus the horizontal lift defines a map 7r-1(a;o) —>• T _ 1 ( X I ) which we call the parallel transport of the fibre above xo to the fibre above x±. It can be proved that (i) this map is an isomorphism 7r -1 (:ro) —> ^~1(xi) and that (ii) it is independent of the parametrization of the curve £(s). [KN] p.70. Suppose next that the curve £(s) in X is closed, that is x\ = XQ. Then the parallel transport is an isomorphism of 7r -1 (:ro) to itself. By considering all piecewise differentiable closed curve through a;o, it is easy to see that the parallel transports form a group of automorphisms of 7r -1 (:ro), called the holonomy group $(uo) through u 0 , which can be identified with a subgroup of the structure group G. It can be shown that if X is connected, then all holonomy groups through any given u 6 P are conjugate to one another and are hence isomorphic. Therefore, if we are concerned only with the abstract holonomy group for a given connection and not which particular subgroup of G it is, then we can omit the reference to UQ. We can thus consider simply the holonomy $(C) of a closed curve C in X as an element of G. Dictionary 4 holonomy <—> phase factor (Compare Lecture 1.) N o t e . The curvature at a given point can again be thought of as the holonomy of an infinitesimal closed looop through that point. Flat
connections
Suppose X is connected. If X is not simply connected, then a flat connection may give rise to nontrivial holonomy. In fact, there is a 1-1 correspondence: gauge equivalence classes "1 1-1 f conjugacy classes of irreducible 1 of flat connections J 1 representations of iri(X) ->• G J 384
Let fixX be the space of closed piecewise differentiable loops in X, called the loop space of X. For convenience, we always consider loops starting and ending at a fixed point xo- Given any connection w, the holonomy $ defines a map
which satisfies the composition law. This means that given two loops C\ and C2, We can compose them to give a third loop C\ 0C2, by first going round C\ for s = 0 ->• 7r and then going round C2 for s = 7r ->• 27r, for instance. Then we have $(CioC2) = $(C2)*(Ci), where the product on the right-hand side is group multiplication. The converse problem: given a map <J>: fi1 —> G satisfying the composition law (and some other obvious conditions), can one define a connection of which $ is its holonomy? This problem has great importance for Yang-Mills theory and is to a large extent solved. However, the known proofs are all hard and not totally rigorous. We shall not present them here. It is, however, interesting to see why the problem is of importance to Yang-Mills theory. I mentioned in Lecture 1 that Yang proved that the Dirac phase factor describes gauge theory exactly, in the sense that there is a 1-1 correspondence between {<&(C)} i—> {physical configurations YM}. This is in contrast to the variables A^ which depend on gauge (which means coordinate bundle in the language of this Lecture), or the variables FM„, which cannot distinguish all physically different situations. However, we have to be extremely careful in not confusing this with the concept of redundancy. A moment's thought will tell us that not all maps $:&
->-G
come from a connection, and further study will reveal the fact that adding the composition law (and certain other obvious conditions) is not enough. In other words, we have to impose constraints on the variables $. The situation can be summarized as follows. If we use the variables A^, then the physically relevant objects are equivalence classes of A^ under gauge equivalence. If we use the variables $ ( C ) , then we have to find the relevant subset by imposing constraints. Depending on the problem at hand, it is 385
sometimes easier to deal with the quotient space (the case of A^) or to deal with a subspace (the case of $(C)). I shall now give a rough non-rigorous description of what the main constraint is. Given a closed loop C, we can make a (^-function variation at any point s on the curve. As the height of this (5-function goes to zero, we can then define the loop derivative <£M(s) of any loop-dependent quantity. Remembering that $(C) is an element of G, we define the Lie algebra-valued quantity F M (C, s) as the logarithmic derivative of $(C):
Ffl(C,s)=l-*-1(C)(6fi(sMC)). This can be illustrated by the sketch in Figure 5. In asmuch as this quantity carries the phase factor from one loop to a neighbouring loop, it is
Pa
Figure 5. Sketch of loop derivative.
like an infinitesimal phase transport and can indeed be regarded as some sort of "connection" in a coordinate bundle over f2 1 (X). We can even go further and consider its holonomy, this time of a closed loop in f2 x (X), which means a closed surface S in X. The constraint we are after is that this connection F^iC^) be flat. We note that this does not necessarily imply that the corresponding holonomy is trivial. Any such nontrivial holonomy can be interpreted as a nonabelian magnetic monopole charge, as we shall see.
386
4
Gauge group and charges
4-1
Locally isomorphic
semi-simple
Lie
groups
So far, we have been rather vague about the exact gauge group that occurs in a particular Yang-Mills theory. A particularly interesting, but often neglected, aspect is the different choices of Lie groups which correspond to the same Lie algebra. In asmuch as the Lie algebra can be identified (as a vector space) with the tangent space at the identity, it is clear that groups which are locally isomorphic (that is, quotients by discrete subgroups) have the same Lie algebra. In fact, for semi-simple groups, among all locally isomorphic groups there is one which is simply connected and which is the universal cover of all the others. These latter can then be obtained from the universal cover group by factoring out by various subgroups of its (discrete) centre. Some examples will make this clear. 1. Consider the matrix groups 5 0 ( 3 ) and 5f/(2). The Lie algebra of 5 0 ( 3 ) consists of 3 x 3 skew matrices, and we can choose as generators: /O Y1
=
00\
o o i , \o-ioy
/
y2 =
001\
/
010\
o o o , y3 = - i o o I , V-iooy \ 000/
which satisfy the commutation relations [Yi,Yj) = eljkYk. The Lie algebra of SU(2) consists of the trace-free skew hermitian 2 x 2 matrices, and we can choose as generators: where <7; are the Pauli matrices <7l =
(io)' *2=(°~o)'
(o-i)-
They satisfy [Xi,Xj] = eijkXkHence the two Lie algebras are isomorphic. The groups are not isomorphic, only locally isomorphic. In fact, there is a 2-1 map SU{2) —> 50(3) in such a way that 517(2) is a double cover of 5 0 ( 3 ) . This map can easily be worked out explicitly (Exercise). 387
More generally, we can think of SU(2) as the unit sphere S3 in l 4 , by identifying SU(2) with unit quaternions (SU(2) £ Sp(l)). Then the 2-1 map corresponds to identifying antipodal points of S3. Furthermore, this discrete quotient is by the centre Z2 of SU(2), and this discrete group is the fundamental group of 5 0 ( 3 ) . 2. Very similar considerations apply to SU(N), with centre Z#- In the case of SU(3), we have altogether 2 locally isomorphic groups: SU(3) and S£/(3)/Z 3 . In the case of SU{6) we have 4: SU(6), 5f/(6)/Z 2 ,5t/(6)/Z 3 ,
SU(6)/Z0,
where Z2 and Z 3 are subgroups of the centre Z63. The group SU(2) x 1/(1) is a double cover of (7(2), the covering map being given by multiplication as follows. First embed U(l) ->• U{2) by ia e
feia 0 \ ^ ^ 0 eia J •
Then ((ab\ \\cd)
5C/(2) x 17(1) ^ 4 U(2) (eia 0 \ \ (aeia beia\ ia iQ 'V 0 e )j ' " ^ Vce deia J "
II
II
/
9
We see immediately that (/, g) and (—/, —g) have the same image in U{2). 4-2
Specification
of the gauge
group
Recall (Lecture 1) that gauge invariance comes about as the invariance of the wavefunction of a charged particle under the action of a group G, so that to specify G one has to examine all the charged particles occurring in the theory, in other words, its spectrum. Start with electromagnetism. Under a phase rotation, ip n- eieAi[), so that we can parametrize the circle group U(l) corresponding to the phase by [0,27r/e]. Next suppose there are charges e i , . . . , eft in the theory; then tpr H> eierA-ipr, r = 1, ...,&. If the charges are commensurate, that is, if there exist e and integers nr such that
388
then again we can parametrize the f/(l) by [0,2-ir/e], because if A changes by any integral multiple of 27r/e, the wavefunctions corresponding to all the charges will be unchanged. If, however, there is at least one pair of charges whose ratio is irrational, then we no longer have U{1) as a gauge group. In fact, charge quantization is equivalent to having U{1) as the gauge group of electromagnetism. On the other hand, if we consider pure electromagnetism without charges, then the only relevant gauge transformation are those of A^: A^ n-> Af, + dMA, so that the group will just be the real line given by the scalar function A(x). Similar considerations apply to nonabelian theory. In the vast majority of cases, from the physics point of view, one knows the Lie algebra, and then one needs to inspect the spectrum to get the correct Lie group. One must bear in mind that this implies, for any given Yang-Mills theory, that if in future the spectrum is changed for any reason, one may have to consider another Lie group instead. Consider first a pure Yang-Mills theory without charges, so that the only gauge transformation one needs to consider is on the gauge potential A^x):
A„^SA^S-1--dllSS-1.g Let G be the universal cover of all the groups corresponding to the given Lie algebra. Then the effects on A^ of S and 7 5 , where 7 is an element of the centre Z of G, are identical. Hence the correct gauge group must be G/Z, which is in an obvious sense the smallest of all the possible groups. So in the example (1) we considered, the group is 50(3) and not SU(2). On the other hand, if the su(2) theory contains particles with a 2component wave function tp = {ipi, i = 1,2}, then
i/)^Si>, s e su(2) and the effect of S and —S are not identical. Hence in this case the correct gauge group is indeed SU(2) and not 5 0 ( 3 ) . These considerations can also be cast in terms of representations. Charged particles in a Yang-Mills theory are in certain representations of the gauge group. What we are saying is the known result that the collection of all representations determines the group. In the above case, the gauge potential is in the 3-dimensional adjoint representation and the 2-component i/> is in the 2-dimensional spinor representation. In the absence of the spinor representation, the group is 5 0 ( 3 ) , but when spinors are present, the group must be 389
SU(2). This representation theory is entirely identical to the theory of spin and angular momentum in quantum mechanics. We now apply these considerations to the Yang-Mills theories occurring in particle physics. For these we suppress the gauge couplings g for convenience. 1. Strong interaction. Because we postulate the existence of quarks, which are in the 3-dimensional fundamental representation of SU(3), we conclude the gauge group is indeed SU(3). 2. Electroweak interaction. means 'weak isospin'):
The particles are of 2 types (where 'flavour'
(a) SU(2) flavour doublets with half-integral weak hypercharge—T3 = ± i , Y = §, k odd; (b) SU{2) flavour singlets or triplets with integral weak hypercharge— T 3 = 0 , 1 , Y = k. Under a gauge transformation generated by the generators T3 and Y, we have for the two types of particles: (a)
(exp 2-KITS) ip = (exp in) ip = — ip
(b)
(exp2niY)ip = (expin)ip — —ip (exp 2niT3) ip = (exp in) ip = tp (exp 2niY) ip = (exp in) ip — ip,
so that the resultant we conclude that in (/, g) = (—/, —g), so is 17(2). However, if in future
action of T3 + Y in both cases is the identity. Hence the group SU(2) x U(l) we should identify pairs that the correct gauge group for electroweak theory we either discover or postulate more particles, e.g.
(c) SU(2) flavour doublets with integral weak hypercharge, and/or (d) SU(2) flavour singlets or triplets with half-integral weak hypercharge, then the effect of (/, g) and ( - / , —g) on these particles are distinct, and in that case the correct group is SU{2) x {7(1). 3. Standard model. Similar considerations of the known/postulated spectrum, as given in Lecture 2, will show that we should have a 6-fold identification in SU(3) x SU(2) x (7(1), where the following 6 triplets should be identified: (c,/,y), (cci ,/,3/j/i), (cc2 ,f,yy2) ,(c,ff,yy), 390
(cci ,ff,yyyi),
{cc2
Jf,yyy2),
where c, / and y are elements respectively of SU(3), SU(2) and with:
U(l),
cr - e x p — ^ A 8 , r = 1,2; / = exp27riT3j
yr = exp 4-irirY, r — 1,2; y = exp 67riy, and '10
0'
A8 = 4 = | 01 0 | , v^loO-2i'
T3=(J_J), ^ " V O - i y
Y= *
6'
with obvious embeddings in SU(5) and abuse of notation (same symbol for generators of different representations). 4-3
Charges
and
monopoles
We have known for a long time what the electric charge is. There are several equivalent ways of defining or describing it. For our purpose here, we shall consider it as giving a nonvanishing right hand side to Maxwell's equation: where the current j M is given in one of two ways 1 : .p _ j e J " d r ( d y / d r ) classical quantum ~ I1etpj^ip The quantity e here in fact plays a double role: (a) it is the electric charge—it determines how the charge interacts with the field, and (b) it is the coupling constant—it fixes the strength of the interaction. In Yang-Mills theory, a non-abelian electric charge (sometimes referred to generically as a "colour electric charge") can also be thought of as giving a non-vanishing right hand side to the Yang-Mills equation:
1 In the above I have introduced the gamma matrices 7**, which are important ingredients in Dirac's theory of the spin | particles and which provide a prosaic way of using Clifford algebras. For lack of time I shall not expand into the subject, but they are treated in depth in Hijazi's lectures. See also the lectures of Langmann.
391
where j ^ = gipj^tp f° r t n e quantum particle. [In Yang-Mills theory, one does not usually concern oneself with classical charges.] But here the two roles (a) and (b) are quite distinct. The wavefunction ip is in a certain representation of the gauge group G, and how the particle interacts with the gauge field is determined by the representation, the interaction being given by the covariant derivative. The coupling constant g, on the other hand, is the numerical factor which fixes the strength of the interaction. We see that both abelian and nonabelian electric charges occur as nonvanishing currents. In order to make a practical distinction between these and the topological charges we shall discuss next, let us call them electric charges, or simply charges when there is no confusion. There is another type of charges called monopoles. They are typified by the magnetic monopole as first discussed by Dirac in 1931. To understand them better, let us look at Maxwell's equations both in the 3-vector and 4-vector notations: div E = p curl B - dE/dt = 3
curl E +
div B = 0 \ dB/dt = 0 /
dvF»v = - j " v °v F» = 0.
Here the charge density p and the electric current J together form the 4current j ^ . We also define the dual field tensor *FV,V by
where e^vpa is the totally skew symbol with the convention that g 0123 = — 1. At the classical particle level, these equations simply tell us the experimental fact that magnetic charges, called magnetic monopoles, do not exist in nature. If, on the other hand, we are concerned with quantum particles, then the Bohm-Aharonov experiment (Lecture 1) tells us that we have to introduce the vector potential A^ bearing the relation with the field FM„ as
Simple algebra will tell us that this implies 5 l/ *F /i " = 0 as above. Hence we conclude that: 3 monopole ==> A^ cannot be well defined everywhere. The result is actually stronger. Suppose there exists a magnetic monopole at a certain point in spacetime, and without loss of generality we shall consider 392
a static monopole. If we surround this point by a (spatial) 2-sphere £, then the magnetic flux out of the sphere is given by
/ / B • da = [f
B da + [f B da,
where T,N and S s are the northern and southern hemispheres intersecting at the equator S. By Stokes' theorem since F^ has no components F0i = JE», //
B da = i
JJBN
Ads
JS
//
B da = (b
J JT.S
Ads,
J-S
where —5 means the equator with the opposite orientation. Hence §s + $_s = 0. But this contradicts the assumption that there exists a magnetic monopole at the centre of the sphere. Hence we see that if a monopole exists, then A^ will have at least a string of singularities leading out of it. This is the famous Dirac string. The more mathematically elegant way to describe this is that the principal bundle corresponding to electromagnetism with a magnetic monopole is nontrivial, so that the gauge potential A^ has to be patched (i.e. related by transition functions). [Recall the collection of local 1-forms uia.] Consider the example of a static monopole of magnetic charge e. For any (spatial) sphere Sr of radius r surrounding the monopole, we cover it with two patches TV, S: (N) : 0 < 9 < 7T, 0 <
(5) : 0 < 9 < -K 0 <4>< 2TT, and define in each patch: A(N) A
l
_ ey ~r(r+z)'
A(S)
—
A
~
l
Sy
r(r-z)'
ex AN) r(r+z)' ^ 3
# ) _ ^2 A(S) A
2
— «x
-
r(r-z)>
&(s) — n
A
3
N
In the overlap (containing the equator), A^ ^ and A^ transformation A(N)
_ A(S)
= a_Aj
A =
- (\. ~ U>
2eta.n-1(y/x)
~
U
'
are related by a gauge = 2ecf>.
Notice that A\ has a line of singularity along the negative z-axis (which is the Dirac string in this case). Similarly for Af'. Furthermore, the corresponding field strength is: E = 0,
B = er/r3. 393
If we now evaluate the 'magnetic flux' out of Sr, we have ff
B • da =
J JSr
(AW
- Als))dx»
= 47re,
./equator
in other words, in the presence of a magnetic monopole the last two Maxwell's equations are modified: curl E + dB/dt = J J
J
"
'
with -M _ / e[dT^zrfi(x |_ e^j^ip
~ ^ ( T ) ) classical quantum.
How are the charges e and e related? Well, the gauge transformation S = e leA relating A^ ' and must be well-defined, that is, if one goes round the equator once: (j> = 0 —> 2ir, one should get the same S. This gives: 2ee(27r) = 2mr, ee = n / 2 .
neZ
=$•
(Dirac quantization condition)
In particular, the unit electric and magnetic charges are related by ee = 1/2. So in principle, just as in the electric case, where we could have charges e,2e,..., here we could also have magnetic charges of e , 2 e , — In other words, both charges are quantized. Another way to look at this fact is to consider the classification of principal bundles over S2. The reason for these topological 2-spheres is that we are interested in enclosing a point charge. For a nontrivial bundle, the patching is given by a function S defined in the overlap (the equator), in other words, a map S1 -4 U{\). What this amounts to is a closed curve in the circle group U(l). Now curves which can be continuously deformed into one another cannot give distinct fibre bundles, so that one sees easily that there exists a 1-1 correspondence between 1
I- {principal U(l) bundles over S? 2i2 } {homotopy classes of closed curves in f/(l)}= TTI(U(1)) = Z. Hence we recover Dirac's quantization condition. So for electromagnetism, there are two equivalent ways of defining the magnetic charge:
1. du*F»v = - f
ane/0 394
Charges abelian nonabelian
Monopoles
dv*F»v = -f-
DVF»V = - j ^
?
Table 4. Definitions of charges
2. an element of 7ri(£/(l)) S Z. We also note that both give us the fact that these charges are (A) discrete (quantized) and (B) conserved (invariant under continuous deformations). We now want to define the magnetic monopoles in the nonabelian case. For simplicity, these are sometimes referred to as "colour magnetic monopoles". [Note that there is another kind of monopole which is a solitonic solution and not a fundamental charge as these are, as we shall explain briefly later.] For several (subtle) reasons the obvious expression (see Table 4) JVF"" =
-f
does not work. The quickest way to say it is that *F^U in general has no corresponding potential A^ and so cannot describe the quantum monopole. We shall come back to this later. But we just saw that in the abelian case there is another equivalent definition, and that is, a magnetic monopole is given by the gauge configuration corresponding to a nontrivial U(l) bundle over S2. This can be generalized to the nonabelian case without any problem. Moreover, this definition automatically guarantees that a nonabelian monopole charge is (A) quantized and (B) conserved. We invoke the following classification result. Proposition. The equivalent classes of nontrivial G bundles over S2 are in 1-1 correspondence with the elements of ir\(G). [The proof is very similar to the abelian case. See Steenrod's book, in the Bibliography.] Definition. A nonabelian monopole for gauge group G is given by an element
ofm(G). 4-4
Examples.
1. J T i ( t f ( l ) ) = Z . 2. ni(SU(N))
= 0 ==> no monopoles. 395
Figure 6. Surface swept out by the one-parameter family of loops &.
3. 7Ti(50(3)) = Z2 =>• 3 the vacuum and one type of monopole only. Charges can be denoted by a sign -I- or —. 4. 7Ti(5f/(3)/Z3) = Z3 =>• charges are given by the cube roots of unity l,o;, w2.
Example of SO (3) monopole charge — On a 2-surface S ~ S2 enclosing the monopole, choose a family of closed curves spanning it, as illustrated in Figure 6. £t(s):s = 0->-27r,
t=
0->2n,
with 6(0)
= 6(2TT) = P0,
&(*)
We shall work in SU(2), so that 396
= & w (s)
=
P0.
Figure 7. A curve representing an 5 0 ( 3 ) monopole
monopole charge ~ class of curves going half-way round In other words, we consider the holonomy to be an element of SU(2). Without loss of generality, choose the base point Po to be on the equator, corresponding to the loop £tc (s)> as in Figure 7. Starting at t — 0, &N(£o) = 1, the phase factor $ N (£<) traces out a continuous curve until it reaches t = te. Then one makes a patching transformation and goes over to $ s ( & ) . From te onwards this again traces out a continuous curve ending in 1 for t = 2-K. In order that the curve V so traced out winds round only half way in SU{2) while being a closed curve in 50(3) we must have
*JV(6.) = -* S (6.). So the holonomy of the closed loop £t m ^ S corresponding to the flat connection i*)t(£, s) is — 1, which is the monopole charge.
397
5
Action principle and symmetry breaking
So far we have not discussed dynamics at all, that is, nothing much is said about the time evolution of the gauge system. These are given, in the classical and first quantized cases, by equations of motion, wich are normally obtained by the first variation of the functional of the fields called actions. Various actions will describe various physical systems. We shall study some of them. 5.1
Maxwell
equations
and minimal
coupling
The Maxwell action is usually given as:
A°F = -\j
F^F^d^x.
This is the free field action, that is, it corresponds to a freely propagating electromagnetic field. Recall that FMl/ = dvAp, — d^Au. Then A% = - \ f(dvA„
- d^)F^d4x
= -\
f(d,A^F^d4x.
Varying with respect to A^ we get dvF»v = 0. This, we recall, is the first pair of Maxwell's vacuum equations in covariant notation. The second pair, in this situation, can be considered as kinematics, because by the definition of F^ it is an identity: dpF^ + d^p
+ dvFPiL = 0,
or equivalently
with *F^ = -yv^Fpa. On the other hand, if we have a free (classical) particle in free space, then we define its free action A°M -m M =
idT'Y
where T is the proper time, the integral is along the worldline of the particle, with ^jf-~af- = - 1 - Varying with respect to the worldline Y^{T), we get rf2yM
398
If we simply add the two actions together, we will not of course get any interaction. One way to introduce interaction is to add an "interaction term", using the minimal coupling assumption: f
dY^(r)
At = - e J ^(y( T ))^LilAd T . So we have the total action: Am.c. = A°F + A°M + AI. Varying with respect to A^(x) and y ( r ) we get dvF^{x)
= - e fdTdY^Th4(x
- Y(T))
m^Z = _ e F ^ ( y ( T ) , f ^ . dr* dr The first is Maxwell's equation in the presence of an electric current-density j ^ , and the second is the Lorentz equation for a charge moving in an electromagnetic field. For the quantum particle, which we assume to be a Dirac particle (i.e. with spin | ) , then we replace the particle action by
A°M = I'd*x${x)(id^
-m)rP(x),
and the interaction term by Ai = -ie
/
ip^A^ip,
which on variation with respect to A^ and ip give the following equations of motion: d„F>^{x) = {id^
- m)i)(x) =
-ei>{x)^ip{x) -eAftix^ipix).
These are then the quantum equations, the second being the well-known Dirac equation. 5.2
Yang-Mills
equation
We can do the same for Yang-Mills theory. In the free theory, we have the same action:
A°F = -\ j 399
F^F^x,
where now FM„ = dvA^, — <9MA„ + iglA^, Av]. As before, varying with respect to Afj, we get the equations of motion, this time the Yang-Mills equation: DVF^
= 0.
Here the covariant derivative is DM = <9^ — ig[A^, ]. Again, in the presence of a colour electrically charged Dirac particle (a 'quark' for example), one can introduce an interaction term by the hypothesis of minimal coupling: •A-i = -i9 /
^A^ip.
We obtain equations which are the analogues of the abelian ones:
0„F"" = -gfrfxl), (id^
~ ™)ip =
-gA^ip.
Classical analogues, called the Wong equations, exist but since in applications particles are quantum, we shall not derive them here. 5.3
Wu-Yang
criterion
What about the dynamics of monopoles? Let us go back to the abelian theory first. We know that in the presence of a magnetic monopole, the potential A^ has to be patched, say by overlapping northern and southern hemispheres on any sphere surrounding the monopole. We immediately face several difficulties. (a) Varying patched A^; (b) As the monopole moves, the patching moves with it, thus depending on say Y(T); (c) What is the interaction term? Wu-Yang
criterion:
Equations of motion are obtained by varying the free field and free particle action, under the constraint that there exists a magnetic monopole. In other words, the interaction comes from the constraint. Intuitively, this is quite reasonable. Around the monopole, the field configurations A^ have to be such that we have a nontrivial bundle. As the monopole moves in spacetime, the field configurations have to rearrange themselves to maintain this topological condition, hence there is mutual influence, that is, interaction. 400
So the prescription is:
A 0 = A0F + A°M to be varied under the constraint:
where we can insert either the classical or the quantum expression for the current. Using the method of Lagrange multipliers, we in fact vary the auxiliary action:
A' = A°+ f
X^dSF^+f).
