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j - ^ d + epeq -i V2~BpBq + -^i_a p s g J • (75) (u) and a = cr(u) are arbitrary in u, with e^da = dX and K(u, z) is arbitrary in u but holomorphic in z. These solutions necessarily lift to six dimensions to give some supersymmetric solutions there. More information about other six-dimensional supersymmetric solutions is has been provided by Marco Cariglia and Oisin MacConamhna. 6 4.5. String
We list the conjugate relations for completeness Sj$
= wpqJpq$
= iy/2 wpqd+6p§q$,
(76)
(77) The variations of the generalized derivative and its conjugate are given by Sj V = uJmn [ Jmn , V ] = - ujmn V m " , 5jV
= w™ [ Jpq , V] = j~= w*> dpdq A .
(78) (79)
Invariance under 50(8) is checked by doing a Sj variation on the cubic vertex, including its complex conjugate. The cubic vertex is 50(8) invariant if a = 1, and the generalized derivative is totally determined V = B+T-^dpdqd^
.
(80)
To obtain this result, we used the antisymmetry of the structure functions, the chiral constraints, the "inside-out" constraints, and performed integrations by parts on the coordinates and Grassmann variables. In this light-cone form, the Lorentz invariance in ten dimensions is automatic once
53
the little group invariance has been established. We have therefore shown ten-dimensional invariance, since the quartic term does not need to be changed. 5. Eleven Dimensions We can now redo the scheme above to construct the eleven-dimensional supergravity. 11 We extend the Lorentz symmetry to SO (9) by adding in the 50(7) which is missing in the four-dimensional case and construct also the the generators for 50(9)/SO(7).The translation generators are also extended. We use the same superfield as in four dimensions and and the only difference in the action is that we have to find corresponding generalized derivatives V and V. They transform together with d and 5 as 9-vectors. We do not give the details here but refer to the forthcoming paper. The final action is then
5=
Jdnxjd80d8e(^^4>
+ f {£* * B - 1 ) " Q ) ( V 2 - 8*n4> V" 3+ 2 ->) + c.c.) + . • • .(81) Again we find that this action has exactly the same structure as the tendimensional one apart from having infinitely many terms. However, those properties like the results of simple one-loop calculations can be taken over from one model to the other. We also know that those properties are very similar.12 It is also natural to use this formalism to study possible symmetries over and above the superPoincare symmetry. This work is in progress. 6. Further Extensions In a series of papers we have studied an extension of the 11-dimensional supergravity theory. 13 It is based on the observation that the Dynkin indices of supergravity multiplet has very interesting properties. 14 They match up bosons vs fermions for all but the largest of the Dynkin indices, I8. This property is also respected by an infinity of triplets of the little group 50(9). These triplets then correspond to higher spins. This result can be understood in the following way. SO(9) is the maximal subalgebra of FA. They have the same rank. For two algebras like this one can associate to every representation in the larger algebra an multiplet in the subalgebra with properties as above. 15 Kostant has constructed a Dirac-like equation
54
the solutions of which are exactly the multiplets above. In the case of 5 0 ( 9 ) we can write the Kostant equation 1 6 as 7
»Ti^
=
0
,
( 82 )
T* are the generators of F
COSMOLOGICAL SINGULARITIES, BILLIARDS AND LORENTZIAN KAC-MOODY ALGEBRAS
THIBAULT DAMOUR Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur- Yvette, France The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of Belinskii-Khalatnikov-Lifshitz and technically simplified by the use of the Arnowitt-Deser-Misner Hamiltonian formalism) that the asymptotic behaviour, as one approaches the singularity, of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space. For certain Einstein-matter systems, notably for pure Einstein gravity in any spacetime dimension D and for the particular Einstein-matter systems arising in String theory, the billiard tables describing asymptotic cosmological behaviour are found to be identical to the Weyl chambers of some Lorentzian Kac-Moody algebras. In the case of the bosonic sector of supergravity in 11 dimensional spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac-Moody group E\o, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space E\o/K(E\o), where K{E\o) is the maximal compact subgroup of E\o.
It is a pleasure to dedicate this review to Stanley friend and a great physicist to whom I owe a lot.
Deser,
a dear
1. I n t r o d u c t i o n a n d O v e r v i e w A remarkable connection between the asymptotic behavior of certain Einstein-matter systems near a cosmological singularity and billiard motions in the Weyl chambers of some corresponding Lorentzian K a c - M o o d y algebras was uncovered in a series of works. 1 " 7 This simultaneous appearance of billiards (with chaotic properties in important physical cases) and of an underlying symmetry structure (infinite-dimensional Lie algebra) is an interesting fact, which deserves to be studied in depth. Before explaining the techniques (notably the Arnowitt-Deser-Misner Hamiltonian formalism 8 ) t h a t have been used to uncover this fact, we will s t a r t by reviewing previous related works, and by stating the main results of this 55
56
Figure 1.
Lorentz space and projection on Poincare disk.
billiard/symmetry connection. The simplest example of this connection concerns the pure Einstein system in D = 3 + 1-dimensional space-time. The Einstein equations (RitviSctfi) = 0) a r e non-linear PDE's for the metric components. Near a cosmological spacelike singularity, here chosen as t = 0, the spatial gradients are expected to become negligible compared to time derivatives {d/dxl -C d/dt); this then suggests the decoupling of spatial points and allows for an approximate treatment in which one replaces the above partial differential equations by (a 3-dimensional family of) ordinary differential equations. Within this simplified context, Belinskii, Khalatnikov and Lifshitz (BKL) gave a description 9 ' 10 ' 11 of the asymptotic behavior of the general solution of Einstein's equations, close to the singularity, and showed that it can be described as a chaotic 12,13 sequence of generalized Kasner solutions. The Kasner metric is of the type ga0{t)dxadx13
= -N2dt2
+ Ait2pidxl
+ A2t2p2dxl
+ A3t2p*dx23,
(1)
where the constants pi obey a ~P2=P21+P22+P2.-(PI+P2+P3)2=0.
(2)
An exact Kasner solution, with a given set of Ai's and pi's, can be represented by a null line in a 3-dimensional auxiliary Lorentz space with coordinates pi,P2,Ps equipped with the metric given by the quadratic form "j?2 above. The auxiliary Lorentz space can be radially projected on the unit hyperboloid or further on the Poincare disk (i.e. on the hyperbolic plane H2): the projection of a null line is a geodesic on the hyperbolic plane. BKL showed that, because of non-linearities in Einstein's equations, the generic solution behaves as a succession of Kasner epochs, i.e., to a a
I n the N — 1 gauge, they also obey pi + P2 + P3 = 1-
57
Figure 2.
Picture of chaotic cosmological behavior for 3 + 1 gravity
broken null line in the auxiliary Lorentz space, or a broken geodesic on the Poincare disk. This broken geodesic motion is a "billiard motion" (seen either in Lorentzian space or in hyperbolic space). The billiard picture naturally follows from the Hamiltonian approach to cosmological behavior and was first obtained in the homogeneous (Bianchi IX) four-dimensional case 14,15 and then extended to higher space-time dimensions with p-forms and dilatons. 3 ' 6,7,18 " 21 Recent work7 has improved the derivation of the billiard picture by using the Iwasawa decomposition of the spatial metric. Combining this decomposition with the ArnowittDeser-Misner Hamiltonian formalism highlights the mechanism by which all variables except the scale factors and the dilatons get asymptotically frozen. The non-frozen variables (logarithms of scale factors and dilatons) then undergo a billiard motion. This billiard motion can be seen either in Lorentzian space or, after radial projection, on hyperbolic space (see below for details). A remarkable connection was also established1"7 between certain specific Einstein-matter systems and Lorentzian Kac-Moody (KM) algebras. 22 In the leading asymptotic approximation, this connection is simply that the Lorentzian billiard table within which the motion is confined can be identified with the Weyl chamber of some corresponding Lorentzian KM algebra. This can happen only when many conditions are met: in particular, (i) the billiard table must be a Coxeter polyhedron (the dihedral angles between adjacent walls must be integer submultiples of ir) and ii) the billiard must be a simplex. Surprisingly, this occurs in many physically interesting Einstein-matter systems. For instance, pure Einstein gravity in D dimensional space-time corresponds to the Lorentzian KM algebra 4 AED-I which is the overextension of the finite Lie algebra ADS'- for
58
D — 4, the algebra is AE3 the Cartan matrix of which is given by A=
( - 12~l 2 - 2 °\ . \
0-2
(3)
2/
Chaotic billiard tables have finite volume in hyperbolic space, while nonchaotic ones have infinite volume; as a consequence, chaotic billiards are associated with hyperbolic KM algebras; this happens to be the case for pure gravity when D < 10. Another connection between physically interesting Einstein-matter systems and KM algebras concerns the low-energy bosonic effective actions arising in string and M theories. Bosonic string theory in any space-time dimension D is related to the Lorentzian KM algebra DED-3'5 The latter algebra is the canonical Lorentzian extension of the finite-dimensional algebra DD-2- The various superstring theories (in the critical dimension D = 10) and M-theory have been found3 to be related either to Ew (when there are two supersymmetries in D = 10, i.e. for type IIA, type IIB and M-theory) or to BEW (when there is only one supersymmetry in D = 10, i.e. for type I and II heterotic theories), see the table below. A construction of the Einstein-matter systems related to the canonical Lorentzian extensions of all finite-dimensional Lie algebras An, Bn, Cn, Dn, G2, F4, EQ, £7 and Es (in the above "billiard" sense) is presented in Ref. [5]. See also Ref. [23] for the identification of all hyperbolic KM algebras whose Weyl chambers are Einstein billiards. The correspondence between the specific Einstein-three-form system (including a Chern-Simons term) describing the bosonic sector of 11dimensional supergravity (also known as the "low-energy limit of Mtheory") and the hyperbolic KM group -Bio was studied in more detail in [6]. Reference [6] introduces a formal expansion of the field equations in terms of positive roots, i.e. combinations a = £j n1 cx.i of simple roots of £10, ctj, % — 1 , . . . , 10, where the nl's are integers > 0. It is then useful to order this expansion according to the height of the positive root a — Sj nl an, defined as ht(a) = £ j n \ The correspondence discussed above between the leading asymptotic evolution near a cosmological singularity (described by a billiard) and Weyl chambers of KM algebras involves only the terms in the field equation whose height is ht(a) < 1. By contrast, the authors of Ref. [6] managed to show, by explicit calculation, that there exists a way to define, at each spatial point x, a correspondence between the field variables g^(t,x), A^x(t,x) (and their gradients), and a (finite) subset
59
of the parameters defining an element of the (infinite-dimensional) coset space Eio/K(Eio) where K(Ei0) denotes the maximal compact subgroup of £10, such that the (PDE) field equations of supergravity get mapped onto the (ODE) equations describing a null geodesic in EIOJK(EIQ) up to terms of height 30. This tantalizing result suggests that the infinitedimensional hyperbolic Kac-Moody group Eio may be a "hidden symmetry" of supergravity in the sense of mapping solutions onto solutions (the idea that E\o might be a symmetry of supergravity was first suggested by Julia long ago 24 ' 25 ). Note that the conjecture here is that the continuous group Eio(M.) be a hidden symmetry group of classical supergravity. At the quantum level, i.e. for M theory, one expects only a discrete version of £10, say i?io (Z), to be a quantum symmetry. See [26] for recent work on extending the identification of [6] between roots of £ 1 0 and symmetries of supergravity/M-theory beyond height 30, and for references about previous suggestions of a possible role for EIQ. For earlier appearances of the Weyl groups of the E series in the context of [/-duality see [27,28,29]. A series of recent papers 30 " 34 has also explored the possible role of -En (a nonhyperbolic extension of £10) as a hidden symmetry of M theory. It is also tempting to assume that the KM groups underlying the other (special) Einstein-matter systems discussed above might be hidden (solution-generating) symmetries. For instance, in the case of pure Einstein gravity in D = 4 space-time, the conjecture is that AE3 be such a symmetry of Einstein gravity. This case, and the correspondence between the field variables and the coset ones is further discussed in [7]. Rigorous mathematical proofs 17 ' 35 ' 16 are however only available for 'non chaotic' billiards. In the remainder of this paper, we will outline various arguments explaining the above results; a more complete derivation can be found in [7]. 2. G e n e r a l M o d e l s The general systems considered here are of the following form
S[gMN,d>,A^]= J 1 2
dDx^i
V^ 2^(„
l +
l)|
R(g) - d.'M<, rK4>p(P) e r
5M,
M1-Mp+1r
p(p)M1-Mp+1
(4)
Units are chosen such that IQTTGN = 1, GM is Newton's constant and the space-time dimension D = d + 1 is left unspecified. Besides the standard
60
Einstein-Hilbert term the above Lagrangian contains a dilaton b field <j> and a number of p-form fields A^ ... M (for p > 0). The p-form field strengths F^ = dA^ are normalized as F
= (P +
^^MI^MI-M^!]
=
9
± P permutations . (5) As a convenient common formulation we adopt the Einstein conformal frame and normalize the kinetic term of the dilaton 4> with weight one with respect to the Ricci scalar. The Einstein metric gMN has Lorentz signature (—h • —h) and is used to lower or raise the indices; its determinant is denoted by g. The dots in the action (4) above indicate possible modifications of the field strength by additional Yang-Mills or ChaplineManton-type couplings. 36,37 The real parameter Ap measures the strength of the coupling of A^ to the dilaton. When p — 0, we assume that Ao ^ 0 so that there is only one dilaton. M\-MP+1
M.4...MP+1
3. Dynamics in the Vicinity of a Spacelike Singularity The main technical points that will be reviewed here are the following • near the singularity, t —> 0, due to the decoupling of space points, the Einstein's (PDE) equations become ODE's with respect to time. • The study of these ODE's near t —> 0, shows that the d diagonal spatial metric components ugu" and the dilaton <j> move on a billiard in an auxiliary d + 1 dimensional Lorentz space. • All the other field variables {gij,i ^ j,-Ai1...jp,7rtl-,,lp) freeze as t-t0. • In many interesting cases, the billiard tables can be identified with the fundamental Weyl chamber of an hyperbolic KM algebra. • For SUGRAn, the KM algebra is _Ei0. Moreover, the PDE's are equivalent to the equations of a null geodesic on the coset space E10/K(EW), up to height 30. 3.1. Arnowitt-Deser-Misner
Hamiltonian
formalism
To focus on the features relevant to the billiard picture, we assume here that there are no Chern-Simons and no Chapline-Manton terms and that the The generalization to any number of dilatons is straightforward.
61
curvatures F ( p ) are abelian, F^ = cL4(p). That such additional terms do not alter the analysis has been proven in [7]. In any pseudo-Gaussian gauge and in the temporal gauge (g0i = 0 and A0i2...ip = 0, Vp), the ArnowittDeser-Misner Hamiltonian action 8 reads 9ij,^A^A^..Jp,ni^
J dx* J dfiz L^ + Wpi + ^ £ <)" JP ^., P - H(6) V
where the Hamiltonian density H is
H = NH,
(7)
H= K+M ,
(8)
p
X = -gR
+
gg*%^^
+
Yt2^^\3^-i^FWil'"ir+l
• (10)
and iZ is the spatial curvature scalar. iV = N/y/gW is the rescaled lapse. The dynamical equations of motion are obtained by varying the above action with respect to the spatial metric components, the dilaton, the spatial p-form components and their conjugate momenta. In addition, there are constraints on the dynamical variables, H~0
("Hamiltonian constraint"),
(11)
Hi ss 0
("momentum constraint"),
(12)
("Gauss law" for each p-form),
(13)
fj(p) h~1 * ° with
Hi = -2«ii{j + n,di4> + J2 h ^'J°F£-iP
-
p
where the subscript \j stands for a spatially covariant derivative.
(14)
62
3.2. Iwasawa
decomposition
of the spatial
metric
We systematically use the Iwasawa decomposition of the spatial metric g^j and write d
0=1
where Af is an upper triangular matrix with l's on the diagonal. We will also need the Iwasawa coframe {0a}, 9a=Maldxi,
(17)
as well as the vectorial frame {ea} dual to the coframe {0a},
e 0 =AP a A
(18)
where the matrix A / \ is the inverse of Afai, i.e., NaiNlb = 8%. It is again an upper triangular matrix with l's on the diagonal. Let us now examine how the Hamiltonian action gets transformed when one performs, at each spatial point, the Iwasawa decomposition (16) of the spatial metric. The kinetic terms of the metric and of the dilaton in the Lagrangian (4) are given by the quadratic form
G^d^d/T
d I d \2 a 2 a = J > / 3 ) - £ d/3 \ + dcj>2, p = (/3a, 0). 0=1
\<J=I
(19)
/
The change of variables (#,j —> j3a,Nai) corresponds to a point transformation and can be extended to the momenta as a canonical transformation in the standard way via
Tr^y = Y, *«F + E Pt^ai a
•
(20)
a
Note that the momenta
P\ = J £ - =Ye^b-^^jNhAfib oNai
(21)
^
conjugate to the nonconstant off-diagonal Iwasawa components Hai are only defined for a < i; hence the second sum in (20) receives only contributions from a < i.
63
3.3. Splitting
of the
Hamiltonian
We next split the Hamiltonian density H (7) in two parts: Ho, which is the kinetic term for the local scale factors /3M = (/3a, (j>), and V, a "potential density" of weight 2, which contains everything else. Our analysis below will show why it makes sense to group the kinetic terms of both the offdiagonal metric components and the p-forms with the usual potential terms, i.e. the term M in (8). Thus, we write H = Ho + V,
(22)
with the kinetic term of the /3 variables H0 = \ G^n^
,
(23)
where GM" denotes the inverse of the metric Gav of Eq. (19). In other words, the right hand side of Eq. (23) is defined by
G^x^^E^-^ifE^j +*h where 7rM = {na^^) i.e.
( 24 )
are the momenta conjugate to (3a and
= 2G^-^.
(25)
The total (weight 2) potential density, v is naturally split into a "centrifugal" part Vs linked to the kinetic energy of the off-diagonal components (the index S referring to "symmetry,"), a "gravitational" part VG, a term from the p-forms, £ ) p Vp, which is a s u m of an "electric" and a "magnetic" contribution and also a contribution to the potential coming from the spatial gradients of the dilaton V$. • "centrifugal" potential Vs = l52e-Wb-^(pibAf"J)\
(27)
a
• "gravitational" (or "curvature") potential VG - -gR
= I £
e-^^\Cabcf
- E
e"W/3)Fa ,
(28)
64
where aa6c(/?) = ^ / 3 e + / ? a - / ? 6 - / ? c ,
a^b,b^c,c^a,
(29)
e
and d6a = -^CabcebA9c
(30)
while Fa is a polynomial of degree two in the first derivatives d(3 and of degree one in the second derivatives d2j3. • p-form potential V(p) = V g ) + V - ^ ° ,
(31)
which is a sum of an "electric" VfL and a "magnetic" V,m?gn contribution. The "electric" contribution can be written as „—Xp(t> _ _ _Jl"'Jj>_
•oel
=
_L
^
e -2e„ 1 ...« I ,(/J)^o,-Opj2 )
&ii&2>*" , a
a
ai
a2
(32)
p
a
where £ i"°P = Af hM h---Af "jpiT^-^ the "electric wall" forms,
, and e0l...0j>(/3) are
e0l...Op(/3) = /3»i+ . . . + /?<*+ ^ 0 .
(33)
And the "magnetic" contribution reads, Vmagn = K
(P)
e
"
0
2(p + l)!
P(P)
H
F(P)JI-J'P+I
Ji-^+i-1
oi,02,-" ,ap+i
where .F 0 l ... a p + 1 = Afhai • • • Njv+1 ap+lFh...jp+1, "W-op+iCS) a r e the magnetic linear forms
m a i ... 0 p + 1 (/?)=
£
^-y
and the
(35)
6^{ai,a2,'--«p+i}
• dilaton potential V^ = ggVdiWj* =
^e-Ma(/3)(A^i9.(A)2!
(36) ( 3 7 )
65
where /xa(/3)
= £>e-/?a.
(38)
e
3.4. Appearance
of sharp walls in the BKL
limit
In the decomposition of the hamiltonian as H = Wo + V, Ha is the kinetic term for the /3M's while all other variables now only appear through the potential V which is schematically of the form V(/3", dx0", P,Q)
= J^cA{dx^,
P, Q) exp ( - 2wA(J3)),
(39)
A
where (P,Q) = {Nai,Pia,£ai-a*,Fav..ap+1). Here wA((3) = wA)1^ the linear wall forms already introduced above: symmetry walls : wf6 = (3b — /3a;
are
a
(3 + Pa - f3b - (3C, a ^ b, b / c, c ^ a,
gravitational walls : aabc(P) = ^ e
e
electric walls : e0l...0p(/3) = /3 01 + • • • + /3 a " + -Ap, /?e - /T 1
magnetic walls : mai...ap+1 (/?) = ^
/J"^ 1 - | A P 0 .
e
In order to take the limit t —> 0 which corresponds to /?M tending to future time-like infinity, we decompose /3M into hyperbolic polar coordinates (p,7^), i.e. P» = Pr, (40) where 7M are coordinates on the future sheet of the unit hyperboloid which are constrained by a^-f-f = r^
= -i,
(41)
and p is the time-like variable defined by p2 = -GMUp»p"
= - W
> 0,
(42)
which behaves like p ~ — lnt —» +oo at the BKL limit. In terms of these variables, the potential term looks like J2cA(dx/3»,P,Q)p2exp(-2pwA(1)). A
(43)
66
The essential point now is that, since p —> +00, each term p2 exp ( — 2PWA(I)) becomes a sharp wall potential, i.e. a function of WA{I) which is zero when WA{J) > 0, and +00 when WA(J) < 0. To formalize this behavior we define the sharp wall 9-function c as f0
if x < 0,
I +00 if x > 0 . A basic formal property of this 0-function is its invariance under multiplication by a positive quantity. Because all the relevant prefactors CA{dx(3^iP->Q) a r e generically positive near each leading wall, we can formally write lim
CA(dx^, Q, P)p2 exp ( - pwA{n)\ = cA(Q, P)Q( - 2 ^ ( 7 ) )
p—*oo
= 6(-21^(7)),
(45) 7
valid in spite of the increasing of the spatial gradients. Therefore, the limiting dynamics is equivalent to a free motion in the /3-space interrupted by reflections against hyperplanes in this /3-space given by WA(P) = 0 which correspond to a potential described by infinitely high step functions V(/3,P,Q) = ^ e ( - 2 ^ ( 7 ) ) -
(46)
A
The other dynamical variables (all variables but the /3M's) completely disappear from this limiting Hamiltonian and therefore they all get frozen as t->0. 4. Cosmological Singularities and Kac—Moody Algebras Two kinds of motion are possible according to the volume of the billiard table on which it takes place, i.e. the volume (after projection on hyperbolic space) of the region where V = 0 for t —> 0, also characterized by the conditions, wA((3)>0
VA
(47)
Depending on the fields present in the Lagrangian, on their dilatoncouplings and on the spacetime dimension, the (projected) billiard volume c
O n e should more properly write 0<x>(x), but since this is the only step function encountered here, we use the simpler notation Q(x).
67
Figure 3. Sketch of billiard tables describing the asymptotic cosmological behavior of Einstein-matter systems. is either finite or infinite. The finite volume case corresponds to neverending, chaotic oscillations for the /3's while in the infinite volume case, after a finite number of reflections off the walls, they tend to an asymptotically monotonic Kasner-like behavior, see Fig. 3. In Fig. 3 the upper panels are drawn in the Lorentzian space spanned by ((3^) = (j3a,<j>). The billiard tables are represented as "wedges" in (d + 1)dimensional (or
68
Some of the walls stay behind the others and are not met by the billiard ball. Only a subset of the walls WA(P), called dominant walls and here denoted {w,(/3)} are needed to delimit the hyperbolic domain. Once the dominant walls are found, one can compute the following matrix
where Wi.Wj = G^w^Wjy. By definition, the diagonal elements are all equal to 2. Moreover, in many interesting cases, the off-diagonal elements happen to be non positive integers. These are precisely the characteristics of a generalized Cartan matrix, namely that of an infinite KM algebra (see appendix). As recalled in the introduction, for pure gravity in D space-time dimensions, there are D — 1 dominant walls and the matrix Aij is exactly the generalized Cartan matrix of the hyperbolic KM algebra AED-\ S AQ^_3 = J 4 ^ 1 3 which is hyperbolic for D < 10. More generally, bosonic string theory in D space-time dimensions is related to the Lorentzian KM algebra DED 3 ' 5 which is the canonical Lorentzian extension of the finitedimensional Lie algebra Do-2- The various superstring theories, in the critical dimension D = 10, and M-theory have been found3 to be related either to E\o (when there are two supersymmetries, i.e. for type IIA, type IIB and M-theory) or to BE\o (when there is only one supersymmetry, i.e. for type I and II heterotic theories), see the table. The hyperbolic KM algebras are those relevant for chaotic billiards since their fundamental Weyl chamber has a finite volume. The precise links between a chaotic billiard and its corresponding KacMoody algebra can be summarized as follows • the scale factors /3M parametrize a Cartan element h = ]Cn=i /?MV> • the dominant walls Wi(f3),(i = l,...,r) correspond to the simple roots on of the KM algebra, • the group of reflections in the cosmological billiard is the Weyl group of the KM algebra, and • the billiard table can be identified with the Weyl chamber of the KM algebra. 5. EXQ and a "Small Tension" Limit of S U G R A n The main feature of the gravitational billiards that can be associated with the KM algebras is that there exists a group theoretical interpretation of the billiard motion: the asymptotic BKL dynamics is equivalent (in a sense
69 Table 1. This table displays the Coxeter-Dynkin diagrams which encode the geometry of the billiard tables describing the asymptotic cosmological behavior of General Relativity and of three blocks of string theories: B2 = {M-theory, type IIA and type IIB superstring theories}, B\ = {type I and the two heterotic superstring theories}, and Bo = {closed bosonic string theory in D = 10}. Each node of the diagrams represents a dominant wall of the cosmological billiard. Each Coxeter diagram of a billiard table corresponds to the Dynkin diagram of a (hyperbolic) KM algebra: Eio, BE\o and DE\oTheory
Corresponding Hyperbolic KM algebra
Pure gravity in D < 10
a\
at
at
03
M-theory, IIA and IIB Strings
01
ma
03
<*4
«s
<*6
"7
<*B
"9
type I and heterotic Strings
Oi
orj
03
1x4
as
Ore
07
<*8
<*9
closed bosonic string in D = 10
Ol
oj
Q3
Q4
Q5
Q6
<*T
<*8
to be made precise below), at each spatial point, to the asymptotic dynamics of a one-dimensional nonlinear cr-model based on a certain infinitedimensional coset space G/K, where the KM group G and its maximal compact subgroup K depend on the specific model. As we have seen, the walls that determine the billiards are the dominant walls. For the KM billiards, they correspond to the simple roots of the KM algebra. As we discuss below, some of the subdominant walls also have an algebraic interpretation in terms of higher-height positive roots. This enables one to go beyond the BKL limit and to see the beginnings of a possible identification of the dynamics of the scale factors and of all the remaining variables with that of a nonlinear cr-model defined on the cosets of the KM group divided by its maximal compact subgroup. 6 ' 7 For concreteness, we will only consider one specific example here: the relation between the cosmological evolution of D = 11 supergravity and 6 a null geodesic on EW/K{E\Q) where KE\0 is the maximally compact subgroup of EIQ. The cr-model is formulated in terms of a one-parameter dependent group element V = V(t) € Ew and its Lie algebra value derivative
"(*) : = ? V _ 1 W
e e
io-
( 49 )
70
The action is
s?w = J ^yK m (0KmW>,
(so)
with a lapse function n(t) whose variation gives rise to the Hamiltonian constraint ensuring that the trajectory is a null geodesic. The symmetric projection VSym-= T(V + VT)
(51)
is introduced in order to define an evolution on the coset space. Here (.|.) is the standard invariant bilinear form on Eio', vT is the "transpose" of v defined with the Chevalley involution13 as vT = —u>(y). This action is invariant under E\Q, V(t) -» k(t)V(t)g,
where
k £ KEW,
g £ E10.
(52)
Making use of the explicit Iwasawa parametrization of the generic Eio group element V = KAN together with the gauge choice K = 1 (Borel gauge), one can write V(t)=expXh(t)-expXA(t), with Xh(t) = habKba and XA(t) = ±AabcEabc
+ ^ „ 1 . . . a . £ a i - 0 a + ^Aao]ai...a8Ea°^-a»
+ ••• .
Using the Ew commutation relations in GL(10) form together with the bilinear form for E\Q, one obtains up to height 30 e , nC = \{gacghd - gabgcd)gab9cd + +l±DAai...a6DAai-a°
±±DAaia2a3DAa^a°
+ ±l,DAaolai...a8DAa°^-a»,
(53)
where gab = eacebc with e\ = d
(exph)ab,
T h e Chevalley involution is defined by u)(hi) = — ha ui(ei) = —ft; (jj(fi) = — e;. We keep only the generators Eabc, £ ° i - ° 6 and E a o | a i " - a s corresponding to the Eio roots a = 5 Z w i a » with height £V m < 29 (oti are simple roots and rn integers). e
71
and all "contravariant indices" have been raised by gab. The "covariant" time derivatives are defined by (with dA = A)
DAai„Me
: = uAai...ae
-^'At 1 |a2...a9 '
=
^J^a1\a2...ag
+ 10yl[aia2a3aj4a4O5a6], + 4 2 ^ 4 ^ a i a 2 a 3 C7j4 a4 ... ag )
—42oA( a i a 2 a 3 A a i ,.a9) + 280A^aia2a3Aaia5a6dAarasag^.
(54)
Here antisymmetrization [...], and projection on the £ = 3 representation (...), are normalized with strength one (e.g. [[...]] = [...]). Modulo field redefinitions, all numerical coefficients in (53) and in (54) are uniquely fixed by the structure of E\Q. In order to compare the above coset model results with those of the bosonic part of D = 11 supergravity, we recall the action ssugra
=
jdnx^ZGR(G)
-^F
a M 5
F*^
The space-time indices a, ft,... take the values 0 , 1 , . . . , 10; £ 0 1 - 1 0 = + 1 , and the four-form J- is the exterior derivative of A, T = dA. Note the presence of the Chern-Simons term T AT AA in the action (55). Introducing a zero-shift slicing (Nl = 0) of the eleven-dimensional space-time, and a time-independent spatial zehnbein 6a(x) = Eai(x)dxl, the metric and fourform T = dA become ds2 = Ga0 dxa dx? = -N2(dx0)2 T=
^Foabcdx°A0aAebAec
+ Gab9ae\
(56)
+ ^Fabcd 6a A9b A9C A0d.
We choose the time coordinate x° so that the lapse N = VG, with G := det Gab (note that x° is not the proper time f T = J Ndx°; rather, a;0 —> oo as T —> 0). In this frame the complete evolution equations of D = 11
f
In this section, the proper time is denoted by T while the variable t denotes the parameter of the one-dimensional cr-model introduced above.
72
supergravity read d0(Gacd0Gcb) d0{Gf°abc)
= \GTa^bTbfil5
- ±GFa^5Fa/3l55ab
-
2GRab{T,C),
^eabca^a>b^b°b*F0aia2a3fblb2b3bi
= +
lGJrde[abcc}de
_
GCedefdabc
_
dd(pjrdabc^
f
doFabcd = §Foe[abGecd] + 49[ a ^ r 0 6cd] i
(57)
where a,b £ { 1 , . . . , 10} and a, (3 € { 0 , 1 , . . . , 10}, and Rab(T, C) denotes the spatial Ricci tensor; the (frame) connection components are given by ^GadT bc = Cabc + Cbca — Ccab + dbGca + dcGab — daGbc with Cabc = GadCdbc being the structure coefficients of the zehnbein d9a = \Cabc6b A 6C. (Note the change in sign convention here compared to above.) The frame derivative is da = Ela(x)di (with EaiElb = 5%). To determine the solution at any given spatial point x requires knowledge of an infinite tower of spatial gradients; one should thus augment (57) by evolution equations for daGbc,daJ:'obcd,daJ-'bcde, etc., which in turn would involve higher and higher spatial gradients. The main result of concern here is the following: there exists a map between geometrical quantities constructed at a given spatial point x from the supergravity fields G)lu(x0,x) and A^vp{x°,x) and the one-parameterdependent quantities gab(t),Aabc(t),... entering the coset Lagrangian (53), under which the supergravity equations of motion (57) become equivalent, up to 30th order in height, to the Euler-Lagrange equations of (53). In the gauge (56) this map is defined by t = x° = J dT/^/G and
9ab(t)
=Gab(t,x),
IM0l-a«(t) = DA»\ai...a8{t)
=
-iea-M*b*b*rblb2b3lH(t,x), 3 e a 1 ...a 8 6 l6a ( C! 6 6i62(a;)
+
f^C^]^)) .
(58)
Let us also mention in passing (from [39]) that the Eio coset action is not compatible with the addition of an eleven-dimensional cosmological constant in the supergravity action (an addition which has been proven to be incompatible with supersymmetry in [40]).
73
6. Conclusions We have reviewed the finding that the general solution of many physically relevant (bosonic) Einstein-matter systems, in the vicinity of a space-like singularity, exhibits a remarkable mixture of chaos and symmetry. Near the singularity, the behavior of the general solution is describable, at each (generic) spatial point, as a billiard motion in an auxiliary Lorentzian space or, after a suitable "radial" projection, as a billiard motion on hyperbolic space. This motion appears to be chaotic in many physically interesting cases including pure Einstein gravity in any space-time dimension D < 10 and the particular Einstein-matter systems arising in string theory. Also, for these cases, the billiard tables can be identified with the Weyl chambers of some Lorentzian Kac-Moody algebras. In the case of the bosonic sector of supergravity in 11-dimensional space-time the underlying Lorentzian algebra is that of the hyperbolic Kac-Moody group E\Q, and there exists some evidence of a correspondence between the general solution of the Einstein-three-form system and a null geodesic in the infinite-dimensional coset space E\Q/K(EIO), where K(Eio) is the maximal compact subgroup of Ew Acknowledgments It is a pleasure to thank Sophie de Buyl and Christiane Schomblond for their help in trimming the manuscript and in improving the figures. Appendix A. Kac-Moody Algebras A KM algebra Q(A) can be constructed out of a generalized Cartan matrix A, (i.e. an r x r matrix such that i) An = 2, a = 1, ...,r; ii) —Aij € N for i ^ j and Hi) A^ — 0 implies Aji — 0) according to the following rules for the Chevalley generators {hi, e%, ft}, i — 1,..., r: l e ii Jj\
=
\JXi,&j\ = [hiijj]
=
"ijili, J\.ijCj, ~-™-ijJj,
[hi,hj] — 0. The generators must also obey the Serre's relations, namely ( a d e i ) 1 - A y e j = 0,
(ad/O1^/;^,
74
and the Jacobi identity. Q(A) admits a triangular decomposition G(A) = n-@hen+,
(A.l)
where n_ is generated by the multicommutators of the form [fix, [/j2,...]], n + by the multicommutators of the form [e^, [ej2,...]] and h is the Cartan subalgebra. The algebras Q{A) build on a symmetrizable Cartan matrix A have been classified according to properties of their eigenvalues • if A is positive definite, Q{A) is a finite dimensional Lie algebra; • if A admits one null eigenvalue and the others are all strictly positive, Q(A) is an Affine KM algebra; • if A admits one negative eigenvalue and all the others are strictly positive, G(A) is a Lorentzian KM algebra. A KM algebra such that the deletion of one node from its Dynkin diagram gives a sum of finite or affine algebras is called an hyperbolic KM algebra. These algebras are all known; in particular, there exists no hyperbolic algebra with rang higher than 10. References 1. T. Damour and M. Henneaux, Phys. Rev. Lett. 85, 920 (2000) [hepth/0003139]; see also a short version in Gen. Rel. Gray. 32, 2339 (2000). 2. T. Damour and M. Henneaux, Phys. Lett. B488, 108 (2000) [hepth/0006171]. 3. T. Damour and M. Henneaux, Phys. Rev. Lett. 86, 4749 (2001) [hepth/0012172]. 4. T. Damour, M. Henneaux, B. Julia and H. Nicolai, Phys. Lett. B509, 323 (2001) [hep-th/0103094]. 5. T. Damour, S. de Buyl, M. Henneaux and C. Schomblond, JEEP 0208, 030 (2002) [hep-th/0206125]. 6. T. Damour, M. Henneaux and H. Nicolai, Phys. Rev. Lett. 89, 221601 (2002) [hep-th/0207267]. 7. T. Damour, M. Henneaux and H. Nicolai, Class. Quant. Grav. 20, R145 (2003) [hep-th/0212256]. 8. R. Arnowitt, S. Deser and C. W. Misner, The Dynamics Of General Relativity, gr-qc/0405109. 9. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 19, 525 (1970). 10. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Sov. Phys. JETP 35, 838 (1972).
75 11. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, Adv. Phys. 31, 639 (1982). 12. E.M. Lifshitz, I.M. Lifshitz and I.M. Khalatnikov, Sov. Phys. JETP 32, 173 (1971). 13. D.F. Chernoff and J.D. Barrow, Phys. Rev. Lett. 50, 134 (1983). 14. D.M. Chitre, Ph. D. Thesis, University of Maryland, 1972. 15. C.W. Misner, in D. Hobill et al., eds., Deterministic chaos in general relativity, p. 317 (Plenum, 1994) [gr-qc/9405068]. 16. T. Damour, M. Henneaux, A.D. Rendall and M. Weaver, Annales Henri Poincare3, 1049 (2002) [gr-qc/0202069]; 17. L. Andersson and A.D. Rendall, Commun. Math. Phys. 218, 479 (2001) [grqc/0001047]. 18. A.A. Kirillov, Sov. Phys. JETP 76, 355 (1993). 19. A.A. Kirillov and V.N. Melnikov, Phys. Rev. D52, 723 (1995) [grqc/9408004], 20. V.D. Ivashchuk, A.A. Kirillov and V.N. Melnikov, JETP Lett. 60, 235 (1994) [Pisma Zh. Eksp. Teor. Fiz. 60, 225 (1994)]. 21. V.D. Ivashchuk and V.N. Melnikov, Class. Quant. Grav. 12, 809 (1995). 22. V.G. Kac, Infinite Dimensional Lie Algebras, Third Edition (Cambridge University Press, 1990). 23. S. de Buyl and C. Schomblond, Hyperbolic Kac Moody algebras and Einstein billiards, hep-th/0403285. 24. B. Julia, Report LPTENS 80/16, Invited Paper Presented at the Nuffield Gravity Workshop, Cambridge, England, June 22 - July 12, 1980. 25. B. Julia, in Lectures in Applied Mathematics, AMS-SIAM, vol. 21 (1985), p. 355. 26. J. Brown, O.J. Ganor and C. Helfgott, M-theory and E\Q: Billiards, Branes, and Imaginary Roots, hep-th/0401053. 27. H. Lu, C.N. Pope and K.S. Stelle, Nucl. Phys. B476, 89 (1996) [hepth/9602140], 28. N.A. Obers, B. Pioline and E. Rabinovici, Nucl. Phys. B525, 163 (1998) [hep-th/9712084]. 29. T. Banks, W. Fischler and L. Motl, JHEP 9901, 019 (1999) [hep-th/9811194]. 30. P.C. West, Class. Quant. Grav. 18, 4443 (2001) [hep-th/0104081]. 31. I. Schnakenburg and P.C. West, Phys. Lett. B517, 421 (2001) [hepth/0107181]. 32. I. Schnakenburg and P.C. West, Phys. Lett. B540, 137 (2002) [hepth/0204207]. 33. F. Englert, L. Houart, A. Taormina and P. West, JHEP 0309, 020 (2003) [hep-th/0304206]. 34. F. Englert and L. Houart, JHEP 0405, 059 (2004) [hep-th/0405082]. 35. A.D. Rendall and M. Weaver, Class. Quant. Grav. 18, 2959 (2001) [grqc/0103102]. 36. E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Nucl. Phys. B195, 97 (1982). 37. G.F. Chapline and N.S. Manton, Phys. Lett. B120, 105 (1983).
76
38. E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B76, 409 (1978). 39. T. Damour and H. Nicolai, Eleven dimensional supergravity and the E\o/K(Eio) a-model at low Ag levels, invited contribution to the XXV International Colloquium on Group Theoretical Methods in Physics, 2-6 August 2004, Cocoyoc, Mexico; to appear in the proceedings, hep-th/0410245. 40. K. Bautier, S. Deser, M. Henneaux and D. Seminara, Phys. Lett. B406, 49 (1997) [hep-th/9704131].
GENERALIZED HOLONOMY IN M-THEORY
A. BATRACHENKO, M.J. DUFF, JAMES T. LIU and W.Y. WEN Michigan Center for Theoretical Physics Randall Laboratory, Department of Physics, University of Michigan Ann Arbor, MI 48109-1040, USA E-mail: {abat,mduff, jimliu,wenw}@umich. edu
In M-theory vacua with vanishing 4-form F^, one can invoke ordinary Riemannian holonomy H C Spin(10,1) to account for unbroken supersymmetries n = 1, 2, 3, 4, 6, 8, 16, 32. However, in the presence of non-zero f(4), Riemannian holonomy must be extended to generalized holonomy H C SL(32,R) to account for more exotic fractions of supersymmetry. The resulting number of M-theory vacuum supersymmetries, 0 < n < 32, is then given by the number of singlets appearing in the decomposition of the 32 of SL(32,R) under H C SL(32,R).
1. I n t r o d u c t i o n M-theory not only provides a non-perturbative unification of t h e five consistent superstring theories, b u t also embraces earlier work on supermembranes and eleven-dimensional supergravity. 1 It is regarded by m a n y as the dreamed-of final theory and has accordingly received an enormous amount of attention. It is curious, therefore, t h a t two of the most basic questions of M-theory have until now remained unanswered: i) What are the symmetries of M-theory? ii) How many supersymmetries can vacua of M-theory preserve? While t h e issue of hidden symmetries of M-theory is an i m p o r t a n t one in its own right, here we focus on the more direct task of counting supersymmetries preserved by any given M-theory vacuum state. In the supergravity limit, the equations of M-theory display the maximum number of supersymmetries N = 32, and so n, the number of supersymmetries preserved by a particular vacuum, must be some integer 0 < n < 32. Furthermore, physical arguments restricting maximum spin and spacetime dimension give us confidence t h a t this upper limit of 32 supersymmetries extends t o the full M-theory as well. To proceed, however, we content ourselves with working in the super77
78
gravity limit. In this case, in vacua with vanishing 4-form F( 4 ), it is well known that n is given by the number of singlets appearing in the decomposition of the 32 of Spin(l,10) under H C Spin(l,10) where H is the holonomy group of the usual Riemannian connection DM
=
9M
+
(1)
^AB-
I^M
This connection can account for vacua with n = 0, 1, 2, 3, 4, 6, 8, 16, 32. Vacua with non-vanishing F^ allow more exotic fractions of supersymmetry, including 16 < n < 32. Here, however, it is necessary to generalize the notion of holonomy to accommodate the generalized connection that results from a non-vanishing F^ VM = DM-
^(TMNpQR
- 86MTpQR)FNPQR.
(2)
As discussed in Ref. [2], the number of M-theory vacuum supersymmetries is now given by the number of singlets appearing in the decomposition of the 32 of Q under H c Q where Q is the generalized structure group and 7i is the generalized holonomy group. Discussions of generalized holonomy may also be found in Refs. [3,4]. In subsequent papers by Hull 5 and Papadopoulos and Tsimpis 6 it was shown that Q may be as large as SL(32,R) and that an M-theory vacuum admits precisely n Killing spinors if and only if SL(31 - n,R) K (n + l ) R ( 3 1 - n ) 2 U C SL(32 - n,R) K nR (32 -™\
(3)
i.e. the generalized holonomy is contained in SL(32 — n,R)ixnR ( 3 2 - ? l ) but is not contained in SL(31 - n,R) ix (n + l ) R ( 3 1 - n ) . Here we expand on the use of generalized holonomy as a means of classifying M-theory vacua, and provide some examples of n = 16 and n = 8 generalized holonomy groups. 2. Holonomy and supersymmetry The number of supersymmetries preserved by an M-theory background depends on the number of covariantly constant spinors, VMt
= 0,
(4)
called Killing spinors. It is the presence of the terms involving the 4-form F(4) in (2) that makes this counting difficult. Hence we first consider vacua for which F(4) vanishes. In this case, Killing spinors satisfy the first order integrability condition [DM, DN]Z = \RMNABTABe
= 0,
(5)
79
where RMN is the Riemann tensor. The subgroup of Spin(10,1) generated by this linear combination of Spin(10,1) generators TAB corresponds to the holonomy group H of the connection % . We note that the same information is contained in the first order Killing spinor equation (4) and second-order integrability condition (5). One implies the other, at least locally. The number of supersymmetries, n, is then given by the number of singlets appearing in the decomposition of the 32 of Spin(10,1) under H. In Euclidean signature, connections satisfying (5) are automatically Ricci-flat and hence solve the field equations when Fu\ = 0. In Lorentzian signature, however, they need only be Ricci-null so Ricci-flatness has to be imposed as an extra condition. In Euclidean signature, the holonomy groups have been classified.7 In Lorentzian signature, much less is known but the question of which subgroups H of Spin(10,1) leave a spinor invariant has been answered.8 There are two sequences according as the Killing vector VA = eT^e is timelike or null. Since v2 < 0, the spacelike VA case does not arise. The timelike VA case corresponds to static vacua, where H C Spin(10) C Spin(10,1) while the null case to non-static vacua where H C ISO(9) C Spin(10,1). It is then possible to determine the possible n-values and one finds n = 2, 4, 6, 8, 16, 32 for static vacua, and n = 1 2, 3, 4, 8, 16, 32 for non-static vacua. 9 ' 10 ' 11
2.1. Generalized
holonomy
In general we want to include vacua with F^ ^ 0. Such vacua are physically interesting for a variety of reasons. In particular, they typically have fewer moduli than their zero F^ counterparts. 12 Now, however, we face the problem that the connection in (2) is no longer the spin connection to which the bulk of the mathematical literature on holonomy groups is devoted. In addition to the Spin(10,1) generators TAB, it is apparent from (2) that there are terms involving TABC and TABCDE- In fact, the generalized connection takes its values in the Clifford algebra SL(32,R). Note, however, that some generators are missing from the covariant derivative. Denoting the antisymmetric product of k Dirac matrices by T^k\ the complete set of SL(32,R) generators involve { r ^ . r ^ . T ^ . r ^ . r ^ } whereas only { r ( 2 ) , r ( 3 ) , r ( 5 ' } appear in the covariant derivative. Another way in which generalized holonomy differs from the Riemannian case is that, although the vanishing of the covariant derivative of the spinor implies the vanishing of the commutator, the converse is not true, as discussed below
80
in section 2.2. This generalized connection can preserve exotic fractions of supersymmetry forbidden by the Riemannian connection. For example, M-branes at angles 13 include n = 5, 11-dimensional pp-waves 14,15 ' 16 ' 17 include n = 18, 20, 22, 24, 26, squashed N(l,l) spaces 18 and M5-branes in a pp-wave background 19 include n — 12 and Godel universes 20 ' 21 include n — 14, 18, 20, 22, 24. These various fractions of supersymmetry may be quantified in terms of generalized holonomy groups. Here generalized holonomy means that one can assign a holonomy Ti C Q to the generalized connection appearing in the supercovariant derivative V where Q is the generalized structure group. The number of unbroken supersymmetries is then given by the number of H singlets appearing in the decomposition of the 32 dimensional representation of Q under H C Q. For generic backgrounds we require that Q be the full SL(32, R) while for special backgrounds smaller Q are sufficient.5 To see this, we write the supercovariant derivative as VM = DM + XM
(6)
for some other connection DM and some covariant 32 x 32 matrix XM- If we now specialize to backgrounds satisfying XMe = 0,
(7)
then the relevant structure group is G C Q. Consider, for example, for the connection D arising in dimensional reduction of D = 11 supergravity. One can show2 that the lower dimensional gravitino transformation may be written
<% = b&
(8)
in terms of a covariant derivative A* = 3M + ^af}l«0
+ Qv,abTab + ±eiaeibekcd^ijkTabc.
(9)
Here j a are Spin(d — 1,1) Dirac matrices, while Ta are Spin(ll — d) Dirac matrices. In the above, the lower dimensional quantities are related to their D —
81
11 counterparts (EMA,^M dsfn) = A'^dsl
Qf = e ^ d ^ ,
,AMNP)
through
+ gijdy'dyf,
P»ij = ea{idllej)a,
ijk = Aijk.
(10)
The condition (7) is just <5A, = 0 where Aj are the dilatinos of the dimensionally reduced theory. In this case, the generalized holonomy is given by H C G where the various G arising in spacelike, null and timelike compactifications are tabulated in Ref. [2] for different numbers of the compactified dimensions. These smaller structure groups are also the ones appropriate to more general Kaluza-Klein compactifications of the product manifold type, i.e. without a warp factor.5 2.2. Integrability
conditions
Yet another way in which generalized holonomy differs from Riemannian holonomy is that, although the vanishing of the covariant derivative implies the vanishing of the commutator, the converse is not true. Consequently, the first order integrability condition alone may be a misleading guide to the generalized holonomy group H.22 To illustrate this, we consider Freund-Rubin 23 vacua with F(4) given by
where [i = 0,1,2,3 and m is a constant with the dimensions of mass. This leads to an AdS 4 x X7 geometry. For such a product manifold, the supercovariant derivative splits as VI1 = D„+ m 7M 75
(12)
and ^ m = A n - \mTrn,
(13)
and the Killing spinor equations reduce to V^{x)
=0
(14)
and
Vmv(y) = o.
(15)
82
Here e(x) is a 4-component spinor and rj(y) is an 8-component spinor, transforming with Dirac matrices 7^ and r m respectively. The first Killing spinor equation is satisfied automatically with our choice of AdS4 spacetime and hence the number of D = 4 supersymmetries, 0 < N < 8, depends upon the number of Killing spinors on X 7 . 2 4 They satisfy the first order integrability condition [Dm,T>n]v = \CmnabTabV
= 0,
(16)
where Cmnab is the Weyl tensor. Owing to this generalized connection, vacua with m ^ 0 present subtleties and novelties not present in the m = 0 case. 25 As an example, consider the phenomenon of skew-whiffing.26'27 For each Freund-Rubin compactification, one may obtain another by reversing the orientation of X7. The two may be distinguished by the labels left and right. An equivalent way to obtain such vacua is to keep the orientation fixed but to make the replacement m —> —m thus reversing the sign of F 4 . So the covariant derivative (13), and hence the condition for a Killing spinor, changes but the integrability condition (16) remains the same. With the exception of the round S7, where both orientations give N = 8, at most one orientation can have N > 0. The squashed S7 provides a non-trivial example: 28,26 the left squashed S7 has JV = 1 but the right squashed S7 has N = 0. Other examples are provided by the left squashed JV(1,1) spaces, 18 one of which has N = 3 and the other N = 1, while the right squashed counterparts both have N = 0. (Note, incidentally, that N = 3 i.e. n = 12 can never arise in the Riemannian case.) All this presents a dilemma. If the Killing spinor condition changes but the integrability condition (16) does not, how does one give a holonomy interpretation to the different supersymmetries? The resolution to this lies in the higher order integrability conditions. 25 ' 22 In particular, for FreundRubin vacua, a second order integrability condition may be written as [Dp, [Dm, Vn}}V = \{DpCmnabYab
+ 2mCmnpaTa)r,.