The next question is, what are the variables? If we use A^, we still have the troubles (a) and (b). To answer this, let us look at the simpler problem of pure electromagnetism. Previously, to get the free Maxwell equations we varied A°F with respect to A^. Now we could also have varied with respect to F^, provided we compensate for the fact that there are more FM„ than A^, the former with 6 degrees of freedom and the latter with 4. Suppose now we restrict ourselves only to those F^ satisfying d„*i7V"/ = 0, then we shall be able to recover A^ (at least in a local region). This is clear in the language of forms, because dv*F»v = 0 <-—>dF = 0 and in flat spacetime the Poincare lemma will tell us that there exist a 1-form A such that F = dA. So in other words, the sets of variables {Afl(x)} and {F^v{x) with du*F'1'J — 0} are entirely equivalent. So suppose we form the action A' = A°F+ j
X^{x){dv*F^)
and vary with respect to F^, we shall indeed get from
A = - I JiF^F'"' -
\eW\pdaF^),
giving Fnv
=$• duF»v
=
2ei"vi"Tdp\(T
= 0,
which is the desired Maxwell's equation. So we see that the two derivations are entirely equivalent. 401
Coming back to the point monopole, the constraint is rfV dT—— S(x -
Y(T)).
/ We see that away from the monopole worldline, we have again dv*F^ = 0 -*=> 3AM. Along the monopole worldline, we know already that no A^ exists. Hence in the constrained action principle, we are justified in using F^ as variables. Hence the Wu-Yang criterion gives us the dynamics as follows: d^F^" = 0 m
«q^
=
dJpi,v
=
no electric charges
_c*F""(y(T))4£Iil
_£jdT!!12Lill6(xT_Y(T))
constraint
These are identical to the dual of the equations of motion of an electrically charged particle, as expected—we shall study electromagnetic duality in more details later on. What about nonabelian theory? In principle, the Wu-Yang criterion can be applied to nonabelian monopoles. In fact, it looks highly plausible that it can be applied to any topological charges to find their dynamics. The difficulty in the case of the nonabelian monopole is to write down the constraint. Recall the charge is defined as an element of 7Ti(G). This programme has in fact been carried out using loop space variables. Since Yang-Mills theory is not symmetric under the Hodge star * (as we shall show) the equations thus obtained are new. Unfortunately it is a bit too lengthy to present here. What I shall do is to indicate to you how to use the Wu-Yang criterion in the pure Yang-Mills case to re-obtain the Yang-Mills equation, just as I did above for the Maxwell case. Recall the constraint for the existence of A^, in terms of loop variables is that the 'connection'
FM(f|S) = i$- 1 (0(*„( a )*(0) is flat, that is, its 'curvature' vanishes: G W 6 « ) = 8»(sWZ,s)-8ll(8)Fv(t,s)+ig[Fll{t,s),Fv(t,s)]
= 0.
Next we want to write the Yang-Mills action A£ in loop variables. It turns out that, modulo uncertainties about measures in function spaces,
AS =
-^Js^Jrd8Tr(F^,8)F"U,s))(tLr1, 402
where N is an infinite normalization factor, the explicit expression for which need not bother us here. So by the Wu-Yang criterion we form the constrained action with the Lagrange multipliers L^: A' = A°F + j&zjds
Tr(X^(e, s)G M „(e, a)).
Because GM„ is skew, we can choose without loss of generality LP" to be skew as well. Varying with respect to F^ we get F " = -N^tfulf
-
ig[Fv,L1"']).
Write this as a loop covariant derivative V -Ni2{VuL^).
F* = Then PMF" =
-N^V^LT)
= ~i2{[D^Vv]L^) = 0
(•.• V»> skew)
(-.- F,, is flat).
But V^F^ = 6fj.F^ - ig[FM,F'*] and the commutator term vanishes, from which we obtain 5ll(s)F"(Z,a)=0. This we refer to as the Polyakov equation, and is shown by Polyakov [P] to be equivalent to Yang-Mills equation: DVF»V - 0. So we see that again the Wu-Yang criterion gives us the right dynamics. N o t e . We have used the Wu-Yang criterion quite extensively, and have recovered all the known equations for interaction between the gauge field and charges, and have also obtained some new equations as well, notably for the nonabelian monopoles. 5.4
Symmetry
breaking
Prom Lecture 2 we learned that we need to consider more complicated gauge theories, that is, those that exhibit symmetry breaking as in electroweak theory. 403
We shall now look at the action for electroweak theory. For simplicity and to make the symmetry breaking mechanism more transparent, we shall omit the charges (i.e. leptons). To the Yang-Mills action A% we now add the Higgs action: AH = J(Dp<j>iyi4> + V(4>))dix1 where <j> is an SU{2) doublet of complex scalar fields
with weak hypercharge Y = — \. Hence the covariant derivative is D^
= (9M - \92T
• WM + | s i W >
where W ^ represents the 3 components of the SU(2) weak isospin gauge potential, and Y^ the U{1) weak hypercharge potential, and
V{
2
- f|0| 4
(A>0).
If fi2 > 0, then the scalar field <j> has mass /J, and the vacuum (or ground state) corresponds to
|^o| - -V/A = r, ? 0. We now choose a gauge such that
In this way, the vacuum corresponds to a particular direction in the space of su(2) © u(l) and once this choice is made, the physics will no longer be invariant under the whole of the U(2) group. This is symmetry breaking. In fact, since (j) is a complex vector in C2 (although we sometimes call it a spinor), there will be a phase rotation left over after fixing a directin as above, and this constitutes the 'little group' C/(l) corresponding to electromagnetism. For a quantum field theory, we look at quantum excitations around the vacuum, that is,
404
where a(x) € K, which is a gauge choice. Hence
n*
(fiqig22 ( (
( °0 \\
W WH
W W},
l~il-iW},'
»
from which D^D^
„2„2
= i(a M( 7) 2 + ^ l ( ( ^ ) 2 + ( W M 2 ) 2 ) + ^ ( 5 2 W M 3 + 5 l y M ) 2 + c u b i c + q u a r t i c .
Now define: A,= *
92
z,=
*
9i
5i 92
with 9\92
e •
.
V9i + 92' or in terms of the Weinberg angl<^ sin# W
5i =
—:
V5i+5f
we have Ali = - sin 0W Wl + cos 6W Y"M ZM = cos 9W Wl + sin 6W *V As far as the particle spectrum is concerned, cubic and quartic terms are unimportant. They can be either got rid of by redefining fields or they represent self-interactions. So we concentrate on the quadratic terms. Recall the Klein-Gordon Lagrangian - ( d M < O T + m2
- M2 - MW2
- M2 - Mw
-
92f]
M2 Mz
——,
-
92
^ „
-
M
W
2 Q
,
I I cos^ t>w cosz (?w (and also the Higgs field a becomes massive) while the abelian vector field Ali = - sin 9W Wl + cos 6W YM has no mass term and hence remains massless. This can easily be identified as the photon (especially if we consider the lepton terms as well). 405
6
Electric—magnetic duality
Electric-magnetic duality, where it exists, is an important concept in theoretical physics. 1. As a symmetry of nature, we should study it. Also since it is discrete, it should be relatively easy. 2. As a result of this symmetry we need study only half of the phenomena. 3. Dirac's quantization condition (abelian and nonabelian respectively) says ee = 27r,
gg = 4iv.
This should hold even under renormalization. We have therefore a correspondence which relates weak coupling (where perturbation expansion is good) to strong coupling (where perturbation expansion is bad). 4. 't Hooft's theorem (see later) leads to a mechanism for confinement (of quarks) via duality. 6.1
Abelian
theory
We recall that the duality operator (*) is defined by: *F
-
_I
T?P°
f
the sign being the consequence of Minkowski signature (H ). Duality, as the name implies, is such that we recover the original field tensor (up to sign) if we do the operation twice: *(*F) =
-F.
In terms of the electric field E and the magnetic field B these tensors can be represented in matrix form:
• fit/
0 Ei E2 E3\ —Ei 0 — B3 B2 —E2 -83 0 —B\ \-E3-B2 Bi 0/
( °
(
*F —
-Bi
-B3 \-B3
B\ £?2 #3 ' 0 E% —E2 E% 0 Ei E2 -Ex 0/
So we see that under *: E -> B , B -> - E . The question is: is the theory invariant under this duality? In vacuo, the Maxwell equations are: (1) dvFlu'=0, (2) dw*F^ = 0. 406
Hence we conclude that the theory is indeed invariant in this case. But we can go back even further and see that the derivation of these equations is also invariant. In this connection we note that
A = -\JF^F^ = \ j *F^*F»», so that the action is invariant (the — sign being of no significance as it does not affect the dynamics). Applying the Wu-Yang criterion, we can use either (1) or (2) as constraint and obtain the other as equation of motion, so that we end up with the same equations in both cases. Recall that •A. —
"g I yi'iiu^4
2^Mt/P°' P °" M 1 '/'
leading to F^v =
2e^p(TdvX^,
which implies dvF»v = 0
This is clear by looking at Chart I, both columns 2 and 3. Next, in the presence of a magnetic monopole, we have: (1) dvF»v
= 0
(2) dv*F»v =
-T-
If we now look at Chart II, column 1 or 2, we see that: (2) as constraint ~> (1) as equation of motion. Dually, in the presence of an electric charge, we have (1) dvF»u = (2) d^F^
-p
= 0.
And now Chart II, column 3 or 4, tells us: (1) as constraint ~-> (2) as equation of motion. Remarks 1. Duality: E +* B, e
2. A dual potential Ay, emerges, which is just the Lagrange multiplier. Away from charges and monopoles, we have both A^ and A^. In pure theory, neither A^ nor A^ appears in the equations, but in the presence of charges/monopoles, the Lagrange multiplers cannot be eliminated and the potentials appear explicitly, as demanded by the Bohm-Aharonov experiment. 3. The duality goes deeper, as linking physics with geometry. Ay exists as potential for F^ (F = dA)
d/F"" = 0 {dF = 0)
Pojncar^
1
t
Principal A^ bundle trivial
No magnetic monopole e
GEOMETRY
PHYSICS
Gauss
Dually, we have exactly the same picture: A^, exists as potential for *Ftll/ PmncarS
(*F = dA)
6.2
d„F^ = 0 (d*F = 0)
I
I
Principal A^ bundle trivial
No electric charge e
GEOMETRY
PHYSICS
The star operation
in Yang-Mills
theory
In nonabelian theory, we have the same star operation: *F
— -If
408
T?P<*
Gauss
Let us now try to fill in the boxes as in the abelian case. We find in the direct picture: A^ exists as potential for F^ (F = DAA)
Df*Fnv
o {DAF = 0)
Bianphi
I Principal A^ bundle trivial
=
?
No magnetic monopole g
definition
And in the dual picture: A^ exists as potential for *Fll„ (*F = DAA)
Gu-Yang
D^v =0 (DA*F = 0)
tYM
t Principal A^ bundle trivial
No electric charge g
All these go to show that the star operation in nonabelian theory does not give us the desired electric-magnetic duality, unlike the abelian case. In fact, we have a stronger result, in the following counter-example discovered by Gu and Yang. T h e Gu—Yang counter-example Gu and Yang phrase their example in terms of DA*F, but we can equally think in terms of DAF. Since in general DAF — 0 =£$• dF = 0, there is really no reason to suppose that F is in any sense exact. Furthermore, we are not asking F to be exact, but we want the existence of A for which F = DA A. So the existence of a gauge potential in nonabelian theory has very little to do with the usual Poincare lemma. Counter-example. Let G = SU(2). Take an explicit 'hedgehog potential' A\ = €ijkXkg{r),
409
AJ0 = 0,
(abandoning our summation convention temporarily) with g a function of the radius r satisfying (gr)" + ~{gr)' - (1 + r2g)fe + rg2) = 0. r r It can easily be verified that for such a potential the gauge field satisfies the Yang-Mills source-free equation: D^F**" = 0. Previously, Wu and Yang found numerical solutions to the above differential equation for g, depending on a real parameter c > 0, with c g(r) —>•
1 as r —> 0.
Now let Vi = F^
VJ = Fil,
Vi = F{2,
and consider Vi,V2, V3 as 3 vectors in 3-space. Then it can be proved that V\,V2,V3 linearly indept -o g(r) ^ -5- for any a. Hence for the given solution, the vectors are linearly independent. Suppose for a contradiction that A^ exists as a potential for *F*1V'. Then its Bianchi identity is
This, together with the Yang-Mills source-free equation, implies [Afi-Au,F»»]
= 0,
where for convenience we have absorbed the coupling constant g into A^, g into Ap. Notice that AQ may not be zero, but it does not contribute because F0l/ = 0. Write now U3k = Aj. — A3.. Then in 3-space notation, the commutator equation can be written as: U2 A V3 - U3 A V2 = 0, Ui A V3 - U3 A Vi = 0, d A V2 - U2 A Vi = 0. We now claim that: V lin. indept. =*>• Ui = 0. 410
In fact, all the quadruples (U2, U3,V2, V3) etc. are coplanar. This implies, say, U2 + U3 = a2V2 + CX3V3 U1+U2 = /3iV3 + p2V2, which in turn implies <*2 =02,
etc.
Hence U1 + U2 =aiVi +a2V2, U2 + U3 = a2V2 + 0:2^2, U1 + U3 = aiVi+a3%, and therefore all a» are equal, say to a. This gives Ui=aVu V* ==> a(V2 AV3) = a{V3 AV2) =4> a = 0, which justifies our claim. We therefore conclude: Ai = A{,
Wk,j.
Now F32 = JFIO = 0,
but on the other hand *^32 = -F32 = —Vi 7^ 0, which is a contradiction. C l a i m . Hence nonabelian Yang-Mills theory is not dual symmetric under *. 1. A^ need not exist, 2. the dual of Yang-Mills equation is not Bianchi identity. 411
6.3
Generalized theory
electric-magnetic
dualtiy for
Yang-Mills
We saw that electromagnetic duality in Maxwell theory is both important and useful. So we want to salvage the situation as regards to nonabelian theory. There are two ways of going about it. (A) Generalize the concept of duality, that is, modify it in the nonabelian case. (B) Enrich the theory, for example, make it supersymmetric, so as to enlarge existing symmetries. We shall briefly talk about both. Take (A) first. We seek a dual transform (~) satisfying the following properties: 1. (
)~~=±(
),
2. electric field F^ <-^-»- magnetic fields FM„, 3. both A^ and A^ exist as potentials (away from charges), 4. magnetic charges are monopoles of A^, and electric charges are monopoles of i M , 5. ~ reduces to * in the abelian case. So far, we are only able to express this new dual transform in terms of loop variables. I cannot go through the construction here but those interested can refer to our papers, especially recent reviews. Suffice it to say that the above 5 points are indeed satisfied, and we have full duality as depicted below:
A^ exists
extended Poincare'
t Principal A^ bundle trivial
loop space formula DVF»V = 0 (YM)~
t definition
GEOMETRY
No magnetic charge PHYSICS
412
Dually, we have
Afj, exists
6.4
(loop space formula)" D^F^ = 0 (YM)
extended Poincare
I
t
Principal A^ bundle trivial
No electric charge
GEOMETRY
PHYSICS
't Hooft's
definition
theorem
and its
consequences
In a quantum gauge field theory, the phase factor $(C) = Psexpig
/
Jc
A^dx"
is an operator in a Hilbert space. Let A(C)=ti${C), which is still an operator. This is the more usual definition of the Wilson loop. In a gauge theory with gauge group corresponding to the Lie algebra su(N), 't Hooft introduced abstractly the operator B(C) dual to A(C), by the commutation relation A(C)B(C)
=
B(C')A(C)exp{2mn/N),
where n is the linking number between the two spatial loops C and C". He describes these two quantities as: • A(C) measures the magnetic flux through C and creates electric flux along C • B(C) measures the electric flux through C and creates magnetic flux along C So they play dual roles in the sense we have been considering. However, there was no "magnetic" potential available at the time, so that the definition of B(C) was not explicit, only through the commutation relation above. 413
But in the construction mentioned in the last subsection (which I did not give explicitly), the magnetic potential A^ exists, so that one can actually prove the commutation relation. This has been done about 2 years ago. 't Hooft's Theorem. / / the Wilson loop operator of an SU{N) theory and its dual theory satisfy the commutation relation given above, then: SU(N) confined <£=^ SU(N) SU(N) broken <=» SU(N)
broken confined
Note that the second statement follows from the first, given that the operation of duality is its own inverse (up to sign). The theorem does not hold for a U{1) theory, where both U(l) and U(l) may exist in a Coulomb phase, that is, with long range potential (~ 1/r). The statement is phrased in terms of phase transition, and has profound implications. It has been a cornerstone for attempts to prove quark confinement ever since. I may add that we have exploited 't Hooft's theorem in the reverse way. Given that SU(3) colour is confined, we deduce that dual colour is broken, which we have identified as the 3 generations of particles as observed in nature. There are many consequences of such a hypothesis, not only in particle physics, but also in nuclear and astrophysics. Coming back to the commutation relation, I wish to show you how to prove it in the abelian case, just to give you a taste of what is involved. The nonabelian case is too complicated to treat here. In the abelian case, we do not need the trace, hence A{C) = $(C), B{C) = $(C"), and the $ are genuine exponentials. So if we can show the following relation for the exponents, we shall have proved the required commutation relation: it
2imi.
Using Stokes' theorem the second integral becomes -ie
if
*Fijdaij = ie f /
E{da\
where dZ,c< = C.
For simplicity, suppose the linking number n — 1. Then the loop C will intersect £
Using the canonical commutation relation for Ai and Ej [Ei(x),Aj(x')}
= i5ijS(x - x')
we get ie (b Aidx1, ie I Ejda-' Jc J JT.C,
= iee = 2iti
by Dirac's quantization condition. Hence we have shown explicitly in the abelian case that our definition of duality coincides with 't Hooft's. The same is true in the nonabelian case. 6.5
Magnetic
monopoles
from symmetry
breaking
We looked at electroweak symmetry breaking in detail: U(2) -» U(l). We can also do similar breaking with SU(2) ->• U(l). Again we can choose the Higgs field in the (
1
J direction, and the residual symmetry will again be a
17(1), this time generated by the generator T3. Now SU(2) being simply connected, there are no nontrivial bundles over S2. However, there can be nontrivial reductions to the U(l) subgroup. Topologically, this can be seen by looking at part of the exact sequence of homotopy groups, obtained from the principal bundle: U{1) -»• SU{2) ->• SU{2)/U(l)
~ S2,
whence -»• ir2(SU{2)) -»• 7r 2 (5C/(2)/[/(l)) ^ 7n([/(l)) -»• 7n(5l7(2)) - •
II
II
0 0 The boundary condition of the Higgs field (j> at infinity will determine the nature of the reduced bundle (more precisely, its first Chern class), that is, the homotopy class of the map: S2 -> SU(2)/U(1) ~ S2, the first S2 being the sphere at infinity. This is precisely given by •K2{SU{2)/U(\)) ~ Z, which by the exactness of the above, is isomorphic to the magnetic charges of the residual C/(l), namely 7Ti({/(l)). Let us look at an example, the residual charge 1 magnetic monopole. It is a particular solution of the Yang-Mills-Higgs equations we saw before. Inserting the asymptotic condition we get a solution, for large r, similar to the Wu-Yang potential we had: F$i = 0, Ffj =
^eijkrk, 415
all others = 0
=> -Bfc = —o> A
er that is, a magnetic field in radial direction at infinity, which is why this is referred to by Polyakov as the "hedgehog solution". Such a solution is called a 't Hooft-Polyakov monopole. In the (unphysical) limit studied by Prasad and Summerfield,
101 -» 1 \ exact solutions exist for A = 0 = /*. These are finite energy solitonic solutions. There is a parameter occurring in the behaviour of \
6.6
Seibery-Witten
duality
The second way to study electric-magneitc duality is to exploit the duality between the electric and magnetic charges which occurs as a result of symmetry breaking from a nonabelian Yang-Mills theory. In the models so far studied, supersymmetry is a necessary ingredient. Supersummetry is a hypothetical symmetry between fermions and bosons, and has tremendous theoretical and mathematical appeal to a lot of physicists. We cannot discuss it here for lack of time (and expertise!). The dual symmetry I shall outline below works for all N = 4 and some N = 2 supersymmetric Yang-Mills theories, with gauge group SU(2)—this can be generalized. The bosonic part of the action is
A=-J^TrFfluF^
-^TrD^D^-VW
+
jj^TrF^*F^,
where the last term corresponds to the second Chern class or instanton number. This is a topological term, and the coefficient in front of it is the so-called ^-vacuum angle. Experimentally it is very nearly 0. 416
By giving a nonzero vacuum expectation value 77 to the Higgs field 4>, we effect the symmetry breaking SU(2) -* U(l). There are solutions which are BPS monopoles. In fact they are dyons, with both electric and magnetic charges. Their masses satisfy the Bogomolny bound: r.2
M
' = <**»->i£U,H0(»)
where 9 T= —
+ ?
An ~,
_ n Qm = - ,
_,
. nO. Qe=e{m+—).
One sees that the mass formula is invariant under f m\ (a b\ (m V n ) / " ( :\ c! d / U so that this theory of charges and monopoles are invariant under the group 5L(2,Z). In the particular case when 6 = 0, the generator 5 of 5L(2,Z) corresponding to a = d, b = c = —1, that is r t-4 — 1/r, induces T !-»•
ar + b CT + d
e2 47r e2 -— i-> —5= — , n *-> m, m J 47T
e
47T
H->
—n
and we recover the usual electric-magnetic dualtiy with ee = 4n. This also goes under the name of S-duality. Explicit solutions are constructed by making use of a certain holomorphic function of r occurring in the theory having to do with the metric on the moduli space. This duality is found to be a symmetry of the quantum field theory. Seiberg and Witten also considered supersymmetric Yang-Mills theories in which the dual symmetry is only partial, in the sense that the spectrum of dyons in one theory is found to match the dual spectrum (electric ++ magnetic) of another theory, perhaps with a different gauge group. The whole subject has been intensely studied in recent years, with many ramifications into string theory, membrane theory and M-theory. They are definitely outside the scope of these lectures.
417
U[Vj invariance {A„ i-> A,, + fyA)
(F derivable from potential .4)
L •rp pit -— - I *fpvt)0 f FPO
•40 = - l k / V
extremize w.r.t. A„
dvF^ = 0 (no el. charge)
3/F"" = 0 (no mag. charge)
LEMMA
F^^l^^X'
LEMM
lo-j
A0=A°
extremize w.r.t. F„„
POINCA
extremize w.r.t. 'Fu„
'F.^-iK^idfy-dn")
Fliv — Ot/Afi ~~ OpAv
"pi/ — Oi/jifi — OjiJij/
I'F derivable from potential A)
A„ = iirXf, (F derivable from potential .4)
POINCARE
(no el. charge)
(7(1) invariance
(A„nA„ + di,\)
U(l) invariance (ApHAp + dpA)
.
ELECTRIC CH
MAGNETIC CHARGE
Defining constraint
Potential A for F patched
Defining const hF"" = -4i (Maxwell eq
- ) ' (if per r
(Gauss' law)
/ i*=ifdT*g-S{x-Y(T))
j» =
^ = -ifc/W" A%-mJdr
A=Aa+fX„^F'"'+'kii'i extremize w.r.t. F^^V
JO _ _J_ C*p
AF = A°F
^
^ A° =
/ = e^i,
i^i>
AQ=A% + $i>{ip-m)ii>
A=.A° + J\„
^
AQ=A°F~j4>(i?-m)i,
extremize w.r.t. *Fttv,il>
\*F'"' = -4ir€l"">e
F"" = iTTC>""^daK Sijj
&"F^ = 0 (Gauss' law
9"F,„ = 0 (Maxwell eq.) (i ft — m)ip = —Ijfcfy (dual Lorentz eq.)
*F derivable from potential/!
Ay — 4TTA^
(dual Dirac eq.)
[/(I) invariance Ap^Ap+dpA ip i-> exp ie A ^
Afl = 47rA„ (Dirac eq.)
F derivable t/(l) invariance from potentia An *->A„+d„\ \j> i-V exp !>.A $
References [KN] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, Interscience, New York, 1963. [S] N. Steenrod, The topology of fibre bundles, Princeton University Press, Princeton, 1974. [O] T. Okubo, Differential Geometry, Marcel Dekker Inc., New York and Basel, 1987. [P] A.M. Polyakov, in Nucl. Phys. B164 (1980) 171-188 Further bibliography (see also Ocampo's lectures) General
Gauge
theory
• IJR Aitchison and AJG Hey, Gauge Theories in Particle Physics, Adam Hilger, 2nd edition, 1989. • AP Balachandran et al., Gauge Symmetries and Fibre Bundles, Lecture Notes in Physics # 188, Springer, Berlin, 1983. • Chan Hong-Mo and Tsou Sheung Tsun, Elementary Gauge Theory Concepts, World Scientific, Singapore, 1993. • Tsou Sheung Tsun, Symmetry and symmetry breaking in particle physics, in Proc. 8th EWM General Meeting, Trieste 1997, hep-th/9803159. • TT Wu and CN Yang, Concepts of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D12 (1975) 3845-3857. Further
interests
Those who are interested in our recent work on duality may like to read: • Chan Hong-Mo and Tsou Sheung Tsun, Nonabelian Generalization of Electric-Magnetic Duality—A Brief Review, hep-th/9904102, RAL-TR1999-014, invited review paper, International J. Mod. Phys. A 1 4 (1999) 2139-2172. • Chan Hong-Mo and Tsou Sheung Tsun, The Dualized Standard Model and its Applications—an Interim Report, hep-ph/9904406, RAL-TR1999-015, invited review paper, International J. Mod. Phys. A 1 4 (1999) 2173-2203.
420
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
A N I N T R O D U C T I O N TO S E I B E R G - W I T T E N T H E O R Y
Departamento
HERNAN OCAMPO de Fisica, Universidad del Valle, Cali, Colombia, hernan@univalle. edu. co
We give an elementary introduction to Seiberg-Witten theories in N=2 supersymmetry. Starting with a description of electromagnetic duality and the different forms of monopole, we continue with an introduction to supersymmetry to arrive at the non-perturbative solution of the Moduli space.