(17)
Note that the SO(7) generators Tab, augmented by presence of Ta, together close on SO (8). (This is also directly evident in the covariant derivative (13).29) Hence the generalized holonomy group satisfies H C SO(8). Furthermore, second order integrability distinguishes between left and right squashing, as (17) is no longer inert under m —> —m.
83
We now ask how the 8 of SO(8) decomposes under H. In the case of the left squashed S7, H = SO(7)~, whereupon 8 —*• 1 + 7 and TV = 1, but for the right squashed S7, H = SO(7) + , so that 8 -> 8 and N = 0. Prom the first order integrability condition alone, however, we would have incorrectly concluded that H = G2 C SO (7), for which 8 —> 1 + 7 and hence that both orientations give N = l.22 2.3. Higher order
corrections
Another context in which generalized holonomy may prove important is that of higher loop corrections to the M-theory Killing spinor equations. As shown in Ref. [30], higher loops yield non-Riemannian corrections to the supercovariant derivative, even for vacua with F^ = 0, thus rendering the Berger classification inapplicable. Although the Killing spinor equation receives higher order corrections, so too does the metric, ensuring, for example, that H = G2 Riemannian holonomy 7-manifolds still yield N = 1 in D — 4 when the non-Riemannian corrections are taken into account. This would require a generalized holonomy H for which the decomposition 8 —> 1 + 7 continues to hold. 3. Generalized holonomy for n = 16 We now turn to a generalized holonomy analysis of some basic supergravity solutions. These results were presented in Ref. [31], where we refer the reader to for additional details. We first note that the maximally supersymmetric backgrounds (n = 32), namely E 1 ' 1 0 , AdS 7 x S 4 , AdS4 x S7 and Hpp, all have trivial generalized holonomy, in accord with (3). However, only flat space may be described by (trivial) Riemannian holonomy. Somewhat more interesting to consider are the four basic objects of Mtheory preserving half of the supersymmetries (corresponding to n = 16). These are the M5-brane, M2-brane, M-wave (MW) and the Kaluza-KIein monopole (MK). The latter two have F(4) = 0 and may be categorized using ordinary Riemannian holonomy, with H C Spin(10,1). 3.1. The
M5-brane
The familiar supergravity M5-brane solution 32 may be written in isotropic coordinates as ds2 = Hs1/3dxl
+
Fijki = eijkimdmHr,,
H25/3dy2, (18)
84
where H$(y) is harmonic in the six-dimensional transverse space spanned by {y l L and e^/m = ±1- While the transverse space only needs to be Ricci flat, we take it to be E 5 , so as not to further break the supersymmetry. A simple computation of the generalized covariant derivative on this background yields \Tf}P+H-3'2diH,
ViL = dil-
Vi = di + § i y P+d,- In H - \T{5)di In H,
(19)
where P^ = ±(1 ± r< 5) ) is the standard 1/2-BPS projection for the M5brane and T^ = ^€ijkimX'i:'klm. All quantities with bars indicate tangent space indices. To obtain the generalized holonomy of the M5-brane, we examine the commutator of covariant derivatives. Denning MMN
= [VM,VN],
(20)
we find that MM„ = 0, so that the holonomy is trivial in the longitudinal directions along the brane. On the other hand, the transverse commutator Mij yields a set of Hermitian generators T^ = — ^TjjP^. It is easily seen that they generate the SO(5) algebra [Tij,Tki] = i(SikTji - duTjk - SjkTu + SjiTik).
(21)
As a result, the transverse holonomy is simply SO(5) + , where the + refers to the sign of the M5-projection. Turning next to the mixed commutator, MM», we see that it introduces an additional set of generators, K^i = T^P^. Since r^P^" = PgTp, it is clear that the K^ generators commute among themselves. On the other hand, commuting K^ with the SO(5)+ generators Tij yields the additional combinations K^ = T^P^ and K^j = r ^ j P 5 + . Picking a set of Cartan generators T 12 and T34 for SO(5)+, we may see that the complete set {K^, K^i, K^ij} has weights ±1/2. As a result, they transform as a set of 4-dimensional spinor representations of S0(5)+. We conclude that the generalized holonomy of the M5-brane is HM5 = SO(5)+K6M4(4).
(22)
Note that the additional commutations required to close the algebra arise naturally from the higher order integrability conditions.
85
3.2. The
M2-brane
Turning next to the M2-brane, its supergravity solution may be written as 3
ds^H^dxl
+
H^dy2,
H2
A similar examination of the commutator of generalized covariant derivatives, (20), for this solution indicates the presence of both compact generators Tij = — | r y P 2 + and non-compact ones K^ — T^jP^. Here, P£ = £(1 ± r<2)) where T™ = ± 6 ^ ^ is the M2-brane projection. Furthermore, the coordinates on (23) correspond to a 3/8 longitudinal/transverse split. Hence the transverse holonomy in this case is SO(8) + . To obtain the generalized holonomy group HM2, we must first close the algebra formed by Ty and K^. Upon doing so, we find the additional generators K^k = T ^ j P ^ . As in the M5 case, we may see that the set {KnifKpijk} form eight-dimensional representations of SO(8)+. Working out the precise representations, 31 yields the M2 generalized holonomy W M2 = SO(8)+ K 12K 2(8s) . 3.3. The
(24)
M-wave
We now turn to the pure geometry solutions. The wave (MW) is given by 33 ds2 = 2dx+dx~
+ Kdx+2
+ dy2,
(25)
where K(y) is harmonic on the nine-dimensional Euclidean transverse space E 9 . In a vielbein basis e + = dx+, e~ = dx~ + \Kdx+, el = dyl, the only non-vanishing component of the spin connection is given by w + t = \diK e+. Thus the gravitational covariant derivative acting on e is given by D+ = d+ + \diKT-Ti,
D- = d-,
Di = di.
(26)
The only non-vanishing commutator of covariant derivatives is given by M+i = -fadjKT-Ti,
(27) l
so we may identify the generalized holonomy generators as T = r _ I V Since Tt = 0, these nine generators are mutually commuting, and the MW generalized holonomy is HMW
=
K •
(28) 16
In addition to being a subgroup of SL(16,M) K 16R , this may also be viewed as a subgroup of ISO (9) appropriate to backgrounds with a null Killing vector.
86
3.4. The
M-monopole
The final basic M-theory object we consider is the Kaluza-Klein monopole, which is given by the Euclidean Taub-NUT solution 34 ds2 = dx\ + H(dr2+r2dfll)
+ H~l{dz - qcos9d
(29)
where dfi2. = d92 + sin2 9 d
(30)
4. Some n — 8 examples Having looked at the basic objects of M-theory, we now turn to intersecting configurations preserving fewer supersymmetries. 35 ' 36,37 While large classes of intersecting brane solutions and configurations involving to branes at angles have been constructed, we will only examine some of the simple cases of orthogonal intersections yielding n = 8. 4.1. Branes with a
KK-monopole
It has often been noted that the basic supergravity p-brane solutions are not restricted to having only flat Euclidean transverse spaces. This indicates, in particular, that the M5 and M2 solutions of (18) and (23) demand only that the transverse space spanned by {y} is Ricci flat. Of course, this Ricci flat manifold must still be supersymmetric in order to preserve some fraction of supersymmetry. A simple example would be to replace E 4 with a Taub-NUT configuration in four of the transverse directions to the brane. For the M5 case, the resulting M5/MK solution has the form38 ds2 = H~1/3dxl
+ H*/3[dy2 + H6(dr2 + r2dQ22) + H^l{dz
- q6 cos9d<j))2}. (31) Here, the M5-brane is delocalized along the y direction, so the harmonic functions have the form H5 = l+q
87
embedding of the self-dual connection in SO(4) leads to explicit SU(2) generators T ^ K ) = - § r a S P + where P± = ± ( l ± r 1 2 3 4 ) and a, b,... = 1 , . . . , 4. On the other hand, as shown in section 3.1, the 50(5)+ generalized holonomy of M5 in the transverse directions involve the P§ projection, and is generated by T^ = - § I \ j P 5 + , where i, j , . . . = 1 , . . . 5. As a result, the transverse holonomy of this M5/MK configuration arises as the closure of
7f 5) andTi b MK) Working this out, we obtain the transverse holonomy group S0(5)+ x SU(2)_ where ± refers to the embedding inside the D structure group S0(5)+ x S0(5)_. The additional M5 mixed commutator generators {K^,K,j_i, K^ij} now transform under both S0(5)+ and SU(2)_. The generalized holonomy of this M5/MK configuration is HMS/MK = [S0(5)+ x SU(2)_] x 6 R 2 ( 4 - 1 )+( 4 - 2 ).
(32)
For the M2-brane, the eight-dimensional transverse space may be given a hyper-Kahler metric, 38 which is generically of holonomy Sp(2). However, we only consider the product of two independent Taub-NUT spaces, with holonomy Sp(l) x Sp(l). Provided both are oriented properly with the M2, this yields a single additional halving of the supersymmetries, leading to n = 8. The transverse holonomy of this solution corresponds to the embedding S0(8) x SU(2) x SU(2) C S0(8) x S0(4) x S0(4) c S0(8) x SO(8) C SO(16), where SO(16) is the D structure group. The complete generalized holonomy group is «M2/MK/MK
= [S0(8) x SU(2) x SU(2) x 3K (8s ' 2 ' 2) ] x eR 2 * 8 " 1 ' 1 ).
(33)
With only a single Taub-NUT space, the generalized holonomy is instead H M 2 / M K = [S0(8) x SU(2) x 3M2<8-2)] x eR 2 ^ 8 ' 1 ). 4.2.
Branes
(34)
with a wave
For solutions with an extended longitudinal space, it is possible to turn on a wave in a null direction along the brane. We consider the M2/MW and M5/MW combinations, both of which preserve a quarter of the original supersymmetries. For the M2/MW combination, the supergravity solution is given by 36 ds2 = H22/3(2dx+dx~
F+-Zi = di±-. tl2
+Kdx+2+
dz2) +
H\/2,dy2,
(35)
88
Here, both K and H2 are harmonic on the eight-dimensional overall transverse space; the wave is delocalized along the z direction. If H2 is turned off, the solution reverts to the MW solution of (25), however with dependence on only eight of the nine directions transverse to the wave. The resulting holonomy would be R 8 . Combining this with the M2 generalized holonomy, (24), must yield a larger group that is nevertheless contained in SL(24,R) x 8R 24 . The generalized holonomy algebra is formed by the closure of the MW algebra, generated by T-T^, and the M2 algebra, generated by — ^T-QP^ and Tp^P^ where fi denotes one of the longitudinal coordinates, +, — or z. This closure has been worked out in Ref. [31], and the corresponding generalized holonomy group is W M 2/MW = [SO(8) x SL(16,R) x R (8 ' 16) ] x
8R(8>i)+(i,i6)_
(36)
The generalized holonomy analysis for the M5/MW solution 36 ds2 = H~1/3(2dx+dx-
+Kdx+2+
dz42) +
H2/3dy2,
Fijkl = CijlclmdmHs,
(37)
is similar. Here the functions H$ and K are harmonic on the fivedimensional overall transverse space. This corresponds to a superposition of a M5-brane with a delocalized wave, where the latter has R 5 holonomy. Closing the holonomy algebra over the M5 and MW generators yields the generalized holonomy H M 5 /MW = [SO(5) X SU*(8) X 4R<4'8>] X 8R2(4,1)+2(1,8)_
(3g)
Note that SU*(8) ~ SL(4,HF), and the latter is built out of multiple copies of the five-dimensional real Clifford algebra Cl(5,0)_|_ ^ GL(2,H). 4.3.
Other
examples
Additional pure geometry backgrounds may be constructed by combining a wave with a Taub-NUT space. An n = 8 example is given by 36 ' 39 ds2 = dx+dx~ + K dx+2 + dy'52 + H6(dr2 + r2dti2,) + H^1 (dz - q6 cos 6 d<j))2, (39) where K = q^/r + qy/y3 and HQ = 1 + q§/r. Since the transverse space is a direct product of E 5 with Taub-NUT, the generalized holonomy has the direct product form HMW/MK = R 5 x (SU(2) x R2^).
(40)
89
Finally, there are numerous examples of overlapping or intersecting brane configurations involving multiple M2 and/or M5 branes. Various fractions of supersymmetry may be preserved by placing branes at appropriate angles. Here, we only consider the orthogonal intersection of M2 and M5 on a string, given by 36 ' 37 ds2 = H;2/3H~1/3dxl
+ H^H^dw,2
+
H-2/3H2^3dz2 +H^H25^dy42,
Fnuzi = eiivdi—, Fijkz = €ijkidiH5. (41) rl2 The full holonomy algebra is obtained by the closure of the M5 and M2 holonomies, given by (22) and (24), respectively. A slight complication arises in that the individual generators work on different relative transverse directions for the M5 and M2 branes. By taking the non-compact generators of one of the branes (e.g. the 12R2(8s) for the M2) and commuting with the transverse holonomy generators of the other (in this case the 50(5)+ for the M5) we end up filling ,up all of SL(24,1R). As a result, we find that the M2/M5 generalized holonomy fills all of the maximally allowed case for n = 8, namely WM2/M5 = SL(24, R) K 8R 24 .
(42)
5. Discussion As we have seen, the generalized holonomy of M-theory solutions takes on a variety of guises. Our results are summarized in table 1. We make note of two features exhibited by these solutions. Firstly, it is clear that many generalized holonomy groups give rise to the same number n of supersymmetries. This is a consequence of the fact that while H must satisfy the condition (3), there are nevertheless many possible subgroups of SL(32 - n,R) tx nK ( 3 2 _ n ) allowed by generalized holonomy. Secondly, as demonstrated by the plane wave solutions, 31 knowledge of H by itself is insufficient for determining n; here H = R 9 , while n may be any even integer between 16 and 26. What this indicates is that, at least for counting super symmetries, it is important to understand the embedding of H in Q. In contrast to the Riemannian case, different embeddings of TC yield different possible values of n. Although this appears to pose a difficulty in applying the concept of generalized holonomy towards classifying supergravity solutions, it may be
90 Table 1. Generalized holonomies of the objects investigated in the text. For n = 16, we have H C SL(16,R) x 16IRi6, while for n = 8, it is instead T-l C SL(24,E) X 8R 2 4 . The plane wave solutions are discussed in Ref. [31]. n 32
18,.. . ,26 16
Background
Generalized holonomy
Ei,io
{1}
AdS 7 x S 4 AdS 4 x S7 Hpp plane waves M5 M2 MW MK M5/MK M2/MK/MK M2/MK M2/MW M5/MW MW/MK M2/M5
R9 SO(5) x 6R 4 W SO(8) x 12R 2 ( 8 ») R9 SU(2) [SO(5) x SU(2)] x e R ^ 4 ' 1 ' ^ 4 . 2 ) [SO(8) x SU(2) x SU(2) x 3R<8».2'2>] x 6R 2 ( 8 -i'i> [SO(8) x SU(2) x 3R2(8<"2>] x 6R2(8<"D [SO(8) x SL(16, K) X R(8-16>] X 8R( 8 - 1 )+( 1 - 16 ) [SO(5) x SU*(8) x 4R<4'8>] x 8R 2 ( 4 .i>+ 2 (!. 8 ) E 5 x (SU(2) x R2<2>) SL(24, R) x 8R 24
possible that a better understanding of the representations of non-compact groups will nevertheless allow progress to be achieved in this direction. While the full generalized holonomy involves several factors, the transverse (or D) holonomy is often simpler, e.g. SO(5) for the M5 and SO(8) for the M2. The results summarized in table 1 are suggestive that the maximal compact subgroup of H, which must be contained in SL(32 — n,IR), is often sufficient to determine the number of surviving supersymmetries. For example, the M2/MK/MK solution may be regarded as a 3/8 split, with a hyper-Kahler eight-dimensional transverse space. In this case, the D structure group is SO(16), and the 32-component spinor decomposes under SO(32) D SO(16) D SO(8) x SU(2) x SU(2) as 32 -> 2(16) -> 2(8,1,1) + 2(1,2,2) + 8(1,1,1) yielding eight singlets. Similarly, for the M5/MW intersection, we consider a 2/9 split, with the wave running along the two-dimensional longitudinal space. Since the D structure group is SO(16) x SO(16) and the maximalcompact subgroup of SU*(8) is USp(8), we obtain the decomposition 32 -> (16,1) + (1,16) - • 4(4,1) + (1,8) + 8(1,1) under SO(32) D SO(16) x SO(16) D SO(5) x USp(8). This again yields n = 8. Note, however, that this analysis fails for the plane waves, as R 9 has no compact subgroups. A different approach to supersymmetric vacua in M-theory is through the technique of G-structures. 40 Hull5 has suggested that G-structures may
91
be better suited to finding supersymmetric solutions whereas generalized holonomy may be better suited to classifying them. In any event, it would be useful t o establish a dictionary for translating one technique into the other. Ultimately, one would hope to achieve a complete classification of vacua for the full M-theory. In this regard, one must at least include the effects of M-theoretic corrections to the supergravity field equations and Killing spinor equations and perhaps even go beyond the geometric picture altogether. It seems likely, however, t h a t counting supersymmetries by the number of singlets appearing in the decomposition 32 of SL(32, R) under H C SL(32, E ) will continue t o be valid.
Acknowledgments We have enjoyed useful conversations with Lilia Anguelova, Jianxin Lu, J u a n Maldacena, Paul de Medeiros, Malcolm Perry, Hisham Sati, Dimitrios Tsimpis and Oscar Varela. This work was supported in p a r t by the US Department of Energy under grant DE-FG02-95ER40899.
References 1. For a review, see M. J. Duff, The world in eleven dimensions: supergravity, supermernbranes and M-theory (I.O.P. Publishing, Bristol, 1999). 2. M. J. Duff and J. T. Liu, Hidden spacetime symmetries and generalized holonomy in M-theory, Nucl. Phys. B674, 217 (2003) [hep-th/0303140]. 3. M. J. Duff and K. Stelle, Multimembrane solutions of d = 11 supergravity, Phys. Lett. B253, 113 (1991). 4. M. J. Duff, M-theory on manifolds of G^ holonomy: the first twenty years, hep-th/0201062. 5. C. Hull, Holonomy and symmetry in M-theory, hep-th/0305039. 6. G. Papadopoulos and D. Tsimpis, The holonomy of the supercovariant connection and Killing spinors, JEEP 0307, 018 (2003) [hep-th/0306117]. 7. M. Berger, Sur Jes groupes d'holonomie homogene des varietes a connexion affine et des varietes riemanniennes, Bull. Soc. Math. France 83, 225 (1955). 8. R. L. Bryant, Pseudo-Riemannian metrics with parallel spinor fields and non-vanishing Ricci tensor, math.DG/0004073. 9. B. S. Acharya, J. M. Figueroa-O'Farrill and B. Spence, Planes, branes and automorphisms. I: Static branes, JHEP 9807, 004 (1998) [hep-th/9805073]. 10. B. S. Acharya, J. M. Figueroa-O'Farrill, B. Spence and S. Stanciu, Planes, branes and automorphisms. II: Branes in motion, JHEP 9807, 005 (1998) [hep-th/9805176]. 11. J. Figueroa-O'Farrill, Breaking the M-waves, Class. Quant. Grav. 17, 2925 (2000) [hep-th/9904124].
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12. M. J. Duff, B. E. W. Nilsson and C. N. Pope, Compactification of D = 11 supergravity on K3 x T3, Phys. Lett. B129, 39 (1983). 13. N. Ohta and P. K. Townsend, Supersymmetry of M-branes at angles Phys. Lett. B418, 77 (1998) [hep-th/9710129]. 14. J. Michelson, (Twisted) toroidal compactification of pp-waves, Phys. Rev. D66, 066002 (2002) [hep-th/0203140]. 15. M. Cvetic, H. Lu and C. N. Pope, M-theory pp-waves, Penrose limits and supernumerary supersymmetries, Nucl. Phys. B644, 65 (2002) [hepth/0203229]. 16. J. P. Gauntlett and C. M. Hull, pp-waves in 11-dimensions with extra supersymmetry, JHEP0206, 013 (2002) [hep-th/0203255]. 17. I. Bena and R. Roiban, Supergravity pp-wave solutions with 28 and 24 supercharges, Phys. Rev. D67, 125014 (2003) [hep-th/0206195]. 18. D. N. Page and C. N. Pope, New squashed solutions of d = 11 supergravity, Phys. Lett. B147, 55 (1984). 19. H. Singh, M5-branes with 3/8 supersymmetry in pp-wave background, hepth/0205020. 20. J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis and H. S. Reall, All supersymmetric solutions of minimal supergravity in five dimensions, Class. Quant. Grav. 20, 4587 (2003) [hep-th/0209114]. 21. T. Harmark and T. Takayanagi, Supersymmetric Godel universes in string theory, Nucl. Phys. B662, 3 (2003) [hep-th/0301206]. 22. A. Batrachenko, J. T. Liu, O. Varela and W. Y. Wen, Higher order integrability in generalized holonomy, hep-th/0412154. 23. P. G. O. Freund and M. A. Rubin, Dynamics of dimensional reduction, Phys. Lett. B97, 233 (1980). 24. E. Witten, Search for a realistic Kaluza-Klein theory, Nucl. Phys. B186, 412 (1981). 25. P. van Nieuwenhuizen and N. Warner, Integrability conditions for Killing spinors, Commun. Math. Phys. 93, 277 (1984). 26. M. J. Duff, B. E. W. Nilsson and C. N. Pope, Spontaneous supersymmetry breaking by the squashed seven-sphere, Phys. Rev. Lett. 50, 2043 (1983). 27. M. J. Duff, B. E. W. Nilsson and C. N. Pope, Kaluza-Klein supergravity, Phys. Rep. 130, 1 (1986). 28. M. A. Awada, M. J. Duff and C. N. Pope, N = 8 supergravity breaks down to N = 1, Phys. Rev. Lett. 50, 294 (1983). 29. L. Castellani, R. D'Auria, P. Pre and P. van Nieuwenhuizen, Holonomy groups, sesquidual torsion fields and SU(8) in d = 11 supergravity, J. Math. Phys. 25, 3209 (1984). 30. H. Lu, C. N. Pope, K. S. Stelle and P. K. Townsend, Supersymmetric deformations of G2 manifolds from higher-order corrections to string and Mtheory, JEEP 0410, 019 (2004) [hep-th/0312002]. 31. A. Batrachenko, M. J. Duff, J. T. Liu and W. Y. Wen, Generalized holonomy of M-theory vacua, hep-th/0312165. 32. R. Gueven, Black p-brane solutions of D = 11 supergravity theory, Phys. Lett. B276, 49 (1992).
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33. C. M. Hull, Exact pp-wave solutions of ll-dimensional supergravity, Phys. Lett. B139, 39 (1984). 34. S. K. Han and I. G. Koh, N = 4 supersymmetry remaining in Kaluza-Klein monopole background in D = 11 supergravity, Phys. Rev. D 3 1 , 2503 (1985). 35. G. Papadopoulos and P. K. Townsend, Intersecting M-branes, Phys. Lett. B380, 273 (1996) [hep-th/9603087]. 36. A. A. Tseytlin, Harmonic superpositions of M-branes, Nucl. Phys. B475, 149 (1996) [hep-th/9604035]. 37. J. Gauntlett, D. Kastor and J. Traschen, Overlapping branes in M-theory, Nucl. Phys. B478, 544 (1996) [hep-th/9604179]. 38. J. P. Gauntlett, G. W. Gibbons, G. Papadopoulos and P. K. Townsend, Hyper-Kaehler manifolds and multiply intersecting branes, Nucl. Phys. B500, 133 (1997) [hep-th/9702202]. 39. E. Bergshoeff, M. de Roo, E. Eyras, B. Janssen and J. P. van der Schaar, Intersections involving monopoles and waves in eleven dimensions, Class. Quant. Grav. 14, 2757 (1997) [hep-th/9704120]. 40. J. P. Gauntlett and S. Pakis, T i e geometry of D = 11 Killing spinors, JHEP 0304, 039 (2003) [hep-th/0212008].
ORIENTIFOLDS, BRANE COORDINATES AND SPECIAL GEOMETRY
R. D ' A U R I A Dipartimento di Fisica, Politecnico di Torino C.so Duca degli Abruzzi, 24, 1-10129 Torino, Italy E-mail: riccardo. dauriatSpolito. it S. F E R R A R A CERN,
Theory Division, CH 1211 Geneva 23, Switzerland; INFN, Laboratori Nazionali di Frascati, Italy; University of California, Los Angeles, USA E-mail: sergio.ferrara3cern.ch M. T R I G I A N T E
Dipartimento di Fisica, Politecnico C.so Duca degli Abruzzi, 24, 1-10129 E-mail: mario. trigianteQpolito.
di Torino Torino, Italy it
We report on the gauged supergravity analysis of Type IIB vacua on K3 x T2 /Z2 orientifold in the presence of £>3 — £>7-branes and fluxes. We discuss supersymmetric critical points correspond to Minkowski vacua and the related fixing of moduli, finding agreement with previous analysis. An important role is played by the choice of the symplectic holomorphic sections of special geometry which enter the computation of the scalar potential. The related period matrix A/- is explicitly given. The relation between the special geometry and the Born-Infeld action for the brane moduli is elucidated.