Contents 1 Introduction
422
2 Electro-Magnetic Duality
423
2.1 2.2 2.3 2.4 2.5
The Dirac Monopole The Bosonic part of the Georgi-Glashow model 't Hooft-Polyakov Monopole The Bogomol'nyi Bound and the BPS States The Witten effect
3 Supersymmetry
434
3.1 Super symmetric Quantum Mechanics 3.2 Supersymmetric Field Theory in 1+1 Dimension 3.3 In 4 Dimensions 4 The Seiberg-Witten Analysis 4.1 4.2 4.3 4.4
424 426 428 430
Moduli space Duality Seiberg-Witten analysis Solving the Monodromy Problem
References
436 439 441 443 443 443 445 447 448
421
1
Introduction
Recently more importance has been given to finding theories which allow nonperturbative solutions. A relevant development in this direction was initiated by the work of Seiberg and Witten [SW1,SW2] on N=2 supersymmetric Yang-Mills theory. There are many developments and review articles in this field. The aim of this course is to offer an introduction to the material handled in these numerous works. Duality plays a very important role since SeibergWitten theory showed that it can be interpreted as a quantum symmetry. In the first part, we present the electromagnetic duality including the introduction of different forms of monopole and its consequences in the different quantization conditions. We follow review articles by Alvarez-Gaume and Hassan [AH] and Figueroa-O'farrill [FIG] as well as classical books like that by Cheng and Li [CHL]. We will not discuss the problem of singularities which are presented in the lectures by Tsou Sheung Tsun [TSOU] in this volume. We then introduce supersymmetry following books like [-BVF], [Fr] and [DF], from which we shall borrow the notation commonly used in supersymmetry. Finally we present the theory of Seiberg and Witten and the description of the moduli space for the SU(2) symmetry as an elliptic curve. In this last part we follow the original articles, the reviews mentioned above and that written by Lerche [L].
422
2
Electro-Magnetic Duality
In this section we show the consequences of the symmetrization of the Maxwell equation in different formulations. Let us begin with the Maxwell equations V • E = 4np , V x B - dE/dt =j V • B = 0 , V x E + dE/dt = 0 .
(1)
This set of equation in absence of charges p = 0 , j = 0 is invariant under a transformation E-> B B-S--E, which is a discrete symmetry, called duality symmetry. It is clear that the symmetry is broken in presence of charges. To restore the symmetry Dirac proposed the introduction of a monopole density a and a monopole current k in the following form V • E = 4?rp , V x B - dE/dt = j* V • B = CT , V x E + dE/dt = - k ,
(2)
where the transformation set is enlarged with
p -» a J-» k^ a -¥ -p k-> - J . The Maxwell Eq. (1) can be also written in terms of the potentials <j> and A. The absence of monopole density allows to write B = V x A and the absence of a monopole current E = —V(^> — dA/dt. In presence of a monopole A can not be defined globally, as we will discuss later. The electromagnetic equation can be written in a covariant form when we put the potentials in a 4-vector A11 = {(j), A } and define the field strength tensor F^u as Foi = d0Ai - diA0 = -doA* - diA° = Ei , Fij = diAj - djAi = -(diA> - d>Ai) = eijkBk i
k
oi
.
Note that B = -\e^ Fjk = - | c ' " ' F M V and E = Foi, (where c«fc = 1 if i = 1, j = 2, fc = 3 and all even permutation of them, = — 1 for any odd permutation and = 0 in other case). The choice of the potential AM has the gauge freedom {A^ -> A^ — A* + d M x). We also need a dual of the strength field tensor F*v =
l
ie^P5Fps 423
,
where e0123 = 1. Now the covariant expression of the Maxwell equations is: duF^
= -f
dvFllv = 0 ,
,
(3)
where j ^ = {p,j}- The new expression for the duality symmetry of the Maxwell equations without charges is F ->• F, and F -»• —F. Introducing the monopole the equation becomes 0„F"" = -f
,
dvFi™ = k» ,
and the symmetry is enlarged with the transformations f -> Jfc" , and 2.1
The Dirac
A" ->• - j M .
Monopole
The inclusion of monopole at first sight destroys the possibility of a gauge theory, but Dirac proposed a very ingenious way to allow for a potential A in presence of monopole. By Stoke's theorem, it cannot be well-defined in a neighborhood around the charge an hence in any sphere around the charge, A must have a singularity. By continuity, the singularities are on a line starting in the charge and ending in a monopole of opposite charge or at infinity. This construction is called the Dirac string. But if we consider the monopole as a point their magnetic field should be radial of the form B TOOn = gr/Airr3, where g is magnetic charge of the monopole. In this formulation Dirac introduced a solenoid along the string such that B = Bmon
+ Bsoi , and
V •B = 0.
For example, if the solenoid is chosen along the z-axes Bsoi — g0(—z)8(x)5{y)z, (where 6{x) = 1 for x > 0 and = 0 for x < 0), it is possible to find a well-defined vector potential A, with which one can formulate electromagnetism in a covariant way. By quantizing this electromagnetic theory with Dirac monopole one finds a condition for the charges. The Schrodinger equation for a particle interacting with an electromagnetic field is:
where D = V + ieA with e = q/h. This equation is gauge invariant under the transformations A -»• A + Vx and tp -¥ exp{-iex)ip • 424
But the wave function in this last expression is not single-valued. In the Aharonov-Bohm experiment the presence of the string will affect the phase factor. If one considers the variation of the phase factor along a closed curve r , encircling the string, which is the boundary of a surface S, we have e <j> Asol
• d\ = e / B s o ;
ds-eg.
The conditions for a single-valued wavefunction is known as the Dirac charge quantization condition eg = 2irn . There is another semi classical way to derive a charge quantization condition. Let us consider a particle with electric charge q in presence of the magnetic field B = gr/Anr3 of a monopole. The particle experiences a force m r = qr x B. Therefore the change of the angular momentum of the electric charge is given by d, - ^ - ^ qg -. ,u -N d.qgr^ — (mr x r) = mr x r = - — - r x (r x r) = — (). dt 47rr 3 dt 4irr Hence, the total conserved angular momentum of the system is s L, qg r J = r x mr — . 4TT r The second term is the angular momentum of the electromagnetic field, calculated from = / d3xr x (E x B) Assuming that the total angular momentum and the particle angular momentum are quantized, we find again the Dirac quantization condition 99 1 — = -nn% . 4TT 2 Now, let us consider particles carrying electric and magnetic charges called dyons. The calculation of the interaction between two dyons with charges (qi-,9i) and ((72,52) is carried out in a similar way. One splits the electric and magnetic field as E = Ei + E 2 and B = B \ + B 2 and finds the DiracSchwinger-Zwanziger quantization condition 9192 - 929i 1 = A ^ 2ni2 • Note that this condition is invariant under an 50(2) transformation written as (q + ig) - • e^iq + ig). 425
In 1974, 't Hooft ['tH] and Polyakov [Pol] independently discovered that the bosonic part of Georgi-Gashow model admits finite energy solutions that look like Dirac monopole from afar. In contrast with the Dirac monopole, these solutions are everywhere regular and do not requires the introduction of a magnetic charge source. 2.2
The Bosonic
part of the Georgi-Glashow
model
The Georgi-Glashow model describes the electromagnetic interaction. Here we will consider only the bosonic part of this model with a rotation (50(3)) symmetry. The Lagrange density of the model is given by L = - \&»
• a„„ + \D*$ • Dj
- V(<j>) ,
(4)
where the gauge strength field GM„ = d^W^ — <9„GM - eW M x W „ . Here the vector notation means the component in the Lie algebra
(5)
The Higgs potential V(
V(
(6)
where <j>2 = cf> • <j) and A is assumed non negative. The Lagrangian density L is invariant under the 50(3) gauge transformations
4>-*$'=g(x)4> W „ -J- W V = 9(x)-Wmug{x)-1
+ \d„g{x) g{x)-1
,
(i)
where g(x) is a £-dependet rotation, ie. a possibly rc-dependet 3 x 3 orthogonal matrix in 5 0 ( 3 ) with unit determinant. The classical dynamics of the fields is determined from the Lagrangian D^"
= -e$ x Di*$,
D"D^
and from the Bianchi identity D^G
=0, 426
= -\(
a2)$,
where G = ^e^^GpsThe canonical conjugate momentum to the gauge field W ^ and the Higgs
=
_QOi ^ j j
=
Doj
Defining Bl by Gy = -e^jtB" = +CijfeBfe , we can write the energy density as H = iE< • Ei + i n • n + ^Bi • Bi + l ^ • £>;
(8)
which is manifestly positive-semidefinite and also gauge invariant. We call vacuum the configuration for which the energy density vanishes. That means G M , = 0 , Dj=0,
V(4>)=0.
(9)
The second two equations define the Higgs vacua. The last one is satisfied if
This leads to the following mass spectrum Mass Charge Ap 0 0 tp MH = ay/2Xh 0 W± Mw = aeh ±eh 427
2.3
't Hooft- Polyakov
Monopole
The energy of a given field configuration is the spatial integral of the Hamiltonian density E = f d3x H. In this model the Hamiltonian density is given by (8). Finite energy means that the integral exists. Hence the field must approach a vacuum configuration asymptotically when r —> oo. This means the Higgs field approaches the Higgs vacuum Mo which is topologically a sphere 0oo(r) = lim <j>(r) £ M0 • 1—KX>
It is well-known that the space of continuous maps from a sphere to a sphere has a infinite number of connected components indexed by an integer, called the degree of the map. Heuristically, the degree is the number of times one sphere wraps around the other. It is a topological number, which is a invariant under continuous deformations and is an element of the homotopy group 7T2(52). Let us relate this with the definitions of a Higgs vacuum ie. 4>vac • 4>vac = a2 ,
(10)
d^vac - eW M x (j>vac = 0 .
(11)
The most general solution of the second equation is given by W M = -~^$Vac X d^vac
+ ~4>Ap, .
(12)
ear a Using this solution, we can calculate the gauge field strength tensor G^ and with it the unbroken part which corresponds to the electromagnetic field strength tensor F^, •*>"
=
^Yvac
• Gfj.v
^
, >
= d^Av - duA^ + J£i$vac • (dpifivac X dv<j>vac) •
The equation of motion of the Higgs satisfies: ^F"" = 0,
dltFi"'=0.
We can now calculate the magnetic charges gs associated to this solution, using Bi = -±eiikFjk, : gx = / B W = — -
-—3 / -eijk$vac
• (dj<j>x dk4>vac)dsl
(14)
Since S is also topologically equivalent to S2,
Hence, we have a topological quantization condition 4?r n 5s =
• (15) e The Ansatz proposed by 't Hooft [tH] and Polyakov [Pol] for a construction of a monopole solution in the Georgy-Glashow model is given by $(v) = £ H(aer) W^ =-eaij£(l - K(aer))
(16)
for some arbitrary functions H and K. Plugging the Ansatz into the energy density expression one finds
E=^I™dZ
(17)
(£ 2 if + httw " H? + ^K2 ~ V2 + K2R2
+ MH2 ~ £2)2) -
which is finite in the boundary condition if: when £ = aer —>• oo K -> 0 £-•1, and when £ -> 0 K -1< 0(0 H < 0(0 sufficiently fast. Notice that, with the above asymptotic conditions -. r ^oo(r) = lim —-H(aer) = ar . r—¥oo CT
With this Ansatz it is possible to show that asymptotically F ^ = -<£ cdotGij is given by
J
J
er 3
and then
B= -I4. 429
2.4
The Bogomol'nyi
Bound and the BPS
States
In the center of mass frame, all the energy of the monopole is concentrated in its mass. Therefore M = / K 3 ( | E i • Ei + | B i • Bj + §11; • 11; + \Di$-Di$+
V(0))
> | / f i 3 ( E ; • Ei + Bj • B* + A«?- A sinO and Bj • Di <j> cosd we find $R3(\\t.i = Di$sind\\2 + \\Bi - A • Ej + cos6 jRJDi(j) • B* > sinO JR3Di
M>\
But now note that fR3Di
JR3
Dj
where q is the electric charge of the solution. Therefore for all angles 9 we have the following bound on the mass: M > agcosO + aqsinO . The sharpest bound occurs when the right hand side is a maximum, which happens for q cosB — g sinO or in terms of the electric and magnetic charges M > ay/q2 + g2 .
(18)
This is known as the Bogomol'nyi bound for the mass. The saturation for static solution with Ej = 0 and D0
(19)
2.5
The Witten
effect
In order to study the different vacua Witten introduced a term in the Lagrangian density which does not affect the equation of motion because it is locally a total derivative and is an integer (called the instanton number) times a parameter 6 which parametrizes inequivalent vacua. This term is
Lagrange density reads
L+Le
= ~i^6^ •6^ + d^6>"' ^
+ D
\ ^' D^~ v{4>) • (21)
The first two terms can be rewritten in a suggestive form defining Q^
=
GU + »G„„ and r = £ + i% - 3 ^ / m ( r ^ . ^ ) .
(22)
Let us see one of the consequences of the presence of this term in the mass bound. For this we consider an operator N which implements a gauge rotation around the <j> direction. Under this transformation a vector v and the gauge field transform as follows: 8v = i ^ x v W , = - ^ .
(23)
At large distance where |^>| = a, the operator e2nzN is a 27r-rotation around <j> an therefore exp (2iriN) = 1. Elsewhere, the rotation angle is 2n\
"-/'-fe^sM-
(24)
Using the expression above as well as J0 = 0. we find N =
/ aeJR3
—-Di$. Sd0Wi
Since E» = G 0 i and l/2e 0 i o ^G a / 3 = B{ then 431
(25) K
N = i - / Et • Dt$+ - 4 - / Bt • Dj$= q- + ae JR3
sn^a JR3
e
~g
87H
(26)
For the 't Hooft-Polyakov monopole, eg=-47r, hence the integrality of N means that Q = ne + — .
(27)
The bound mass of a dyon with charges (q,g) given in (18), can be now written using the quantization condition for monopole charge g = nm4ir/e and for the electric charge q = nee + nme9/2ir M > 47ro2n* • A(T) • n ,
(28)
where n = {ne^rimY and
<29)
•M-E^Ur*.')-
From here one can read the SL(2, Z)-invariance when A transforms as follows: A{T)
-»• A(M • T) = ( M - 1 ) ' • A(T) • M-1
,
(30)
and n-> M - n , which is related with the Montonen-Olive duality conjecture.
432
(31)
3
Supersymmetry
In nature particles are classified according to their statistics into Fermions and Bosons. Consequently it is natural to divide the Hilbert space into two subspaces H — H+ © H~. Supersymmetry is the symmetry between Bosons and Fermions, in this case the algebra must include commutators and anticommutators, building a graduated Lie algebra [5W]. The extension of the Lie algebra to a graded algebra allowed Haag, Lopuzanski and Sophnius to avoid the Coleman Mandula no-go theorem and to combine space-time operators and internal symmetry operators in the same algebra. The algebra of Supersymmetry includes hermitian operators Qi, i = 1 , . . . , N, which map H+ into 7i~ and vice-versa. These operators, like any symmetry operators, must commute with the Hamiltonian operator QiH - HQi = 0 . Since the Hamilton Operator is the time translation generator, in order to build a relativistic quantum field theory Q must also commute with the momentum operator P. In four dimensions the sypersymmetry algebra is [BW, DF] {Qa,Qh} {QaM [Pll,Qa] [P»,Pv]
= = = =
2 1 ^ , {Qa,Qi,} = 0, [Pv,Qi,]=o, 0,
,-_, [il)
where the Latin indices (a, b,..., a, b,...) run from one to two and denote the spinors indices. The Greek indices (/i, u,...) run from one to four and denote the space-time indices. T is a map from the time-like cone V to its spinor space S (spin V) and its dual S* given by the Pauli Matrices T\i = 1,2,3 and the Identity. This is the usual vector representation in the spinor algebra. One way to formulate a Supesymmetry theory is to look for a realization of this algebra in local coordinates and their vector fields, i. e. a local parameterization of the tangent bundle, this method is called superspace mechanism. In Supersymmetry the usual space must be extended to a superspace which in local coordinate is parameterized by the usual coordinates xl with i = 1 , . . . ,n and a set of Grassmann coordinates 9l ((91)2 = 0). When the number of 0 coordinates and spacetime coordinates are the same we call it N = l superspace. In other cases, it is called extended superspace. When the number of 6 coordinates are twofold it is called N=2 superspace, fourfold N=4 superspace and so on. A function on superspace is usual written as a 433
polynomial F(x,9) = f(x)
+ f^x)
• 9 + f(x)(9)2
+ ... + fn(x)(9)n
,
(33)
where the fl{x) are functions under the usual spacetime coordinates, and are called the components of the superfunction F(x,9). In four dimensions, we can array the spacetime coordinates, as a vector in the spinor representation i.e. in a complex 2 x 2 matrix yab — V^x^ as follows: y
li _ x ^ x 1 2 '
,.22 y „12 y ,.2i y
_ x°-x' ~ 2 ' _ x2+ix3 — 22 3 ' _ x -ix ~ 2
(34)
the Grassmann coordinate as a complex set 9a,9b, where (0a)* = 9a and the vector fields denoted by <9M = g ^ , da = -^ and db = - J j . This set of elements obey the commutation and Leibniz rules condensed in the following differential algebra calculus [x»,x»}=0 [d^x"]=6», {9\9b} = 0 {da,9»} = 6ba, [9\x»] = 0 [da,d»} = 0, [da,x»]=0 [d»,9a} = 0, {9\8b}=0 {da,9b} = 0.
(3g)
All other relations then follow from complex conjugation. Usual integration can be also extended by a linear and translation invariant integral in the 9 coordinates: f 9ad9b = 6ab ,
f d9 = 0
(36)
In this algebra we can find a representation of the supersymmetry algebra in the form
Qa = da + T»Jbd»
Qk-d^ + ^Ki,^ There is a second set of operators 434
(37)
ebvv:dv,
5a = -^rgga
ab "
satisfying the following relations {Da,Di>} = -2Ty, {Da,Db}=
0
(39)
={Dh,Db}.
The operators D and Q anticommute
{Da,Qb} = {Da,Qb} = {Dh,Qb} = { AM 0 J = 0 . Now let us develop some simple supersymmtric models. 3.1
Supersymmetric
Quantum
Mechanics
A simple example of a supersymmetric model can be written for one particle dimension. To built it let us begin with the description of one dimensional model in Classical Mechanics. In this case the superspace coordinates are (t,9), and the superpoint particle is described by the superfield $(i,0) = 0 ( t ) + 0 ^ ( t ) -
(40)
The supersymmetry operators are of the form Q = dff + 0dt
and
D = de - 6dt
with the relations {Q, Q} = 2q2 = 23t
and
{£>, D} = 2D2 = -2dt .
The supersymmetric transformation can be written by introducing a Grassmannian parameter rj, in order to have a scalar variation Sn = —r]Q. Under this transformation the superfield transforms as 6^
= ( - i j Q ) * = -rnl>(t) + erjdt^t)
,
(41)
hence, we can write the transformation of the components field as (42) 6n<j> = -r)ip(t)
and
8^
= r)dt(t> = r\<$> .
A supersymmetry invariant Lagrange function can be found from the super435
space action S = J dtdOL , with L = \D$D(D§).
After the 6 integration we find
S = Jdt(±
(43)
In the Lagrange function only the kinetic energies appear. To introduce a potential we could try adding a function of the superfield in the superlagrangian V(*(t,0)) = VW) + fy{t)) = VW)) + %\»=omt)
(44)
but the term V{
da+0bT°ahat
(45)
Da = da- 6»r°abdt with the algebra
(46) {Qa,Qb} = Sab2dt
and {Da,Db}
= -6ab2dt ,
and the superfields are
$(t,e\e2) = <j>{t) + e1xp1{t) + d2^{t) + e1e2F.
(47)
The action for N=2 now becomes
S = J dtd6ld62 l-eabDa$Db$ ,
(48)
which after carrying out the 0-integrations, yields S=
f d t ^ + ip^+TPrti 436
+ F2).
(49)
Here appears a new boson variable F without a kinetic term called auxiliary field in physics. The Euler-Lagrange equations yields F = 0. We can now add an interaction term
sint = f dtd01e2 w($(t, e\e2)),
(50)
with an arbitrary function of the superfield $ , which will maintain the supersymmetric invariance. Expanding and carrying out the ^-integration we find
The equation of motion for the auxiliary field now gives
Inserting the interaction in the action we have:
\j>2 + ±i;alPa - ±V(4>)2 + ^ l \ ^
S = fdt
m
frfo .
(53)
More generally, for a superparticle in a n-dimensional manifold M, with metric gap the Lagrange function reads:
L = ±4>°4>a + ypai>Q , where the cj>a and ipa are the bosons and fermion point particle coordinates at t. A canonical quantization of this model [DIJ] is based on [4>n,P(j>p\ = iS^ and {4>a, Pii>p } = i5a0 and the superstates are functions in the Hilbert space given by the expansion n
*(0, VO = £ *». • • •a" ( M a i • • • 1>ak >
( 54 )
k—o
the supersymmetry transformation are given by Sv4>a = -r]ipa, a
6r,ipa = VPa .
a
(55)
So that Q = —ii/) Pa = — ip -grz acting in the state functions. We find that Q2 = 0 and its cohomology gives the de Rahm cohomology of M. 437
In the case N=2 we have two supercharges Q\ and Q2. Complexifying them Q = (Qi + iQ2)/2 shows that Q and Q can be interpreted as d and d* and the ground states
IQH* >= 0 | * > = 0 , hence are harmonic differential forms. 3.2
Supersymmetric
Field Theory
in 1+1
Dimension
Let us construct an N = l supersymmetric model with the superspace mechanism. We begin by identifying the two-dimensional superspace a reduction of the four dimensional one by setting x2 = x3 = 0 and real fermion coordinates 0a = Ba. We shall work in the light cone coordinates described by x+ = x1 + x2 and x~ = x1 — x2. In this model a superfield can be described by
$(x, e\e2) =
(56)
and the supersymmetry generators are Qi = di+ 6ld+ Q2=d2 + 0 2 3_ Di=di6ld+ D2 = d2- 02d- •
(57)
The kinetic part of the Lagrange function is of the form
S= I' dxd91dP^eabDa$Db$
(58)
and the interaction can be introduced, like in the one dimensional case, by the superpotential W($). After the integration of the fermionic coordinates and solving the auxiliary field with the Euler-Lagrange equations (F = — W(
dx[-a+f - ^ ( 0 ) Vi *h] • (59) TThe conserved supersymmetric current can be calculated and its charges are Qi=J
dxd-H>\ - V{(j>)il>2
( 6 °)
By a change of coordinates of the ^-space ^_ = — ipi, tp+ — rp2 we find the usual chiral charges [OW] 438
Q_ = fdx | d-H>- + V(
(61)
Witten and Olive showed a Bogomol'nyi bound for this model that we will reproduce in this lectures. The supersymmetric algebra relations for these charges have an extension: Q+ Q_ + Q_ Q+ = J dx 2V(<j,)d(f>/dx ,
(62)
which can also be written Q+ Q- + Q-Q+=
[ dx d/dx {2H{<j>)) = Z ,
(63)
where H{<j>) is a function such that H'(
dx — (2a2X(j)--X(j>3) -00
.
(64)
"
&•£
Z vanishes in a topologically trivial state, has a positive value in a kink state, a negative value in an antikink state. Our modified algebra is now Q\ = P+, Ql=P-, Q+Q- + Q-Q+ = Z, and has an interesting consequence. From the algebra we find P + + P _ =Z+(Q+-Q_)2 P++P-
, 2
= -Z + (Q+ + Q_)
2
But (Q+ + Q-) > 0, so P++Pat rest, P+ — P_ = M, implies M>
(65) .
> \Z\. For a single particle the mass M
\\Z\
(66)
called the Bogomol'nyi bound or BPS bound. This equation is saturated by the states \a > such that (Q+ - Q-)\a > = 0 or ( Q + + Q-)\a > = 0 439
which is satisfied at least classically by the soliton and antisoliton <>/ (d<j>/dt — 0 , and d(f>/dx = ±V(
In 4
dimensions
In four dimension the procedure is more complicated but in principle it follows the same mechanism. It is known to work better in a subspace of the superfields [BW] called scalar or chiral field (£»$ = 0) or vector field (V+_=_V). It is also possible to write a scalar field from a vector field (Wa = \DDDaV) and with them in a special gauge choice (Wess-Zumino gauge) we have in a component the Yang-Mill field strength part arise as a component. The class of renormalized Lagrangians may be restricted by a global invariance called R-invariance. R acts on chiral multiplets as follows: R ${x,6) = R $+(x,6)
e2ina
= e-2ina$+(x,eia9)
,
where n £ N and a 6 R. In N = l Supersymmetry the full Lagrangian can be written as: L = ±Im {Tr r / d26W_aWa) + J d20d26 Tr &e-2V$ 2
2
(67)
+ J d 6W + J d 0W . In components the expansion after ^-integration looks like L = --^F«VF^ + ^F*UF^ - £ A V ^ A ° + £sDaDa +(D^4>yDii<j> - itpa^D^ - Da^Ta(j> iy/2^Ta\^
+iV2i>Ta\f\a + F}FZ + ^F + 0 F /
(68)
where W denote the scalar component of the potential and the Ta are the generators in the group. The auxiliary fields F and Da can be eliminated by equations of motion to get the scalar potential
^ = EH^II 2 -^ 2 (^» 2 -
(69)
i
In N=2 SUSY we have to introduce a second copy of the field and the two fields can be organized in a multiplet under N = l . The most general Lagrangian can be constructed using the chiral superfield $ 440
1 L = —Im-
L=~Im
([d20FTSWraWsa
~ Tr
a A^I \f d(ftzQd^T{^)
or
+ 2 f d26d29(&e2gV)rFs($))
,
(70)
where T is a holomorphic function, J 7 S ($) = dF/d$s, .7>s($) = d2Tj'd$Td$s, * is a chiral field in N = l and T is referred to as the N=2 prepotential. From here we can read a Kahler structure with a Kahler prepotential Im&Ts($). Writing in components the electromagnetic and scalar part with a potential T = T(j>2 yields L = ±Tr(-^F^F+g2^F^
+ (D^D>*
.
(71) In the case of a SU(2) gauge group the potential is V(
441
(72)
4
The Seiberg-Witten Analysis
In this section we show the analysis of Seiberg-Witten using duality and the BPS bound of the mass (M> %/2||a(ne +i/a nm)\\) to describe the Moduli space for the SU(2) symmetry as an elliptic curve. 4.1
Moduli
space
The scalar potential in N=2 has a Higgs vacuum in [(/>, ft] = 0, which implies that <j> takes values in the Cartan subalgebra of the gauge group <j> —
In the case of SU(2) the range is one. The Weyl invariant parametrization of the vacua is given by u = (Tr
Duality
To find the duality transformation we consider the gauge terms in the bosonic part of the action.These terms can be written: ^Im
Jr{a){F
+ iF)2 = ^Im
jr{a){F2
+1FF) .
Now, as in the above first part, we regard F as an independent field and implement the Bianchi Identity dF=0 by introducing a Lagrange multiplier vector field VD which is coupled with a monopole:
£ J V»^>°dvFpa
±Re J(FD - iFD)(F + iP) ,
= ±.JFDF=
where Foy,v = d^Vov — duVo^- Adding this term to the gauge field action and integrating over F, gives the dual theory:
a W (-;) ^ + i^»2 = l W ("?)« + i# " F ">' 442
Thus the effect of the duality transformation is to replace a gauge field which couples to electric charge by a dual gauge field which couples to magnetic charges, and at the same time, transforms the gauge coupling: r ->
TD
=
1 T
.
The action is also invariant under the transformation r -» r + 1. Both transformations generate the SL(2, Z) group, which is identified to the full duality group. This group acts on r by: ar + b CT + a where, ab-cd=l and a, b, c, d 6 Z. In the above N = 2 supersymmetric formulation let r(
Im J dtehWtf
= Im f d46hD((t>D)(j>D ,
is also invariant. The action of this group on
('a b\ ('4>D
\cd)
V0
The transformations r —> — 1/r and r —> r + 1 are implemented by the matrices
which are the generators of SL(2,Z). Let us see how this invariance can be interpreted in the Moduli space. Prom the scalar part of the Lagrange action we interpret the vacuum space as a Kahler space where the metric is ds2 = ga da1 do? = Im -—^— da1 da3 . J
(73)
ddiOCLj
Now we can use the abpve identification to write ds2 = Im ddD da — ~- (dao da — da dao) ,
(74)
which is invariant under SL(2,Z). Prom here we can identify a and ajr> as coordinates on a space X ^ (E on which we can choose a symplectic form 443
w = Im dao A da. The functions (a£>(u),a(w)) gives a map from M to X. In other words, they determine a section of X regarded as an SL(2, Z) bundle over M. In terms of the coordinate u the metric on the moduli space takes the form , o r dap da , ,_ i (dao da dap da\ , ds = Im — — dudu — — - I — — ——— duu . du du 2 \ du du du du J 4-3
Seiberg-Witten
.__. (75)
analysis
In the N=2 Supersymmetry when we choose a vacuum <j> = \ aaz with a ^ 0 the SU(2) is broken into a U(l) symmetry and by the Higgs mechanism W± acquires a mass. It is clear that this theory only has a meaning outside a neighborhood of u — 0, since at this point the W± become massless. This tells us that u = 0 is a singular point in the classical moduli space Mc • The other singularity is at u = oo where the coupling is weak because of asymptotic freedom. It is known that a N=2 supersymmetric Yang-Mills theory is not exact but the quantum corrections are under complete control [SEI]. It is governed by the holomorphic prepotential T and it receives all corrections at the one loop level. As one could expect from a mathematical point of view, the non triviality of the metric arises from a (logarithmic) singularity of F: H4>) = \TC
l
-H^\
,
(76)
where A is a scale factor. FVom this we can compute in the region of weak coupling, near to u = oo r(u)=rc + ^ln[^].
(77)
This implies that if we loop around u = oo in moduli space, the logarithm will produce an extra shift of 27rz and thus r -> r - 4 . Prom this it is clear that r and T are multivalued function, moreover Ira T(U) is a metric must be positive definitive but it is a harmonic function therefore if it is globally defined it must be constant. The question is how many and what kind of singularities the exact quantum moduli space should have, and what the physical significance of these singularities might be. Seiberg and Witten [5H^1] proposed two singularities at u = ±A 2 and the classical singularity at the origin disappears. This proposition is very difficult to prove rigorously and it was postulated on the grounds of an analysis of the strong behavior guided 444
by the 't Hooft ideas about confinement. Like in classical u=0 singularity, the strong coupling singularities in the quantum moduli space were attributed to certain excitations with the gauge bosons becoming massless as well. The BPS bound is a powerful tool to handle soliton masses. Like in the 1+1 dimensional supersymmetric model above, the bound is given by the central charge of the algebra in question: m 2 > \Z2\
(78)
For N=2 supersymmetric Yang-Mills theories, it take the form Z = qa + gaD
(79)
where q, resp. g are the magnetic, resp. electric quantum numbers of the BPS states under consideration. From the expression for the quantum T at M = A 2 , a / 0 but QD = 0 so that a monopole hypermultiplet with charges (q,g)=(±l,0) would be massless. On the other hand at u = —A2 obtain from u —> —u and is similar to that obtain at u — A2 replacing ao in FD^D) by a/j — 2a.
The key idea is to patch together the local information in a consistent global way to fix the theory. The local term determines the local monodromy M around a given singularity, and this acts on the section as
(\ aDa(u) }UA J -> M (\ aDa{u) ^>\ J . In particular, the asymptotic behavior at large ||a|| is given by
V«(«))-{
vW2
) •
m
The monodromy at oo is calculated by a loop on the u-plane around u=0. We get In u + 2iri, hence aD -> -ad + 2a , a ->• -a.
(S1>
The monodromy matrix implements this M no =
-1 2 0 -1
(82)
The magnetic and the electric quantum numbers of BPS states undergo a change (q,g) -¥ (-q - 2g, —g) so that the mass formula is unchanged. For the other monodromies at M^ and M_A2 we use other strategies. Under the 445
transformation the mass bound must be invariant therefore the charges (q,g) transform under the inverse matrix i.e if
a
a)-(zti°) to ©-(v;)' (83>
where aS — @j = 1. Now we choose the parameters a, /?, 7, 6 in order to have g=0 and q = l after the transformation. So we can identify the matrix in the form
Since we assume that the singularity are only at u = 00 and u = ±A 2 , MA2(g,q)M_A2(g',q')
= M00
(85)
f-12\
(86)
A simple solution to this equation is given by 1 0\
,,
where the monopole (q,g) = (0, ±1) and the dyons (q,g) = (—2,1) becomes massless 4-4
Solving the Monodromy
Problem
In Mathematics the problem of finding multivalued functions a(u), CLD{U) with this monodromy at the singularities has a unique solution and it is called the Riemann-Hilbert problem. We can consider a(u), ao(u) as certain periodic integrals related to some surface X, which can be seen as a spectral surface. The matrices M±A2,oo generate a subgroup T 0 (4) of SL(2,Z) built from matrices of the form r 0 (4) = { ( ° M eSL(2,Z),b
=
0mod4}.
The quantum moduli space can thus be viewed as the upper half-plane modulo the monodromy group: M S JJ+/r0(4) . The appearance of a subgroup of the modular group suggests that one can formulate the manifold as a toroidal Riemann surfaces of an elliptic curve. This is algebraically characterized by 446
y2(x, u) = n ^ - * ( « , A)) = (x2 - u)2 - A4 .
(87)
i=l
The period matrix of the torus is interpreted as the gauge coupling. Prom the theory of Riemann surface, the periodic matrix is defined by =
_ w(u)
U>D(U) — f w, JB
w(u)
T{U)
,
(88)
where =
Ja
Here w = -A—7^y and a, 0 are the canonical basis homological cycles of the torus. From the relation r = dap/da we find _
daD(u)
_
da(u)
which implies O,D{U) = f Xsw, JB J/3
a(u)
= f Ac Ja
where 1 o dx Xsw = -7^-x2— . v/27r y(x.u)
(89)
The four zeroes of y (x,u) = 0 are ei = -y/u + A2 , e 2 = -Vu — A2 e% = —yju — A2 , e4 = y/u + A2 The singularities in the quantum moduli space arise when the torus degenerates, and this happens when any two of the zeros e^ coincide. The calculation for other groups can be read in [KLT, L]. References [AH] Alvarez-Gaume, L. and Hassan, S.F., Introduction to S-Duality in N=2 Supersymmetric Gauge Theories, Fortsch.Phys 45 (1997) 159-236, hep-th 9701069 447
[BW] Bagger, J. and Wess, J., Supersymmetry Supergravity, Princeton Univresity Press 1998 [CHL] Cheng, T., Li, L., Gauge Theory of Elementary Particle Physics, Oxford University Press, 1989 [DF] Deligne, D., Freed, S., Supersolutions, Quantum Fields and Strings: A Course for Mathematicians, Am. Math Society and Institute for Advanced Study 1999 [DIJ] Dijkgraaf, R., Fields, String and Duality, Les Houches 1995 [FIG] Figueroa-O'Farrill, J.M. ,Electromagnetic Duality for Children, Draft, 1997. [Fr] Freund, P., Supersymmetry, Cambridge Monographs on Mathematics Physics 1988 [KLT] Klemm, A., Lerche, and Theisen, S., Nonperturbative Effective Actions of N=£ Supersymmetric Gauge Theories, Int.J.Mod. A l l (1996)19291974, hep-th 9505150 [L] Lerche, W., Introduction to Seiberg-Witten Theory and its String Origin, Nucl.PhysProc.Suppl 55B (1979) 83-117; Fortsch. Phys. 45 (1997) 293340, hp-th 9611190 [OW] Olive, D. and Witten, E., Supersymmetry Algebras That Include Topological Charges, Phy. Lett. B78 (1978) 97-101. [Pol] Polyakov A. M., Particles spectrum in the quantum field theory, J E T P Lett. 20 (1974) 194-195 ['tH] 't Hooft G., Magnetic monopoles in unified gauge theories, Nucl. Phys B79 (1974) 276-284. [TSOU] Tsou Sheung Tsun, Concepts in Gauge Theory Leading to Electromagnetic Duality, In this volume [SEI] Seiberg, N., Supersymmetry and Non-Perturbative Beta Functions, Phy. Lett B206 (1988) 75-80. [SW1] Seiberg, N. and Witten, E., Electromagnetic Duality, Monopole Condensation, and Confinement In N=2 Supersymmetric Yang-Mills Theory, Nucl. Phys, B426 (1994) 19, hep-th 9407087. [SW2] Seiberg, N. and Witten, E., Monopole, Duality and Chiral Symmetry Breaking in N~2 Supersymmetric QCD , Nucl. Phys. B431 (1994) 484, hep-th 9408099.
448
SHORT COMMUNICATIONS
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
REMARKS ON DUALITY, ANALYTIC TORSION A N D G A U S S I A N I N T E G R A T I O N I N A N T I S Y M M E T R I C FIELD THEORIES ALEXANDER CARDONA Laboratoire de Mathematiques Appliquees Universite Blaise Pascal (Clermont II) Complexe Universitaire des Cezeaux 63177 Aubiere Cedex, France. E-mail:[email protected] From a path integral point of view (e.g. [Q98]) physicists have shown how duality in antisymmetric quantum field theories on a closed space-time manifold M relies in a fundamental way on Fourier Transformations of formal infinite-dimensional volume measures. We first review these facts from a measure theoretical point of view, stressing the importance of the Hodge decomposition theorem in the underlying geometric picture, ignoring the local symmetry which lead to degeneracies of the action. To handle these degeneracies we then apply Schwarz's Ansatz showing how duality leads to a factorization of the analytic torsion of M in terms of the partition functions associated to degenerate "dual" actions, which in the even dimensional case corresponds to the identification of these partition functions.
Introduction Antisymmetric field theories are generalizations of electromagnetic theory where the potential 1-form is replaced by a fc-form. Some remarkable facts arising in electromagnetism are also observed in general antisymmetric theories, notably T-duality on which we will focus here. In electromagnetic theory this type of duality corresponds to the observation that electric and magnetic fields in the theory are interchanged under transformations taking solutions of field equations into solutions of the Bianchi identity, particles into topological defects, weak couplings into strong couplings, etc. (for a review see [095]). Consider a theory of antisymmetric tensors on a n-dimensional spacetime manifold M equpped with a Riemannian metric. Let ojili2„_ih be a fc-tensor field on M, consider the fc-form Wfc = Wi-li2...ikdx%1
A dx12 A • • • A
dxlk,
for 0 < k < n, and the Euclidean Action of the theory denned by S(u>k) =
(dkuk,dkuk), 451
(1)
where dfc denotes the exterior derivative on the space Ctk of fc-forms on M and the inner product (,) : Ctk x Clk ->• H is denned by Hodge-star operation on Clk, namely (ofc,j8fc)= /
akA*pk.
(2)
JM
This generalizes electromagnetic theory, where the potential is described by a 1-form A, the electromagnetic field by its exterior derivative (F = dA), and where by gauge invariance of the theory we mean the invariance of F under "gauge" transformations on A of the form A^A
+ dx,
(3)
X being an arbitrary function (0-form) on M. Following [<398], T-duality in the case of antisymmetric field theories is the statement that two different theories (defined by two different actions S and S*) give rise to the same generating function, being therefore (at the quantum level) physically equivalent. As in [Q98] [W99] and many other references on this topic, in this paper we focus on the identification of partition functions, hoping to complete the discussion on the level of generating functions in some later work. By partition function we understand the formal object Z(S) = f exp{-kS(a)}
[Va]
(4)
where k denotes a (positive) constant (including Planck's and coupling constants), [Da] denotes a formal measure on the space of all the fields a and S(a) the classical action of the theory under consideration. Looking for a dual version of a theory means looking for a different action, called dual action (on a different set of dual fields), giving rise to a dual partition function. Starting from a given action (i.e a given theory), a standard procedure to obtain a dual action (i.e. a dual theory) is the so-called gauging of the global symmetry of the original theory [<398][W99]. This requires introducing new variables into the original action in such a way that integrating them out we can recover the original theory and integrating out the original variables of the action we find the dual one. Unlike in [Q98], we only consider local symmetries (namely of the type (3)) of the classical action, having left aside global symmetries because of the acyclicity assumption (see section 1). The very presence of local symmetries leads to degenerate actions. Forgetting about degeneracy of the classical action, as was pointed 452
out in [
1
The Geometric Setting
Consider a closed (i.e. compact and without boundary) n-dimensional Riemannian manifold M, and let p be a representation of the fundamental group of M on an inner product vector space V. Let E(p) be the vector bundle over M defined by p, and consider the space of fc-forms on M with values in E(p), for 0 < k < n, i.e. C°°-sections of the vector bundle AkT*M
Finally, we assume that the complex 0 _ > 0 ° _ ^ . . . . n * - l d±Z}Qk J^
nk+l
d
±t} . . . 0 n J^
0>
(5)
is acyclic, i.e. all the de Rham cohomology groups of the complex are trivial (Hh(M,p) = {0}, 0 < k < n). This representation of m(M) will be fixed through all the paper and no specific reference to it will be given (in the notation) in the sequel. Let us focus on the space of A;-forms
n*-1 d±4 nk ^L nk+\ where d*k denotes the formal adjoint to dk, and on the Hodge decomposition
nk = nk®n'k'
(6)
where tt'k = Im dk-\ = Ker dk and Q'k = Im d^ — Ker d*k_l, as follows from our asumption of acyclicity. Accordingly, wk £ Qk splits into ojk = oj'k © w^' where u>'k = dfc-iWfc-i 6 &'k and u)k = d*kujk+\ G Wk, for some uik-i € fi fc_1 , uik+i enk+1. Consider the functional S0 : nk ->• IR uk
(->• <S0 (w fc ) = (wfe, ojk),
(7)
on fc-antisymmetric tensor fields. Then, from the decomposition (6) and dkdk-i — d*k_^dk = 0, it follows that S0{uk) = {dk-iLJk-i @d*kuk+i, dk-iOJk-x ®d*kuk+i) = {dk-iUk-i, dk-iuk-i) © (4wfc + i, dfcWfc+i)Thus, we find a canonical decomposition of iS in terms of two degenerate action functionals, namely
So(wfc) = s(w f c _i)es*(w f c + i),
(8)
<S(wfc_i) = (dfc-iWfc-i, dk-iUJk-i)
(9)
where
and 5*(w fc+ i) =
(10)
which are degenerate on fi fc_1 and Qk+1, respectively. The functionals <S(wfc_i) and <S*(u>fc+i) are degenerate but, by restriction on the respective domains, the maps
dk: ni -+ n'fc+1
(ii)
d*k : fi'fc+1 -+ ill
(12)
and are isomorphisms, giving rise to the bijective maps d*k_idk-\
: ftk-i "*• ^fc-D
Thus, the functionals S K ^ ) = (dfc-iW^!, d t - l C l )
(13)
5*(wt+1) =
(14)
and are non-degenerate on n^.'_1 and flj t + 1 , respectively. These spaces are the ones we shall be working with in section 2 in order to have partition functions of non-degenerate actions. The identification between two dual antisymmetric field theories involves identifying formal integrals, which we will interpret as gaussian integrals since they are defined using quadratic actions. In section 2 we study the "equivalence" between two such partition functions from a measure theoretical point of view in the case in which the action functionals involved are not degenerate and, following Quevedo [Q98], we give the heuristic path integral interpretation of duality. The case of degenerate action functionals will be studied in section 3. 2 2.1
Duality and Gaussian Measures on Antisymmetric Tensor Fields Some
Facts about Gaussian
Measures
A characteristic function on a topological vector space E is a continuous (on every finite dimensional subspace of E) function x satisfying N
Y,
«i*fcX(fc-&)>0
j,k=l
455
for ak E <E, £,•• € E (j, k = 1, ...,N). In a finite dimensional vector space E, with inner product (,), Bochner's theorem assures a one-to-one correspondence between characteristic functions and measures [Y85]. In particular, to the function x(0 = e x p | - i ( e , o } there corresponds a unique Borel measure on E, called Gaussian Measure and denoted by fi, such that X ( 0 = / exp{*(£,)}dn(4>) JE
and fi(E) = 1. In infinite dimensions, starting from a characteristic function X on a topological vector space E, one typically ends up with a measure with support in a larger space. Even in the case of a Hilbert space %, the corresponding measure to a characteristic function lies in some HilbertSchmidt extension of H. Bochner's theorem holds in the case of continuous characteristic functions on a nuclear Hilbert space (a topological vector space whose topology is defined by a family {|| • | | a } of Hilbertian semi-norms such that Va 3a' : || • || tt is Hilbert-Schmidt with respect to || • \\a>) [GV64]. The case we are dealing with is that of a Hilbert Space H (with inner product (, )-u) and, for a > 0, where we consider the characteristic function Xa.o(£)=expj-^
(15)
where G is a positive bounded operator on H, corresponding to the infinite dimensional gaussian measure (with covariance G) formally written d/i„,o(0" = " - ^ - « p { - ! < G - V . 0 > w } p t y ] ,
(16)
(where Za,a is a constant such that /i o , G (?0 = 1) the support of which lies in a Hilbert-Schmidt extension of %. All this can be sumarized in the single equation Xa,a(0=
exp{i(^^)«}djUa>GW,
(17)
in where the left hand side involves G (see (15)) and the right hand side involves G~l (see (16)). This generalizes the very well known relation
e x p j - ^ A - 1 ^ ^ ! =k{detA)-*
J ^expU(x,y)-^(Ay,y)\dy, 456
(18)
where x,y £ IR™, k is a constant, A denotes a positive matrix and (,) denotes the inner product in this space. Equation (17) defines the function Xa,G as the Fourier Transform of the gaussian measure /j,aiG, which we will denote by
2.2
Gaussian
Measures
and
Duality
Consider the acyclic complex (5) and Hodge decomposition (6) on the space of fc-forms. We take T-Lk = L2(ilk), where the closure is taken with respect to the L 2 -hermitian product (,) denned by the Riemannian metric on M and the inner product structure of E(p), and we consider the decomposition Kk — H'k® It'l induced by (6). For a, b > 0 consider the gaussian measures fia on flh and fj,b on Q'k defined by the characteristic functions fia(ak) = e x p | - - ( a f c , a f c ) |
(19)
and
ftfok) = exp{-^i£X>}-
(20)
P r o p o s i t i o n 1 Let a, b > 0, then [
d/4fafe)/Wfc) = /
<Wafc)%K).
(21)
Proof. By (27), /2 o (0 = / exp{i(^
dn'b(n'k)i2'a(ri'k) = \
dfj,'b(n'k)
= j = /
exp{i(r]'k,ak)}dfia(ak) JQk
JU'k
JWk
dfia(ak)
/
exp{i{n'k,a'k)}dnb(ri'k)
dna{ak)V!b{a'k).
a Let us see that equality between the partition functions corresponding to the action functionals (13) and (14) can be seen as the heuristic limit of (21) when b goes to infinity. Let e be a positive real number and take b = - . Then, /
dna(ak)ii'i{<x'k
and taking the limit e —> 0 (in the sense of distributions) of the gaussian characteristic function juj, we find a Dirac delta function forcing a'k to vanish. The corresponding limit of the associated gaussian measure on Sl'fc is heuristically (proportional to a) Lebesgue measure on that space. Thus, if we write the formal expression (all these calculations are formal, the measures are of course ill-defined "Lebesgue measures" on L2 spaces of forms) /
ftfok)
VW\ " = " I
< M « * ) SK = 0],
(22)
we find, by using of the formal relation (16), j a i exp { - | ( ^ , ^ ) } [DV'k] " = " J^ exp | - l < a f e , a f c > } S[a'k = 0}[Vak]
" = »^exp{~-^(^',a'fe')}[^'fc']. Now let us do the change of variables defined by the maps (11) and (12), rj'k = dk-\0Jk_l and a'k' = d*k(jj'k+1, then we find
J&_! /exp {-\§WU)}
PM'-J
(23)
" = " Jk+i j ^ exp | _ ^ K where Jk-i Jk-i
and Jk+i
+ 1
denotes the associated jacobian
) | [Vu'k+1], determinants
:= Wdet(^_ 1 d f c _i) and Jk+X •= y/det(dkd*k).
Finally let us write down the formal calculations usually used to arrive to relation (23); they involve Fourier Transforms, usual properties of gaussian integrals and changing the order of integration [Q98]:
/"exP{-fS0(^)}M J
[Thi'k] J
[Vak]exp | - ^ 5 0 ( a f c ) | exp {i{r,'k,ak}}
J
[Vak]exp{-±S0(ak)}
J
Mexp{t
^[2>afc]exp{-^S0(afc)} J
[P^]exp{i(^,o4)}
458
J
[Vak]exp^~S0(ak)^S[a!k
= 0]
which, after the change of variables defined by the maps (11) and (12), is equivalent to (23). Let us make a few comments on this computation which, very formal, gives the gist of the dualization procedure.
although
1. Hodge decomposition in the case of an acyclic complex splits the space of fc-antisymmetric tensor fields (6) and then, through isomorphisms (11) and (12),
nk^n'U®^k+i2
(24)
k
The L scalar product on Q then gives rise to two (non degenerate) actions <S and S* ((13) and (14)), on tt'k_1 and ftj.+1 respectively, which are related by a Fourier transform. The non-degeneracy in the actions comes from the fact we restrict ourselves to C i ^ n ^ ^ i .