1. I n t r o d u c t i o n We report on the four dimensional gauged-supergravity description of a class of K3 x T 2 / Z 2 orientifolds1'2 in the presence of D3-D7 space-filling branes and three-form fluxes.3'4 In superstring and M-theory compactifications, which, in the absence of fluxes have an iV-extended local supersymmetry, the low-energy dynamics is encoded in an effective supergravity theory with a certain number of matter multiplets, which describe the degrees of freedom of both bulk and 94
95
brane excitations. In particular in N = 2 supergravity in D = 4 vector and hypermultiplets are described by special and quaternionic geometries, describing the moduli space of these theories. When fluxes are turned on the effective supergravity theory undergoes a mass deformation which is encoded in a "gauged supergravity", whose general scalar potential is given in section 3. This potential is completely fixed by the underlying scalar geometry, the period matrix, the special geometry of the vector multiplets and by the Killing vectors of the gauged isometries of the quaternionic and special geometries. A correct choice of the period matrix (explicitly given in Appendix B) and of the Killing isometries, would then allow to reproduce the flux vacua with any number N = 2,1,0 of rigid supersymmetry in flat space. 5 " 14 This is indeed the case and the agreement is found, for the particular choices of fluxes considered, with the compactification analysis of Tripathy and Trivedi.3 These results are especially relevant because the supergravity effective potential can be further generalized to incorporate other perturbative or non-perturbative results 15 which may further stabilize the other moduli and lead to satisfactory inflationary cosmologies.16"20 Prom the point of view of the four-dimensional N = 2 effective supergravity, open string moduli, corresponding to D7 and D3-brane positions along T 2 , form an enlargement of the vector multiplet moduli-space which is locally described, in absence of open-string moduli, by:4
fsuq.m
vwr).
x
fsu(i,i)\ vwr)t
fsu(i,i)\
l~uorJu •
m (1J
where s, t, u denote the scalars of the vector multiplets containing the K3volume and the R-R four-form on KS, the T 2 -complex structure, and the IIB axion-dilaton system, respectively3-: S = C( 4 )
t = — 922 U = Cm)
a
iVol(tf 3 ), y/detg i
, 922
ie*\
(2)
We notice that in [22] the imaginary parts of u and t were chosen to be positive. This however is inconsistent with the positivity domain of the vector kinetic terms which requires s,t,u to have negative imaginary parts. Indeed Im(s) and Im(u) appear as coefficients in the kinetic terms of the D7 and -D3—brane vectors.
96
where the matrix g denotes the metric on T2. When D7-branes moduli are turned on, what is known is that SU(1, l)s acts as an electric-magnetic duality transformation 21 both on the bulk and D7-brane vector field-strengths, while the SU(1,1) U acts as an electricmagnetic duality transformation on the D3-vector field-strengths. Likewise the bulk vectors transform perturbatively under SU(1, l ) u x SU(1, l)t while the D3-brane vectors do not transform under SU(1, l ) s x SU(1, l ) t and the D7-brane vectors do not transform under SU(1, l)u x SU(1, l)t. All this is achieved starting from the following trilinear prepotential of special geometry: 22 F(s,t,u,xk,yr)
= stu--sxkxk
--uyryT
,
(3)
where xk and yr are the positions of the D7 and D3-branes along T 2 respectively, k = 1 , . . . , n 7 , r = 1 , . . . , TI3, and summation over repeated indices is understood. This prepotential is unique in order to preserve the shiftsymmetries of the s, t, u bulk complex fields up to terms which only depend on x and y. The above prepotential gives the correct answer if we set either all the k x or all the yr to zero. In this case the special geometry describes a symmetric space: (mhH) V U(l)
x
)
s
fSU(M)\
l
,, X
SO(2,2 + n 7 ) SO(2)xSO(2 + n 7 ) '
V
SO(2,2 + n 3 )
U(l) J u S O ( 2 ) x S O ( 2 + n 3 ) '
U
'
W
fc_
^
X
'°'
(5)
For both x and y non-vanishing, the complete Kahler manifold (of complex dimension 3 + n 3 + n 7 ) is no longer a symmetric space even if it still has 3 + n 3 + n 7 shift symmetries15. Note that for xk — 0 the manifold is predicted as a truncation of the manifold describing the moduli-space of T 6 / Z 2 N = 4 orientifold in the presence of D3-branes. The corresponding symplectic embedding was given in [24]. For yr — 0 the moduli-space is predicted by the way SU(1,1) S acts on both bulk and D7 vector fields. Upon compactification of Type IIB theory on T 2 , the D7-brane moduli are insensitive to the further K3 compactification and thus their gravity coupling must be the same as for vector multiplets coupled to supergravity in D = 8. Indeed if 2 -I- n vector °The prepotential in Eq. (3) actually corresponds to the homogeneous not symmetric spaces called L(0, n-j, 713) in [23]. We thank A. van Proeyen for a discussion on this point.
97
multiplets are coupled to N = 2 supergravity in D = 8, their non-linear er-model is: 25 ' 26 SO(2,2 + n) . v ¥ + SO(2)xSO(2 +' n) - l .
(6)
Here R + denotes the volume of T 2 and the other part is the second factor in (4). Note that in D = 8, N = 2 the R-symmetry is U(l) which is the U(l) part of the D — 4, N = 2 U(2) R-symmetry. The above considerations prove Eq. (4). Particular care is needed 27 when the effective supergravity is extended to include gauge couplings, as a result of turning on fluxes in the IIB compactification.28 The reason is that the scalar potential depends explicitly on the symplectic embedding of the holomorphic sections of special geometry, while the Kahler potential, being symplectic invariant, does not. In fact, even in the analysis without open string moduli, 4 it was crucial to consider a Calabi-Visentini basis where the SO(2,2) linearly acting symmetry on the bulk fields was SU(1, l ) u x SU(1, l) t . 2 9 - 3 0 In the case at hand, the choice of symplectic basis is the one which corresponds to the Calabi-Visentini basis for yr — 0, with the SU(1,1) S acting as an electric-magnetic duality transformation, 4 but it is not such basis for the D3-branes even if the xk = 0. Indeed, for xk = 0, we must reproduce the mixed basis used for the T 6 /Z2 orientifold 31,32 in the presence of D3-branes found in [24]. We note in this respect, that the choice of the symplectic section made in [20] does not determine type IIB vacua with the 3-form fluxes turned on. It does not correspond in fact to the symplectic embedding discussed in [4], [24] and [27]. The problem arises already in the absence of branes. This report is organized as follows: in section 2 we review the gauged supergravity description of the model and briefly discuss the moduli stabilization of different vacua. In section 3 we discuss the scalar potential in the presence of D3-D7 moduli. In section 4 we consider the relation between the N = 2 special geometry corresponding to the D3-D7 system and the Born-Infeld action, taking into account the Chern-Simons terms describing the couplings among bulk and brane moduli. The final section is devoted to conclusions. Appendices A and B contain some relevant formulae for the scalar potential and the period matrix A/AS
98
of the special geometry describing the bulk-brane coupled system of vector multiplets.
2. N = 2 and N = 1 supersymmetric cases. 2.1. N = 2 gauged
supergravity
We consider the gauging of N = 2 supergravity with a special geometry given by Eq. (3). Let us briefly recall the main formulae of special Kahler geometry. The geometry of the manifold is encoded in the holomorphic section Q = (XA, Fs) which, in the special coordinate symplectic frame, is expressed in terms of a prepotential Jr(s,t,u,xk,yr) = F(XA)/(X0)2 = A F(X /X°), as follows: n = {XA, FA = dF/dXA).
(7)
In our case T is given by Eq. (3). The Kahler potential K is given by the symplectic invariant expression: K = -\og[i(XAFA-FAXA)\
.
(8)
In terms of K the metric has the form g^ = dtdjK. The matrices UAS and 7^AS a r e respectively given by: UAE = eK ViXAVjXE
giJ = - i l m ( A ^ ) - 1 - eKX~AXs
TJAZ = hMI o ( / - l ) ' r , where ft =
fe^
,
; hA» = (^ffj
. (9)
For our choice of T, K has the following form: K = - •log[-8 iog[-8(Im(s) (Im(sjIm(t)Im(u) \m{t)Lm{u) - - ~ Im(s) (Im(a;)fc ) 2 1
2Im(u)(lm(yyy)],
(10)
with Im(s), lm(t), Im(u) < 0 at xk = yr = 0. The components XA, Fs of the symplectic section which correctly describe our problem, are chosen by performing a constant symplectic change of basis from the one in (7) given
99
in terms of the prepotential in Eq. (3). The symplectic matrix is
(A-B\
U A) V2
/ 1 0 0 0 0 0 0-1-10 -10 0 0 0 0 0 1-10 0 0 0 0 \/2 \ 0 0 0 0 0
0\ 0 0 , 0 0 y/2j
/0 1 0 0 0 0\ 1B =
V2
000000 0 10000 000000 000000
(11)
\0 0 0 0 0 0/
The rotated symplectic sections then become 1
u
x = 7=2' 1
2
1
x = V2 k
X
k
Ft
xl
•),
~k\2
(1+t; r
•
)
,
X6
t t —u
~7f
r
=x , X = y , s ( 2 - 2 i u + (a; fc ) 2 )+u(7y r ) 2
F0 = F2
k\2 (x«)
-tu-
- 2 s (t + u) + (yr)2
2V2
2%/2
(2 + 2 i i - {xk)2) - • \yr)X\2 2V2 Fr = -uyr -sx
2 s {-t + u) + (yr)2
•F«
=
2A/2
(12)
A
Note that, since dX /ds = 0 the new sections do not admit a prepotential, and the no-go theorem on partial supersymmetry breaking 33 does not apply in this case. As in [4], we limit ourselves to gauge shift-symmetries of the quaternionic manifold of the K3 moduli-space. Other gaugings which include the gauge group on the brane will be considered elsewhere. We will also consider particular assignments of the gauge couplings which give X2 = X 3 — 0, i.e. t = u = —i. Other choices for the gauge couplings, allowing u ^ -i are possible and we shall discuss some cases here. 2.2. N — 2 supersymmetric
critical
points
In the sequel we limit our analysis to critical points in flat space. The N — 2 critical points demand VXA — 0. This equation does not depend on the special geometry and its solution is the same as in [4], i.e. 52, #3 / 0, g0 = gx = 0 and e™ = 0 for a = 1, 2, were the Killing vectors gauged by
100
the fields A^ and A3^ are constants and their non-vanishing components are k2 = 32 along the direction qu — Ca=1 and k% = g$ along the direction ca,m = 1,2,3 &nd a = I,..., 19 denote qU = Ca=2 T h e 2 2 fidds cm^ the Peccei-Quinn scalars. The vanishing of the hyperino-variation further demands: k\XA
= 0 =• X2=X3
= 0 <s> t = u, l+t2
= ^ - .
(13)
Hence for N = 2 vacua the D7 and D3-brane positions are still moduli while the axion-dilaton and T 2 complex structure are stabilised. 2.3. N = 1,0 critical
points
The N = 1 critical points in flat space studied in [4] were first obtained by setting go, gi 7^ 0 and g2 = g3 = 0, with k% = g0 along the direction qu = Cm=l
a n d ku = gi
a l o n g t h e d i r e c t i o n qu =
Cm=2
Constant Killing spinors. By imposing Se2 f = 0 for the variations of the fermionic fields / we get the following: From the hyperino variations: 5e2CAa = 0 => e^ = 0 m = l , 2 ; a = l , . . . , 1 9 , <5e2 C = ()=>• vanishing of the gravitino variation.
(14)
The gravitino variation vanishes if: + ig1X1=0.
S22 = -goX°
(15)
From the gaugino variations we obtain: 5t2 {\l)A = 0 =» e% VXA (diXA + (diK)XA)axA2
= 0,
(16)
the second term (with diK) gives a contribution proportional to the gravitino variation while the first term, for i = u, t, xk respectively gives: ~g0daX0 ~g0dtX°
+ ig1duX1 +
= Q,
ig1dtX1=0,
-godxkX°
= 0, (17)
r
for i = y the equation is identically satisfied. From the last equation we get xk = 0 and the other two, together with S22 = 0 give u = t = - i , 9o = 0 i .
101
So we see that for N = 1 vacua the D7-brane coordinates are frozen while the D3-brane coordinates remain moduli. This agrees with the analysis of [3]. If go jt gx the above solutions give critical points with vanishing cosmological constant but with no supersymmetry left. More general N = 1,0 vacua can be obtained also in this case by setting 92, 53 7^ 0. The only extra conditions coming from the gaugino variations for N = 1 vacua is that e ^ 1 , 2 — 0. This eliminates from the spectrum two extra metric scalars eg =1 ' 2 and the Ca=i,2 axions. These critical points preserve N = 1 or not depending on whether |go| = |pi| or not. We can describe the N = 1 —* N = 0 transition with an N = 1 noscale supergravity 34,35 based on a constant superpotential and a non-linear sigma-model which is U ( l , l + n3) SO(2,18) l U ( l ) x U ( l + n3) SO(2) x SO(18) ' ' where the two factors come from vector multiplets and hypermultiplets, respectively. This model has vanishing scalar potential, reflecting the fact that there are not further scalars becoming massive in this transition. 4 We further note that any superpotential W(y) for the D3 brane coordinates would generate a potential 36 term eK Kyy dyWdvW,
(19)
which then would require the extra condition dyW = 0 for a critical point with vanishing vacuum energy. The residual moduli space of K3 metrics at fixed volume is locally given by ( j SO(17) • We again remark that we have considered vacua with vanishing vacuum energy. We do not consider here the possibility of other vacua with nonzero vacuum energy, as i.e. in [20].
2.4. More general
vacua
There are more general critical points defined by values of t, u different form — i and depending on ratios of fluxes. Let us give an instance of this for the N — 2 preserving vacua. Consider the situation with generic flux / P A , p = (m, a), A = 0 , . . . , 3, which corresponds to the charge-couplings: V „ C = 0/iC*, + / * A ^ .
(21)
102
For a N = 2 vacuum, for the vanishing of the gravitino and gaugino variations, we need V% = 0, where Vl=u>lkl=u>$fl.
(22)
From the hyperino variations we have k\XK
= fAXK
= 0.
(23)
We take A = 2, 3 with / | ) 3 ^ 0 for p = a, (a = 1 . . . , 19) and / £ = 0 otherwise. The hyperino variation then is: fa2X2 a
a
Setting f 2 = af
+ fa3X3=0.
(24)
we obtain
3
f%(aX2+X3)
= 0,
(25)
that is Xl=l+tu-^=-a
=
( 2 6 )
-J^-
The condition V\ = 0 on the other hand implies eI„/°2,3=0, a
(27)
a
but since / 2 = a / 3 then the above equation is equivalent to the following single condition exafa2
= 0,
(28)
namely it fixes only one triplet of metric moduli. This vacuum preserves N = 2 supersymmetry with one massive vector multiplet corresponding to a combination of A^ and Az. Moreover condition (26) fixes the T2 complex structure modulus in terms of the axiondilaton and the xk moduli of the D7 brane coordinates. Note that in the previous solution 22 X2 = X3 = 0, u = t, t2 = - 1 + ^ y - and xk were still unfixed. For a = 0 or oo we get X3 or X2 vanishing which corresponds to the example given in [3]. 3. The potential The general form of the N = 2 scalar potential is: V = 4 eKhuvkuAkvE XA ~XE + eKgij k\k3s XA +eK(UAE
-3eKXAXE)VZT>Z,
I
r
(29)
103
where the second term is vanishing for abelian gaugings. Here huv is the quaternionic metric and k^ the quaternionic Killing vector of the hypermultiplet cr-model. The scalar potential, at the extremum of the e^ scalars, has the following formc: 3
14>0K
V = 4e^e
1 2
/
A
Y,(9A) \X f + Ugl+g\){t - t ) ( ( u - « ) AA=0
Z
V
l(xk- -xk)2\ 2 (t--i) ) (30)
From the above expression we see that in the N = 2 case, namely for go = 9i — 0, the potential depends on yr only through the factor eK and vanishes identically in yr for the values of the t, u scalars given in (13), for which X2 = X3 = 0. If g0 or g\ are non-vanishing (N = 1, 0 cases) the extremisation of the potential with respect to xk, namely dxkV — 0 fixes xk = 0. For xk = 0 the potential depends on yT only through the factor eK and vanishes identically in yr for t = u — —i.
4. Special coordinates, solvable coordinates and B.I. action The prepotential for the spatial geometry of the £>3 — Dl system, given in (3), was obtained in [22], by using arguments based on duality symmetry, four dimensional Chern-Simons terms coming from the p-brane couplings as well as couplings of vector multiplets in D = 4 and D = 8. A similar result was advocated in [37,38] by performing first a K3 reduction to D = 6 and then further compactifying the theory to D = 4 on T2. The subtlety of this derivation is that the naive Born-Infeld action derived for D5 and D9 branes in D = 6 gives kinetic terms for the scalar fields which, at the classical level, are inconsistent with TV = 2 supersymmetry. This is a consequence of the fact that anomalies are present in the theory, as in the D — 10 case. The mixed anomaly local counterterms are advocated to make the Lagrangian N = 2 supersymmetric in D = 4. Therefore the corrected Lagrangian, in the original brane coordinates is highly non-polynomial. In fact the original Born-Infeld, Chern-Simons c
Note that there is a misprint in Eq. (5.1) of Ref. [4]. The term e 2 * eK g0gi(X0Xi X I X Q ) is actually absent.
+
104
naive (additive) classical scalar action \ds' + crddr\2 (s' - s')2
s = s
Idu' +
tfdtfl2
(u' - u')2
\dryr
xi = al+tbl
\t' ddr + dd
+
(s' - s') {f - V) WdV + daH2 («' - u') (*' - V)
; u' =u-1-bixi ; yr = 6'
\dt ' | 2 (f - t')2 '
; t' = t,
tdr ,
(31)
has a metric which was shown 38 to be Kahler with Kahler potential K
-log
r
:-.r\2
{s-s)(t-t)--(y
- l o g (u-u)(t-F)--(xi-xi)2 = -logYSK-log(l
+
+ log(t - 1 )
XA —^),
(32)
ISG
where X« =
(x*-x*)2(yr-yr) 4(t-t)
r,r\2
YSK = (a - s)(t - t)(u -u)-l-{u-
u)(yr - f)2
- l ( s - 5) (a* - x')2 , (33)
where here and in the following summation over repeated indices is understood. Therefore the correction to the scalar metric in the brane coordinates is:
dpd?AK = d p 0 ? log(l + •£*-)
(34)
ISG
It is clear that the classical brane coordinates are not good "supersymmetric" coordinates, in that the corrected action is not polynomial in them. Prom the fact that the combined system is a homogeneous space, we indeed expect that suitable coordinates exist such that the quantum corrected (N = 2 supersymmetric) action has a simple polynomial dependence on them, including the interference term. Such coordinates do indeed exist and allow to write the combined Born-Infeld action and supersymmetric Y$K differs by a factor —t from the special geometry formula obtained from the prepotential in (3).
105
counterterms, in a manifest supersymmetric way. Modulo field redefinitions, these coordinates reduce to the standard brane coordinates when either the D3 or the £>7-branes are absent, in which cases the homogeneous space becomes a symmetric space. This parametrization in terms of "supersymmetric" coordinates, corresponds to the solvable Lie algebra description of the manifold first introduced by Alekseevski, 39,40 which we shall discuss in what follows. In Alekseevski's notation the manifold under consideration is of type K(n3,n?) which can be written as: K{n3,n7)
Y±,Z±),
= W(ga,ha,
d i m ( y ± ) = n 3 ; dim(Z ± ) = n7 ,
(35)
where 123 and n7 denote the number of D3 and D7-branes respectively. Our identification of the scalar fields with solvable parameters is described by the following expression for a generic solvable Lie algebra element: Vaha + 6tgt + 0ugu + 6sg3 + yr±Y±
Solv = { Y,
+ zl±Zf}
,
a—t,u,s
6t = et+yr+yr-+zi+zi-,
(36)
where (yr+ ,yr~) and (zl+, zl~) are related to the real and imaginary parts of the D3 and D7-branes complex coordinates along T 2 . The non trivial commutation relations between the above solvable generators are: [huY±}=1-Y±
; [/ l t ) Z ± ] = i z
[h.,Y±]=±±Y± [g.,Y-]
[gu,Z~] = Z+ ,
[Yr , Yf] = 5rs gt ; [ha,ga\=
,
[hu,Z±]=±±Z±,
;
=y+ ;
+
±
[Zf, Zj] = 8ij gt ; r, s = 1 , . . . , n 3 i, j = 1 , . . . , n7 ,
9a ; a = t,u,s.
We exponentiate representative: L = e<>s9s
the solvable
(37) algebra
using the following
eV'-Y- e„-+y+ eeu9u eJ-z;
e^zf
e§t 9t eV"ha
_
coset-
(38)
The order of the exponentials in the coset representative and the particular parameter 6t used for gt, have been chosen in such a way that the axions 6S, 8t, 6U, yr+, zl+ appear in the resulting metric only covered by
106
derivatives. The metric reads: ds2 = (d<pa)2 + e- 2 «" Uet + ^deu(z~)2 +e-2^d92u
+ e-2,fi°d62 + e'^-^
+e-Vt-'p°{dyr+
+ d6syr-)2
+
+ \d6s{y-f
{dzi+ + dduZ^f
+
dyrA
e~^+^{dzi-)2
e-f'+^idy1-)2,
n-j
(z+) 2 = ^ > < + ) 2 ;
+ z*"dz i+ + yr~
«3
{y+)2^(yr+)2.
i=l
(39)
r=l
Identifying the axionic coordinates 6S, 9t, 6U, yr+, zl+ with the real part of the special coordinates s, t, u, yr, xl, and comparing the corresponding components of the metric one easily obtains the following relations between the solvable coordinates and the special coordinates:
t = 6t~l- (e*» +\e^ x^z^+'-e^z'-
{z-f + \e*" (y~)2
; yr=yr++l-e*°yr-.
(40)
Note that the classical B-I+C-S action (31), with no interference term in the D3 (c, d) and Dl (a, b) brane coordinates is still described by a homogeneous manifold spanned by the following 2 713 + 2 717 + 6 isometries: u —> e u u ; Su = UQ + alQb%, s —> eXs s ; 5s = so + cr0dr , t -> eXt t ; St = t0, cr —» e "2 c r ; 5cr =todr , a* -» e _!k 2 _t a* ; <5a* = OQ + t0 bl, V -> e ^ * 6* ; 56* = bQ .
(41)
The underlying homogeneous space is generated by the following rank 3 solvable Lie algebra {T£, Tg, T£, T^, hs, ht, hu, gs, 9t, 9u} whose non triv-
107
ial commutation relations are: [TZ,TI}=5rsgs,
ra,n]=5^9u-
[n, gt] = Ta ; [Tdr, gt] = Tl, [ha, ga]= 9a,
a =
lh„TS) = ±TS; [K,Ti]
=
l
-Tl-
s,t,u, [h3,Tr]=l-T;,
[hu,Ti]
l
=
\ht,Trd} = -\Trd;
\huTl\
[huTl]=-l-Ti;
[ht,12\=±7i,
=
-Ti,
\Tl, (42)
where the nilpotent generators have been labeled by the corresponding axionic scalar fields. This space is not a subspace of the original quaternionic space, but it becomes so if we set either a, b = 0 and exchange the role of s and t or if we set c, d = 0 and exchange the role of u and t. The amazing story is that the coordinates in D = 4 corresponding to the supersymmetric theory, deform this space into an other homogeneous space generated by the isometries in (37) which corresponds to an ./V = 2 special geometry. The relation between the solvable Lie algebra generators {T^, T6*, T£, TJ, hSl ht, hu , gs, gt, gu} corresponding to the classical coordinates and the solvable generators {Y±, Z±, ha, ga} corresponding to the "supersymmetric" coordinates is the following:
Tl = Zi+ ; Tl = fr- , where Y and Z are the generators with opposite grading with respect to Y and Z respectively. It can be shown that in the manifold K(n3,ri7), Y or Z are isometries only if n-j = 0 or 713 = 0 respectively. Indeed in these two cases the manifold is symmetric and each solvable nilpotent isometry has a "hidden" counterpart with opposite grading. Otherwise the manifold spanned by the classical coordinates and the manifold parametrized by the "supersymmetric" ones are in general different. 5. Conclusions The present investigation allows us to study in a fairly general way the potential for the 3-form flux compactification, in presence of both bulk
108
and open string moduli. In absence of fluxes the D3, D7 dependence of the Kahler potential is rather different since this moduli couple in different ways to the bulk moduli. Moreover, in the presence of 3-form fluxes which break N — 2 —> TV = 1,0 the D7 moduli are stabilised while the D3 moduli are not. For small values of the coordinates xk, yr the dependence of their kinetic term is (for u = t = — i), —(dflyrd!*yr)/lm(s) for the D3-brane moduli, and — {dllxkd^xk) for the D7-brane moduli. This is in accordance with the suggestion of [16]. Note that the above formulae, at x = 0, u = t = — i are true up to corrections 0(Im(y) 2 /Im(s)), since y and s are moduli even in presence of fluxes. The actual dependence of these terms on the compactification volume is important in order to further consider models for inflatons where the terms in the scalar potential allow to stabilise the remaining moduli. Finally, we have not considered here the gauging of compact gauge groups which exist on the brane world-volumes. This is, for instance, required 41 ' 42 ' 20 in models with hybrid inflation.43 This issue will be considered elsewhere. Acknowledgments Work supported in part by the European Community's Human Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time, in which R. D'A. is associated to Torino University. The work of S.F. has been supported in part by European Community's Human Potential Program under contract HPRN-CT-2000-00131 Quantum Space-Time, in association with INFN Frascati National Laboratories and by D.O.E. grant DE-FG0391ER40662, Task C. Appendix A. Some relevant formulae We are interested in gauging the 22 translations in the coset SO(4,20)/(SO(3,19) x 0(1,1)). Let us denote by L the coset representative of SO(3,19)/SO(3) x SO(19). It will be written in the form:
L=(il+ef)hn
-eT
I m
T
a
"" , V (l + e T e)2 /
(A.1) v
'
where e = {e a}, e = {e m} , m = 1,2,3 and a = 1,...,19, are the coordinates of the manifold. The 22 nilpotent Peccei-Quinn generators are
109
denoted by {Zm, Za} and the gauge generators are: tA = fmAZm
+ haAZa ,
(A.2)
the corresponding Killing vectors have non vanishing components: k™ = fmA and k\ = haA- The moment maps are: VXA = V2 (e+ {lTx)*m
fmA + e* ( L " 1 ) * , h%) ,
(A.3)
where
(A.4)
The potential has the following form: V = 4 e 2 * (fmA
fmE
+ 2 eame\
fmA fns
+2 e 2 * {UAE - 3 LA LE) (fmA +2 [(1 + e e T ) 5 ] V « fmiA
+ haA haE) LA LE
fmE
+ eamean
fmA
haE) + enaenb haA hb^j
fns .
(A.5)
In all the models we consider, at the extremum point of the potential in the special Kahler manifold the following condition holds: (UAS — 3 LA LE) |Q fm(A ha£) = 0 . As a consequence of this, as it is clear from (A.5), the potential in this point depends on the metric scalars e™ only through quadratic terms in the combinations emahaA and eamfmA. Therefore V is extremised with respect to the e™ scalars once we restrict ourselves to the moduli defined as follows: moduli:
emahaA
= eamfmA=0.
(A.6)
The vanishing of the potential implies {UAE - LA LE) ,0 fm{A
fmE) + 2 {LA LE)
|Q
h\A haE) = 0 -
(A.7)
Furthermore, one may notice that, as in [4], the following relations hold in all the models under consideration: (UAE - LA LE)
|0
r
i A
fmE) = (LA LE)
|0
h\A haE) = 0.
(A.8)
Our analysis is limited to the case in which the only non-vanishing / and h constants are: f \ = 9o I / 2 i = 9i ; hl2 = 92 ; h23 = g3 ; h2+k3+k h
2+n +r
?
3+n7+r
= gl.
= g\ , (A.9)
110 A p p e n d i x B . T h e m a t r i x A/" Using the special geometry formula (9) it is possible to compute the matrix TVAE f ° r any choice of the symplectic section, including those cases for which no prepotential exists. For the sake of simplicity we will suppress the indices k and r in xk and yr by considering the case 713 = nj = 1. Moreover we will express the complex coordinates in terms of their real and imaginary parts: s = si+is2;
t = ti+it2;
u = ui+iu2;
yi+iy2 (B.l) Let the Dl and D 3 brane vectors correspond to the values A = 4 , 5 respectively. We shall list below the independent components of the real and imaginary p a r t s of M: 1
Re(A0o,o = si-
-uiyi 2
+
(— 1 + t\ui
x = xi+\x2;
u2 ( - 2 + 2 t i u i -xi2) — 5 2*2^2 — 2:2 — ^xi2)
(2tiu22
y =
2/12/2
+ x2 (—2u2X\ + u\x2))
V22
(-2t2u2+x22)2 2 + 4 (h + M-,I ) W22/2) 2/1 (. - 2 i 2- U2- „2/1 . +x2 yi ---. L Re(Af)o,l = * -;—g + 2 8*2 "2 — 4X2 8*2 "2 - 4X2^ 2 2/2 2 (2u2 (2-2ti (ti+2ui)+xi2^ + 4 (i 2 +ui)
u2x1x2^
2 2
A{-2t2u2+x2 ) (2-2ui
( 2 t i + « i ) + x i 2 ) z 2 2 2/2 2 4(-2t2W2 + K 2 2 ) 2
1 «2 (2*1 MI - x i 2 J yi2/2 2 Re(A0o,2 = ^"12/1 + 2 lyl -2t2u2 + x22 m u i - 5 xi2)
(2t\u22
+x2
(-2u2xi
+ u\
x2)j
2/2
2
(-2i2U2+ai22)2 2/1 ( - 2 t 2 « 2 2 / i + a ; 2 2 2 / l + 4 (
8*2'"2 — 4X2 2
-(u2
( « 2 (2 + 2 i ! 2 - 4 * i M i + a ; i 2 ) + 2 ( - i i + w i ) 2:12:2) 2/22) 2 ( - 2 t 2 w 2 + a;22)2
, 2 ^2 + 2 u i ( - 2 i i + u i ) + a:i 2 ) x2 2 „2/2
4(-2t2M2 + ^22)2
Ill
Z12/2 ( ~ 4 <2 U22 2 / l + 2 < i U2 2 2/2 + " i £2 2 2/2) Re(A0o,4
1
V^(-2i2"2 + " 2 ^ 2 2/2 ("-2 s i 2:2 2/1 +
(
2 _
V2(-- 2 £2 U2
2 i i u\ + 3 x ]
+ X22)2
+ 2>12J 2/2
2t2U\ U2J/1 — Ml XQ ! 2 2/1 + «2 ( 2 - 2 t i u i Re(A0o,5
%/2 ( 2 i 2 U 2 - Z 2 2 ) ( t i + w i ) ( 2 u 2 2 + a;2 2 ) 2/22
Re(AA)i,i = «1
(-2£2U2+x22)2
2/1 ( 2 £2 " 2 2/1 - 2J22 2 / 1 - 4 (£1 + u i ) U2 2/2J Re(A0i,2
8*2 " 2 - 4 Z 2 2 w 2 ( " 2 ( 2 + 2 t i 2 + 4 t i u i - X ! 2 ) - 2 (£1 + u i ) xi 2:2) 2/2 2 2(-2£2«2+a;22)2 (2 + 2 « i (2£1 + u i ) - x i 2 )
x22y22
4 ( - 2 £ 2 u 2 + X22)2 (2uiu22-t1X22) Re(AT)i,3
2/22
(-2£2«2+a;22)2 \2u2
£1 + 2 (£1 +u\)
Re(A0i,4
J 2/2
V / 2 ( - 2 £ 2 w 2 + X22)2 A/2 [2t2U2yi
-X22yi
Re(Af)i,5
Re(AA) 2 , 2
112x2 + x\X2
- 2 (£1 + t i i ) W22/2J
4£2«2 — 2X22 1 U
u2 ^2 + 2 £1 MI - xi2J
2
-«1-2 12/1
+
2 £2^2 - X2 2
f 1 + £iui - \ x \ \
\2t\U22
+X2
2/12/2
2
{—2u2X\ + U1X2)) 2/2
( - 2 £ 2 u 2 + X22)2 2/1 (2 £2 u 2 2/1 - Z2 2 2 / 1 + 4 ( - £ 1 + u i ) U2 2/2J Re(A^) 2 ,3 =
+
8 £2^2 — 4a;2 2 2
2
U2 ( u 2 ( - 2 + 2 £ x - 4 £ i MI + x i ) + 2 ( - £ 1 + wi) xi 2(-2t2u2 f2 + 4 £ i w i - 2 u i
2
-xi
2
)
4(-2£2M2+a;22)2
+
X2 2 2/2 2
X22)2
x2) yi
112 a;i2/2 ( - 4 i 2 « 2 yi + 2 f i « 2 2/2 + wia;2 2/2) -p -5 + V 2 ( - 2 t 2 u 2 + a;22)
Re(A^)2,4 =
U2X2V2 (2x1x22/1 - 3 xi2 2/2 + 2 (2/2 + *i «i 2/2) J U2 + X2 ) 2
+U2 (2 + 2 t i ^ i — a! 2 J y2
—2 £2 n\U2V\ + u\X2 yi Re{Af)2 5
'
=
V2 (2t2U2-X2>) 2
Re(A^)3,3 = - s i +
2
— 2
(h-Ui) (2U2 -X2 ) —± - ^ (-2t2"2+X22)
V2
( 2«2 2 xi + 2 (-£1 + ui) «2 »2 - x\ X22) J/22
M^hA
=
-p-
^2
^ 2 ( - 2 t 2 U 2 + Z2 2 ) \[2 ( 2 t 2 u 2 2 / i - £ 2 2 2 / l + 2 {-t\ + w i ) «2 2/2) Re(AT)3,5 = — ^ 5—5 *3:1X2 3/22 —2 (-2i2U2+z22) _2U2JC12/2 . — o
-.A „ Re(7V)4,4 = - s i R
Re(AT)4,5-
4M2
- 2 £2 1*2+£2
Re(AT)5,5 = - U l
(B.2)
As far as Im(A/") is concerned, its independent entries are:
, ,»A S2(4 + 4 ( t l 2 + t 2 2 ) ( U l 2 + U 2 2 ) + X 1 4 + X 2 4 + 2 X 1 2 ( 2 + X22)) , lm(A/) 0 ,o = — —5 V 8t 2 U 2 - 4X2^ S2t2 ( —2mxiX2 + M2 (xi — X2) ( l l + X 2 )) 2 t 2 « 2 — X2 2 $2*1 (2ll2XlX 2 + Ul (2 + Xi 2 - X 2 2 ) ) 2<2«2 — X2 2 2 ( 2 t m 2 2 + X2 ( - 2 M 2 X I + 1x1x2)) V2
yi_ (
2 yi2yi+
-2t2u2+x2*
(4«i 2 (ui 2 - u 2 2 ) + 4t 2 2 («i 2 + u 2 2 ) - 4tiU! (2 + X!2) + (2 + x i 2 ) 2 ) y22
-u2
4(-2t2«2 + x22)2 X2 ( t 2 " l 2 X 2 + M2 ( - 2 t l U a a i + ( l l l l l + t2M2 + XI 2 ) X 2 )) j/2 2 (-2t2U2 +X22)2 -X2
3
(4tllXl + U2X2) i/2 2 4 ( - 2 t 2 u 2 + X2 2 ) 2
113
Im(7^) 0 ,i =
s2 ( - 2 ( t 2 + u 2 ) x i x 2 + ui ( - 2 + 2ti 2 + 2t 2 2 - x i 2 + x 2 2 ) ) —-^———x —+ it2U2
— 2X2
s 2 tl ( - 2 + 2«i 2 + 2« 2 2 - x i 2 + x 2 2 ) 4t2U2 — 2X22 2 V2 {t2 U1U2V2 ~ tlU23V2 + t2 (2«232/l + U2X22yi (-2t2u2 + x 2 2 ) 2 - ((ti +u1)u2
UlX22y2))
( - 2 + 2tim - x i 2 - x 2 2 ) y 2 2 ) 2(-2t2U2+X22)2
X2 (2tl 2 2 + X 2 2 ) j/2 (X2V1 - Xij/2) 2(-2t2«2 + x22)2 r ,KK Im(Af)o,2 =
S2 ( - 4 + 4 t l 2 ( u i 2 + « 2 2 ) + 4 t 2 2 ( u i 2 + « 2 2 ) + X J 4 + 2 X 1 2 X 2 2 + X 2 4 ) x —.—: 8t2U2 — 4X2 i S2 ( t i ( 2 M 2 X I X 2 + Ml ( x i 2
- X22))
+ t2 ( 2 M I X I X 2 + M2 (~X X 2
2*21*2 — X2
+ X22)))
2
V\ ( - 2 t 2 M 2 2 i / l + 4tiU 2 2 3/2 + 2uiX 2 2 j/2 + "2X 2 (x 2 j/l - (u2 ( 4 - 4 ( t i 2
4xij/2))
4t 2 U 2 — 2X2 2 + t 2 2 ) m 2 + 4 ( t i — t 2 ) («i + <2)«2 2 + 4 t m i X ! 2 - Xj4) 2
4(-2t2U2+X2 ) X2 ((*2 ( " l 2 + U22)
, +
2
+ M 2 Xl 2 ) X2 + t i l i 2 ( - 2 M 2 X 1 + MlX 2 )) j / 2 2 (-2t2«2 +X22)2
X 2 3 ( 4 m X l + M 2 X 2 ) j/2 2
4(-2i2u2+x22)2 , ,.« s 2 ( 2 ( - t 2 + « 2 ) x i x 2 + u i (2 + 2 t i 2 + 2 t 2 2 + x i 2 - X 2 2 ) ) Im(A0o,3 = r: * ^—o — + 4t 2 u 2 - 2x2^ - (»2ti (2 + 2m 2 + 2« 2 2 + x i 2 - x 2 2 )) + (2u 2 2 - x 2 2 ) ylV2 it2u2 — 2x 2 2 2 - (m ( - 2 t 2 x 2 + u2 (2 + 2ti 2 + 2t 2 2 + x i 2 + x 2 2 )) y22) 2{-2t2u2 + x 2 2 ) 2 2 - ((xix 2 ( - 2 « 2 + x 2 2 ) - t m 2 (2 + 2u!2 - 2M 2 2 + x t 2 + x 2 2 )) y 2 2 ) 2(-2t2«2+X22)2 3 .... s2(xi -2(t2u1+t1u2)x2 + x1(2-2t1u1 +2t2u2+x22)) t 2 ^ ' V2(2t2u2 -X2 ) y2 ( - 4 t 2 M 2 2 X 2 y i + 2tt2X2 3 1/l + 2tlM 2 2 X2j/2 + " l X 2 3 y 2 ) + V2(-2t 2 «2 + x 2 2 ) 2 2 2 2 u2xi (2 - 2 t m i + x i - 2x 2 ) y2 V2(-2t2u2 + x 2 2 ) 2 . ,.,, V2 (-2t2u22y1 + 2t\u22y2 + u\x22y2 + u2x2 (x2yi - 2xi2/2)) Im(A/)o,5 = — ^—5 4t 2 u 2 - 2x 2 2 2 2 2 2 s2 ( t i + t2 + 2 t i u i + u i + « 2 + x 2 2 ) T ,,,, Im(JV)i,i = — + 2t2u2 — x2* ( ( - t 2 2 - (tl + Ml)2) U2 + U23 + (t 2 + U2) X 2 2 ) V22 (-2t 2 «2 + Z2 2 ) 2
y22)
114
S 2 ( - 2 (t 2 + U2) XlX 2 + Ul (2 + 2 t l 2 + 2 t 2 2 - X!2 + X2 2 ))
, ,s
41 2 u 2 - 2 x 2 2 s 2 t i (2 + 2 « i 2 + 2 u 2 2 - x i 2 + x 2 2 ) 4 t 2 « 2 — 2X2 2 2
' l / 2 + t 2 (2M2 3 J/1 + U2^2 2 j/1 ~ MiX 2 2 M 2 )) J/2 (t 2 MlU2y2 - *lU233/2 (-2t2M2 + ^ 2 2 ) 2 - ("21/2 ( ( - * ! - Ul) (2 + 2 t i m - X l 2 - X 2 2 ) 1/2 + 2u 2 X2 (X 2 1/1 - X1J/2))) 2(-2t2u2 + x 2 2 ) 2 3
- {X2 y2 (^21/1 ~ Elite)) 2(-2t2u2 + x 2 2 ) 2 _ 32 ( t l 2 + t 2 2 ~ U l 2 - U 2 2 ) (2t 2 U2 ~ X2 2 ) ~ («2 ( t j 2 + t 2 2 - Ml 2 + U 2 2 ) ~ t 2 X 2 2 ) J/ 2 2 (-2t3«2 +X22)2 ,.,. Im(A/)i,4 =
\ / 2 s 2 ( ( t l + U i ) x i + (t 2 + U 2 ) x 2 ) , — ; +
T
— 2t 2 U2 + X2^ ( - 2 (tl + Ml) U 2 Xl + 2M22X2 + X 2 3 ) 1/22 %/2(-2t 2 M 2 + X 2 2 ) 2 V2(2u22+X22)t/2 j —f 4t2U2 — 2X2 2
, ,.A lmAOi,5 = Ira(A/")2,2
_ s2 (4 + 4 t t 2 ( m 2 + M 2 2 ) + 4 t 2 2 ( m 2 + u 2 2 ) - 4 x i 2 + x i 4 + 2 x i 2 x 2 2 + x 2 4 ) 8t2M 2 — 4X2 2 «2 ( t l (2u 2 XlX2 + Ml ( - 2 + X l 2 - X 2 2 ) ) + t 2 (2uiXlX 2 + U2 ( ~ X l 2 + X 2 2 ) ) ) 2t2U2 — X22 ui (2t 2 M2 2 i/i - 4 t i u 2 2 i / 2 - 2 u i x 2 2 j / 2 + u 2 x 2 ( - ( x 2 y i ) + 4 x n / 2 ) ) 4t2U2 — 2X2 2 -M2 ( 4 t l 2 (Ul - M2) (Ul + U2) + 4 t 2 2 (Ml 2 + U 2 2 ) ~ (2 + 4tlU X - X 2 ) ( - 2 + X l 2 ) ) U22 4(-2t2u2+x22)2 2
2
2
- X 2 (8tlU2 Xi - 4 (t 2 ( M I + U 2 ) + U2 ( t l U l + X l 2 ) ) X2 + 4 u i X l X 2 2 + U 2 X 2 3 ) J/22 4(-2t2M2+X22)2 Im(7V) 2 ,3 _
S2 ( 2 t i 2 M i + 2 ( - t 2 + U 2 ) x i X 2 + M l ( - 2 + 2 t 2 2 + 1 1 2 - X 2 2 ) + t l ( 2 - 2 M I 2 - 2 u 2 2 - Xl 2 + X 2 2 ) ) 4t2U2 — 2X2 2 2
M2 ( 2 t 2 m + 2 t m 2
2
+ (ti - m ) (2 + 2tiMi - x i 2 - x 2 2 ) ) j/2 2 2(-2t2u2 + x22)2
- JV2 ( x 2 ( - 2 U 2 2 + X 2 2 ) (x2yi ,.,, s2 ( x i 3 Im(A/) 2 ,4 = T
- Xij/ 2 ) + 2t 2 (2«2 3 j/l - M2X22j/l +
U1X22y2)))
2(-2t2U2+X22)2 - 2 ( t 2 u i + tiU2) x 2 + x i ( - 2 - 2 t i u i + 2t 2 M 2 + x 2 2 ) ) -=— — + V^(2t2M2 - X22)
V2M2X2J/2 ( - 2 t 2 U 2 M l + X 2 2 J/1 + tlM 2 J/ 2 ) (-2t2U2 + X 2 2 ) 2 ( M I X 2 3 + u 2 x i ( - 2 - 2 t m i + x i 2 - 2 x 2 2 ) ) 1/22 %/2(-2t2u2+X22)2
115
Im(Af) 2 , 5 Im(A03,3
\ / 2 (2t2U22yi
- 2tlM 2 2 j/2 - MlX22j/2 + «2X2 ( - (x 2 j/l) + 2 x i j / 2 ) ) 4t2«2 - 2x22
82 ( t i 2 + t 2 2 - 2 t m i + m 2 + « 2 2 - x 2 2 ) 2 t 2 « 2 — X2 2 (u2 ( ( t l ~ Ml) 2 + (t 2 ~ " 2 ) (<2 + " 2 » + ( - t 2 + U2) X 2 2 ) j / 2 2 (-2t2U2 + X22)2
Im(./V)3,4
•«/2s2 (tlXl - MlXl + (t 2 - Ua) X 2 ) 2t2u2 — x22 ( —2tlU 2 Xl + 2tt 1 U 2 Xl — 2U 2 2 X2 + X2 3 ) J/22 \/2{-2t2u2
Im(7V) 3 , 5 Im(AT)4,4 Im(Af)4,5 Im(AT)5,5
V2(-2«2
2
+ x22)2
2
+X2 )J/2
4t2«2 — 2x22 s 2 ( 2 t 2 " 2 - x 2 2 ) ( 2 t 2 u 2 + 2 x x 2 + x 2 2 ) + 2 u 2 ( - i i + x 2 ) (xi + x 2 ) j/2 2 (~2t2u2 2M2X2j/2 2t2U2 — X22 u2
+x22)2
(B.3)
References 1. G. Pradisi and A. Sagnotti, Phys. Lett. B216, 59 (1989); A. Sagnotti, Phys. Rept. 184, 167 (1989). 2. J. Polchinski and Y. Cai, Nucl. Phys. B296, 91 (1988); J. Dai, R.G. Leigh and J. Polchinski, Mod. Phys. Lett. A 4 , 2073 (1989). 3. P.K. Tripathy and S.P. Trivedi, JHEP 0303, 028 (2003). 4. L. Andrianopoli, R. D'Auria, S. Ferrara and M.A. Lledo, JHEP 0303, 044 (2003). 5. T.R. Taylor and C. Vafa, Phys. Lett. B474, 130 (2000). 6. P. Mayr, Nucl. Phys. B593, 99 (2001); P. Mayr, JHEP 0011, 013 (2000). 7. G. Curio, A. Klemm, D. Lust and S. Theisen, Nucl. Phys. B609, 3 (2001). 8. M. Berg, M. Haack and B. Kors, Nucl. Phys. B669, 3 (2003). 9. J. Louis and A. Micu, Nucl. Phys. B635, 395 (2002). 10. G. Curio, A. Klemm, B. Kors and D. Lust, Nucl. Phys. B620, 237 (2002). 11. G. Dall'Agata, JHEP 0111, 005 (2001). 12. G.L. Cardoso, G. Curio, G. Dall'Agata, D. Lust, P. Manousselis and G. Zoupanos, Nucl. Phys. B652, 5 (2003); G. L. Cardoso, G. Curio, G. Dall'Agata and D. Lust, JHEP 0310, 004 (2003). 13. K. Becker, M. Becker, K. Dasgupta and P.S. Green, JHEP 0304, 007 (2003). 14. D. Lust, S. Reffert and S. Stieberger, hep-th/0406092. 15. S. Kachru, R. Kallosh, A. Linde and S.P. Trivedi, Phys. Rev. D68, 046005 (2003). 16. J.P. Hsu, R. Kallosh and S. Prokushkin, JCAP 0312, 009 (2003). 17. K. Dasgupta, C. Herdeiro, S. Hirano and R. Kallosh, Phys. Rev. D65, 126002 (2002).
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18. S. Kachru, R. Kallosh, A. Linde, J. Maldacena, L. McAllister and S.P. Trivedi, Towards inflation in string theory, JCAP 0310, 013 (2003). 19. C.P. Burgess, R. Kallosh and F. Quevedo, JHEP 0310, 056 (2003). 20. F. Koyama, Y. Tachikawa and T. Watari, Phys. Rev. D69, 106001 (2004). 21. M.K. Gaillard and B. Zumino, Nucl. Phys. B193, 221 (1981). 22. C. Angelantonj, R. D'Auria, S. Ferrara and M. Trigiante, Phys. Lett. B 5 8 3 , 331 (2004). 23. B. de Wit and A. Van Proeyen, Commun. Math. Phys. 149, 307 (1992). 24. R. D'Auria, S. Ferrara, F. Gargiulo, M. Trigiante and S. Vaula, JHEP 0306, 045 (2003). 25. A. Salam and E. Sezgin, eds., Supergravities In Diverse Dimensions. Vol. 1, 2 (North-Holland, Amsterdam, 1989). 26. L. Andrianopoli, R. D'Auria and S. Ferrara, Int. J. Mod. Phys. A 1 3 , 431 (1998). 27. C. Angelantonj, S. Ferrara and M. Trigiante, JHEP 0310, 015 (2003); Phys. Lett. B 5 8 2 , 263 (2004). 28. J. Polchinski and A. Strominger, Phys. Lett. B388, 736 (1996). 29. L. Andrianopoli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P. Fre and T. Magri, J. Geom. Phys. 23, 111 (1997). 30. A. Ceresole, R. D'Auria, S. Ferrara and A. Van Proeyen, Nucl. Phys. B444, 92 (1995). 31. A.R. Frey and J. Polchinski, Phys. Rev. D65, 126009 (2002). 32. S. Kachru, M. Schulz and S. Trivedi, JHEP 0310, 007 (2003). 33. S. Cecotti, L. Girardello and M. Porrati, Phys. Lett. B145, 61 (1984). 34. E. Cremmer, S. Ferrara, C. Kounnas and D.V. Nanopoulos, Phys. Lett. B133, 61 (1983). 35. J.R. Ellis, A.B. Lahanas, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B134, 429 (1984); J.R. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B247, 373 (1984); A.B. Lahanas and D.V. Nanopoulos, Phys. Rept. 145, 1 (1987). 36. S. Ferrara and M. Porrati, Phys. Lett. B545, 411 (2002). 37. S. Ferrara, R. Minasian and A. Sagnotti, Nucl. Phys. B474, 323 (1996). 38. I. Antoniadis, C. Bachas, C. Fabre, H. Partouche and T.R. Taylor, Nucl. Phys. B489, 160 (1997). 39. D.V. Alekseevskii, Izv. Akad. Nauk USSR Ser. Mat. 9, 315 (1975); Math. USSR Izvesstvja, 9, 297 (1975). 40. S. Cecotti, Commun. Math. Phys. VIA, 23 (1989). 41. E. Halyo, Phys. Lett. B387, 43 (1996). 42. R. Kallosh, N = 2 supersymmetry and de Sitter space, hep-th/0109168. 43. A.D. Linde, Phys. Rev. D49, 748 (1994).
C O N S I S T E N T R E D U C T I O N S OF HIGHER DIMENSIONAL GRAVITY A N D S U P E R G R A V I T Y THEORIES
G.W. GIBBONS D.A.M.T.P., Cambridge University, Wilberforce Road, Cambridge CBS OWA, U.K.
I review some recent work on consistent reductions of gravity and supergravity theories. An important distinction is made between reductions on a group manifold ('De-Witt reductions') for which reduction is unproblematic, and reductions on a coset space ('Pauli reductions'), which are in general inconsistent. It is emphasised that only special conspiracies between the gravity and matter sectors allow consistent Pauli reductions.In some cases, one may obtain a Pauli reduction by taking a Kaluza style U(l) quotient of a De-Witt reduction. Of special interest are models derived from string theory. That based on the bosonic string allows consistent reduction on S3 = SO(4)/SO(3) in all dimensions. A truly remarkable example of a consistent reduction is the six-dimensional supergravity model of Salam and Sezgin. This allows a supersymmetric monopole reduction on S2 to give a supersymmetric coupling of gravity, 5(7(2) Yang-Mills and an axion and a dilaton in four dimensions with a flat supersymmetric Minkowski vacuum. Subject to the assumption of maximal spacetime symmetry, the ground state is unique. The model may be obtained by a consistent reduction from ten-dimensional supergravity on the non-compact riemannian hyperboloid Ji ' followed by a Kaluza reduction and chiral truncation.