(25)
Thus, the field u)k £ flk splits into wfe = dfc-iw^! ©
(26) ^k-i^'k+i-
2. In the process of taking the Fourier Transform, the coefficient of the quadratic action is inverted (a H> a - 1 ) , a fact often observed in duality and typical for Fourier transforms of gaussian functions. A strong coupling can thus be turned into a weak coupling [£>98]. 3. Finally, if we consider Hodge star duality on the complex, through the relation djw fe+ i = (-l)"*+»+ 1 *d„_ f c _ 1 *w fc+1> we recover the usual "moral" of duality in antisymmetric fields [Q98]: a (k—l)-rank antisymmetric tensor field (the "gauge potential" u>k-i) is dual to a (n—k— l)-rank antisymmetric tensor field (r)n-k-i = 459
*Uk+i) or, in "brane" language, a (fc-2)(electric)-brane is dual to a (n—k—2) (magnetic)-brane. In fact observe that {d*kuJk+i,d*kLOk+i) — (*d„_fc_i *wfe+1,*dn_/fc_i *w fc+ i) = *2(dn_fc_177n_fc_1,dn_fc_177„_fc_1) where * 2 denotes a ± sign depending on k and the dimension of M. i,From a physical point of view this formal computation tells us that if we consider an antisymmetric field theory modelling physical fields by fc-forms, given by the action (7) then (in this acyclic case) we find two possible "potentials" associated to that field: w^'_x and co'k+1, the first one for the exterior differential dk-i, the second one for d£ (see (26)). Writing the partition function of the theory with respect to one or the other give us "dual" formulations of the same theory. 3
Duality and the Analytic Torsion of the de R h a m Complex
Unlike in the previous section, we now want to take into account local symmetries of the type (3) and "dual ones" obtained replacing d by d*. Thus we now consider the degenerate actions (9) and (10) computing their corresponding partition functions and we show how from this point of view duality leads to a factorization of the analytic torsion of the space-time manifold. The analytic torsion of a Riemannian manifold M is a topological invariant defined by some spectral properties of the Laplacian operators acting on spaces of differential forms on M. These properties are a consequence of the one to one correspondence ilk -A- Cl'k+1 used previously, and its "dual" fik+1 -4 flk (both of them defined in the acyclic case), together with the Hodge star duality map. In this section we will study the relation between two dual antisymmetric tensor field actions, their partition functions and the analytic torsion of the space-time manifold on which such fields are defined. We will use zeta-regularization techniques [G95] and an Ansatz introduced by Schwarz to define the partition function associated to a degenerate action functional [579]. 3.1
Zeta-Regularized Determinants Riemannian Manifols
and Analytic
Torsion
on
Let us take again a closed n-dimensional Riemannian manifold M and the acyclic de Rham complex (5) on it, with the Hodge decomposition (6) of the space of (E(p)-valued) fc-forms on M. The Laplacian operator on fc-forms, 460
Afc = dk-id*k_1 + d*kdk, is a positive selfadjoint elliptic operator, and its determinant can be defined by the zeta-function regularization method as, detcAfc=exp{-Ckfc(0)},
(27)
where the zeta-function is given by
and the sum is over all the eigenvalues A of A*. Indeed, it can be shown using properties of elliptic operators on closed manifolds that this function is analytic for s € (D with Re(s) » 0, and extends by analytical continuation to a meromorphic function on
G £kW,dkojk
= 0} = £k(X) n fl'fc
and
s'k\\) •.= {Wfc e £ fc (A),4_ lWfc = o} = £k(\) n ni Let wfc e Sk(X), and suppose wfc € ilk. Then dku>k e ^' fc+1 and Ak+1dkujk
= dkd*kdkuk = dkAkuk
= Xdkojk,
so dk maps £k(X) bijectively into £'k+i(X): giving us a bijective correspondence between (non-zero) eigenvalues (and their corresponding eigenvectors) of the operators d*kdk and dk+id*k+1. 461
The zeta-regularization techniques used to define the determinant of the Laplacian operators can also be used to define the "regularized determinants" of the maps dk-idl^ and d*kdk- Prom the decomposition (6) of each space fifc, and the nilpotency of the d*k and d& operators, it follows that the set of eigenvalues of the Laplacian operator A^ is the union of the eigenvalues of dk-idl_1 and d%dk- Since the eigenvalues of dk-\d*k_l and d%_xdk-\ are the same, the set of eigenvalues of A^ is the union of the d^_1dj;_i and d*kdk eigenvalues. So, if we define the zeta-function associated to the operators <4-idfc_i and d*kdk by
and
where A" and A' denote (non zero) eigenvalues of dk-id*k_x tively, it follows that, for 0 < k < n, CA„(S)
an
d d*kdk, respec-
= Cfc-i
(32)
Hence &td>(*) =
CA.(S)
- Ub-A«) +
CA,_2(S)
- • • • + (-l) fc CA 0 (^),
(33)
and from the properties of the zeta-function of the Laplacian, it follows that C<2fc_id*_ and Cd*dk are well defined and analytic for s € C with Re(s) » 0, and extend by analytic continuation to meromorphic functions on C, regular at the origin. Moreover, using the fact that £k(\) = £'k+1(X), we find Cdl_1dk-As) = Qkdl(s).
(34)
It is clear now that we can write det c A fc =exp{-CA f c (0)}
= exp{-Ct_ld._1(0)-^.(ifc(0)} = det^dk-idl^)
det c (c^d fc ),
where det^(dk~idl_1) and det^(d^dfc) are defined in a similar way as the zeta determinant of Laplacian operators (equation (27)), using (30) and (31), respectively. 462
3.2
The Partition Function (Following Schwarz)
of a Degenerate
Action
Functional
Let us come back to the action S0(ujk) = (wk, w^) on Qk and its decomposition (8), S0(u>k) = <5(wfc-i) © <S*(u>fc+i), induced by Hodge decomposition of Clh, into the degenerate action functional ((9) and (10)), <S(wfc_i) = (dl^dk-iuik-i,
Wfc_i)
and <5*(wfc+i) = (dkd*kujk+i, Wfc+i), on ft*"1 = n'fc_x © ft'^ (= Ker (4_ 1 d f c _ 1 ) © Ker (dj^dk-i)- 1 -) and nk+1 = il'k+1 © fi'fc'+1 (= Ker {dkd%) © Ker (cW^)- 1 ), respectively. In section 2 we were dealing with non degenerate actions S and S*, since we had restricted ourselves to fij.' and fij. + 1 . As we pointed out there, the degeneracy leads to some formal volume of an infinite dimensional space, and the non degeneracy condition restrict us to look at only a part of the complex (5), namely (25). Schwarz suggested an Ansatz, inspired from the well known Faddeev-Popov procedure, to "compute" this volume and give a meaning to the partition function of a degenerate action functional [579]. Following Schwarz's method, provided we can associate a chain of vector spaces and maps, called resolvent, to the degenerate action, then the partition function associated to that action can be defined in terms of (regularized) determinants of the maps appearing in the resolvent. In our particular case, as we will see, this means to consider the whole complex (5) and not only a part of it, as we done in the non degenerate case. In the case of S(wk-i) the resolvent is the given by (for details about the definition of the resolvent associated to a degenerate functional see e.g. [579] or [BT91])
o-^n°A-
d
±3 nfe-2 d-^ n;.! d'k^1 o,
(35)
and we define the partition function associated to that action (and resolvent) as fc-i Z{S) = d e t c K - W f c - O ^ I I l d e t c(dfc-i-i)| ( ~ 1 ) i + 1 . 3=1
463
(36)
Notice that when k = 1, Z{S) gives back the usual Ansatz to compute the partition function (compares with (4) and (18)) Z(5)" = " /
exp{-i<do/,do/>}[X>/]" = " ( d e t A 0 ) - i .
In the same way, taking the resolvent associated to S*(u>k+i), 0 —> n» ^
. . . di±? fi*+2 di±? n ^ + 1 ^
0,
(37)
we define the associated "dual" partition function by Z(S*) = dek(dkdl)^
H
I detC(^+,)|(-1)J+1 -
(38)
Here, det^(dfc) and det^(d^.) are defined by |det<(dfc)|:=y/de^(44), |det c (dj)|:=^/det c (d f c dj) and, as we remarked in the previous section, d e t c ( 4 d & ) = det(
(dkdl).
Therefore, the two "dual" partition functions are given by fc
Z[S) = n [**<(<**-;<**-:;)]L=¥L
(39)
3=1
and n—fc—1
Z(S*)=
I I [det c (d fc+i diE +i )]
.,l I=i
^.
(40)
3=0
3.3
Analytic
Torsion
on Riemannian
Manifolds
and
Duality
The relation between the analytic torsion of the manifold M and the partition function of an antisymmetric field theory defined on it, by the method discussed above, was pointed out by Schwarz ([579], see also [5T84]), studying quantization of antisymmetric tensor field theories defined by the degenerate action (1) on flk. It has been also used in the context of Topological Quantum Field Theories [W89][BT91][ASQ5]. Schwarz shows that the Hodge star duality map and (34) imply a factorization of the analytic torsion T(M) in 464
terms of the two partition functions, corresponding to the actions S(tOk-i) and S(u>n-k-i)In this section we want to relate the partition functions Z{S) and Z{S*) ((36) and (38)), corresponding to the two "dual actions" {dk-iWk-i,dk-iWk-i) and (d£u>fc+i,d£wfc+i), with the analytic torsion of the manifold M on which these antisymmetric field theories are formulated. Such a relation is clear if we look at the splitting in the de Rham complex (5) induced by the two resolvents, (35) and (37), associated with the partition functions of the dual theories defined by the given actions, namely O-^fioi^...^
n f c _!
dj^ nk g_
fifc+1
£+,
£_,
fin
^ _Q
(41)
Observe that, from (39) and (40), Z(S) S
Z ( H
, =
r
lo de
,
, „: , ,,<-D*
^ k(dod0)}
,
.,
. ,„, , . 1 (-i) t ~ 1
» +log[det c (dfd 1 )]
»
+•••
+ log[det c (dJ_ 1 d f c _i)]T -log[det c (d f c dJ)]Tr - log [ d e t c ( 4 + i 4 + 1 ) ] 2 and from (27) and definition (29)
log[det c (d n _id;_ 1 )] < ~ 1 2"" ,
T(M)=expfi2(-l)*&t(0)J then, logT(M) = i £ ( - 1 ) ^ ( 0 ) fc=0
= £>g(det c d j d * ) ^ fc=0
so,
Z(S)Z(S*r1 =T(M)^1^~k~\
(42)
Thus, we can say that two dual actions yield a factorization of the analytic torsion of the space-time manifold (coming from the splitting (41)) in terms of their corresponding partition functions. Hence in even dimensions we get the expected identification of the partition function with its dual. Note that the analytic torsion is a topological invariant of M, but there is no reason for Z(S) and Z{S*) to have this property. 465
Acknowlegments This is the written version of two talks given in the Summer School on Geometrical Methods in Quantum Field Theory, Villa de Leyva (Colombia), July 1999, and at the CIRM colloquium Families of Operators and their Geometry, Marseille (France), June 2000. The author wishes to thank S. Paycha, T. Wurzbacher, D. Adams and S. Rosenberg for many helpful discussions. The author is also indebted to the referee for helpful comments and suggestions. References [AS95] Adams, D. and Sen, S. Phase and Scaling Properties of Determinants Arising in Topological Field Theories. Phys. Lett. B 3 5 3 , 495 (1995). [BT91] Blau, M. and Thompson, G. Topological Gauge Theories of Antisymmetric Tensor Fields. Ann.Phys. 205, 130 (1991). [D98] Dijkgraaf, R. Fields, Strings and Duality, in "Quantum Symmetries", Les Houches Session LXIV. Elsevier, 1998. [GV64] Gel'fand, I.M. and Vilenkin, N. Generalized Functions Vol. 4- Academic Press, 1964. [G95] Gilkey, P. Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. CRC Press, 1995. [095] Olive, D. Exact Electromagnetic Duality, hep-th/9508089. [Q98] Quevedo, F. Duality and Global Symmetries. Nucl. Phys. B (Proc. Suppl.) 61 A, 23 (1998). [RS71] Ray, D.B. and Singer, I.M. R-Torsion and the Laplacian on Riemannian Manifolds. Adv. Math. 7, 145 (1979). [R97] Rosenberg, S. The Laplacian on Riemannian Manifolds. Cambridge University Press, 1997. [S79] Schwarz, A. The partition Function of a Degenerate Functional. Comm. Math. Phys. 67, 1 (1979). [ST84] Schwarz, A. and Tyupkin, Y. Quantization of Antisymmetric Tensors and Ray-Singer Torsion. Nucl. Phys. B242, 447 (1984). [W89] Witten, E. Quantum Field Theory and the Jones Plynomial. Comm. Math. Phys. 121, 351 (1989). [W99] Witten, E. Dynamics of Quantum Field Theory, in Deligne, P. et al. Quantum Fields and Strings: A Course for Mathematicians, Vol. 2. American Mathematical Society, 1999. [Y85] Yamagushi, Y. Measures in Infinite Dimensional Spaces. World Scientific, 1985.
466
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
MULTIPLICATIVE A N O M A L Y FOR T H E C-REGULARIZED DETERMINANT CATHERINE DUCOURTIOUX Laboratoire de Mathematiques Applique.es Universite Blaise Pascal (Clermont II) Complexe Universitaire des Cezeaux 63177 Aubiere Cedex, France. E-mail:c.ducour@ucfma. univ-bpclermont.fr We study new regularized determinants on elliptic pseudo-differential operators. They are of the form " expTr^Log", where Tr® is a weighted trace. We express the obstruction preventing the ^-determinant from being of this type, this leading us to a general formula for the multiplicative anomaly of the ^-determinant.
1
Introduction
In this paper we study determinants of elliptic classical pseudo-differential operators, acting on sections of some finite rank vector bundle E based on a closed manifold M. Two types of determinant are involved: determinants associated to weighted traces and the well known ^-regularized determinant. Using an auxilary operator Q, one can define a pseudo trace Tr® which we call a weighted trace. This weighted trace is a linear form on the algebra of classical pseudo-differential operators. It coincides with the ordinary trace on trace class operators but, in general, it is not tracial i.e. Tr^(AB) jt TrQ(BA). By determinant associated to a weighted trace Tr®, we mean a functional det® which is defined by a formula of the type: " detQ = expTr^Log". We call this determinant a weighted determinant. In general, weighted determinants do not satisfy the multiplicative property i.e. det®(AB) ^ det® (A)det® (B). However for operators of the type I + K, with K of trace class, weighted determinants coincide with the Predholm determinant and the multiplicative property holds. By restricting to operators for which one can use the same determination of logarithm, we show that the weighted determinant defi satisfies the multiplicative property as long as these operators are taken in a subalgebra on which the weighted trace Tr® is tracial. On the other hand, the ^-regularized determinant det$, introduced by Ray and Singer [RS] in 1971 in a geometric context, also gives rise to a non trivial 467
multiplicative anomaly: F((A,B)
:=
detc{AB)/detc(A)det((B),
previously studied by M.Kontsevich and S.Vishik, and independently by K.Okikiolu (see [KV1,2] and [01,2]). We relate the weighted determinants to the C-regularized determinant, this leading us to a general formula for the multiplicative anomaly of the ^-regularized determinant, which extends formulae established by M.Wodzicki ([W], [K]) for commuting operators, and by K.Okikiolu ([02]) for operators with scalar leading symbols. From now on, we consider a C°° closed riemaniann manifold M of finite dimension d and a hermitian vector bundle E -^ M of finite rank n based on M. We shall denote by T(M, E) the set ofC°° sections ofE. The space T{M, E) is endowed with the hermitian product: < s,p > = JM< s(x),p(x) >BX dvol(x), where vol is the riemaniann volume measure on M and <,>EX is the hermitian structure on the fiber Ex of E over x£M. 2
Preliminaries
In this section, we introduce pseudo-differential operators (PDOs) on T(M, E), we define complex powers and logarithms, and we describe a topology on classical PDOs. We refer the reader to the appendix for notations and basic facts concerning PDOs on an open subset of ]Rd. • Pseudo-differential operators acting on sections of a vector bundle Let (U, (f>, u) be a local chart of E where (U, <j>) is a local chart of M and u a local trivialization giving rise to a diffeomorphism •K-l{U)-><j){U) x
(0(7T(Z)),U(Z)).
Then, <> / and u induce two maps: Pu : C°°(4>(U)$n) - • T(U,E),
say that A\, is A read in the local chart ([/, <j>,u). Invariance under a change of coordinates (see Appendix) shows that these definitions are independent of the choice of the local chart. Since the manifold M is compact, by using a partition of unity one can compose PDOs. We shall denote by PDO(M,E), resp. CL(M,E), resp. CL(_i)(M,.E) the algebra of PDOs acting on T(M,E), resp. classical, resp. odd-class classical. • Topology on CL(M,E) Let (Ui,
B\N)
:= Z^^OP(a%-),
A\ := B™ + E[N).
For N > d + m,
^
N < d+m
has a kernel K\ {x,y) which is c ~ - ^. We endow the vector space CLord<m(M,E) of classical PDOs of order less than m, with the Prechet topology given by the following semi norms labelled by multiindices S, 7 and integers j > 0, N > d + m:
\\M5,yj,N := maxi[supx^^1\^-m\\D5xDja^_j(x,0\\ SU
Px,yeVt,\5'\+\y\
+
Dl'KlN\X>V)\\)-
One can check that this topology does not depend on the choices we made. • Leading symbols and elliptic operators Let A e CL(M,E). The first homogeneous component of its symbol (which is only locally defined) not identically equal to zero can be globally defined on T*M and is called the leading symbol crL{A) of A. For x € M, £ 6 T*M, (TL(A)(X,£) is an endomorphism of the fiber Ex of E. If O-L(A)(X,^) is invertible for £ ^ 0, one says that A is elliptic. We endow the set of leading symbols with a norm: lkz,(4)|| := We denote by Ell(M,E)
supx€MSupii\=1\\aL(A)(x,0\\End(Er.)the set of elliptic operators in 469
CL(M,E).
• Admissible elliptic operators Let A € Ell*ord>0(M,E). For 6 G H, we denote by Lg the half line {z €
[01]) Let A € Ell*0'^dd^i{M', E) with a spectral cut Lg and a positive order m. Let To be the contour in
Al:=±
\z{A-\I)-'d\
[
where Xz is defined by determinations of complex logarithms given by the contour Fg, and where (A — A / ) - 1 is the resolvent of A . The integral makes sense since ||(i4 - A / ) _ 1 | | = 0(1/|A|) when |A| -> +00 (see [Sh] p 77), and since / °° f$id\ is finite for Rez < 0 . Using the semi group property and the fact that Ag~k = (^4_1)fc for any positive integer k, the definition of complex powers can be extended to the whole complex plane. For any z £ C, A# G Ellord=mz(M,E) and for any k e 2Z and s £ E , the map: {Rez < k} -»
L(Hs(M,E),Hs-mk{M,E))
z - > A% is a holomorphic function, where L(Hs(M,E),Hs~mk(M,E)) denotes the space of bounded linear operators from the Hilbert space HS(M, E) of sections 470
of E of Sobolev class Hs to the space Hs-mk (M, E) of sections of E of Sobolev class Hs~mk . Following [KVl,2] and [01], we now define logarithms. Definition 1 For A 6 Ell*o^0{M,
E) with spectral cut Lg we set
Log0A :=
DzAze\z=0.
The operator LoggA is a PDO but it is not a classical one as the next proposition shows: Proposition 1 [KVl], [01]: LogeA is a PDO of order e for any e > 0. In some local chart, the symbol of LoggA is of the form: o-(LogeA)(x,£) = mlog\£\I + o-£(x,£), where O~Q is a classical symbol of zero order. Because the term mlog|£|7 behaves "badly" under change of coordinates, aft is not a symbol of a PDO. But for two operators A,B £ Ell*'adm(M, E) with resp. Lg1, Lg2 as spectral cuts, then aft — a^ is a symbol of a PDO and we have: ho
^A
Logoff
~ojA)
ojBY
CL
°"**W>E>-
Let us specify that if the orders of A and B are both even integers and if A and B are both of odd-class, then - ^ [KV1,2]).
^m~
€ CL
(-i)(M>E)
(see
If an operator A € Ell*-adm(M,E) is of zero order, then the spectrum of A is bounded. In this case, we can define complex powers and logarithms of A directly using a Cauchy integral formula. The logarithms of A belong to (M,E). In what follows, we denote by Q an operator of Ell*0'^d^j{M, E) and we denote by q its order. When we take logarithms, we shall drop the mention of the chosen determination. 3
Weighted traces
Concerning weighted traces, we refer to [MN], [CDMP]. In order to study weighted traces and weighted determinants, let us introduce the Wodzicki Residue ([W], [K]) by its local formula, which we take as a definition. 471
Definition 2: Let A G CL(M,E). res(A) =-—J d (27r)
The Wodzicki Residue of A is: / JMJS'M
tr a-d(x,£)d£
dx,
where a_d is the homogeneous component of degree — d of a symbol of A. This definition is independent of the chosen local chart. Prom this expression of the Wodzicki Residue, it easily follows that the Wodzicki Residue vanishes on the algebra CL(_i)(M,E) whenever M is odd dimensional. Let us now give a fundamental result on meromorphic extensions of traces for holomorphic family of classical PDOs. For that, let us recall a few facts about traces and trace class operators. A PDO A is of trace class if and only if o(A) < —d. In that case, the Schwartz Kernel A(x, y) of A is a continuous section and the trace of A is given by Tr(A) = JM TrA(x, x)dx. Theorem 1 [KVl]:Lei / be an holomorphic function on
= - — U ^ r e s (*(£(*,-)))•
Proof. In what follows, we consider a local chart of E, and we identify the domain of this chart with the corresponding open subset of IR . For every s £ € , let X)j>o "/(«)- ^ e * n e f ° r m a l symbol of B(s) and let B\S,K_. be a PDO of formal symbol &/A\_ • These PDOs can be denned with the help of a cut-off function ip. Let N £ IN* and let us set:
3=0
On the set {s €
Let 0 < j < N - 1, and let s € C such that Ref(s)
< j - d.
The kernel of By) •. • restricted to the diagonal is given by:
£#)_;,•(*, *) = j
MWM-jfrZW-
Going to polar coordinates gives: ^)_J.(«,*) = / o - ^ ( r ) r / W - ^ - 1 d r 4 | = 1 6 W ) _ / * , 0 ^ . We can split the first integral /0°° V(r)rfW-i+d-1dr Ji *(r)rfM->+d-xdr Both
+ f™ tf (r)r'W--»' + d - 1 dr.
complex s
functions
: +d 1
J ip(r)r^ ^~ ' ~
defined
by
Jif|=i ^ W - j ^ ' ^ ) ? ^
an(
^
dr are analytic on(D. /
The integral J^° tp(r)r^s^~:i+d~ldr we have:
is equal to / 1 °°r-^ i, )~- 7+d_1 dr and hence
Then - B ^ N (^ a ; ) c a n ^ e e x t e n d e d in a meromorphic function with a simple pole at Sj = f~x(j — d) and the residue at this point is given by:
Taking traces yields the theorem. Applying Theorem 1 to the holomorphic family A~s, where A G Ell*0'raf^(M,E), leads to the (-function of A, £A{s) := Tr{A~s), which is holomorphic at s = 0. Applying Theorem 1 to the holomorphic family AQ~S, where A 6 CL(M, leads to a linear functional which we now define. Definition 3: Let A £ CL(M,E) TrQ(A)
of order a € ft. The expression:
:= Tr(,4Q- s ) - — resM)
is called the Q-weighted trace of A. In a similar way, we define the Q-weighted trace of logarithms. 473
E),
Proposition 2: Let A € Ell*o'*dd™0{M, E) of order a and let e > 0. The function s -> Tr((LogA)Q~s) is holomorphic in the domain {Re(s) > ^±£-} and can be extended to a meromorphic function on the whole complex plane with a simple pole at s = 0. The residue of which is: Ress=QTr((LogA)Q-s)
= -res{LogA
- -LogQ).
Proof. For Re(s) > ^-, the operator (Log.A)Q~s is of trace class and we have: Tr((LogA)Q-s) = Tr((LogA-±LogQ)Q-s) + ±Tr((LogQ)Q-s). From s the definition of Log, it follows that j^Q~ = (LogQ)Q~s and since Tr is continuous on trace class operators, we have Tr ((LogQ) Q~s) = CQ{S)- Hence Tr((LogQ)Q~s) admits a meromorphic extension to
:= Tr((Log^)Q~ s ) - —res(Log.4 - -LogQ) qs q
s=0
The obstruction preventing TrQ from being tracial and the dependence of Tr® in relation to Q can be expressed in terms of a Wodzicki Residue. Proposition 3 : 1. Let A,B e CL(M,E),
then [logQ,A]B e CL{M,E)
TrQ[A,B] 2. Let A G CL(M,E)
=
and:
--res([logQ,A]B).
and let Q i , Q 2 e Ell*o?dd™0{M',E), then
Tr^(A)-TrQ*(A)=res((logQ2/o(Q2)-logQ1/o(Q1))A).
The assertion 1. was first proved independently in [CDMP] and in [MN]. The second one can easily be deduced from Theorem 1 and was proved in [KV1]
and [02]. 474
This proposition provides us with two examples of subalgebras of CL(M, E) on which TrQ is tracial: - the subalgebra CL®(M, E) of operators which commute with Q modulo a regularizing operator (it is easy to check that it is an algebra since the product of a PDO by a regularizing operator is a regularizing operator). - the subalgebra CL(_i)(M, E) of odd-class operators when M is odd dimensional, Q is of odd-class and of even integer order. In that case, Tr® is independent of Q and is denoted by Tr^~^ as in [KV1,2]. The first assertion follows from the fact that the Wodzicki Residue vanishes on regularizing operator, as its local expression shows. The second one relies upon the fact that the two residues in Proposition 3 are then taken on oddclass operators and hence vanishes in odd dimensions. The following result will be used later. Proposition 4: Let (At)ten be a family in CL(M,E) der. If (At) is continuous w.r. to t then limt-n,res(At) limt^toTrQ(At) = TrQ(Aio), where fe € R .
of constant or= res(A^) and
Proof. Let m be the constant order of the family (At)t&R.- W.r.t. the topology on CLord<m(M, E) described above, the first result is obvious. The second one relies upon the expression of the meromorphic extension of Tr(AQ~s) given in the proof of Theorem 1. 4
Determinants associated to weighted traces
Definition 5: Let A £ Eiro^d^(M,E). The determinant associated to the Q-weighted trace Tr®, also called the Q-weighted determinant, is defined by: detQ(A) :=
expTrQ(LogA).
It coincides with the usual determinant when M is reduced to a point and with the Predholm determinant for suitable operators as the next proposition shows. Proposition 5: Let K 6 CL(M,E) of order strictly less than —d. Then the Q-weighted determinant of I + K is independent of Q and is equal to the Fredholm determinant of I + K. Proof. Since K is of trace class, Log(7 + K) is also of trace class and via Hadamard's Theorem, we have detpTed(I + K) = expTrhog(I + K).
475
• Multiplicative anomaly P r o p o s i t i o n 6:Let A be a subalgebra of CL(M, E) on which Tr® is tracial and such that AD Ell*(M,E) is a group. Let (At) be a differentiable one-parameter family of operators in A D Ell^d-I^j(M, E), of constant order, and such that there exists a same spectral cut contained in a common Agmon-Nirenberg angle for all At. Then: jtTrQ(LogAt)=TrQ(AtAT1). Proof. Let us denote by A the Agmon-Nirenberg angle for all At and let be ( 6 l . We need to show that for any A £ Sp(At), A 6 A, Tr® [(At - Xiy'AtiAt
- A/)- 1 ] = Tr® [At(At - A / ) " 2 ] .
Then we conclude by integrating by parts. 1
(At - A/)" ] = Tr^ [(At - \I)~lAt(At
lim^o TrQ [{At - A/)" (^f^) A/)
-1
By Proposition 4, we have 1
-
and an analogous expression for the r.h.s.. These traces being limits
of traces of operators in A, the proposition follows from the fact that Tr® is tracial on A, switching the order of the terms in the l.h.s. using the tracial property. T h e o r e m 2 [D]: Let A be a subalgebra of CL(M,E) satisfying the assumptions of Proposition 6 and such that if A € - 4 n £ / / * ; S ( M , £ ) , then A* e A for any t G 1R. Let A,B € An EU*ord>0{M,E) of integer orders and such that their leading symbols are sufficiently close to positive definite self-adjoint ones. Then A, B, and AB admit a same spectral cut and: detQ{AB)
=
detQ(A)detQ{B).
Proof. Let F® be the multiplicative anomaly of deft denned as F^. Let A,B e An Ell*ord>0(M,E) of integer orders, B of order 1. We consider the family At := (AB-O(-AA B°^A\ 0 < t < 1. If A and B have their leading symbols sufficiently close to positive definite self-adjoint ones, then A, B, AB et At for all t are admissible and admit a same spectral cut contained in a same Agmon-Nirenberg angle. The assumptions on A imply that At € A D Ell*ord>0(M', E). We have ^ l o g FQ{At,B) = 476
j-tTrQ(Log(AtB) - ±Tr<*{LogAt). Since AtB(AtB)-1 = AtAj\ by Proposition 6, we have ^ l o g F^{AUB) = 0 for all 0 < t < 1. One can check that FQ(B°(A\B\ = 1, hence F^(A,B) = 1. If o(B) > 1, the multiplicative property still holds. Since detQ{ABl'°^B)) = detQ(A)detQ(B1/°^), we have detQ{AB2/°^)
= de^iAB^^B1/"^)
detQ(A)detQ(B2/°W)
=
and by iterating this argument we obtain detQ(AB)
=
detQ(A)detU'\Q\\(B).
Examples: The two subalgebras CLQ(M,E) and CL^_^{M,E) in section 3 satisfy the assumptions of Theorem 2. 5
described
(-regularized Determinant
In this section we essentially relate the weighted determinants to the (,regularized determinant, and thus we deduce a general formula for the multiplicative anomaly of the (-regularized determinant. We also explain how weighted determinants extend M.Kontsevich and S.Vishik's ^-determinant for zero order odd-class operators when the manifold is odd dimensional. Definition 6 : For any operator A G Ell*0^™0{M, E), the (-regularized determinant of A is : de*c(A):=exp(-C^(0)). The next result is fundamental for the sequel. Proposition 7 : For any operator A G Ell*0'"£™Q(M, E) of order a, TrA(LogA)-TrQ(LogA)
= _-Lres((LogA - -LogQ)2).
A proof can be carried out by deforming A, differentiating the two sides of the equality and using Proposition 3. This is a particular case of a more general result given in [02]. Using Proposition 7 one can check that in the odd case, i.e. odd-class operators and odd dimensional manifold, the trace Tr^~^ (defined at the end of §3) extends to logarithms of even integer order. Let be A e Ell*o'^™0(M,E). We recall that detQ(A) = expTrA(LogA) (see the last remark in §3). Let us assume that A is an odd-class operator with 477
even integer order and that M is an odd dimensional manifold. Then we have detQ(A) = expTr^XLogA). In [KV1.2], M.Kontsevich and S.Vishik have noted that for any operator A € Ell*OTd=Q(M, E) with leading symbol sufficiently close to a positive definite self-adjoint one, the ratio det^(AC) /det^(C) does not depend on the odd-class positive self-adjoint operator C of even integer order chosen in Ell*ord>Q(M,E). This comes from the fact that since TA'1') is tracial, we have Tr^1^(Log(AC)) = Tr^1XLogA)+Tr^1XLogC). For such an operator A, they define : det(A) := det^(AC) / det^(C). In fact, det(A) = expTr^XhogA). It follows that any weighted determinant detQ on zero order operators, with Q of odd-class and even integer order, extends the determinant of M.Kontsevich and S.Vishik denned in the odd case. Now since again det^(A) = expTrA(LogA),
we obtain:
Corollary 1 [D]: For any operator A € Ell*o'"dd™0{M, E) of order a, detc(A) detQ(A)
6XP
1 ~2o [res((L°S-4
_
- L o g<9) 2 )] | •
Corollary 1 can be understood as the obstruction preventing det^ from being of the form expTr® Log. The expression of this obstruction confirms that there is a quadratic non linearity hidden in the definition of the ^-determinant, as M.Kontsevich and S.Vishik pointed out in [KV1]. Corollary 2 [D]: Let A,B <E Ell*>adm(M,E), of resp. orders a > 0, b > 0, such that their principal symbols are sufficiently close to positive definite selfadjoint ones. The multiplicative anomaly of the (^-determinant is given by: logFQ{A,B)
-^LogiAB))2)
= j-res^LogA
-
+ j-bres((LogB
- J^LogiAB))*)
(1)
+ TrAB (Log{AB) - LogA - LogB^j. In particular, if A and B commute, then we have:
^^
5 ) =
2^T6)
r e S
(
L O g (
^"
a ) ) 2
)-
(2)
Proof. We write LogF c (/l,fl) = TrAB {Log(AB)) - T r ^ L o g ^ ) B Tr (LogB) = TrAB(Log(AB)-LogA-LogB)+TrAB (LogA)-TrA (LogA) + 478
TrAB(LogB)-TrB(LogB). Applying Proposition 7 to the two last differences leads to (1). If [A,B] = 0, then one can check that Log(AB) = Log/1+ LogB. Hence we deduce (2) from (1) by a direct calculus. In [01], K.Okikiolu established a Campbell-Hausdorff formula for operators with scalar leading symbols: if A,B e Ell*0'"£^{M', E) admit scalar leading symbols and satisfy some technical assumptions then Log(AB) — E £ f C
+ \£\)m-\s\
< Cy,s(K)(l
Mx € Jjf.Vf £ H d .
We shall denote 5(17) := U m e R S a ( [ / ) , S-°°(U) := n m g R S a ( l 7 ) . The relation: <TI ~ (72 O-CTI— <J2 6 S~°°(U) defines an equivalence relation on S(U). A symbol a of order m (which can be a complex number) is called a classical symbol if there exists an asymptotic expansion 00
°{X, 0 ~ Y, ^(Oo-m-j (X, 0 3=0
479
(Al)
where tp is a cut-off function i.e. t/j e C°°(]R d ), ip(£) = 0 for |f| < 1/2, V'(C) = 1 for |£| > 1, and am-j(x,£) is a positively homogeneous function on U x Hd of degree flea - j in f, i.e. o-a_j(a;, ff) = tm-jaa-j{x, f) V* € M + . In (Al), the symbol ~ means that, for every integer TV > 0, a — E ^ 1 V ' ( 0 < 7 m _ j ( x , 0 G 5 m - w (C/). We shall call E ^ o ^ - i a f o r m a l expansion of a and we shall note: a ~ Ej*Lo am-jNotice that Property (Al) implies that for |f| > 1, crm-j{x,0 E Sm-j(U) ([Di7], p i 14). Furthermore, if another classical symbol a' is given by the same homogeneous components as a, then a eta'. On the set of symbols, there is a composition law denoted by * and defined by: Q!'
a
In particular, if o, b are two classical symbols with formal expansion: a ~ E^=o a m - j > & ~ E ? l o kn-j, then a * b is also a classical symbol and the homogeneous component of a * b of degree m + n — j is given by: (a * 6) m + n _,- = ambn-j
+
2J
i~ | a | —
D^am-kD"bn-i.
k+l+\a\=j,l<j
• Odd-class classical symbols Following Kontsevich and Vishik [KV1], we shall say that a classical symbol lies in the odd-class if the positively homogeneous components o~m—j are
moreover homogeneous i.e.: am-j(x,t£) = tm~:'arn-.j(x,^) Vt 6 JR. From the composition law formula, it follows that the composition of two odd-class classical symbols is also an odd-class classical symbol. • Classical pseudo-differential operators To a symbol er € S(U) we can associate an operator Op(a) defined by: Op(a) : C?W)
-HC°°(tf) u -»• (x -»• Op(a)«(a;) = /
e t e - c o-(a;,f)«(0df) •
where C°°(U), resp. C%°(U) denotes the space of complex valued smooth functions, resp. with compact support, in U. As usual, we shall say that an operator is regularizing if it admits a C°° kernel. 480
If a € S°°(U) then Op(cr) is a regularizing operator on U and the converse is true (see [Tl], p35). A pseudo-differential operator A on U is an operator which can be written in the form: A = Op{a) + R where R is a regularizing operator and a belongs to S(U). The symbol a is called a symbol of A and the order of a is called the order of A. We denote a symbol of A by a{A) and the order of A by o(A). Notice that a(A) is defined modulo S~°°(U). If o{A) is classical then A is called a classical PDO . An odd-class classical PDO is a classical PDO such that its symbol lies in the odd class. Notice that ordinary differential operators provide example of classical PDOs in the odd class. We shall say that an operator A is a matricial PDO of order m if A is a matrix with all terms given by PDO of order m. Let us recall classical results on the composition of PDOs. A PDO A on U is called properly supported if for every compact K C U, the set {(x,y) e SuppKA : x € K or y G K} is compact. A properly supported PDO A on U is defined on C°°(U) and it admits a unique symbol given by: a(A)(x,£) = e-^(x)Ae^(x), where e^(x) := elx^. PDOs and properly supported PDOs are related as follows: Let A be a PDO. Then A can be written as a sum: A = AQ + Ai where Ao is a properly supported PDO and Ai is a regularizing PDO. In general, an operator of the form Op(a) is not properly supported. Now, if A,B are two properly supported PDOs on U, then the composition AB is a properly supported PDO and a{AB) = a{A) * o-(B). Furthermore, if A is a PDO and R is a regularizing operator, then AR and RA are both regularizing operators. From this result and the composition law formula, it follows that the set of PDOs, resp. classical, resp. odd-class classical PDOs, on U are algebras. Finally let us recall the classical result on change of coordinates: Let x • U —l x(U) be a diffeomorphism and let A be a PDO on U. Let A\ be the operator on C%°(x(U)) defined by: Ami •= [i4(uiox)]ox - 1 . Then A\ is a PDO and a symbol of A\ is given by:
Y,haiA)la){x'X'{x)Tl)-D^eXp{ix'^z)^z^
^iXv.^xM ~ a
481
where a(A)^(x,0
:= Df*{A)(x,G)
andX*{z)
:= x{z) - x{x) - x'{x){z - x).
References [CDMP] A. Cardona, C. Ducourtioux, J.-P. Magnot, S. Paycha, Weighted Traces on Algebras of Pseudo-Differential Operators and Geometry on Loop Groups, Preprint (2000) [D] C. Ducourtioux, PhD thesis in preparation [Di7] J. Dieudonne, Elements d'analyse 7, Gauthier-Villars (1978) [EVZ] E. Elizalde, L. Vanzo, S. Zerbini, Zeta-Function Regularization, the multiplicative Anomaly and the Wodzicki Residue in Commun. Math. Phys. 194 613-630 (1998) [K] Ch. Kassel, Le residu non commutatif [d'apres Wodzicki], Seminaire Bourbaki 41eme ann. 708 Asterisque 177-178 (1989) [KVl] M. Kontsevich, S. Vishik, Determinants of elliptic pseudodifferential operators, Max Planck Preprint (1994) [KV2] M. Kontsevich, S. Vishik, Geometry of determinants of elliptic operators in Functional Analysis on the Eve of the 21st Century Vol. I 173-197 (ed. S.Gindikin, J.Lepowski,R.L.Wilson)Progress in Mathematics (1994) [MN] R. Melrose, V. Nistor, Homology of pseudo-differential operators I. Manifolds with boundary, Preprint: funct-an/9606005 (Oct. 96) [01] K. Okikiolu, The Campbell-Hausdorff theorem for elliptic operators and a related trace formula, Duke Math. Journ. Vol. 79 687-722 (1995) [02] K. Okikiolu, The multiplicative anomaly for determinants of elliptic operators , Duke Math. Journ. Vol. 79 723-750 (1995) [RS] D.B. Ray, I.M. Singer, R-Torsion and the Laplacian on Riemannian Manifolds, Adv. Math 7 145-210 (1971) [Se] R.T. Seeley, Complex powers of an elliptic operator in Proc. Sympos. Pure Math. 10, 288-307, Amer. Math. Soc. (1968) [Sh] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer Verlag (1987) [Tl] F.Treves, Introduction to pseudodifferential and Fourier Integral Operators Vol.1, The University Series in Mathematics, Plenum Press, New York (1980) [W] M. Wodzicki, Non commutative residue in Lecture Notes in Mathematics 1289 Springer Verlag (1987)
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Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
ON COHOMOGENEITY ONE RIEMANNIAN MANIFOLDS S.M.B. KASHANI School of Sciences, Tarbiat Modarres University P.O.Box 14155-4838 Tehran-Iran E-mail:[email protected]. ac.ir In this paper after giving a short introduction to the subject "Riemannian Gmanifolds", we concentrate on cohomogeneity one Riemannian manifolds. We study "revolution hypersurfaces" and characterize such hypersurfaces completely. This acticle follows a previous paper [Ka] of which it uses some of the results.
0 Introduction Let (Mn,g) be a Riemannian manifold of dimension n, let G be a closed Lie subgroup of Iso (M)=the Lie group of isometries of M. Suppose that G acts smoothly on M. In the theory of Riemannian G-manifolds one studies the topological and/or geometric properties of M, M/G (the orbit space), G and the orbits of the action of G on M with the help of each other. A good reference for G-manifold theory is [B]. Riemannian G-manifolds, which are a generalization of homogeneous spaces, form a beautiful and important subject in mathematics. They also have many applications in mathematics and physics, including Riemannian submanifold theory, especially isoparametric hypersurfaces, and submanifolds (see [CD]) invariant theory, differential equations and variational problems. In general if G acts "nicely" on M i.e., if M admits a section (see [PT]) and one wants to study some objects defined on M, if the objects are invariant under the action of G, then one can consider M/G and study the induced objects on M/G. This is in general a (much) simpler problem. For a detailed discussion of the applications one should see [PT]. The congugacy class of a closed subgroup H of G is denoted by (H) and is called a G-orbit type. The orbit G.x is said to be of type (H) if (Gx) = (H), Gx is the isotropy subgroup of G at x. M(#) C M denotes the union of all orbits oftype(.f7). Among all orbit types (H) with M(Hy^ <j>, there is a unique ([/) such that for H < G with M(#) ^ cf>, there is a g in G with the property that gUg~1 C H. It follows that G/U has maximum dimension and (for that dimension) the maximum number of components. An orbit G.x for which (G x ) = (U) is called principal orbit. The nonprincipal orbits are called singular orbits. The union of principal orbits (resp. singular orbits) is denoted by Mreg (resp. 483
Ms). We have that M = Mreg U Ms and that Mreg is an open dense subset of M. If the dimension of a principal orbit of Mn is n — k, then M is called cohomogeneity k (Riemannian) manifold. In this case, if G.x is a singular orbit then dim G.x < n — k; if dim G.x = n — k, G.x is called an exceptional singular orbit. Riemannian G-manifolds, especially cohomogeneity one Riemannian manifolds have been studied recently by many mathematicians, including Bredon [B], Palais & Terng [PT], Mostert [M], D.V& A.V. Alekseevsky [A], P. Podesta & A. Spiro [PS]. Following F. Podesta & A. Spiro, I want to concentrate on one problem in the theory of cohomogeneity one Riemannian manifolds. I want to study a family of hypersurfaces of the Eulidean space called revolution hypersurf aces. There is a natural family among all cohomogeneity one hypersurfaces of the Euclidean space Rn+1 for which the Lie group G can be considered as a subgroup of Iso (K n + 1 ). But one should note that not all cohomogeneity one hypersurfaces of Rn+1 for which G can be considered as a subgroup of Iso (R n + 1 ) is a revolution hypersurface. For Examples of such hypersurfaces see 2.9 of [PS] and Examples 2.2 & 2.3. The motivation to study revolution hypersurfaces is a well-known theorem of S. Kobayashi [K] which says that: T h e o r e m : Any compact homogeneous Riemannian manifold Mn isometrically immersed as a hypersurface of Wn+1 is congruent to a sphere Sn (i.e. there exists a n / i G Iso ( l n + 1 ) such that h(M) = Sn). This theorem has been generalized by other authors in different directions. Nagano and Takahashi in two papers ([NT],[T]) gave a complete classification of homogeneous hypersurfaces of Riemannian space forms without the assumption of compactness. Their list is as follows: Sm x E n _ m or Hn or Sn or Sm x Hn~m. A. Ros proved in [R] that any compact hypersurface of constant scalar curvature is congruent to a sphere. Recently F. Podesta & A.Spiro inspired by the subject of G-manifolds want to generalize Kobayashi's theorem by weakening the homogeneity assumption and consider compact cohomogeneity one Riemannian hypersurfaces of E " + 1 , they characterize the family of revolution hypersurfaces among all compact cohomogeneity one hypersurfaces. I want to consider revolution hypersurfaces of Rn+1 without the assumption of compactness. The main result of the paper is Theorem 2.1 and Examples 2.2 & 2.3. 484
1 Preliminaries Definition 1.1. Let (Mn,g) and (M ,h) be two Riemannian manifolds, a smooth function / : (M, g) —> (M,h) is called an isometric immersion if hf{x){Txf(v),Txf(w)) = gx(v,w) \/x e M,Vv,w e TXM. In this case / ( M ) is called an isometrically immersed Riemannian submanifold of M. / ( M ) is locally a submanifold of M, thus as far as local notions are concerned one can consider f(M) with the induced metric, as a Riemannian submanifold of M. In differential geometry it is common to make no distinction between M,f(M) and consider M (itself) as a (local) submanifold of M . So the extrinsic geometry of f{M) in M is attributed to f(M) or M or even / equally. Definition 1.2. Let Mn C M™ be a Riemannian submanifold, let NM be the normal bundle of M, let £ 6 T(NM) be a (local) normal vector field, let D be the Levi-Civita connection of M, let X &){(M) D e a (local) tangent vector field, then A% defined by A^X — —(Dx€)T = ~ (tangent component of DxQ is called the shape operator tensor of M along £. Remark 1.3. A^X is a tensor field on M, so for each x 6 M, we have a map A:TM —> TM, denned by A(v) = -{DX0T VX eX(Af),X(a;) =v. A is a self-adjoint linear map called the shape operator of M along £. Definition 1.4. If m — n = 1, then the rank of A at x £ M is called the type number of the hypersurface M at x. If / : (Mn,g) —> (M , h) is an isometric immersion, then rk(f)x is the rank of the shape operator of f(M) in M at x. Definition 1.5. An (immersed) hypersurface Mn of Rn+1 is of revolution (or a revolution hypersurface) if there exists a line I C IRn+1 such that the rotation group Gi S So(n), of rotations around the line I, leaves M invariant. Definition 1.6. A cohomogeneity one Riemannian manifold Mn is called umbilical if each principal orbit is totally umbilic in M, i.e., the shape operators of the principal orbits are of the form \Idn-i, where the A's are smooth functions on the orbits. When M is a cohomogeneity one manifold under the action of G, then it is well-known (see [A]) that M/G £ M^S1, [0,oo) or [0,1]. If k : M —> M/G = 0 is the projection, G.x = the orbit of x is called principal (resp. singular) if k(x) € fi° (the interior of fl), (resp. k(x) is in the boundary of fl). Note that these notions are in agreement with the above definitions. The 485
point x is called regular (resp. singular). So when the orbit space M/G is E or S1 then all orbits are regular, if M/G is the interval [0, oo) there is just one singular orbit and if M/G is [0,1], there are two singular orbits. We set Mreg to be the set of all regular points of M, MTeg is an open dense subset of M. If x G M, there exists a neighborhood U of x such that (U, g) is locally isometric to (G/H x I,gt + dt2). Here H is the isotropy subgroup of G at x, I is an open interval of E, gt is a family of G-invariant metrics on G/H depending smoothly on t G / , dt2 is the standard metric of E. 2 Main results Theorem 2.1 ([Ka]): Let Mn(n > 4) be a complete Riemannian manifold with nonnegative curvature and of cohomogeneity one under the action of a compact connected Lie subgroup of Iso (M) and let / : M —> Rn+1 be an isometric immersion such that / is 1 — 1 on the set of points where rk(f) < 1. Then M is a revolution hypersurface iff M is an umbilical cohomogeneity one Riemannian manifold. Proof. See [Ka] for the complete proof of the theorem. Example 2.2 ([Ka]). Let M = E " # S n C Rn+1 (the smooth connected sum of Rn,Sn), the imbedding <j>: M —• E n + 1 is defined by ,, ,. _ J x if the class [a;] has just one point,
n
d(S — E) and x (identified with (x,0)), x' are the same vector as points of E " + 1 . Here D
is the interior of the disk {x e E n : ||a;|| < 1} and
o
E is the interior of the closed lower hemisphere i.e., E = {x 6 Sn : x = ( # i , . . . ,xn+i),xn+i < 0}, d stands for the boundary. Now we consider M with the induced metric of E n + 1 . G(=the rotations in E n + 1 around the line / passing through the center of Sn and orthogonal to Rn x {0} C Rn+l, so Gi = So(n). M C E n + 1 is a revolution hypersurface, by definition 1.5, since GL{M) C M. If [a;] G M either [x] = x G Rn then the orbit Gt.x is Sn-l{\\x\\) or [x] = x e Sn, then G,.x = S?-^+i)(y/l - x2n+1), i.e., the n+1 (n — l)-sphere with center (0,:r„+i) G E where x = (xi,... ,xn+i) and radius J\ - x2n+1. Both 5 n _ 1 (lkll) a n d ^o'L+i)^1 ~~ xn+i) a r e t o t a l l y bilic in M , so again by theorem 2.1, M is a revolution hypersurface.
486
um
"
E x a m p l e 2.3 ([Ka]). Let M = Sn x Rm C E n + m + 1 , n > l , m > 2, G = 5o(n + l) x So(m) with the action (hi,h2){x,y) = (hix,h2y) V(hi,h2) £ G, V(a;,j/) E M. M is a cohomogeneity one hypersurface that is not of revolution since it does not satisfy definition 1.5. To see this, take any line I C ]R n + m + 1 and consider G;, take any point (x,y) E M — I, then Gi = So(n + m) leaves the (n + Tridimensional plane P through (x, y) and orthogonal to I invariant and it acts transitively on the (n+m — 1) -dimensional sphere S in P with center Pnl. So certainly there is a point (x' ,y') E 5 with IIJE'II ^ 1 and there is a g E G; s.t g{x,y) = {x\y'), but (x',y') 0 M , so M is not invariant under G/. If (x,y) E M is an arbitrary point, its orbit G.(x,y) is 5™ x 5 m _ 1 ( | | y | | ) , so
for ||y|| ? 0 the shape operator of G.(x,y) in M at (x,y) is —• TTTfldm-l L MJ and the shape operator is identically 0 for (x, 0) E M. Hence the principal orbits are not totally umbilic in M. Thus theorem 2.1 shows as well that M is not a revolution hypersurface. Acknowledgments I would like to thank the referee for his suggestions and corrections. References [A] Alekseevsky, D.V. & A.V.: Riemannian G-manifolds with onedimensional orbit space. Ann. Global Anal. Geom. 11 (1993), 197-211. [B] Bredon, G.E.: Introduction to compact transformation groups. Academic Press, N.Y., London 1972. [CD] Cecil, T.E and Ryan, P.J.: Tight & taut immersions of manifolds. Pitman, Boston, 1985. [Ka] Kashani, S.M.B.: Cohomogeneity one revolution hypersurfaces of the Euclidean space. Southeast Asian Bull, of Math. (1999)23:633-642. [K] Kobayashi, S.: Compact homogeneous hypersurfaces. Trans. AMS. 88 (1958), 137-143. [KN] Kobayashi, S. &: Nomizu, K.: Foundations of differential geometry, Vol 1,11, Wiley-Interscience, New York, 1963-1969. [M] Mostert, P.S.: On a compact Lie group acting on a manifold. Ann Math. 65 (1957), 447-455. [NT] Nagano, T. & Takahashi, T.: Homogeneous hypersurfaces in Euclidean spaces. J. Math. Soc. Japan 12 (1960) 1-7. 487
[O] O'Neil, B.: Semi Riemannian geometry with application to relativity. Academic Press, 1983. [PT] Palais, R.S. & Terng, C.L.: A generalized theory of canonical forms. Trans. AMS 300 (1987), 771-789. [PS] Podesta, F. & Spiro,A.: Cohomogeneity one manifolds and hypersurfaces of the Euclidean space. Ann. Global Anal. Geom. 13 (1995), 169-184. [R] Ros, A.: Compact hypersurfaces with constant scalar curvature and a congruence theorem. J. Diff. Geom. 27 (1988), 215-220. [T] Takahashi, T.: Homogeneous hypersurfaces in spaces of constant curvature. J. Math. Soc. Japan 22 (1970), 395-410.
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Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
A D I F F E R E N T I A B L E CALCULUS O N T H E SPACE OF LOOPS A N D CONNECTIONS MARTIN R E I M S Math
Department.
State E-mail:
University of New York. CP [email protected]
11794-3651.