1. Introduction This is an account of a series of papers written over the last couple of years with Marco Cariglia, Mirjam Cvetic, Rahmi Giiven, Hong Lii, and Chris Pope about dimensional reduction. It is a pleasure to speak about this topic at this occasion honouring Stanley Deser, not only because his many contributions to theoretical physics are not confined to the homely realm of 4 dimensions, but even closer to home so to speak, because the subject has a certain family interest. In fact, obtaining theories in 4 dimensions as reduction of theories in higher dimensions has a long history and plays a vital role in the activity we now refer to as String/M-Theory. By now, one may distinguish many types of reduction. In this talk I will focus on consistent reductions.1 If x are the 117
118
lower dimensional coordinates and y the higher dimensional coordinates, then, roughly speaking, a consistent reduction is denned such that there exists a set of (*) Higher dimensional equations in dx,dy on a higher dimensional manifold M* (*•) Lower dimensional equations in dx on a lower dimensional manifold and we demand as a minimal requirement that Every solution of
(irk) provides a solution of
(*).
This property allows oxidation or uplifting. Thus what is required as an ansatz for solutions of (*) in terms of solutions of (**). Additional requirements might be that • (**) and/or (*) arise as the Euler Lagrange equations of some action functionals 5** and 5*, • (**) and (*) arise as the Euler Lagrange equations of same action functional and to obtain (**) one substitutes the ansatz into 5* and varies, i.e. "J**
=
"->* | ansatz •
(1/
• The lower dimensional equations (*•) contain a finite number of fields determined by the properties of a background solution and its isometry group G. Consider, as an Example Reduction on a Group Manifold. Here G acts by isometries on a higher dimensional manifold M* simply transitively so that locally M* is a bundle over the lower dimensional spacetime M** with fibre G. In the case of the Einstein equations, This is known to be consistent in the weak sense, if if the ansatz is the most general consistent with the action of G. For unimodular groups, such that Cab b = 0, one may obtain (*•) by substitution into the Einstein-Hilbert action. For non-unimodular groups, Ca b b ¥" 0, no lower dimensional 5** exists and necessarily
»->** T1 l--1* I ansatz*
(2)
119
The basic Example is G = U(l) and pure gravity for which everything works. If we come down from five dimensions, our ansatz is ds2 = e^(dy
+ 2Alldx»)2 + e~^gliVdxlidxv.
(3)
A suitable action is given by substitution in the Einstein action and is s
" = / ( f - Vd(7)2 - y2aV*Fl)V=-gd4x
+ ••-,
(4)
where the ellipsis indicates boundary terms. The background is S1 x E 3 ' 1 with its flat metric. The ansatz shares the U{1) symmetry of the background and the four dimensional fields comprise the graviton, the graviscalar a and the gravi-vector A^, i.e. the gauge field associated to the U(l) isometry group. The generalization to the case U(l)n follows the same pattern and one gets the full complement of n = dimG gauge bosons. For obvious reasons, we call this a Kaluza Reduction. Note that to set a — 0 would definitely be inconsistent (unless A^ = 0). This point was either not fully realized or disregarded in the past by, for example Klein, and (almost always) by Kaluza. It was not properly appreciated until the 1950's with work by for example Lichnerowicz and by Thiry. Our next case of a group reduction is the Example when G is compact and semi-simple and hence unimodular. An appropriate ansatz is ds2 = <w(A m + 2Am)(Xn + 2Am) + A^g^dx^dx"
,
(5)
where Am are left-invariant one forms on G, Am are g-valued one forms on M** = M+/G and jmn(i) defines, for each x a general left-invariant metric on G. As in the Kaluza case, a consistent set of field equations is obtained by substitution in the Einstein action with Yang-Mills equations coming from a Lagrangian of the form -\gmn{x)FllvFnltvyr=9,
(6)
where Fm are the Yang-Mills curvatures of Am. We remark that • Typically the background has for gmn the bi-invariant metric on G which has isometry group G x G. Thus the ansatz shares only some of the symmetries of the background.
120
• Moreover we only get the gauge bosons associated to the isometry group GL of the general ansatz. The history of reductions on a group manifold is quite complicated with many partial and some false results. We propose calling them DeWitt Reductions1 although, to be strictly accurate his treatment was incomplete. The main points are that the ansatz is GL invariant, the gauge group is G and the numbers of scalars is ^(n + 1) with n — dimG and the extension to matter systems is straightforward. 2. Coset Reductions We turn now to the case when G acts multiply transitively. The basic Example is G = 50(3) but the internal space is 5 2 = 5 0 ( 3 ) / 5 0 ( 2 ) , so that topologically. M5'1 = 5 2 x M 3 ' 1 .
(7)
These was first explored by Pauli who astutely realized that substitution in the 6-dimensional Einstein-Hilbert action is not in general justified. We propose calling reductions of this type Pauli-Reductions.1 The basic observation here is In general, Pauli Reductions are inconsistent Of course one may always make a G-invariant ansatz where G acts by isometries multiply transitively with principal orbits G/H say. However the aim of a Pauli reduction is to achieve more. It is that (**)
should contain all the generators of the group
G
3. A Popular Ansatz Typically one tries ds2 = 9mn(ct>(x)),y)(dym - K™{y)Al(x)dx»){dyn
-
+ W2{
K^{y)Abv{x)dx'J) (8)
where 9mn(0, y)dymdyn
+ W2(0, y)glll/dx'idx'/
(9)
is a G invariant background with Killing fields K™(y)-*p^, a = 1 , . . . , dim G, m = 1 , . . . dim G/H and the scalar fields <j> are taken to vanish in the background. We remark that
121
• No general ansatz is known for the scalar fields deformations 9mn{
^ y/det(gmn(0,y))
'
where m = dimM*. • It is known that deformations must be through G non-invariant configurations. • It is interesting to compare the popular ansatz with the consistent De-Witt ansatz. In this case H — id and K™da are the right invariant vector fields on G which generate left actions. In more detail, the DeWitt ansatz is ds2 = gab(x)(Xa + Aa){Xb + Ab) + (det gab{x))-^gliu{x)dx'"dx'/,
(11)
with A° = X^n(y)dym the left-invariant one-forms on G. Then 9mn(x, y) = gab{x)\am(y)\bn{y),
(12)
with KT(y) = L™(y),
(13)
where L™ are a dual basis to A° which generate right translations on G, not left translations as one might have expected. The conclusion is clear. Although for a De-Witt reduction the generic isometry of the ansatz is lefttranslation GL the gauge group is in fact GR. For a Pauli reduction, in general, the ansatz has isometry group but, if it works, the gauge group is again GR. Note also that the symmetry of background for a reduction on a group G is GL x GR and so if we followed the general pattern we might expect a Pauli reduction to give all the gauge bosons of G x G. As mentioned above, for pure Einstein theory, and without extremely special matter, Pauli reductions are certainly inconsistent. No general procedure is known for obtaining a consistent ansatz without checking consistency by substituting into the field equations.
122
To see what is involved, set (f> = 0 and W = 1. One finds R»v{x) - \gnnuRix)
=
l
-Yab{y)F;a{x)F\°{x)
- \g ^{x)Faa0{x)Fb^
+ \g^{x)R{K{y)),
{x) (14)
where K = G/H and Yab(y) = gmn{y)K™{y)K%{y). The left hand side contains functions of x but the right hand side contains both functions of x and functions of y. If K = G then Y reduces to the bi-invariant metric and no problem arises but for general coset G/H this will not be the case. Obviously we can attempt to cure the problem by setting some of the gauge fields to zero, but this is contrary to the intention behind a Pauli reduction. The removal of inconsistencies requires special internal spaces and special matter distributions. Currently, it is believed that supergravity reductions with ground states • S7 x AdS4 • S4 x AdS7 • S5 x AdS5 are consistent, although it seems that no complete, purely bosonic, analysis is available because of the extremely complicated nature of the ansatz for the scalar fields. In what follows I shall list some further consistent reductions, some of which may be understood symbolically as DeWitt = Pauli o Kaluza Consider pure gravity in D+l dimensions. One can imagine reduction to 3 lower dimensions in two different ways. 1 Firstly might perform a DeWitt reduction on SU{2) to get the gauge bosons of SU(2). Alternatively one might first perform a Kaluza reduction to one lower dimension on U(l) to get a single gauge boson and a single scalar field. One might now embark on a Pauli reduction on S2 = SU(2)/U(l). If there is any justice in the world these two processes should commute. This may be explicitly checked in this case by re-writing the known consistent DeWitt reduction in the form of an initial Kaluza reduction followed by a suitable Pauli type ansatz. The details are rather complicated but the idea should be clear, as should be the moral which is only be means of a very special matter content and a very carefully chosen ansatz, not just for the metric, but also for the two-form arising from the initial Kaluza reduction, (which isextremely non-obvious) is there any chance of this working.
123
4. N e w Examples 4 . 1 . The Bosonic
String
If we ignore quantum effects, the low energy Lagrangian exists in all dimensions, not just twenty six. It takes the form (" denotes higher-dimensional quantities) R-\(dj>)2-\e&**G3*G3,
(15)
where the 3-form G3 is closed G 3 = dB2,
(16)
and
Now any group G admits both a left and a right action by itself and thus it may be regarded as a coset of G x G with respect to G. It turns out that for the very special field content of the bosonic string Cvetic, Hu, Pope and myself were able to show1 that • There is a Pauli Reduction on 5 3 thought of as 5 0 ( 4 ) / 5 0 ( 3 ) giving the full complement of the 50(4) gauge bosons. • There is strong evidence that this works in general for the general case G x G/G. 4.2.
Salam-Sezgin
This is a particular case of a general class of six-dimensional supergravities found by Sezgin and Nishino. They have received some attention recently by phenomenologist working on the large extra dimensions scenario. The Lagrangian is
R - i(9^) 2 - ^ * f f 3 A f f s - ^ A A - 8g2e-l* . Note that the theory has • A 3-form H3 = dB2 + \F2 A Au • an abelian fields strength F2 = dA\, • and a scalar <j> with a positive Liouville type potential.
(18)
124
It was discovered long ago by Salam and Sezgin that this model admits a supersymmetric background of the form S2 x E 3 ' 1
+ monopole
(19)
with the scalar cf> = constant. Chris Pope and I have recently shown2 that there is a completely consistent Pauli reduction yielding the gauge group SU(2) and no cosmological term in 3 + 1 dimensions. If /x* are three Cartesian coordinates defining the unit sphere ^u1 = 1 in E 3 , then the ansatz is ds62 = ei+dsl + e-^9mn{dm
+ 2gAiKD(dm
F2 = 2ge^eabea H3 = H3-
+ 2gAiK™),
(20)
/\ebu.iF\
(21)
2gFi A K?{ea + 2gA^K^),
(22)
$ = -.
(23)
The four dimensional Lagrangian form which the equations can be derived is
R - \(d(t>)2 - \^{^f
- \e~**pi
A
^ + \pi
Api
>
(24)
where we have dualized the three-form H3 = e2^ * da .
(25)
What we have in four dimensions is • SU(2) Yang-Mills Fl coupled to an axion a and a dilaton <j>, but with no potential for the scalar's. • Remarkably, it turns out that there is a completely consistent reduction for the fermion sector, in other words: there is a completely consistent reduction to a supersymmetric theory in 3+1 dimensions. This theory, which is the first example known to us of a realization of Pauli's original idea, has many remarkable properties and is certainly the simplest and most completely constructed example of any known reduction, other than the original example of Kaluza.
125
4.3. Uniqueness
of the Ground
State
The currently favoured Calabi-Yau reductions (which are inconsistent in the sense being used here) are highly non-unique, with many moduli. The Salam-Sezgin model has the significant advantage: if one assumes that the six-dimensional spacetime is a product M6 = M2 x M4 ,
(26)
with M4 being maximally symmetric, i.e., de-Sitter ds4, Anti-de-Sitter AdSi or Minkowski spacetime E 3 ' 1 , and that that two-dimensional 'internal'space M2 is a smooth and closed, then Rahmi Giiven, Chris Pope and I have proved3 that the only solution is that of Salam and Sezgin. In other words Local Lorentz-invariance =» SUSY
(27)
One may also construct solutions with conical defects, representing 3branes which break SUSY. 4.4. Super symmetric
Solutions
Marco Cariglia, Rahmi Giiven, Chris Pope and I 5 have found all solutions of the four-dimensional reduced theory which admit Killing spinor, i.e. we have found all supersymmetric solutions. Interestingly, in the Yang-Mills sector, they contain the non-abelian plane waves discovered by Coleman long ago and whose significance has been somewhat obscure. The solutions take the form ds2 = 2dudv + H{u, x, y)du2 + dx2 + dy2 , Ai = Ai(u,x,y)du,
(28) (29)
with {&l + efyA* = 0.
(30)
Thus one may take Ai = ^(Xi(u,z)+xi(u,z)),
(31)
with x*(u> z) holomorphic in z = x + iy but arbitrary in u and the profile function H is given by H(u, z, z) = K(u, z) + K(u, z) - \e-*{\X\u,
z)\2 + (<£2 + X2)\z\2),
(32)
126
where A = X(u) and
Theory
Origin
In subsequent work with Mirjam Cvetic, 4 the long-standing problem of obtaining the Salam-Sezgin model from String theory was solved. Hitherto, the fact that the potential is positive had proved to be an obstacle, since time independent reductions on compact internal spaces give anti-de Sitter ground states and negative potentials. However, using the techniques outlined above, together with earlier work on sphere reductions, the model was obtained from ten-dimensional supergravity by a two-step consistent reduction. Firstly one descends using a consistent Pauli reduction to seven dimensions on the hyperboloid !H2'2. This is the non-compact riemannian manifold, i.e. with positive definite signature obtained manifold by the isometric embedding into four-dimensional Euclidean space E 4 with metric ds2 = (dX1)2 + (dX2)2 + (dX3)2 + (dX4)2
(33)
of the non-compact surface (X1)2 + (X2)2 - (X3)2 - (X4)2 = 1.
(34)
The metric induced on this 50(2, 2) invariant surface from the 50(4)invariant Euclidean metric on E 4 is positive definite and has isometry group 50(4) n 50(2,2) = 50(2) x 50(2). The seven-dimensional model is an example of a non-compact gauging with gauge group 50(2,2) which is spontaneously broken to to its maximal compact subgroup 50(2) x 50(2) in the ground state. Despite the noncompact gauge group, the scalar fields allow the seven-dimensional model to be 'ghost-free', i.e. for the vector bosons of the gauge group to have positive kinetic energies. To get down to six dimensions one now needs to perform a further Kaluza reduction. However to get the Salam-Sezgin model, which is chiral, it is necessary to perform a further chiral truncation. Remarkably, it again, turns out that this truncation is also consistent and compatible with supersymmetry.
127 In fact this construction is a special case of a more general one 7 in which one may get de-Sitter, as opposed t o anti-de Sitter supergravities from String and M-theory by consistent reductions on non-compact internal spaces.
References 1. M. Cvetic, G.W. Gibbons, H. Lii and C.N. Pope, Consistent Group and Coset Reductions of the Bosonic String, Class. Quant. Grav. 20, 5161 (2003) [hepth/0306043]. 2. G. W. Gibbons and C. N. Pope, Consistent S Pauli Reduction of Sixdimensional Chiral Gauged Einstein-Maxwell Supergravity, Nucl. Phys. B697, 225 (2004) [hep-th/0307052]. 3. G. W. Gibbons, R Giiven and C. N. Pope, 3-Branes and Uniqueness of the Salam-Sezgin Vacuum, Phys. Lett. B595, 498 (2004) [hep-th/0307238]. 4. M. Cvetic, G.W. Gibbons and C.N. Pope, A String and M-theory Origin for the Salam-Sezgin Model, Nucl. Phys. B677, 164 (2004) [hep-th/0308026]. 5. M. Cariglia, G. W. Gibbons, R. Giiven and C. N. Pope, Non-Abelian ppwaves in D = 4 supergravity theories, Class. Quant. Grav. 2 1 , 2849 (2004) [hep-th/0312256]. 6. M Cariglia and O. MacConamhna, The general form of superymmetric solutions of N = (1,0)17(1) and SU(2) gauged supergravities in six dimensions, Class. Quant. Grav. 2 1 , 3171 (2004) [hep-th/0402055]. 7. M. Cvetic, G W Gibbons and C.N. Pope, Ghost-Free de-Sitter Supergravities as Consistent Reductions of String and M-Theory, hep-th/00401151.
ELECTRIC-MAGNETIC D U A L I T Y IN GRAVITY
MARC HENNEAUX Physique theorique et mathematique and International Solvay Institutes, Universite Libre de Bruxelles, Campus Plaint C.P.231 B-1050 Bruxelles, Belgium. and Centro de Estudios Cientificos (CECS), Casilla 1469, Valdivia, Chile CLAUDIO TEITELBOIM Centro de Estudios Cientificos (CECS), Casilla 1469, Valdivia, Chile
We show that duality transformations of linearized gravity in four dimensions, i.e., rotations of the linearized Riemann tensor and its dual into each other, can be extended to the dynamical fields of the theory so as to be symmetries of the action and not just symmetries of the equations of motion. Our approach relies on the introduction of two "superpotentials".
1. Introduction Duality is a fascinating symmetry that has always attracted Stanley's interest. It is a great pleasure to dedicate this article, in which some aspects of the implementation of duality in gravity theory are investigated, to him. We consider here the standard, Einstein gravity. Electric-magnetic duality of conformal gravity has been studied in Ref. [1]. A more detailed account of our results is reported elsewhere.2
1.1. Spin-2 free field equations
and
duality
The linearized Riemann tensor Rx^pa = -R^Xpa = -Rxy-ap fulfills the following identities,
R\v.\po,a] = 0. 128
(2)
129
These identities a guarantee the existence of a symmetric tensor gauge field h^u = hVjl of which Rxppa is the curvature15, R\fiPa = d[\hll]^py
(3)
In the absence of sources, the linearized Einstein equations take the form R»v = 0,
(4) al3
where RpV is the linearized Ricci tensor i?M„ = Rapj3v 7] . It follows that the dual Sxppa = —SpXpa = —Sxpvp of the curvature, denned (in four spacetime dimensions) byc Sx^pa = 2 e\p-ct0R
prT'
(5)
enjoys also the properties Sx[npa] = 0,
(6)
Sx/j.[P(T,a} = 0,
(7)
and Spu = 0,
(8)
a
with S^v = S pau (see e.g. Ref. [7]). This shows that the vacuum linearized Einstein equations are invariant under duality transformations, in which the curvature and its dual are rotated into each other, R
'xp,Pa =
s
cos a
RXHP*
+ sin a SxpP„,
'\tipa = - sin a Rxppa + cos a 5AMP
(9) (10)
It is useful to rewrite the duality transformations in terms of the electric and magnetic components of the Weyl tensor (which coincides on-shell with the Riemann tensor). One defines (see e.g. Ref. [1])
The electric and magnetic tensors £mn and Bmn are both traceless and symmetric on-shell. Thus, they have both 5 independent components, corresponding to the 10 independent of the Weyl tensor. It is easy to verify a
Note that -RAMPO- = RPa\^ is a consequence of (1). T h i s result and similar results used below for deriving the superpotentials and establishing their properties can be viewed as part of a general theory extending the ordinary exterior differential calculus, see e.g. Refs. [3-6]. c We use mostly + ' s conventions for the Minkowskian metric and e 0 1 2 3 = 1 = —£0123b
130
that the transformations (9) and (10) are completely equivalent to £'mn - cos a Smn + sin a Bmn,
(12)
C
(13)
= - sin a Emn + cos a Bmn,
when the equations of motion hold. 1.2. Is duality a symmetry
of the
action?
The question investigated in this paper is: do the duality rotations (9) and (10) define symmetries of the action — and thus of the theory? In Ref. [8], a similar question was asked for the Maxwell theory. It was shown that duality rotations of the field strength Fp_v into its dual *Fllv do define symmetries of the Maxwell action. We show in this note that the same property holds for linearized gravity, described by the Pauli-Fierz action S
\KA = ~\ Id4x (9ph^ dp V - 2 d^
dph"v
p
+2d»h»pd»hfiU-d»h pdtih\).
(14)
By "lifting" the transformations (9) and (10) to the fields hpv, we are able to prove that the action is duality-invariant. We also compute the corresponding conserved charge. The proof of duality-invariance rests on the introduction of two spatial superpotentials leading to a formulation analogous to the double potential formulation of electromagnetism of Refs. [8,9]. In terms of these superpotentials, manifest duality invariance is achieved at the cost of manifest Lorentz invariance. 2. Superpotentials 2.1. Hamiltonian
form of the
action
As in Ref. [8], we work in the Hamiltonian formalism. Any symmetry of the Hamiltonian action is a symmetry of the original second order action when the momenta (which can be viewed as auxiliary fields) are eliminated through their own equations of motion (see concluding section below). When written in Hamiltonian form, the Pauli-Fierz action (14) becomes
Idt
J d3xTTmnhmn
-H-
Id3x
(nH +
nmnn
(15)
131
where 7rmn are the conjugate momenta to the spatial components hmn of the spin-2 field, while n and respectively the linearized lapse and (minus 2 times) the linearized shift. The Hamiltonian H reads H = WmnWmn -l--K2+ 1 drhmn drhmn
- \ dmhmn
drh\
+^dmhdnhmn-^dmhdmh,
(16)
where h = hmm is the trace of the spatial hmn and ir = irmm is the trace of 7rm". The constraints, obtained by varying the action with respect to the lapse and the shift, are 7i = 0 and 7im = 0 with H = dmdnhmn m
- Ah,
(17)
m
U
= 7r "„,
(18)
where A = dmdm is the spatial Laplacian. 2.2. Solution
of the
constraints
In order to exhibit the duality symmetry, we solve the constraints. This can be achieved while maintaining locality of the action principle by introducing "superpotentials". The general solution of the constraint 7im = 0 is TTmn = dpdrUmpnr,
(19)
where the tensor Umpnr has the symmetry properties Umpnr = -Upmnr = jjnrm-p _ _jjmPm_ g y d u a i j z m g jjmpnr m t e r m s 0 f a symmetric "superpotential" Pqs - Psq, jjmpnr
= 6™Pq £nrs
p^^
^0)
this expression can be rewritten 7Tmn = empqenrsdpdrPqs mn
= 5 (AP
(21)
r s
- d d Prs) m n
m r n
+d dP
r
+
n r m
ddP
mn
-d d P-AP ,
r
(22)
where P is the trace of PmnGiven 7rm", the superpotential Pmn is determined up to Pmn
* Pmn "T Om^n ~T Ont,m-
A particular solution is Pmn = -A-lnmn
+ 5mnA-ln.
(23)
132
One easily verifies that (23) reproduces 7rmn when inserted in (22) (on the constraint surface 7rm™n = 0). One may use Pmn instead of 7rm" as fundamental field in the action principle. When this is done, the momentum constraint and its Lagrange multiplier nm drop out from the action principle because the constraint is identically satisfied. Although the expression of nmn is local in terms of Pmn (which is what matters for the locality of the action expressed in terms of Pmn), the inverse transformation is non-local. The momenta 7rm" are not gauge invariant but transform as 7rm™ —> mn •n — dmdnt; + <5m"A£ under the transformation generated by the Hamiltonian constraint. This transformation is simply generated by a conformal change of the superpotential P m n , SPmn = 5mn£. Hence, the total ambiguity in Pmn is
The transformations (24) appear as gauge transformations of the formulation in which Pmn is regarded as the fundamental field. Interestingly enough, these transformations are identical to the gauge transformations of (linearized) conformal gravity, the last term being a linearized infinitesimal conformal transformation. Similarly, one can also solve the "Hamiltonian constraint" "H — 0 in terms of a symmetric superpotential $ m n = $„TO and a vector um as "'mn ~ ^mrs O ™ n ' ^-nrs & ™ m ' ^mun
~r &nurri'
\^)
Given hmn up to a gauge transformation, there is some ambiguity in the superpotential $ m T l , which reads exactly as in (24),
One can also express $ m n non-locally in terms of the metric. A particular solution is $mn = ~
A " 1 (emrs drh\
+ enrs drhsm),
(27)
with um = 1 A " 1 (3 dphpm - dmh).
(28)
We leave it to the reader to verify that this expression leads back to hmn (on Hamiltonian constraint shell) when inserted in (25). When $ m n is used as fundamental field instead of hmn, there is no constraint left and (26) appears as a gauge transformation.
133
Note that the first order constraint Hm = 0 yields an expression for the momenta that involves two derivatives of the superpotential Pmn, while the second order constraint H = 0 yields an expression for hmn that involves only one derivative of $ m „ and um. Accordingly, the Hamiltonian is a homogeneous polynomial quadratic in the second derivatives of both superpotentials. 2.3. Duality
and
superpotentials
It turns out that the duality rotations are simply 50(2) rotations of the superpotentials into each other, P'mn = cos a Pmn + sin a $ m " , mn
(29)
m
$ * * = - sin a P
+ cos a $ " ,
(30)
as it may be verified by a straightforward calculation. 3. Duality invariance of the action 3.1.
Hamiltonian
We now insert the above expressions (22) and (25) in the Hamiltonian. Tedious but straightforward computations give for the kinetic energy density (up to total derivatives that are being dropped) •nij*ij - \*2 = AP« A P i j + \ {dkdmPkm? - 2 dmdiPV dmdkPk3
+ dkdmPkm
AP
- \ (AP) 2 .
(31)
Similarly, the potential energy density becomes (up to total derivatives) i drhmn drhmn - | dmhmn drh\
+ 1 dmh dnhmn - i dmh dmh
= A $ y A $ y + \ (dkdm$km)2 - 2 dmd&v dmdk$kj
+ dkdm$km
- \ (A$) 2 .
A$ (32)
Because the kinetic and potential energies take exactly the same form in terms of their respective superpotentials, one sees that the Hamiltonian is invariant under 50(2)-rotations in the plane of Pmn and $ m " , i.e., the Hamiltonian is duality invariant.
134
3.2. Kinetic
term
The invariance of the kinetic term n h can be checked straightforwardly. Injecting the expressions (22) and (25) into irmn hmn, one gets f dt dsx TT"1" hmn = f dt d3x 2 emrs (dpdqdrPps
- AdrPqs) $ 9 m .
(33)
Because this expression changes sign (up to a total derivative) under the exchange of Pmn with (j)mn, it is invariant under the rotations (29) and (30) (up to a total derivative). This ends the proof of the duality-invariance of the free massless spin-2 theory in four dimensions. 3.3. SO(2)-vector
notations
By introducing 50(2)-vector notations and adding a total derivative to make the kinetic term strictly antisymmetric under the exchange of the superpotentials, one may rewrite the free spin-2 action — with the superpotentials as basic dynamical fields — as S[Zamn] = j dt
f d3x eabemrs
(d"dqdrZaps
- AdrZaqs)
Zb"m - H (34)
with (Z a m n ) = ( P m n ) $""»),
a,b= 1,2,
(35)
and H = Sab Uzaij
AZbij + l- dkdmZakmd*dnZbqn
- 2 5ab (d^Z?
dmdkZbkj
+ dkdmZakm
- i AZa AZb\ ,
AZb\
Za = Zamm. (36)
This expression is manifestly duality invariant because the tensors eab and Sab are 50(2)-invariant. It should be compared with the analogous expression for the Maxwell action considered in Ref. [9], 4. Conserved charge The conserved charge that generates infinitesimal duality rotations is found from the Noether theorem to be
Q = l-Jdzxemrs {(dpdqdrPps -
AdrPqs)P\
-(dpdqdr$ps-Adr$gs)&m}.
(37)
135
It is invariant under the respective gauge transformations (24) and (26) of and $ m n . One may rewrite the conserved charge more suggestively by introducing the curvatures and spin connections of Pmn and $ m n . These are defined by R(P)pqrs
= d[qPp][r,a],
o;(P)M m = dpP\
R($)pqrs
= d[q%][r,s],
w($)"m
=
- d"PPm, ff>*'>m-9'&m.
(38) (39)
In terms of t h e ' curvature two-forms R(P)pq = \R(P)pqrsdxr A dxs, r s R($)pg = 7;R($)pqrs dx A dx and the connection one-forms to(P)pg — oj(P)pqmdxm, w($)"9 = u(
Q = J [R(P)pq A w(Pr
- R($)pq A W(*)P«],
(40)
exhibiting a Chern-Simons structure analogous to that found in the Maxwell case.
5. Conclusions In this paper, we have shown that duality is a symmetry not only of the equations of motion of the free spin-2 theory but also of the Pauli-Fierz action itself. Hence, duality is a symmetry of (linearized) gravity in the standard sense. This was achieved by introducing symmetric superpotentials with gauge symmetries that are exactly the same as those of conformal gravity. We have also computed the canonical generator of duality rotations and found the same Chern-Simons structure as in the spin-1 case. We have explicitly written the duality transformation rules in terms of the electric and magnetic superpotentials and verified duality-invariance only for the reduced action where the constraints have been eliminated, but this is of course sufficient to establish invariance of the original Pauli-Fierz action itself. Although we have not carried it explicitly, we expect the discussion of the duality properties of higher spins gauge fields actions 7 ' 10,11,12 to proceed similarly. Much more challenging would the understanding of how the results can be extended to the full, interacting Einstein theory (in the same Killing vector free context considered here). The inclusion of sources of both electric and magnetic types in the general context is also a very intricate question.
136
Acknowledgments T h e work of MH is partially supported by IISN - Belgium (convention 4.4505.86), by a "Pole d'Attraction Universitaire" and by the E u r o p e a n Commission R T N programme HPRN-CT-00131, in which he is associated t o K. U. Leuven. Institutional support t o t h e Centro de Estudios Cientificos (CECS) from Empresas C M P C is gratefully acknowledged. C E C S is a Millennium Science Institute and is funded in p a r t by grants from Fundacion Andes and the Tinker Foundation.
References 1. S. Deser and R.I. Nepomechie, Phys. Lett. A97, 329 (1983). 2. M. Henneaux and C. Teitelboim, Duality in linearized gravity, submitted for publication. 3. P. Olver, Differential hyperforms I, University of Minnesota report 82-101. 4. M. Dubois-Violette and M. Henneaux, Lett. Math. Phys. 49, 245 (1999) [math.qa/9907135]. 5. M. Dubois-Violette and M. Henneaux, Commun. Math. Phys. 226, 393 (2002) [math.qa/0110088]. 6. X. Bekaert and N. Boulanger, Commun. Math. Phys. 245, 27 (2004) [hepth/0208058]. 7. C M . Hull, JHEP 0109, 027 (2001) [hep-th/0107149]. 8. S. Deser and C. Teitelboim, Phys. Rev. D 1 3 , 1592 (1976). 9. S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim, Phys. Lett. B400, 80 (1997) [hep-th/9702184]; S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim, Nucl. Phys. B520, 179 (1998) [hep-th/9712189]. 10. X. Bekaert and N. Boulanger, Phys. Lett. B 5 6 1 , 183 (2003) [hep-th/0301243]. 11. P. de Medeiros and C. Hull, JHEP 0305, 019 (2003) [hep-th/0303036]. 12. N. Boulanger, S. Cnockaert and M. Henneaux, JHEP 0306, 060 (2003) [hepth/0306023].
R4 T E R M S IN S U P E R G R A V I T Y A N D M-THEORY
PAUL H O W E Department of Mathematics, King's College, London, UK E-mail: phowelSmth.kcl.ac.uk
Higher-order invariants and their role as possible counterterms for supergravity theories in four dimensions are reviewed. The construction of RA superinvariants in string theory and M-theory in D = 10 and D = 11 is discussed.
1. Introduction The title of this contribution to the Deserfest is an appropriate one in view of the fact that the first paper on three-loop (i?4) counterterms in supergravity was written by Stanley Deser, together with John Kay and Kelly Stelle, in 1979.1 It will be recalled that supergravity had been shown to be on-shell finite at one and two loops2 and there were hopes that this would persist at higher loop orders, but these were somewhat spoiled by the above paper, on N = 1 supergravity, and by a follow-up which reported a similar result for N = 2. 3 Any residual hopes that the maximally supersymmetric N = 8 supergravity theory in four dimensions would have special properties were removed by the observation that counterterms can easily be constructed as full superspace integrals, at seven loops in the linearised theory, and at eight loops if one wishes to preserve all the symmetries of the full non-linear theory including E7.4 Subsequently, a linearised threeloop N = 8 invariant was also found.5 In view of the difficulty of carrying out high loop calculations in quantum gravity, and of the success of string theory and M theory, the subject has receded into the background over the years but interest in it has recently been reawakened by the development of new calculational techniques. 6 The implications of this work and its relation to the predictions of superspace power-counting arguments, both in conventional superspace and in harmonic superspace, will be the subject of the next section of the paper. Notwithstanding these recent developments, R4 and other higher order 137
138
supersymmetric invariants are nowadays of most interest in the context of effective field theory actions in string theory and M-theory. Such terms could be interesting both theoretically, especially in the case of M-theory, and in applications to solutions and to beyond leading-order tests of various implications of duality. For such applications one would ideally like to know the complete bosonic part of the effective action. However, these terms seem to be extremely difficult to calculate systematically as will be seen below. Some partial results obtained from string theory and supersymmetry will be outlined in section 3 and the final section of the paper will be devoted to a brief exposition of a superspace approach to the problem in eleven dimensions.7 This work seems to indicate that there is a unique invariant at this order which is supersymmetric and which is compatible with the M-theoretic input of five-brane anomaly cancellation.
2. Supergravity counter-terms The D = 4, N — 8 three-loop counterterm was first constructed in [5]. A manifestly supersymmetric and SU(8) invariant version was given in [8] as an example of a superaction — an integral over superspace which involves integrating over fewer than the total number of odd coordinates. This type of integration, using conventional superspace and measures which carry SU(N) representations, turns out to be equivalent to integration in certain harmonic superspaces where the number of odd coordinates is reduced. 9 The three-loop counterterm has a very simple form in this language. In D — 4 iV-extended supersymmetry harmonic superspaces are obtained from Minkowski superspace by adjoining to the latter a coset space of the internal SU(N) symmetry group, chosen to be complex.10 This coset can be thought of as parametrising sets of mutually anticommuting covariant derivatives (Ds and Ds). The simplest example is in N = 2 where we can select one Da and one D& to anticommute a , and the ways this can be done are parametrised by the two-sphere CP 1 . For higher values of N there are more ways of choosing such sets of derivatives and therefore many different types of Grassmann analyticity (or generalised chirality) constraints that one can impose on superfields. In order for covariance with respect to the R-symmetry group to be maintained such G-analytic superfields must be allowed to depend on the coordinates of the coset space. It turns out that the G-analyticity constraints are compatible with ordinary (harmonic) a
a and a denote two-component spinor indices.
139
analyticity on the coset space and that many field strength superfields can be described by superfields which are analytic in both senses. Harmonic analyticity ensures that such fields have short harmonic expansions. To be more explicit, let i = 1 . . . N and / = 1 . . . N denote internal indices which are to be acted on by SU(N) and the isotropy group respectively. We split / into three, / = (r,R,r'), where the ranges l...p and N — (q + 1).. .N of r and r' cannot be greater than N/2; we write u 6 SU(N) as ujl = (url,URl,uril) and similarly for the inverse element ( u - 1 ) / . This splitting is clearly preserved by the isotropy group S(U(p) x U(N - (p + q)) x U(q)), the coset space defined by this group being the flag space F Pi /y_ q of p-planes within (N — „»;
Di = {u-^i'Di
,
(1)
then the derivatives Dar and Dra are mutually anti-commuting. Superfields which are annihilated by these derivatives are said to be G-analytic of type (p,q); the associated harmonic superspace is known as (N,p,q) harmonic superspace. This formalism can be applied to N — 8 supergravity. The supergravity multiplet is described by the linearised field strength Wijki, i = 1...8. The superfield Wijki is totally antisymmetric and transforms under the seventy-dimensional real representation of SU(8). It obeys the constraints Wijkl = \^klmn^Wmnm,
(2)
DaiWjklm = D^Wjklm}-!
(3)
DaWjklm
(4)
= ~SlD2WMm]n,
the third of which follows from the other two. Note that this superfield defines an ultra-short superconformal multiplet. The same multiplet can be described by an analytic superfield W on (8,4,4) harmonic superspace: W := e r s U W « t W W i j k i .
(5)
It is not difficult to see that the constraints DarW
= Dr^W = 0,
r = 1...4, r' = 5 , . . . 8 ,
(6)
together with analyticity on the internal coset, are equivalent to the differential constraints above, while the reality condition can also be formulated in harmonic superspace.
140
Since W depends on only half of the odd coordinates we can integrate it over an appropriate harmonic superspace measure cfyx44 := d4xduds9d85, where du is the usual Haar measure on the coset and where the odd variables are 9ar' := 6m{u~l)ir' and 6? := u/flf. In order to obtain an invariant the integrand must be (4,4) G-analytic and must have the right charge with respect to the £/(l) subgroup of the isotropy group S(U(A) x 17(4)) ~ U(l) x SU(4) x SU(4). The only possible integrand with these properties which can be constructed from W is W4; it gives the harmonic superspace version of the three-loop counterterm in the form9 h-ioop = / dfHA W4 •
(7)
Since W ~ 04Cap15 + S4Cdt^s + • • •, where C is the Weyl spinor, it is apparent that integrating over the odd variables will give the square of the Bel-Robinson tensor. All other possible invariants in linearised N = 8 supergravity which are invariant under SU(8) and which involve integrating over superspaces of smaller odd dimension than conventional superspace were recently classified.11 There are just two, one in (8,2,2) and one in (8,1,1) harmonic superspace. They both have the same schematic form as the three-loop invariant but the measures and the definition of the superfield W differ. They correspond to five-loop and six-loop counterterms respectively: h-ioop = Idn2,2W4~
fd4xd4R4
+ --- ,
h-ioop = f dfn,i W4~
fd4xdeR4
+ --- .
(8)
In order to discuss the relevance of these counterterms to possible divergences in quantum supergravity one has to know how much supersymmetry can be preserved in the quantum theory. 12,13 This means that we have to know the maximum number M < 8 of supersymmetries that can be realised linearly in the off-shell theory. If we stick to standard off-shell realisations, then only M = 4 is allowed for N = 8 supergravity. 14,15 This would then suggest that counterterms should be integrals over sixteen odd coordinates leading to a prediction of the first divergence occurring at three loops. However, recent work by Bern et al. has indicated that the coefficient of the three-loop counterterm is not divergent.6 Some light can be shed on this apparent discrepancy by looking at the maximally supersymmetric TV = 4 Yang-Mills theory which becomes non-renormalisable in higher spacetime dimensions. In this case, Bern et al.6 also found improved UV behaviour compared to the predictions of
141
superspace power-counting which were again based on the assumption that the best one can achieve off-shell is to preserve half of the total number of supersymmetries. 16 However, in the Yang-Mills case we know that there is an off-shell version with N = 3 supersymmetry available in harmonic superspace. 17 If the superspace power-counting is adjusted for this, then the new predictions precisely match the calculational results. 18 This line of reasoning suggests that there should be an off-shell version of N = 8 supergravity with M — 6 supersymmetry which would correspond to the first divergence appearing at five loops. Results obtained in higher dimensions lend strong support to this contention. In [6] it was shown that the maximal supergravity theory in D = 7 diverges at two loops where the corresponding counterterm has the same form as the D = 4 five-loop invariant. The occurrence of this divergence indicates that an off-shell version of the theory with six four-dimensional supersymmetries should exist, whereas if one could preserve seven such supersymmetries this divergence would not be allowed in D = 7. Unless there is a completely different mechanism at work in four dimensions, it therefore seems that the N = 8 theory is most likely to diverge at five loops. 3. Invariants in string theory and M-theory In discussing invariants corresponding to field-theoretic counterterms it is enough to consider the linearised theory. In string theory or M-theory, however, we are more interested in the full non-linear expressions, and these are very difficult to find. Information about particular terms in full invariants has been derived from string scattering amplitudes and sigma model calculations while some other hints have been obtained using supersymmetry and arguments based on duality symmetries. An important point to note is that the linearised invariants of the type discussed in the preceding section cannot be generalised to the non-linear case in any straightforward manner. This is because the superspace measures do not exist in the interacting theory. For example, D = 4, N = 8 supergravity does not admit an (8,4,4) harmonic superspace interpretation in the full theory. Invariants of the R4 type are reasonably well understood in D = 10, N = 1 supersymmetry where complete expressions are known for the bosonic terms. 19 There are two types of supergravity invariant. The first is a full superspace integral of the form20
jdwxdw9Ef{<j>)~ Jd10xe(^R4+
•••),
(9)
142
where E and e denote the standard densities in superspace and spacetime respectively, <j> is the dilaton superfield, and where the RA terms appear in the combination (tsts — |eio£io)-R4 in the notation of, for example, Peeters et al.,27 where these tensorial structures are explained. The second type of invariant can be constructed starting from certain Chern-Simons terms and so we shall refer to them as CS invariants. In the next section we shall see how such invariants can be explicitly constructed. There are two possible Chern-Simons terms in supergravity, B A tri? 4 and B A (trR2)2. Details of all of these invariants up to quadratic order in fermions have been computed. 19 The structures associated with the N = 1 invariants are seen in various combinations in the TV = 2 invariants. In IIA string theory the pure curvature term in the tree level invariant has the form e _2 *((t 8 ^8 _ |eio e io)-R 4 , which resembles the first type of TV = 1 invariant, while there is also a one-loop term, required to cancel the five-brane anomaly, whieh arises from the CS term B A X8, where X8 = trfl 4 - |(tri? 2 ) 2 . This is believed to be associated with a pure R4 term of the form (tsts + geioeio)^ 4 - The tg terms have been computed from string scattering,21>22>23 the tree-level e2 term can be inferred from sigma model calculations, 24 the one-loop e2 term is suggested by one-loop four-point amplitudes, 25 and the one-loop CS term is required for anomaly cancellation.26 In IIB the linearised theory is described by a chiral superfield $ and there is a linearised invariant of the form J d169 3>4 ~ (tsts - |ei 0 eio)-R 4 + • • •. This does not generalise to the full theory in any straightforward manner b , however, due to the structure of the superspace constraints of the full theory. 29 Nevertheless, using component supersymmetry, input from Dinstanton results and SL(2, Z) invariance some conjectures have been made about the scalar structure of some terms in the invariant. Specifically, each term appears multiplied by a function of the complex scalars (r, f) which is a modular form under 5L(2, Z) whose weight is determined by the U(l) charge of the term under consideration. 30 The i? 4 invariant in M-theory has been discussed by various groups using string theory, 31,28 quantum superparticles 32 and supergravity. 33,34 A straightforward supersymmetry approach is rendered even more difficult by the fact that the field strength superfield W has dimension one so that one would require an integral over eight odd coordinates of WA to obtain i? 4 . One result that is known is that there must be a CS term to cancel the For an explicit attempt to do this see [28].
143
anomaly of the M-theory five-brane.35 This CS term has the form C 3 A X$, where C3 is the three-form gauge field in the theory. An alternative approach is to investigate the modified equations of motion rather than trying to compute the invariant directly. Corrections to the equations of motion can be understood as deformations of the on-shell superspace constraints, and the consistency conditions that the Bianchi identities place on these constraints lead, when one takes the into account the possibility of field redefinitions of the underlying superspace potentials, to a reformulation of the problem in terms of certain superspace cohomology groups. This cohomology, called spinorial cohomology, has been studied for various theories in the literature including M-theory considered as a deformation of D = 11 supergravity. 36 ' 7 In certain circumstances spinorial cohomology coincides with pure spinor cohomology which has been used to give a new spacetime supersymmetric formulation of superstring theories. 37 In the final section we shall investigate M-theoretic RA terms in a superspace setting by looking at the Bianchi identities in the presence of the five-brane anomaly cancelling term. It will be argued that the CS term gives rise to a unique invariant which is both supersymmetric and consistent with the anomaly and it will be shown how this invariant can be constructed. The deformations of the superspace constraints are driven by the anomaly term and can be found systematically, at least in principle. 4. M-theory in superspace The component fields of D = 11 supergravity are the elfbein ema, the gravitino, tpma, and the three-form potential c m n p . 3 8 In the superspace formulation of the theory 39 the first two appear as components of the supervielbein EMA, whereas the third is a component of a superspace three-form potential CMNP-C We have a
M
E =dz EM
a
=
dxmEma
= dxm{ema
dO^E^
= 0 + 0(0),
E«=dZ"EM* = \dxmE~a dO^E^
=
+ 0{6)),
^ " + « = d0"(V* + O(0)),
(10)
while Cmnp(x, 6 = 0) = Cmnp. c
Notation: latin (greek) indices are even (odd), capital indices run over both types; coordinate (tangent space) indices are taken from the middle (beginning) of the alphabet.
144
The structure group is taken to be the Lorentz group, acting through the vector and spinor representations in the even and odd sectors respectively. We also introduce a connection one-form Q.AB which takes its values in the Lie algebra of the Lorentz group and define the torsion and curvature in the usual way: TA = DEA := dEA + EBnBA RAB
= dnAB
+ nAcQcB
=
=
^ECEBTBCA,
\EDECRCD,AB.
(11)
From the definitions we have the Bianchi identities DTA = EBRBA and DTA — 0. The assumption that the structure group is the Lorentz group implies that Ra& = Rab = 0 while Rj = \(rb)a0Rab.
(12)
The equations of motion of supergravity in superspace are implied by constraints on the torsion tensor. In fact, it is enough to set Ta/3C = -i(lC)a(3 to obtain this result. non-zero are
40
TaiP = ~
(13)
The only other components of the torsion which are ({lbcd)fWabcd
+
l
-(iabcd&)fWab«^
,
(14)
and Tab1 whose leading component can be identified as the gravitino field strength. Given these results one can construct a superspace four-form G4 which is closed and whose only non-zero components are Gaffed = —i{lcd)a0,
Gabcd = Wabcd-
(15)
The only independent spacetime fields described by these constraints are the physical fields of supergravity; their field strengths are the independent components of the superfield W. Instead of deducing the existence of a four-form G4 we can include it from the beginning. In this case one can show that there is a stronger result, namely that imposing the constraint GaflyS = 0
(16) 36 41 7
is sufficient to imply the supergravity equations of motion. ' ' Either the geometrical approach or the four-form approach can be used as a starting point for investigating deformations using spinorial cohomology, but we
145
shall choose yet another route by introducing a seven-form field strength G7 as well.7 We then have the coupled Bianchi identities dG4 = 0,
dG7 = \{GAf
+ (3Xs,
(17)
where we have included the anomaly term and where (3 is a parameter of dimension i6. We shall work to first order in (3 which means that we can use the supergravity equations of motion in computing Xs, in other words, Xg is a known quantity. The Bianchi identities can be solved systematically and all of the components of G4, G-j, T and R can be found. Note that the only component of any of these tensors which can be zero at order /? is Gai...ar. However, this solution might not be unique as there could be solutions of the homogeneous equations (i.e. without the X§ term). In principle this question could be tackled using spinorial cohomology but we shall study it indirectly by looking at the action. We briefly describe how one can construct a superinvariant from any CS invariant. If we have a theory in D spacetime dimensions formulated in superspace an invariant can be constructed if we are given a closed superspace D-form LD42'43 This invariant is
1 = JdDx e m i - m D W . m D (*, 6 = 0).
(18)
Under a superspace diffeomorphism generated by a vector field v 5LD = CVLD = d(tvLo)
+ LvdLo = d(LvLo) •
(19)
Identifying the 0=0 components of v with the spacetime diffeomorphism and local supersymmetry parameters we see from this equation that the above integral is indeed invariant. In some situations, notably when we have CS terms available, we can easily construct such closed D-forms. 44,45 Suppose there is a closed (D + l)form W^c+i = dZp where Zp is a potential D-form which we are given explicitly. We can always write WD+I = dKo, where Kp is a globally defined .D-form, because the cohomology of a real supermanifold is equal to that of its body and this this is trivial in degree D + 1 in D dimensions. If we set LD = KD — ZD then LD is closed and hence gives a rise to a superinvariant using the construction described above. Any CS term gives rise to a superinvariant in this manner.
146
In M-theory the appropriate forms are W12 = \G\
+ 3/3G 4 X 8 ,
Z i i = C 3 ( ^ G l + 3/3X 8 ).
(20)
T h e invariant constructed from these forms will include the anomalycancelling CS t e r m and superpartners which will be of R4 type. T h e purely bosonic terms, aside from the CS t e r m itself, come from K 0 l . . . 0 l l , and these are the most difficult t o compute. T h e easiest t e r m to calculate is the lowest non-vanishing term which is -ft'a6c5i...l58- It gives rise to terms with eight gravitinos in the action. To this invariant we could add any invariant coming from a closed form t u . It has been argued on (purely algebraic) cohomological grounds (subject t o some assumptions) t h a t there are no such t e r m s . 7 If this is the case we would conclude t h a t the R4 invariants in eleven dimensions come from CS terms. In principle there could be two of these, since we have b o t h t r i ? 4 and ( t r i ? 2 ) 2 , b u t M-theoretic considerations imply t h a t we have t o choose the linear combination which appears in X$. Finding the explicit form of this invariant is an extremely difficult task. One could s t a r t either by solving the G-j Bianchi identities, or one could t r y to find K\\ directly by solving the equation W12 = dKu. This is currently under investigation.
Acknowledgments T h e recent results on counterterms reported here were obtained with Kelly Stelle whom I t h a n k for many discussions on this subject. I t h a n k Ulf G r a n and Dmitris Tsimpis for ongoing discussions on solving the modified Bianchi identities.