In this short communication we show how a manifold carries a natural geometrical principal bundle from which any gauge theory can be represented. We also report on a natural differentiable calculus (non-local in the manifold) in this geometrical principal bundle inspired from the notion of differentiable spaces. Using the loop transform, we can apply this calculus to describe a G-gauge theory as a non-local differentiable calculus of their connections.
1
Introduction
Let us state the problem of the representation of principal bundles and connections in this way: we know from the theory of principal bundles and connections the following important lemma [£>]: Lemma 1.1 Let n : P —¥ M be a principal G-bundle. Let p : G —¥ H be a morphism of Lie-groups, then there exists a unique principal H-bundle n : W —> M and a morphism $ : P —> W of principal bundles such that ®{.9-P) = P(ff)-^(p)- Moreover given a connection A on P there exist a unique connection AonW such that dp(A) — $*(A). P is called a reduction of W via p and W is called an extension of P via p. So, with this at hand, we ask the following: Given a manifold M, is there a universal principal bundle carrying a universal connection such that any principal bundle and their connections are extensions of it?. The answer as we will see is positive, this will be the universal gauge structure of the manifold. The universal gauge group will be the group of loops of M and the principal bundle the path bundle of the manifold M. I will explain the basics of a differentiable calculus on the path bundle that makes this universal principal bundle similar to a differentiable principal bundle and the group of loops similar to a Lie group. Details can be found in [RS]. 2
The construction of the universal gauge structure.
In this section we introduce the universal gauge structure. The universal reduction will be the so called path bundle of the manifold and the universal gauge group will be the group of loops. It turns out that both structures are 489
subsets of the paths of the manifold. We will develop a differentiable structure on the paths of the manifold, the differentiable structure on the path bundle and the group of loops being it restriction. So we begin by defining the paths of the manifold. First we define the space of parametrized paths of a manifold: Definition 2.1 The space of parametrized paths of a manifold M is Pp = {7 : [0,1] -> M , piece-wise differentiable}, whereby piece wise differentiable means: 3 0 < t\ < ... < tn < I / 7 is differentiable in each [ti, t»+i]. For many purposes (but for many others not) this space is big, because of the parameterization dependence of its elements. So to eliminate this redundancy we define the re-parametrization equivalence relation: ~ i : 7 ~ i ? f> 3 a : [0,1] -> [0,1], piece — wise increasing 7 (<)
diffeomorphism
=?("(*))•
Pu = Pp/ ~ t is the space of unparametrized paths of the manifold. In this space we have a natural operation, which is simply the composition of two paths: o : Pu x Pu | 7 ( 1 ) = ? ( 0 ) -> Pu I [7] o [<;) = [7 o <;] 70?(t)_ 7 ? W -
/7(2t) 0 < < < 1/2 \ ? ( 2 t - l ) 1/2 < * < 1
We see from this that [-y(t)] o [7(1 — t)] is not [e(i) = 7(0)]. In order to define the group of loops we introduce a new equivalence relation ~2First we define a reduction and we note it by c < 73 ? l , a i , . . . ,SmO£n,<;n+l S Pu 1 S < 7 <=> < 7 = Si <Ti ° act\i o Qaj~ . . . o a „ o an x o <jn+1 X <; = ft o . . . o <;n And we define ~ 2 by 7~2
a
"^ ^ < C
This is a non trivial equivalence relation and serves to eliminate the "branches" of a path. We define the paths of the manifold as P — Pu/ ~2We are in a position to define the path bundle and the group of loops [L]. We fix a point o of the manifold. Definition 2.2 The path bundle of the manifold is VM. — {7 € P / 7(0) = o}. 490
Definition 2.3 The group of loops of the manifold is C = {7 e P / 7(0) = 7(1) = 0 } . It is easily seen that £ is a group. The identity being [j(t) — o], the inverse of an element ["f(t)] being [7(1 — t)] and the multiplication being given by o. There are other groups of interest that are quotients of the group of loops, for example the group of hoops. A complete account of the relationship between the group of loops and the group of hoops is given in [S]. The path bundle is a £ principal bundle: II : VM —> M where II is the canonical proyection, i.e it assigns to each path its last point. 3
The differentiable structure
We will use the notion of differentiable spaces in order to give P, C ,VM a differentiable structure; this notion is taken from [C] where it is used in a similar context. So we begin with the definition of differentiable space and latter explain how P, C, VM can be seen as differentiable spaces. Definition 3.1 A differentiable space is a set A together with a family of functions called plots <j> : U C Mn —>• A, U open sets, such that if g : V C Mm - 4 C / C Mn is differentiable, <j> o g is also a plot. Differentiable manifolds offer a first example of such structures and the diffeomorphism group an infinite dimensional example. Examples 1. (M, {0}). A differentiable manifold M, with plots given by all the differentiable functions {(/>} = {
1. The topology of
P,C,VM
A set U € A is open if and only if <j>~l{U) is open V plot. This is the Barret Topology. 2. The family of differentiable functions. f : A —> M is differentiable if and only if f.
~
dx^...dxt
I(X1
'-'X")
for some
Dvfil)
= -&T |x=0
where : (-e, e) ->• P, /
a(xi)
Here a : (—e, e) —»• M is a one parameter plot derived from a curve a such that &'(0) = v and &(Q) = x and a(xi)(t) — a(x\.t).
492
(b) The Loop derivative on VM. and C Definition 3.4 The loop derivative at 7 € II x(x) is the operator which when applied to a function f(j), 7 £ P is
where ^{x^,x,j) = Da;M,a:„ (two dimensional plot) is the parallelogram, taken in a local chart of M, with vertex at x and edges in the directions of the vectors d^ and dv with lengths xM and xv respectively. We refer the reader to figure 1.
Figure 1. On the left the Mandelstam derivative and on the right the Loop Derivative. 4. The tangent and cotangent bundles Following the same idea of substituting the use of differentiable manifolds with differentiable spaces, it can be seen that both, the set of all vectors and the set of all covectors are natural differentiable spaces. Moreover they are locally trivial [RS]. So differentiable tensors of any order are defined. 5. Differential forms and exterior derivative As a particular tensor, a p-differential form is an alternating p- tensor with vectors as entries. We denote them by W(p)(.,...,.). What is rather nontrivial is the existence of exterior derivative. We define it by: 493
n+l
dw(Xi,...,Xn+i)
=
n+ l L
£ ( - i ) i + 1 XiM*!, ...,xit..., xn+1)) + i=l
J2(-i)i+ju>([xi,xj],x1,...,xi,...,xj,...,xn+1)
+ i<3
Xi,... ,Xn+i are vectors fields which at p take the values X\(p),... ,Xn+i(p). Although this is a well defined formula it is not clear that it depends only of X\ (p),..., Xn+i (p). In [RS] it is shown to be true, thus exterior derivative is a well defined alternated tensor. 6. The "Lie" algebra of C As in Lie groups, the "Lie" algebra (TeC, [, ]) is the set of right invariant vectors fields, and the bracket simply the bracket of the fields. 4
Universal connection and curvature
II : VM —> M , II([7]) = 7(1) verifies all the properties of a principal Cbundle except differentiability. A principal bundle is defined by local trivialization and transition functions, a construction we want to extend to VM. Let us see how they are. Suppose {Ua} is a covering of M by contractible open sets. Then a local section over Ua around a; is a plot
o Figure 2. A trivialization over Ua
in a chart h : U ->• M with U a convex neighborhood of 0 and h(0) = We define 6v(D)(f) = D(g). Having now the universal connection one form the curvature is:
II(IT).
n = ds + hs,s] In a chart around a point x of M, we have the vectors fields 3^ which can be raised to T1~1(U) to the horizontal vector field D^j), 7 G II _1 (C/). Now
0(2^, A,) = l-[D^{Dv) - D^DJ = -\s{[D„Du})
- S([D^DU]) + [6(D»),6(D„)}] =
= -\[D^DV]
= - ^ ( 7 ) ,
This shows that the curvature is —1/2 times the loop derivative. The Bianchi identity holds: dfl = [S, n] 5
Holonomy, representation and reconstruction
Let us state clearly what a universal reduction is. Definition 5.1 A differentiable space principal bundle (PU,GU,H,M) and connection 6U is a universal reduction if and only if all differentiable principal G- bundles and connections are extensions of (PU, GU, II, M) for some C°° morphism p : GU -> G.
495
We can now address the main point announced in the introduction, namely the fact that (VM,£,Tl,M) and S form the universal reduction. We first recall the concept of holonomy. Supose (P, G, n, M) is a principal Gbundle with a connection A, both C°°. Given a piece-wise differentiable path 7 in VM and a point p in 7r -1 (0), there is a unique curve 7 in P, such that ^(7) = 7 whose tangent vectors are horizontal at each point. Definition 5.2 If-y 6 C then the holonomy is the element ofG, H(-y), defined &2/7(l) = ff(7)(7(0)). T h e o r e m 5.3 (VM,£,H,M) is a universal reduction. Proof: Let us sketch the proof of that. Suppose (P, G, 7r, M) is a principal G-bundle with a connection A. Construct the C°° holonomy morphism H and take it as p. Over the convex covering Ua define the principal G-bundle by transition functions
A differentiable calculus for connections
The fact that conjugated holonomies H : C -> G are in one to one correspondence with principal G-bundles and connections suggest to think loops and connections as dual to each other. The fundamental idea for the loop approach to quantum gauge theories is to use this kind of non linear duality to go from functions of connections to functions of loops (in fact you have to use the group of hoops[S]) via the loop transform that uses the trace of the holonomy as a kernel *(7)= f
TrHA(1)^{A)DA.
The problem with this integral is the measure DA. It has been successfully 496
denned [A] as a Gaussian measure for the U(l) electromagnetic free gauge theory, and gives a reasonable interpretation of loops as lines of electric flux. With this transform at hand we wish to apply the differentiable loop calculus as a differentiable calculus for functions of connections. But a lot yet remains to be done to achieve this goal. References [A] A. Ashtekar, Loop Representation of the Quantum Maxwell Field, Class. Quant. Grav. 9 (1992) 1121-1150. [B] J. W. Barret, Holonomy description of classical YM theory and GR, Int. J. Theor. Phys. 30 (1991) 1171. [C] K. Chen, Iterated integrals of differentials forms and loop space homology. Ann. Math. 97 (1973) 217. [D] J. L. Dupont, Curvature and characteristic classes, Lecture notes in mathematics, Vol. 640. Springer-Verlag, Berlin-New York, 1978. [GP] R. Gambini, J. Pullin, Loops, knots, gauge theories and quantum gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (1996). [L] J. Lewandowski, Group of loops, holonomy maps, path bundle and path connection, Class. Quant. Grav. 10 (1993) 879-904. [S] P. Spallanzani, Groups of loops and hoops. To appear in Comm. Math. Phys. [RS] M. Reiris, P. Spallanzani, A calculus in differentiable spaces and its applications to loops, Class. Quant. Grav. 16 (1999), 2697-2708. [T] J. N. Tavares, Chen integrals, generalized loops and loop calculus, Int. J. Mod. Phys. A 9 (1994) 4511.
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Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
Q U A N T U M HALL C O N D U C T I V I T Y A N D TOPOLOGICAL INVARIANTS. ANDRES REYES Department, Universidad de los Andes. A.A. 4976 Bogota-Colombia. [email protected]
Mathematics
A short survey of the theory of the Quantum Hall effect is given emphasizing topological aspects of the quantization of the conductivity and showing how topological invariants can be derived from the hamiltonian. We express these invariants in terms of Chern numbers and show in precise mathematical terms how this relates to the Kubo formula.
1
Introduction
Since its discovery [vKDP], the integer Quantum Hall effect (QHE) has been a source for interesting theoretical ideas, often providing the opportunity for an interplay between Physics and Mathematics. One of the first works in which a theoretical justification of the effect was given involved the calculation of the conductivity in terms of the wave function of an electron in a periodic system under the influence of a magnetic field (Thouless et.al. [TKNN]). Here the conductivity was computed from the Kubo formula, a formula for the correlation function describing the response of a system to a weak electric field. Though the quantization of the Hall conductivity was explained in this approach, its stability against small perturbations could not be explained. Later, the work by Avron et.al. [ASS1] [^4555] showed how one can explain not only the quantization of the conductivity but also its stability, using topological arguments. Although these arguments used there only work for a restricted class of models, they show how the conductivity can be seen as a topological invariant, the integral of the first Chern class of a fiber bundle over the Brillouin zone (a compact manifold). In fact, the generalization to more realistic models, for which the Brillouin zone cannot be seen as a manifold anymore, still allows to express the conductivity as a topological invariant (see the work by Bellissard et.al. [BvES] and by Avron et.al. [ASSI]). The purpose of this short communication is to review some of the abovementioned approaches to the theory of the QHE and in particular the topological aspects of the quantization of the conductivity. The plan of the paper is as follows: in Section 2 we summarize the physics 498
background required to describe the QHE and which motivates the mathematical problem treated in the rest of the paper. We express the Hall conductivity cruo.il given by the Kubo formula in terms of the eigenfunctions of the one particle Hamiltonian for our model described below. In Section 3 we use the symmetry of the system to show how topological invariants can be obtained from the hamiltonian. In Section 4 these invariants are expressed in terms of Chern numbers, and they are shown to coincide with the result for the conductivity computed from the Kubo formula. 2
Physics Background
Let us sketch some of the basic ideas from physics motivating the mathematical problem treated in the next sections. The model we use is as follows. We consider electrons on a two dimensional lattice and a homogeneous magnetic field perpendicular to the lattice. This means that we only consider the interactions of the electrons with the lattice atoms but neglect interactions among electrons, a reasonable approximation when the gas density is small. This model allows a rather simple mathematical treatment and already contains the essential features allowing to explain the quantization and stability of the Hall conductivity. Nevertheless, we remark that in order to explain the existence of plateaus it would be necessary to take into account the effect of impurities. The Hamiltonian for one electron in a two dimensional lattice can be written as H = ~ V
2
+ U(Xl,x2),
(1)
where U(xi,X2) (the lattice potential) is a real function periodic in the variables £1 and X2Bloch's theorem [M] states that for an electron in the presence of a periodic potential the solutions to Schrodinger's equation are given in terms of plane waves modulated by periodic functions, with the same periodicity as the potential. This is a consequence of the symmetry of the system. The translation operators commute with each other and with the Hamiltonian, thus forming a complete set of commuting operators. This implies that we can label a physical state by indicating three quantum numbers: the energy band index and the eigenvalues of the translation operators. The latter are phase factors i.e. of the form etk
is therefore not a 2-plane, but a 2-torus, T2. If we now include the effect of a magnetic field implementing the substitution V H 4 V - ^ ^ in Eq.(l), where A is the magnetic vector potential, we find that the hamiltonian is not translation invariant. There is a way to overcome this difficulty, namely to consider, instead of the usual translation group, the so-called magnetic translation group [Z]. The operators given by the representation of this group now take the magnetic field into account. But in order to apply Bloch's theorem, one is forced to assume that the magnetic flux per unit cell, in units of ch/e, is a rational number <\> — p/q. In this case, enlarging the unit cell in one direction by a factor of q, the reciprocal space, now called magnetic Brillouin zone, is again a two dimensional torus. Imposing a rational magnetic flux is a non-physical condition and this is a limitation of the present model. However, this allows a study of the topological aspects of the problem in rather simple mathematical terms. A further simplifying assumption is to consider only one of the following limiting cases: (i) The lattice potential is sufficiently weak compared to tkuc (the cyclotron energy). Here the lattice potential can be treated as a perturbation of the Landau hamiltonian (which considers only one electron under the influence of a magnetic field). (ii) The lattice potential is strong compared to hujc. In this case, approximations such as the tight-binding model [M] can be considered. The reason for this further simplification is that it can be shown that in both cases, finding the solution of Schrodinger's equation reduces to the diagonalization of a hermitian nxn matrix, whose entries are functions defined on the Brillouin zone [77 if ],[#]. (If the (rational) magnetic flux is p/q, then in (i) n = p and in (ii) n = q). In such cases the result for the conductivity of a single band given by the Kubo formula is given by [i?/('],[iJ]:
^• = - ^ / T / M V f c x , 4 ( % ) ] „
(2)
where n
A^(k) := Y, ^> m (*)V t *(j'(*).
(3)
m=l
Here ^^(k) = (ty[j'(k),..., * „ ( & ) ) is the eigenvector corresponding to the th j energy band. Finally, OHM is given by the sum of the contributions Oj of all filled bands. 500
3
Topology and Bloch Hamiltonians
As remarked in the previous section, for a periodic lattice finding the solution of the eigenvalue problem reduces (for particular values of the magnetic field) to the diagonalization of a hermitian matrix whose entries are continuous complex functions on a torus - the Brillouin zone. Hence, in this section we consider functions from the torus into the space of hermitian matrices with non-degenerate eigenvalues. (We restrict to those matrices only, for simplicity). The topological invariants related to the conductivity arise from considering the homotopy classes of functions from the torus into this space of matrices. In this section we closely follow [^4551] and [ASSS]. Let M n (C) denote the space o f n x n complex matrices and let Hn(C) be the subspace of M n (C) consisting of all hermitian matrices with non-degenerate eigenvalues. Proposition 1 There is a homeomorphism Hn(C) ~ Dn x U(n)/U(l)n U{n) denotes the unitary group of order n and
where
Dn := { D i a g ( a i , a 2 , . . . , a n ) | a i < . . . < a n ; a ; 6 E}. Proof. Given A € Hn(C), we can find a unique diagonal matrix DA whose components are the eigenvalues of A, organized in increasing order, since the eigenvalues of A are all real and different. A € Hn(C) also implies we can find UA € U(n) such that DA — U^AUAThis unitary matrix is not unique, but if we consider the quotient U(n)/U(l)n, then its equivalence class [UA] is uniquely defined. Thus we have a homeomorphism g : Hn(C) ^ D A
n
x
U(n)/U(l)n
^(DA,[UA}).
•
Note that since Dn is contractible, g followed by a retraction of Dn to a fixed matrix gives rise to a homotopy equivalence between Hn(C) and U(n)/U(l)n. Now we want to relate the homotopy groups of U(n) and U(l) with those of Hn(C), since this will allow us to interpret homotopy types of maps from the torus T 2 to Hn(C) in terms of bundles over T 2 . Recall that U(n)/U(l)n has a bundle structure, so in order to compute its homotopy groups, we can make use of its corresponding homotopy sequence [S] : • • • -> 7r m (f/(l) n ) H* irm(U(n))
p
^' 7rm(U(n)/U(l)n) 501
^
7r m _i(t/(l)") -» • • • (4)
The proof of the following (standard) result is useful in order to find out the explicit form of the homomorphism im* in sequence (4) when m = 1. Since we are particularly interested in this homomorphism, we give the proof in detail. Proposition 2. For n € IN we have: (i) ^ ( [ / ( r O J - T r x W l ) )
(~Z),
(ii) n2(U(n)) = 0. Proof. The proof relies on the following facts [5] : 7r2(tf(l))=0,
7rm(5n)=0
(for m
and
U(n)/U{n - 1 ) = 5 2 n _ 1 .
Considering the exact homotopy sequence of the fibration U(k)/U(k k = 2 , . . . , n, we have
— 1) for
0 = TraCS2*"1) -> in(U(k - 1)) 4 7T!(£/(*)) -+ Tr^S 2 *" 1 ) = 0, and we see that pu, the homomorphism induced by the inclusion U(k — 1) <-»• t/(fc), is an isomorphism between iry(U(k — 1)) and ni(U(k)) (k = 2 , . . . , n ) . Composing these isomorphisms, we obtain an isomorphism <j>:=pno..-op2:
irx(U(l)) ~ Z —• 7n(t/(n)),
(5)
thus proving (i). Let us now consider the same exact sequence at the 7T2 level: MS™-1)
~> 7T2(£/(fc - 1)) -» 7T2(t/(fc)) -> T ^ S 2 * " 1 ) ,
for A; = 2, . . . , n . When A; = 2, we obtain 7r2(t/(2)) = 0, since 7r2(C7(l)) = 0 = TT 2 (5 3 ). For 3 < k < n, we have T ^ S 2 * - 1 ) = T ^ S 2 * - 1 ) = 0, which yields n2(U(k — 1)) = 7r2([/(fc), and hence proves (ii). O Proposition 3 Up to the isomorphism TT\(U(1)) (4) is given by: i\*(zi,...,zn)=zi-\
~ ni(U(n)), Yzn.
the map «'i* in (6)
Proof. Given an element (zu...,zn) of m(U(l)n) = Z", let f(t) = iei( l ie (e - \... ,e "W) be a continuous map from S1 to U(l)n representing ( z i , . . . ,zn). Let us now define the map / := i o / : 5 1 -4 U(n), where i stands for the canonical inclusion of U(l)n into U(n). If we compose / with 502
det : U(n) -> U(l), we obtain a representative of an element of 7r1(C/(l)) whose image under
'*• °ii*(zi,...,zn)
= zx H
\-zn.
(7)
• Proposition 4. The first two homotopy groups of Hn(C) are given by: 7n(HB(Q)~0 Tr2(Hn{Q) - k e r i u - Z ™ " 1 . Proof. Proposition 1 implies •Kk{U{ri)/U(l)n) ~ irk(Hn(C)). Prom Proposition 3 we know that ti* is surjective. Using sequence (4) we obtain kerpx, = Irmi* = 7Ti(C/(n)), from which Impi* = 0 follows. Hence Ai is injective, since ker A! = Impi*, so iri(U(n)/U(l)n) ~ ImAx = 0. On the other hand from Proposition 2, we have 7r2(f7(n)) = 0. This implies Imp2* = 0, but then ker A 2 = 0. It follows that 7r 2 (£/(n)/[/(l) n ) ~ ImA 2 = kerii*. Finally, from Proposition 3, we know that kerzj* ~ Z n _ 1 . D Let us now establish a relation between the homotopy groups of a sphere and those of a torus. We recall that given a pointed space X with base point xo, the cone of X, denoted C(X), is defined as the topological space obtained from X x I by regarding X x {0} U {a;o} x / as a single point, that is: C{X) = X x I/(X
x {0} U {x0} x I). 503
Similarly,the suspension of X, denoted S(X), is defined to be the quotient space of X x I in which the space (X x {0}) U ({x0} x I) U (X x {1}) is identified with a single point. Let / : X —> Y be a continuous map. There is an inclusion of X into Xxl given by the map x H-» X X {0}. This inclusion is preserved when going to C(X) so that X can be considered as a subset of C(X), via inclusion. Consider the disjoint union of C(X) and Y and define the equivalence relation given (for x e X <->• C(X) and 2 / € F ) b y a ; ~ y - « - ? / = f(x). The quotient space obtained from this equivalence relation is called the mapping cone of / and is denoted by C(f). It can be shown that C(f)/Y = S(X). For / : X ->• Y there is a natural inclusion j : Y c-> C ( / ) . We can construct C(j) and consider the inclusion k : C(f) <-> C(j). The procedure can be iterated to obtain a sequence of maps:
x4
Y4
c(f) A c(j) ^ c(k) ^•••
Since C(j) can be shown to have the same homotopy type as S(X), and similarly C(k) to have the same homotopy type as S(Y), given another space W, we have the following sequence, called the track group sequence [W],[Sp]: • • •-• [Sn+1(X),W]
-> [5"(C(/)),W]-> [Sn(Y),W}->
[Sn(X),W]
-»• • • • (8)
• ••-»• [ S ( n wn ^ [5(x), w] -> [ [r, w] -»• [x, w], where Sn(X) denotes the space resulting from iterating (n times) the suspension operation to the space X. Denoting the set of homotopy classes of functions from the torus T2 into the space Hn(C) by [T 2 , Hn(C)], we have the following Proposition 5. There is a one to one correspondence between [T2,Hn(C)] andTT2(Hn(C)).
Proof. Regarding the unit circle S1 as a pointed subspace of C with 1 as base point, we consider the wedge product of S1 with itself: S1WS1
= {S1 x { l } ) U ( { l } x 5 1 ) .
Define a map / : S1 -> S 1 V S 1 as follows:
f[e
2M
f ( e 8 " M ) , *G [0,1/4] I (l > e «-«), ( 6 [1/4,1/2] '~\(e-Siv\l),te [1/2,3/4]
W
l(l,e-«'*),te[3/4,l]. If we set X = S\ Y = S1 V S 1 and W = Hn(C), we obtain S(X) = S2 and C(f) = T 2 . We also have S2 = T2/{SX V 5 1 ). Denote by q : T 2 -»• 5 2 the 504
corresponding quotient map. In this case, the track group sequence (8) reads: •••-»• [S 2 VS 2 ,tf„(Q] -»• T2(#n(C)) ^ [T\Hn(Q]
-»• [ 5 1 v 5 1 , f f n ( Q ] - • • • •
Since 7Ti(iJ„(C)) is trivial, q* is surjective. The map [S2 V S2,Hn(C)] -> T^iifin(C)) is the null map since it is induced by the suspension of / , which is of degree zero (because / is of degree zero on each circle of the wedge, as can be read from Eq. (9)). Therefore q* is also injective and thus an element of [T 2 , Hn(C)] can be completely specified by an element of n2(Hn(C)). D Summarizing, we have: [T2,Hn(C)]
~ir2(Hn{Q)~teriu = {(*!, • . . , * „ ) € *l(U(l))n
\Z! + --- + Zn = 0}.
(10)
But 7Ti(t/(l)) ~ 7r 2 (B[/(l)), where BU{\) denotes the classifying space for 1/(1)— bundles, that is to say an element of 7Ti([/(l)) determines a unique U{\)— bundle over S2, up to bundle isomorphism [MS], [S]. Since we can equivalently consider its associated line bundle, in this case we have n line bundles subject to a restriction. We can now use the projection q to obtain n induced line bundles over T2. The interpretation is that the sum of this bundles is a trivial bundle. We elaborate this point in the next section. 4
K u b o formula and Chern numbers
We have seen how to obtain topological invariants from particular hamiltonians, implying that they are stable against 'small' perturbations of the latter. The purpose of this section is to give an expression for these invariants in terms of Chern classes and compare it with the expressions in Eq. (2). Consider, as in the previous section, the set [T2,Hn(C)]. For a given a £ [T2, Hn(C)] we can certainly choose a smooth representative / £ a and for x £ T 2 , f(x) is a hermitian matrix all of whose eigenvalues are different. It follows that Pi(x), the idempotent matrix defined by the projection into the ith eigenspace of f(x) in C n , has trace equal to 1 (i = 1 , . . . , n; x £ T2). Furthermore, the matrix valued map x H» Pi{x) is smooth and naturally defines fi-.T2
—• C P " _ 1
x •—•¥*(*)-piteXC1), 2
n
(11)
Since tr pi{x) — 1 and Pi(x) = Pi{x), pi(a;)(C ) is the complex line {y £ C™ : Pi(x)y = y} , considered here as an element of CP™ _1 . 505
Let £( n _i) denote the canonical line bundle over C P n _ 1 and let i and 7r denote respectively the inclusion £( n _i) <-> en and the projection e n —> £(n-i)i where £ n = CP™ -1 x
and
TT, : T(en) -> rOC (n _D).
(12)
n
The trivial connection Vo on e induces a connection on £( n _i) defined by: V := ( i d ® 7 r » ) ° V o o n , 1
(13)
n_1
where id is the identity map on fi (CF ), the space of one forms on the projective space. If s G r(£(„_ 1 )), it follows that V(s) = A
CP"-1
—>n1(CPn"1) , zidzx + • • • + zndzn r L*i,...,*™j—> | Z I | 2 + . . . + | Z B | 2 •
Locally, we may write
'
<14>
,zn(x)] for some smooth func-
+ ••• +
zn(x)dzn(x)
!*!(*)!»+ - + M*)|»
•
(15)
Now, the map y, induces a bundle & = v^C^n-i)) over T 2 . The first Chern class of the (line) bundle &, denoted by ci(£j), is a cohomology class depending only on the isomorphism class of & ([MS]). Prom the properties of Chern classes, it follows that c i t e ) =
(16)
But, noting that (*i(a;),... ,*n(a;)) is the ith eigenvector of f(x) and comparing Eqs. (15) and (16 ) with Eqs. (3) and (2 ), we see that er; as in Eq.(2) can be expressed as o2
Oi = - r /
c
l(&)
if we take / to be the hamiltonian. At this stage we have n line bundles £ i , . . . , £„ over T 2 . To each bundle corresponds an integer number, namely the integral over T2 of j^cife). The sum of these integers must be zero, in accordance with Eq. (10). This can be shown using the fact that all the Chern classes of a trivial bundle vanish and that the property ci(?i 0 • • • © Cn) = c i ( 6 ) + • • • + ci(f„) 506
holds for sums of bundles. To see why the bundle £1 © • • • © £ n is trivial, we recall that each fj was obtained from the map x i-> pi(x). In fact, the following isomorphism of finitely generated projective C°°(M)—modules holds [MT]:
rte)-PiC°°(r2)". From the fact that r ( & ) © • • • © r ( f n ) ~ r ( & © • • • © £„) and the fact that pi © • • • © pn = id, it follows that £1 © • • • © £„ must be a trivial bundle [MT] . Acknowledgments I would like to thank Edwin Langmann and Jean-Yves Le Dimet for their guidance during the process of writing this article. I am also grateful to E.C.O.S. Nord who provided financial support for my stay at the Universite Blaise Pascal in Clermont-Ferrand in the fall of 1999, during which the main part of this paper was written. Finally I am also indebted to the Mazda Foundation for a grant I received during the years 1998/99. References [ASS1] J. Avron, R. Seiler, B. Simon, Phys. Rev. Lett. 5 1 , p. 51-53 (1983) [ASS1] J. Avron, R. Seiler, B. Simon, Commun. Math. Phys.159, p. 399-422 (1994) [ASSS] J. Avron, L. Sadun, J. Segert, B. Simon, Commun. Math. Phys.124, p. 595-627 (1989) [BvES] J. Bellissard, A. van Elst, H. Schulz-Baides, J. Math. Phys. 35, p. 5373-5451 (1994) [H] Y. Hatsugai, J. Phys. Cond. Matter 9, p. 2507-2549 (1997) [HK] Y. Hatsugai, M. Kohmoto, Phys. Rev. B 42, p. 8282-8294 (1990) [KN] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vols. I and II, Interscience, 1963, 1969. [M] O. Madelung, Introduction to Solid-State Theory, Springer Series in Solid-State Sciencies. [MT] I. Madsen, J. Tornehave, From Calculus to Cohomology, Cambridge University Press, 1997. [MS] J. Milnor, J. Stasheff, Characteristic Classes, Princeton University Press, 1974. [Sp] E. Spanier, Algebraic Topology, McGraw-Hill Series in Higher Mathematics, 1966. 507
[S] N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, 1951. [TKNN] D. Thouless, M. Kohmoto, M. Nightingale, M. den Nijs, Phys. Rev. Lett. 49, p. 405-408 (1982) [vKDP] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, p. 494-497 (1980) [W] G. Whitehead, Elements of Homotopy Theory, Springer Verlag, 1978. [Z] J. Zak, Phys. Rev. 134, p. A1602-A1611 (1964)
508
Proceedings of the Summer School on Geometric Methods for Quantum Field Theory edited by H. O. Campo, A. Reyes & S. Paycha © World Scientific Publishing Co.
D E T E R M I N A N T OF T H E D I R A C O P E R A T O R O V E R T H E INTERVAL [OJ] FABIAN TORRES-ARDILA * Deparatmento
de Fisica, Universidad de los Andes, Bogota, Colombia. E-mail: fatorres @uniandes. edu. co
AA
4976,
In this communication we study properties of the determinant of families of boundary value problems over the interval in order to investigate the geometry of the determinant line bundle. In particular, we compare two methods, the first one involving the (^-determinant and the second one a canonically defined determinant and we prove an identification at the level of curvatures obtained from the different methods.The one dimensional case offers a convenient toy model to review the main methods used by [SW1], [SW2], [53] in the higher dimensional case.
Introduction The problem of the computation of the ^-determinant for a family of Elliptic Boundary Value Problems over one dimensional manifolds has received a lot of attention [BFK1,LT]. The main objective of this communication is to show how we can use these analytical tools to understand the geometry of the determinant line bundle of a certain family of Elliptic Boundary Value problems in one dimension. In the case of a finite interval [0, /?] the analytical tools are easier to handle and the one dimensional case offers a toy model to review methods and results on odd-dimensional manifolds of higher dimension [5W1], [5W2], [53]. Specifically, we show how the canonical and C-determinant for a family of Elliptic Boundary Value Problems are related to each other and how this relation is reflected in the geometry of the determinant line bundle. In theorem 1 we identify the canonical and the ^-determinant, while in theorem 2 we identify the curvatures over a special subset Ur of the space that parameterizes the boundary conditions. The results of this communication were derived in the author's master thesis under supervision of Dr. Simon Scott. Detailed proofs of the main theorems can be found in [54], as well as in the the author's master thesis, [T].
"The author would like to thank the organizing committee for their kind invitation to attend the Summer School.
509
1
Geometrical and analytical data
We begin by considering the interval X = [0, /?] and a trivial complex vector bundle £ = X x C™ over X with a metric-compatible connection V. Let il be the space of connections, H = n 1 (X;u(n)), where u(n) is the Lie algebra of the unitary group U(n). We denote by g the group of gauge transformations, 0SC7°°(X;GL(n)). We consider the Dirac operator D = —IVJL that can be written locally as D = -i4- + B(x), B(x)eGL(n). (1) ax Next, we take an element P £ End(C? n ) and define what we will call a boundary value problem. Definition 1. The boundary value problem (BVP) Dp, P £ End(C 2 n ) is the data, DP = D: dom(Dp) -> L2(X;Cn) dom(Dp) = {€ tf^M.C") : P((0),(£)) - 0 £ C2™} Since the boundary of X, Y = dX = {0}|J{/?}, the restriction of the bundle £ to the boundary is isomorphic to C 2 n . A subspace u> of the space of sections of the bundle restricted to the boundary E\y can be seen as a subspace of (C 1 © C™). In this way, we obtain that the space that parameterizes all the boundary value problems is given by the Grassmannian Gr(Cn © C 1 ) , denned by: Gr(C" © C") = | J Grfc,2n(C™ © C n ),
(2)
k<2n
where Grfc i2n (C n ©C71) denotes the space of k-dimensional subspaces of (C™ © C n ). Now, for each k < 2n, we have Grk,2n(Cn ©C") = {P £ End{C2n)
:P2=P,P*=P
and T r P = k} , (3)
or equivalently, each boundary value problem is determined by an orthogonal projection of rank k in C2™. In higher dimensions there is an explicit construction of the parametrix in [52] which shows that Z)u has the analytical properties of an elliptic operator over a closed manifold. In particular, there exists a spectral resolution on the operator and the ^-determinant exists. In dimension one we are going to define a particular subset of boundary conditions which converts the above boundary value problem into an Elliptic Boundary Value Problem [BFK2]. This justifies the fact that we are going to call this an Elliptic Boundary Value problem (EBVP). This special subspace of the set of boundary value problems is given by, 510
Definition 2. We define Ur to be the subspace 0/Gr„ i 2n(C n ®
J.l
Gauge action
f (l + T ' T T 1 (l + T'T)- 1 ! 1 * \ [Til + T^y1 T{1 + T*T)~1T*J ' and
W
EVBPs
Although we are working on a trivial bundle i.e. the moduli space is trivial, the gauge group g = C°°(X, GL{n)) action on the space of connections ilX0->il ( V , 3 ) ^ 3 _ 1 V 5 = Vff, induces a gauge action on the set of EBVP's given by (H x Gr(C" ® C™)) x
0
-> il x Gr(C n ® C 1 )
(DP,g)^DfPu], where u = I „
, „ \ ) € GL(2n) and [PM] = u~lPu.
In particular, over (7r
under a gauge transformation, PT transforms as follows: P
T^
P
g(0YT(g(O))-^Y-
(5)
Finally, let us remark that spec(Dp) = spec(£>?p ,). 2
Determinant line bundle and the canonical determinant
We know from [<5] how to build a determinant line bundle for a family of Fredholm operators {Ab : F? -> F$\b € B,F? and F25 Hilbert spaces} parameterized by a base space B . The complex line Cb is given by Cb = (detKerA;,)*
On the other hand, we are going to describe an alternative construction of the determinant line bundle due to Segal [Seg] which is better suited to deal with the boundary conditions of an EVBP. Given two Frechet spaces V\, V2 and a Fredholm operator A : V\ —> V2 of order zero, the determinant line Det A is the complex line whose points are equivalence classes [Z, A] of pairs (Z, A), A S C, Z — A trace-class. For q = 1 + trace-class : Vi -> V2 with non-zero Fredholm determinant detpq, the equivalence relation is (Zq,X) ~
(Z,XdetFq).
The Segal determinant line bundle of the family of EBVP's parameterized by Gr(Cn 0 C n ) is the line bundle L -> Gr(Cn ©C 1 ) whose fiber over UJ is Lu = Det (-0^). It has the structure of a holomorphic bundle and it can be shown, see for example [52] [Prop. 4.2], that there exists a canonical isomorphism between Quillen's and Segal's determinant line bundles. Hence, given a EBVP Dp we can select a canonical element of its determinant line which is called the canonical determinant, denoted detc(-Dp). Proper calculations [52,54] lead us to the following result, Proposition 1. Consider h{x) : C™ —> C™ the solution to the paralel transport equation Dh = 0,/i(0) = Id, and let h = h(/3). For a given BVP DT the canonical determinant is given by detc(DT)=det{l+T*h). Just as in the case of Quillen's determinant line bundle, we define a canonical metric on the determinant line by
| | d e t i \ , | £ = det c (D*M.
(6)
We define a metric compatible connection and compute the curvature of this metric to find that [T], [54], Proposition 2. Consider Ur C Gr(C™ ©C") i.e the set of boundary conditions w = Graph(T), T invertible, and let Q = (1 + T*T), then the canonical curvature of the determinant line bundle is given by the Kdhler form of the Grassmannian, i.e. Qc = Bd log det Q~1 = dd log det (1 + T T ) - 1 . 3
(7)
Variational calculation of the ^-determinant
We proceed to use the £-regularization method to compute the determinant of a EBVP Dp as well as the curvature of the determinant line bundle over 512
Ur • Following Seeley, for an EBVP Dw we define the C function:
<;DJs) = ±,J\-°(Du-\rld\, where 7 is an appropriate contour which does not contain the spectrum of D^ [Se] i.e. If there exists a ray Rg = {r exp*9 r > 0} such that no element of the spectrum of eigenvalues of Du lies on Rg, then 7 is the contour begining at 00, transversing Rg to a small circle around the origin, and then back along Rg to 00. The ^-determinant is defined by logdet C J D 1 J =ea;p(-C'(0)). where the value at zero is defined by meromorphic continuation. The operator i(D — A ) - 1 is a trace-class operator with kernel [52], L e m m a 2. The Schwartz kernel K\ of i{Du — A ) - 1 is given by:
f7>(x)(l+T*/ l )- 1 r*7>(y)- 1 i c< ! , (
T\(x)(l + T h)
L
TX{y)
x>y
withrx the monodromy 0} the operator i(D w — A ) - 1 , and h — To(/3) the parallel transport of the connection V. Let us remark that K\(x,y,w) jumps by i at the diagonal, so by a simple computation the kernel of D~s is continuous and so D~s is trace-class, hence the associated spectral £ function is well defined. The first theorem can be proved by a contour integral computation [52], but here we give a variational argument which is of some use for the understanding of the higher dimensional case and is valid not only for unitary conditions i.e. T* = T _ 1 , the case treated in [52], but for more general invertible boundary conditions . Specifically, we take T 0 ,Ti e GL(n) such that the EBVP's DPo,DPl are invertible. Let TT, 0 < r < 1 be a path in GL(n) connecting such a pair of EBVP's such that Dpr = DpT is invertible. In order to compute the ^-variation we transfer the r-dependence into the operator via a gauge transformation from C°°(X, GL(n)). Associated to the path we define the path gr = (T*)~lh* € GL(n). Next, fix a smooth non-decreasing function, /:[0,/?]^[0,l], such that f(x) = 0 for x < (3/4 and f(x) = 1 for x > 3/?/4. Hence, we have a smooth path of gauge transformations grj(x) = grf(x)- Under the gauge action we have that DPr and DT x . are equivalent, where DT = g~fDgrf 513
and 9o 0 0 gr It is easily shown that under the choice T\ = h, DT _t = DrPh. Let us note that the gauge equivalence implies that det^ Dpr = det^ DPh and detcDPr = d e t c r > ^ . One can now find the suitable Schwartz kernel Kr(x, y, Ph.) of {D^)-1, which is given by -Kr[x,y,Ph)
- [g;i{x)T{x)il
+
- i T*h)-iT(y)-i9rf{x)
W
x>y-
Using the previous results one can show that Proposition 3. -^logdetcDpr dr
= f tr(Kr{x,x)Dr)dx, Jx
(10)
where Kr(x,y)
= -ig;f\x)h(x)(l
+ Tr*)hr1TT*hh(y)-1gTfiy),
and Dr =
^D\
Before giving a proof of the first theorem, we reformulate the results concerning the canonical determinant for the path of EBVP's DrPh. Lemma 3. With the same notations as before, 1. detc(DrPh)
= det(7 + T*h).
2. £ logdet c D P h = Tr (I + Tr*h)^
j-rT*h.
Sketch of the proof. The first result is obtained using the rules of transformation of the paralel transport under a gauge transformation as well as the rules for the transformation of Tr. The second assertion is obtained from the first one and using the usual formula ^ log det A — Tr A"1 -^A valid for operators A acting on finite dimensional spaces. • The following theorem gives a variational identification of the £ and the canonical approach to the computation of the determinant of an EBVP: Theorem 1. Let DT be the one parameter family of EBVPs defined as above, then £ log detcCI^,.) = ^ logdet c (£> P T J. 514
(11)
Proof. We compute that Vr = 37}(x) UfaGrnx)
F^0111 ( 1 0 )
where G r / ( x ) = £(9rf{x))9rf(x)trace -^logdet c (Dp,.) = - J
+ [B(x),Grfix)]j
tr ih(x){I
a n d usin
grf{x), the
S
(12)
symmetry of the
+ T;h)-xT;hh{x)~l
-^-Grf(x)\
+i / tr {h{x)(I + T;h)-1T;hh(x)~1[B{x),Grf{x)}} Jo
dx dx. (13)
Integrating by parts, the first term of (13) reads - f tr lh(x)(I
+ T;h)-1T;hh(x)-1^Grf{x)\
dx =
- [tr {h(x)(I + Tr*hr1T;hh(x)-1Grf(x)}fx=Q + f
(14)
tr | ^ ( f t ( a ; ) ( / + r ; / i ) - 1 r ; ^ ( a ; ) ~ 1 ) G P / ( s ) J
dx.
The first term of (14) reads - [tr {h{x){I + T;h)-x
T;hh{x)-xGrnx)}fx=ia - t r {h(I + h-^I
= + T;h)~lT*Gr}
1
Tr {{I + T*h)~ T;hGo}
+
(15)
.
l
We have Gr = ±{gT)g7 and gr = {T^h*, and hence GT = -(T*)-l£T*. Since To is independent of r, the second term in (15) vanishes, while the first term is equal to tr j c j + r ; / . ) -
1
^;^ - J^ogdetc^J .
(16)
On the other hand, the second term of (14) is equal to
/
tr
{(-d^h(x)-(I
+
T
*h)'1T;hh{x)-1-
h(x).(I + T;h)-1T;hh{x)-1-^-h(x).h(x)-A
tr {h(x).(I + T:hr1T:hh(xrl[B(x),Grf{x)}}
= -i Jo
515
Grf(x)\
dx.
dx,
(17)
since Dh{x) = 0 implies j^h{x) = iB(x)h(x), which cancels the second term in (13). Combining (13),(14),(15),(16) and (17) completes the proof of the theorem. • 4
Calculation of Quillen's metric
We now proceed to the computation of the ^-curvature of the determinant line bundle. According to Quillen, we need to make the computation of the C-determinant of the associated Laplacian Au of a EBVP Du, A w = D*D W . Since it is a second order operator, in order to compute its determinant we use a method which essentially reduces the problem to the computation of the determinant of an associated first order operator to which we can apply the methods described in the preceding sections. Much of this section is based on results from [LT] and we refer to it for complete proofs. 4-1
The Laplacian
of Du
When dealing with the Laplacian we are faced with a different boundary value problem, because the range of Du does not coincide with the domain of the adjoint boundary problem ££,. First we define the Laplacian of the BVP Du as follows, Definition 3. The Laplacian Ap of the boundary value problem Dw, is defined by: A P = DZDU dom(AP)= { {V G ff2(X;e)P(#),#))
(18) = 0, P'(2ty(0), Drj>m = 0} .
Remark 1. Over Ur, P*, the adjoint boundary condition, has an appropriate expression in terms of T given by: Q-1 PT = ( TQ™ - i1
Q~lT* TTQ~ V I TlT* *
. Q = 1 + T*T-
(19)
More generally we define for an elliptic operator A of order r,
A±±ak£;, ' dx ' k
(20)
k=0
the Elliptic Boundary Value Problem, A P A , with PA £ G r ( C n r ©C""), given by 516
Definition 4. APA=A
dom(i4PA)= _ { € Hr(X; C 1 ) : P(V>(0) • • • ^ - ^ ( O ) , 4>(P) • • • i>(r~l\P))
(21) = 0} •
Furthermore, we define a boundary condition w^, which will play an important role in the subsequent discussion: UA
UVv°)
€ C " r e C " r : Rovo + Rpvp = o\ .
(22)
In order to compute of the determinant of this EBVP, we define an equivalent EBVP A for an operator of order 1 defined on C°°(X; C" r ), which is obtained in terms of the coefficients of A as: /
A =dx±
0 0 0
1 0
0-1 ••• •••
\—aTa,Q —ara,i
0 0 0
\ (23)
• • • —ara,T—\l
The meaning of this equivalence is expressed in the next result, Proposition 4. There exists between the BVP ApA and AUA a canonical isomorphism of the determinant line bundles: DetApA^DetAUA.
(24)
Remark 2. 1. To see the reason for this isomorphism, let us note that given il)eHr{X,Cr-):
where •ij> = (ip,ip^ ••• , ^ r _ 1 ' ) - Hence, one can see that this relation gives a canonical identification KerA = KerA (CokerA = CokerA) . 2. This last result assures us that the curvature over the determinant line bundle is once again the preceding Kahler form. Now, ApA has an inverse (ApA)_1 which has a Schwartz kernel that can be found explicitly [54]. Lemma 4. Let us denote R = RQ + Rph^, then ApA is invertible iff R has an inverse. In this case Aj,1 is an integral operator , ^:L2(/;C")->dom(J4pJ, 517
and the Schwartz kernel is given by given by '-[hA{x){R-lRphA)h-A\y)]^ k{x,y) = < [hA(x)(I-R-iRphA)h^(y)
hr]
^a-l{y) _ ar1(y)
x
(25)
x>y
[l.r]
with hA(x) the parallel transport from 0 to x, of A, and hA = hA(/3). The notation D[i>r] means the matrix r x r located in the top left block of D. 4-2
The ^-function
metric
of the determinant
line
bundle
The essential result that we need in order to apply the above discussion to the Laplacian is [LT] : Proposition 5. Given a BVP Apz, we have: det c vlp( 0 ) = exp(-LIM 0 _ > o o logdet^(z))deti?,
(26)
where R(z) = Ro + RphA(z), R = R(f3) and where LIM means the regularized limit, in the sense of the quoted paper [LT]. Then, we find an explicit expression for the matrices Ro and Rp for the case of the Laplacian Ap = —-^i and its associated first order operator
Lemma 5. The matrices Ro and Rp are given by: ^-{TQ-I
Q-1T
)'R0-\TQ-iT*
0-1
/ '
(
'
Hence, using the results in [LT], we have for R = RQ + Rphp, that the det R of (26) is equal to: Proposition 6. det R = det Q'1 det(l + T) det(l + T*).
(28)
On the other hand, we have for the limit of the same expression, the result: Proposition 7. exp LIMZ-KX, log det R{z)
= 2".
Finally, combining these results we arrive at Theorem 2. det c A P T = 2n log det Q~l det(l + h*T) det(l + T*h) = 2 n e - ^ T » | d e t c £ ) p T |22 , 518
(29)
where we have used the expression e~k^ = l o g d e t Q - 1 with Q = 1 +T*T to express the Kahler potential. Now we are ready to assert the relation between the canonical and £ curvature of the determinant line bundle announced in the introduction. Corollary 1. The Q-curvature of the determinant line bundle is given by the Kahler form of the Grassmannian. 5
Final remark
As stated in the introduction it is remarkable that there exists such an identification between two such different ways of computing determinants of boundary value problems. Now, it is possible to obtain more general results for the 1-dimensional case by using the notion of relative determinant, [F], which allows a very different proof of the same results [54]. Additionally, the same method has been used by Scott for the computation of the curvature of the determinant line bundle in higher dimensions [53].
Acknowledgments I thank an anonymous referee for helpful comments on a preliminary version of this paper, also I am grateful with Sylvie Paycha for her continuous encouragement and lucid observations about this paper. References [BB] Booss, B., Bleecker, D., Topology and analysis, Springer-Verlag, 1985. [BFK1] Burghelea, D., Priedlander, L., Kappeler, T., On the determinant of Elliptic Boundary Value Problems on a line segment, Proc. Am. Math. Soc. 123, 10, 3027-3038, 1995. [BFK2] Burghelea, D., Priedlander, L., Kappeler, T., On the determinant of Elliptic differential and finite difference operators in vector Bundles over S1, Commun. Math. Phys. 138, 10, 1-18, 1991. [F] Forman, R., Functional determinants and geometry, Invent. Math. 88, 447-493, 1988. [LT] Lesch, M., Tolksdorf, J., On the Determinant of One-Dimensional Elliptic Boundary Value Problems, Commun. Math. Physl93, 643-600, 1998. [Q] Quillen, D. G., Determinants of Cauchy-Riemann operators over a Riemann surface, Funk. Anal, i ego Prilozhnya 19, 37-41, 1985. 519
[SI] Scott, S., Splitting the curvature of the determinant line bundle. Proc. Am. Math. Soc. , to appear. [S2] Scott, S., Determinants of elliptic boundary value problems over odd dimensional manifolds, Commun. Math. Phys. 80, 301, 1995. [S3] Scott, S., Relative Zeta Determinants and Relative Determinant Bundles On Manifolds with Boundary , preprint, math.AP/9910148. [S4] Scott, S. G., Torres, F., Geometry of the determinant line bundle in dimension one, preprint. [SW1] Scott, S.G., Wojciechowski, K.,Determinants, Grassmannians and Elliptic Boundary Value Problems, Lett. Math. Phys 40, 135-145, 1997. [SW2] Scott, S.G., Wojciechowski, K. (^-determinant and the Quillen Determinant on the Grassmanian of Elliptic Self-Adjoint Boundary Conditions, C. R . Acad. Sci, t.328, Serie I , 139-144, 1999. [Se] Seeley, R.T., Complex powers of elliptic operators, Proc. Symp. on Singular Integrals, AMS, 10, 288-307, 1967. [Seg] Segal, G.B., The definition of conformal field theory, Preprint. [T] Torres, F. Determinante del operador de Dirac sobre el intervalo, Master Thesis, Mathematics Dept., Universidad de los Andes, Bogota, Colombia, 1998.
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