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22. D.J. Gross and E. Witten, Superstring Modifications of Einstein's Equations, Nucl. Phys. B277, 1 (1986). 23. N. Sakai and Y. Tanii, One Loop Amplitudes and Effective Action in Superstring Theories, Nucl. Phys. B287, 457 (1987). 24. M.T. Grisaru, A.E.M. van de Ven and D. Zanon, Four Loop Beta Function for The N = 1 And N = 2 Supersymmetric Nonlinear Sigma Model in Two-Dimensions, Phys. Lett. B173, 423 (1986); M.T. Grisaru and D. Zanon, Sigma Model Superstring Corrections to the Einstein-Hilbert Action, Phys. Lett. B177, 347 (1986); M.D. Freeman, C.N. Pope, M.F. Sohnius and K.S. Stelle, Higher Order Sigma Model Counterterms and the Effective Action for Superstrings, Phys. Lett. B178, 199 (1986); Q.H. Park and D. Zanon, More on Sigma Model Beta Functions and Low-Energy Effective Actions, Phys. Rev. D 3 5 , 4038 (1987). 25. E. Kiritsis and B. Pioline, On R threshold corrections in type UB string theory and {p,q) string instantons, Nucl. Phys. B508, 509 (1997) [hepth/9707018]; I. Antoniadis, S. Ferrara, R. Minasian and K.S. Narain, R4 couplings in M- and type II theories on Calabi-Yau spaces, Nucl. Phys. B507, 571 (1997) [hep-th/9707013]. 26. C. Vafa and E. Witten, A One loop test of string duality, Nucl. Phys. B447, 261 (1995) [hep-th/9505053]. 27. K. Peeters, P. Vanhove and A. Westerberg, Supersymmetric higher-derivative actions in ten and eieven dimensions, the associated superalgebras and their formulation in superspace, Class. Quant. Grav. 18, 843 (2001) [hepth/0010167]. 28. S. de Haro, A. Sinkovics and K. Skenderis, A supersymmetric completion of the R4 term in IIB supergravity, Phys. Rev. D67, 084010 (2003) [hepth/0210080]. 29. P.S. Howe and P.C. West, The Complete N = 2, D = 10 Supergravity, Nucl. Phys. B238, 181 (1984). 30. M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D59, 046006 (1999) [hep-th/9808061]. 31. A.A. Tseytlin, R terms in 11 dimensions and conformal anomaly of (2,0) theory, Nucl. Phys. B584, 233 (2000) [hep-th/0005072]. 32. M.B. Green, M. Gutperle and P. Vanhove, One loop in eleven dimensions, Phys. Lett. B409, 177 (1997) [hep-th/9706175]; M.B. Green, H.h. Kwon and P. Vanhove, Two loops in eleven dimensions, Phys. Rev. D 6 1 , 104010 (2000) [hep-th/9910055]. 33. J.G. Russo and A.A. Tseytlin, One-loop four-graviton amplitude in elevendimensional supergravity, Nucl. Phys. B508, 245 (1997) [hep-th/9707134]. 34. S. Deser and D. Seminara, Tree amplitudes and two-loop counterterms in D=U supergravity, Phys. Rev. D62, 084010 (2000) [hep-th/0002241]. 35. M.J. Duff, J.T. Liu and R. Minasian, Eleven-dimensional origin of string/string duality: A one-loop test, Nucl. Phys. B452, 261 (1995) [hepth/9506126]. 36. M. Cederwall, B.E.W. Nilsson and D. Tsimpis, Spinorial cohomology and maximally supersymmetric theories, JHEP 0202, 009 (2002) [hep-
149
th/0110069], 37. N. Berkovits, Cohomology in the pure spinor formalism for the superstring, JHEP 0009, 046 (2000) [hep-th/0006003]. 38. E. Cremmer, B. Julia and J. Scherk, Supergravity Theory in 11 Dimensions, Phys. Lett. B76, 409 (1978). 39. E. Cremmer and S. Ferrara, Formulation of Eleven-Dimensional Supergravity in Superspace, Phys. Lett. B 9 1 , 61 (1980); L. Brink and P.S. Howe, ElevenDimensional Supergravity on the Mass-Shell in Superspace, Phys. Lett. B 9 1 , 384 (1980). 40. P.S. Howe, Weyl superspace, Phys. Lett. B415, 149 (1997) [hep-th/9707184]. 41. N. Berkovits, Covariant quantization of the supermembrane, JHEP 0209, 051 (2002) [hep-th/0201151]. 42. R. D'Auria, P. Fre, P.K. Townsend and P. van Nieuwenhuizen, Invariance of Actions, Rheonomy and the New Minimal N = 1 Supergravity in the Group Manifold Approach, Annals Phys. 155, 423 (1984). 43. S.J.J. Gates, M.T. Grisaru, M.E. Knutt-Wehlau and W. Siegel, Component actions from curved superspace: Normal coordinates and ectoplasm, Phys. Lett. B 4 2 1 , 203 (1998) [hep-th/9711151]. 44. I. A. Bandos, D.P. Sorokin and D. Volkov, On the generalized action principle for superstrings and supermembranes, Phys. Lett. B352, 269 (1995) [hepth/9502141]. 45. P.S. Howe, O. Raetzel and E. Sezgin, On brane actions and superembeddings, JHEP 9808, 011 (1998) [hep-th/9804051].
M A R R I A G E OF 4-DIMENSIONAL GRAVITY TO T H E 3-DIMENSIONAL CHERN-SIMONS T E R M
R. J A C K I W Department of Physics, Massachusetts Institute of Technology Cambridge, MA 02139, USA E-mail: jackiwQlns.mit. edu
When 4-dimensional general relativity is extended by a 3-dimensional gravitational Chern-Simons term an apparent violation of diffeomorphism invariance is extinguished by the dynamical equations of motion for the modified theory. The physical predictions of this recently proposed model show little evidence of symmetry breaking, but require the vanishing of the gravitation Pontryagin density.
1. Dedication and Introduction Stanley Deser's activity in two fields has raised him to a SPIRES anointed renown. The two are gravity theory in 4-dimensional space time and ChernSimons as well as gravity physics in three dimensions. After completing his PhD half a century ago in 1953, he began writing on the former topic four years later. Another quarter century went by and he slipped from physical and higher dimensionality onto the plane — into (2+1) dimensional space time, where the Chern-Simons work has earned him his greatest renown. Investigations on planar physics have a long history, as is seen from Abbott's Flatland book. In this dimension one finds interesting mathematical structures, not seen in other dimensions. Also physical systems whose motion is confined to a plane are well described by these models. Finally one can get inspiration for higher-dimensional investigations, as is also indicated by Abbot.
150
151
" O day and night, but this is wondrous strange"
POWTLAND
A ROMANCE ~'XLS> OF M A N Y DIMENSIONS^
)'O,.DC: ^
c*.^Tir« Dimenihtts
Two Diatensiw
D
SPACELAffD
rLATLA!">
Edwin A. Abbott (1884) This Work is Dedicated By a Humble Native of Flatland In the Hope that Even as he was Initiated into the Mysteries Of T H R E E Dimensions
Having been previously conversant With ONLY T W O
So the Citizens of that Celestial Region May aspire yet higher and higher To the Secrets of F O U R F I V E OR E V E N SIX DIMENSIONS
Thereby contributing To the Enlargement of The IMAGINATION
And the possible Development Of that most rare and excellent Gift of MODESTY Exhortation by A b b o t t t o t h e study of various dimensions. His observation t h a t this will contribute "to t h e enlargement of t h e imagination" has been forcefully realized these days, even though his hope for a "development. . . of modesty" has not.
152
As mentioned already, Chern-Simons terms are odd-dimensional entities, interesting to mathematicians 1 and physicists.2 For physicists they are most useful when they are defined on 3-manifolds, and over the last twenty years they have been widely used to model various physical processes in 3dimensional space-time, that is, phenomena confined to motion on a plane, like the Hall effect, or gravitational motion in the presence of cosmic strings. However, these 3-dimensional structures can also be inserted into physical theories in 4-dimensional space-time, and the dimensional mismatch gives rise to kinematical and dynamical violation of Lorentz symmetry, CPT symmetry, etc. The subject of Lorentz and CPT symmetry violation is interesting these days, mainly due to initiating and stimulating work by theorists and experimentalists. The former build plausible extensions of standard theories, with small symmetry-breaking terms; the latter perform more and more precise experiments limiting the magnitude of such possible terms. Thus far, no evidence for symmetry breaking has been found; indeed, conventional symmetries are confirmed, with ever-decreasing uncertainty. Today, I shall describe a Chern-Simons modification of 4-dimensional gravity theory — Einstein's general relativity — and the associated decrease in symmetry. Actually, more than a decade ago, George Field, Sean Carroll and I investigated a Chern-Simons modification of Maxwell's electromagnetic theory. So in order to set the stage for the gravitational extension, I shall first review the Maxwell story. 3 2. Chern-Simons Modification of Maxwell Theory The Chern-Simons term for an Abelian gauge theory on an Euclidean 3space reads CS(4) = ie« f c 2V4 f c = i A . B .
(1)
The first expression is in tensor notation; the second in vector notation, with B being the magnetic field, B = V x A. All indices are spatial [i,j,k : x,y, z\. A related 4-dimensional formula in Minkowski space-time defines the topological Chern-Simons current K" = *F^A„, where *F
,iV
(2)
is the dual electromagnetic tensor. V " = \e^a0Fa0.
(3)
153
It is seen that the Chern-Simons term (1) is proportional to the time t((j, = 0) component of the Chern-Simons current, (2) with the time dependence suppressed. Also the divergence of the topological current is the topological Pontryagin density. d^K" = d^(*F^Av)
= \*F^F^.
(4)
In Chern-Simons modified electromagnetism the Chern-Simons term (1) (with field arguments extended to include t) is added to the usual Maxwell Lagrangian. 1=j d ' x ^ F ^ F ^
+ ^A-BJ.
(5)
Here fi, with dimension of mass, measures the strength of the extension. Formula (5) may be alternatively presented in covariant notation, with the help of an external, constant embedding 4-vector v^.
I = jd4x
(-±
F""FM„ + \ v^F^A^j
,
«/i = ( M , 0 ) .
(6)
In spite of the presence of the vector potential, the action is gauge invariant: Under a gauge transformation it changes by a surface term, since dtl*Ffil/ = 0. This can be made explicit by recognizing that in (6) there occurs the Chern-Simons current K^ (2). Therefore, with the help of (4) and an integration by parts the action acquires a gauge invariant form.
I = Jd4x(~
F^F^
0„0 =
+ \e "F^F^
v
, (7)
The external quantity is now 9, which is taken as 9 = fit so that (5) and (6) are reproduced. Since the explicitly covariant formulations (6), (7) involve external, fixed quantities [a fixed constant embedding vector uM in (6); a fixed function 9, linear in time, in (7)], we expect that Lorentz invariance is lost. Also, since A • B, and *FilvFllv are axial quantities, parity is lost; but C and T are preserved, so CPT is also lost. To confirm these statements, we now look to the solutions of the modified equations of motion. In the electromagnetic equations of motion, which follow from the Chern-Simons extended action, only Ampere's law is modified. - 5
+ V x B = J + /iB.
(8)
154
All other Maxwell equations continue to hold. Also the consistency condition on (8) remains as in Maxwell theory: the charge density p = V • E and the current J must satisfy their continuity equation, as is seen by taking the divergence of (8) and using V • B = 0. The modification that we have constructed is particularly felicitous for the following reasons.
(i) Gauge invariance is maintained, so the photon continues to possess just two independent polarizations. (ii) Eq. (8) is not a radical departure; it has played previous roles in physical theory: in plasma physics one frequently replaces the source current J with a magnetic field B. Of course, we are not working with a collective/phenomenological theory, like plasma physics, rather we are examining the feasibility of (8) for fundamental physics.
To assess the actual physical content of the Chern-Simons extended electromagnetism, and its associated symmetry breaking, we have examined some solutions. We found that in the source-free (J = 0) case, plane waves continue to solve the extended equations. The photon possesses two independent polarizations, (as anticipated from gauge invariance) however they travel at velocities which differ from the velocity of light (thus Lorentz boost invariance is lost — as anticipated) and also the two polarizations travel with velocities that differ from each other (thus parity invariance is lost — as anticipated). The fact that the two photon helicities travel (in vacuum) with different velocities makes empty space behave as a birefringent medium. Consequently linearly polarized light, passing through this birefringent environment, undergoes a Faraday-like rotation, which can be looked for in observations of light from distant galaxies. Much data exists on this phenomenon, and the conclusion is unavoidable: there is no such effect in Nature; /i = 0 is required. This was asserted in our initial investigations, 3 ' 4 and the many other analyses carried out in the intervening years support that conclusion (see e.g. [5]).
155
3. Chern-Simons Modification of Einstein Theory 3.1. Gravitational
Chern-Simons
term in
3-space
The 3-dimensional, gravitational Chern-Simons term can be presented in terms of the 3-dimensional Christoffel connection 3 r ? 2
cs(r) = e^ {\ 3r?9 dfri +13^
3
r« r 3 ry,
(9)
but it is understood that the Christoffel connection takes the usual expression in terms of the metric tensor, which is the fundamental variable. Variation with respect to the metric tensor of the integrated Chern-Simons term results in the 3-dimensional "Cotton tensor", which involves a covariant curl of the 3-dimensional Ricci tensor 3Rlf
•£- J d3xCS(T) = -y/g 3CV = \eimn
3
Dm3Ri + i~j.
(10)
z
C%i is symmetric, traceless and covariantly conserved. It vanishes if and only if the 3-dimensional metric tensor is conformally flat. A related formula gives the 4-dimensional Chern-Simons current K^, 1
n
on—v
r ! 7
(11)
whose divergence is the topological Pontryagin density. 1 d^K" = -*R% "" Kaia, Here RTa)lll
1 = -*RR.
is the Riemann curvature tensor and •R°T*» = I e^R%a0.
*R'JTPJV
(12) is its dual. (13)
[Notation: (i,j, ...) are 3-dimensional, spatial indices, and 3-dimensional geometric entities are decorated with the superscript "3". Undecorated geometric entities are 4-dimensional, and Greek indices label the 4 spacetime coordinates.] Note that unlike in the vector case, the Chern-Simons term (9) is not the time component K°, because the former contains 3dimensional Christoffel entities, while 4-dimensional ones are present in K°. This variety allows various extensions general relativity.
156
3.2. Gravitational
Chern-Simons
4~sPace
term in
In analogy with the electromagnetic formulation (7), we choose to extend Einstein theory by adopting the action 6
=
itc Jd'x {^R ~ \v^) '
v = dA
»
(14)
The first contribution is the usual Einstein-Hilbert term involving the Ricci scalar R. The modification involves an external quantity: 6 in the first equality; <9M0 = i>M in the second equality, which follows from the first by (12) and an integration by parts. Eventually we shall take the embedding vector Vfj, to possess only a time component, and 9 to depend solely on time. So then our modification (14) involves the time component of 4-dimensional Chern-Simons current (11) [rather than the 3-dimensional Chern-Simons term (9)]. The equation of motion that emerges when (14) is varied with respect to g^ is G^
+ C*v
= STTGT^
.
(15)
Here G^v is the covariantly conserved (Bianchi identity) Einstein tensor, Qfj.u = RHV _ lg^vR. We have inserted a source with strength G (Newton's constant) consisting of the matter energy-momentum tensor J7*", which also is covariantly conserved, since we assume matter to be conventionally, covariantly coupled to gravity. C^v is the term with which we are extending the Einstein theory. ^fTg C»v = - £ - \ fd4x Sg»v 4 J = - 1 {vae"^DaRvp
9*RR + vaT*W^
+ ft ~ v).
(16)
CM1/ is manifestly symmetric; it is traceless because *RR is conformally invariant. C " " s first term (involving the curl of RVj is similar to the 3-dimensional y/g3Ct:> (10). Even the second term can be viewed as a generalization from 3-dimensions: it involves only the Weyl tensor part of Riemann tensor, which vanishes in 3 dimensions. [Cotton defined his tensor in arbitrary dimensions d, and his definition is equivalent to ours in d = 3, where it is also given by the variation of the 3-d gravitational Chern-Simons term, as is (10).7 However for d ^ 3, Cotton's tensor does
157
not appear to have a variational definition. Our d = 4 Cotton-like tensor in (16) does possess a variational definition, at the the expense of introducing non-geometrical entities like 6 and uM.] Finally we must examine DliCliV, whose vanishing is a consistency requirement on (15). However, an explicit evaluation (which involves only geometric identities) shows that, unlike in 3 dimensions, C " is not covariantly conserved. Rather Z ^ C " = -r-^= v"*RR.
(17)
Thus the vanishing of *RR is a consistency condition of the new dynamics: every solution to (15) will necessarily lead to vanishing Pontryagin density, QHV + Cnv
=
-torQTiw => *RR
=
o.
(18)
We may derive and understand the expression for the covariant divergence of C " by examining the response of our addition to changes in the coordinates. With the infinitesimal transformation fa" = -F(x),
(19)
we have 5g»v = Dpfv + D„/„.
(20)
The Hilbert Einstein action is of course invariant. To assess the variance properties of our modification, we can proceed in two ways. First observe that *RR is scalar density, so it transforms as S(*RR) = d^(f)1*RR). 9 is an external quantity, therefore we do not transform it. 6ICS = - fd4x
95(*RR) = i J = -i
dAx6dll{f>i*RR) IcfixVpf^RR.
(21a)
Alternatively, we may vary Ics, by varying g^v according to (18), and using the definition (16) for C " . SIcs =
= l\i?x4=i
C<"£V/„ = -2 J dlx V=5 (DeC")U.
(21b)
Equating the two expressions for 5Ics establishes (17), and also demonstrates that *RR is a measure of the failure of diffeomorphism invariance.
158
But *RR vanishes as a consequence of the equation of motion, so is some sense diffeomorphism invariance is dynamically reinstated. For another perspective, consider a variant of our model, where 9 in (14) is a dynamical variable, not an externally fixed quantity. If we postulate that under diffeomorphisms (19) 9 transforms as a scalar, 69 = / " 3 „ 0 = / % ,
(22)
then (21a) acquires the additional contribution j f d4x 66 {*RR) = i / d4x v^ f
*RR,
(23)
which cancels (21a), showing that the Chern-Simons modification with dynamical 9 is diffeomorphism invariant. Now let us look at the equations of motion in this variant of modified gravity: varying g^v still produces (15); varying 9, which now acts as a Lagrange multiplier, forces *RR to vanish, but that requirement is already implied by (15), (18). Thus the equations of the fully dynamical, and diffeomorphism invariant theory coincide with the equations of the non-invariant theory, where 6 is a fixed, external quantity. Formula (21a) shows that when v^ is chosen to have only a time component, v^ = (1//J,,0); equivalently 6 = t/pi, then IQS is invariant under all space-time reparametrizations of the spatial coordinates, and also of shifts in time: / ° = constant, /'arbitrary. Henceforth we make this choice for v^ and 9. 3.3.
Physical
effects of the Chern-Simons
term in 4-d
gravity
We examine some physical processes in the Chern-Simon modified gravity theory. (i) It is important that the Schwarzschild solution continues to hold; thus our theory passes the "classic" test of general relativity. The result is established in two steps. First we posit a stationary form for the metric tensor
*-=(offg>J'
(24)
with time-independent entries. It follows that C 0 0 and C n 0 = C0n vanish. Also one finds that the spatial components reproduce the 3-dimensional Cotton tensor. v/^C'^v^C".
(25)
159
Next, we make the spherically symmetric Ansatz, and find that ClJ' vanishes. Evidently also *RR must vanish on the Schwarzschild geometry, because the modified equations are satisfied. Since the Kerr geometry, carries non vanishing *RR, it will not be a solution to the extended equations. It remains an interesting, open question which deformation of the Kerr geometry satisfies the Chern-Simons modified equations. (ii) Next we perform a linear analysis by expanding the metric tensor around a flat background g^v = rj^ + h^. The purpose of the linear analysis is to determine the propagating degrees of freedom, to study the nature of small disturbances (gravity waves) and to illuminate the construction of an energy-momentum (pseudo) tensor, which is symmetric and divergence-free. Keeping only the linear portions of the Einstein tensor and (7M!/, we verify that both G^" ear and Cj i '" ear are divergence-free: a M G Jinear = Q
=
.^linear
^
This is seen from the explicit formulas. It also follows from the observation that the exact equation D^G^ = 0 implies the above result for G ^ e a r ; moreover, from (17) we see that D^C^U is of quadratic order, hence the above result for G^" ear holds also. It is further seen that the linear portions are invariant under the "gauge" transformation + aMA„ + ^A M .
(27)
In the Einstein theory, one decomposes h^u into temporal parts, and purely spatial parts hlK The latter is further decomposed into its trace, its longitudinal part, and its traceless transverse part, denoted by hl£T. One then finds from the linear equations that, with the exception of hZrfT, all other components of h^ are either non-propagating or can be eliminated by the gauge transformation (27). Only h%T survives and it is governed by a d'Alembertian. Since in 4 dimensions a symmetric, transverse and traceless 3 x 3 matrix possesses two independent components, one concludes that in Einstein's theory small gravitational disturbances are waves, with two polarizations, each moving with the velocity of light (governed by the d'Alembertian). None of this changes where C^" ear is included. Again only hj,T propagates, governed by the d'Alembertian. Explicitly the modified equation for
160
h^T
reads (Sim5jn
+ — e " " " 5nj dp + —eivm =
-16TT
6ni dp) D
h^
G T^r.
(28)
T^ T is the transverse traceless part of the stress tensor. The new terms are the (n~x) contributions; they involve only spatial derivatives. One may consider that the left side of (2.20) involves an operator acting on • h™. OiJmnU
h
TT = -16TTG T$T.
(29a) l
Acting on this equation with the inverse operator V — 0~ effect of the entire extension is to modify the source
shows that the
• K™ = -16TTG V"171^ T^T = -16TTG f$T.
(29b)
Thus we see that in sharp contrast to the electromagnetic case, the Chern-Simons modification of gravity does not change the velocity of gravity waves and there is no Faraday rotation. It is also noteworthy that the reduction to 2 degrees of freedom (2 polarizations) takes place also in the extended theory. Such a reduction of degrees of freedom is considered to be a consequence of gauge invariance, here diffeomorphism invariance, which evidently continues to hold on our modified theory. There does exist a physical manifestation of the extension. Although the velocities of the two polarizations are the same, their intensities differ, due to the modification of the source (T^T —> T^T). One finds for a weak modification (large JJL) that the ratio of the intensity of waves with negative helicity to those with positive helicity is
where to is the frequency. This puts into evidence the parity violation of the modification. Finally we turn to the topic of the energy-momentum (pseudo) tensor. A straight forward approach to this problem in the Einstein theory is to rewrite the equation of motion by decomposing the Einstein tensor GM„ into its linear and non-linear parts, and moving the non-linear terms to the "right" side, summing it with the matter energy-momentum tensor: G^T
= -STT G f TM„ + - ^
G-n-Unear) •
(31)
161
Clearly Gjj" ear is symmetric and conserved, therefore, so must be the right side, which is now renamed as total (gravity + matter) energy-momentum (pseudo) tensor:
J_ G-"- l i n e a r r„„ + — (32) &rG Exactly the same procedure works in the extended theory. We present the equation of motion (15) as G linear
, /^linear
o_/"< / T
i
/^non-linear
i /~mon-linear\ \
/oo\
We have already remarked that the left side is divergenceless. Thus we can identify a symmetric and conserved energy- momentum (pseudo) tensor as
Ti
i_
J_ //^rnon-linear x
, /~mon-linear\
(r>A\
It is striking that this structure is present in a theory that seems to violate Lorentz invariance! In Ref. [8] there is a survey of other gravitational energy-momentum (pseudo) tensors for Einstein's theory that differ from each other by super potentials. In particular there is described a Noether construction with a Belinfante improvement, which also yields a symmetric, conserved energymomentum (pseudo) tensor tied to the Poincare invariance of the Einstein theory. It would be interesting to reconsider this construction in the extended theory and to compare the result to (34). 4. Conclusion Measuring the intensity of polarized gravity waves is not feasible. Thus for present days, our model is only a theoretical exercise. Nevertheless, it shows interesting and unexpected behavior in that an important symmetry — diffeomorphism invariance — is not present in the action, but is restored by the equations of motion. Correspondingly the physical effects of the symmetry breaking are quite hidden. An analogy can be made with the Stiickelberg formalism for massive, Abelian gauge fields. The action
-*m — •
/
"gauge i
m
'
is not gauge invariant —
I m ^d A = 2
M
fmadllA't\,
162
but is seen to be broken by d^A1*. However, the equation of motion dllF'"' + m2Av
= Jv
has as a consequence (for conserved m a t t e r currents) dl/Av equation may be presented in gauge invariant form.
= 0, and the
d^v + m2^~^JA^ = r. B u t an important difference remains. If t h e the mass is promoted to a field, m 2 —> m 2 (a;), and this field is varied, then the resulting equation A^An = 0, is not consequent t o t h e original equation of motion. Indeed it is an unacceptable equation, because it prevents finding non trivial solutions. (One recognizes the Higgs mechanism, in unitary gauge, as providing a kinetic t e r m and a potential for the field "m 2 (o;)", so t h a t its equation of motion becomes dynamically acceptable.)
References 1. S. Chern, Complex Manifolds without Potential Theory, 2nd ed. (Springer, Berlin, 1979). 2. S. Deser, R. Jackiw and S. Templeton, Ann. Phys. (NY) 140, 372 (1982), Erratum-ibid. 185, 406 (1988). 3. S. Carroll, G. Field and R. Jackiw, Phys. Rev. D 4 1 , 1231 (1990). 4. S. Carroll and G. Field, Phys. Rev. Lett. 79, 2394 (1997). 5. M. Goldhaber and V. Trimble, J. Astrophys. Astron. 17, 17 (1996); T. Jacobson, S. Liberati, D. Mattingly and F. Stecker, astro-ph/0309681. 6. R. Jackiw and S.-Y. Pi, Phys. Rev. D68, 104012 (2003); A. Lue, L. Wang and M. Kamionkowski, Phys. Rev. Lett. 83, 1506 (1999). 7. A. Garcia, F. Hehl, C. Heinicke and A. Macias, Class. Quant. Grav. 2 1 , 1099 (2004). 8. D. Bak, D. Cangeini and R. Jackiw, Phys. Rev. D 4 9 , 5173 (1994).
EINSTEIN-.ETHER THEORY
CHRISTOPHER ELING*, TED JACOBSONt'*, and DAVID MATTINGLY § t
Dept. of Physics, University of Maryland, College Park, MD 20742-4111, USA * Institut d'Astrophysique de Paris, 98bis Bvd. Arago, 75014 Paris, France * Dept. of Physics, University of California, Davis, California, 95616, USA We review the status of "Einstein-./Ether theory", a generally covariant theory of gravity coupled to a dynamical, unit timelike vector field that breaks local Lorentz symmetry. Aspects of waves, stars, black holes, and cosmology are discussed, together with theoretical and observational constraints. Open questions are stressed.
1. Introduction Could there be an aether after all and we have just not yet noticed it? By an "aether" of course we do not mean to suggest a mechanical medium whose deformations correspond to electromagnetic fields, but rather a locally preferred state of rest at each point of spacetime, determined by some hitherto unknown physics. Such a frame would not be determined by a circumstance such as the moon's gravitational tidal field, or the thermal cosmic microwave background radiation, but rather would be inherent and unavoidable. Considerations of quantum gravity have in multiple ways led to this question, and it has also been asked in the context of cosmology, where various puzzles hint that perhaps something basic is missing in the standard relativistic framework. Lorentz symmetry violation by preferred frame effects has been much studied in non-gravitational physics, and is currently receiving attention as a possible window on quantum gravity.1 But what about gravity itself? General relativity is based on local Lorentz invariance, so if the latter is violated what becomes of the former? It is hard to imagine, both philosophically and technically, how we could possibly give up general covariance, the deep symmetry finally grasped through Einstein's long struggle. Thus the question that interests us here is whether a generally covariant effective field theory with a preferred frame could describe nature8-. a
More general sorts of Lorentz violation in the gravitational sector are examined in 163
164
The simplest description of such a frame would appear to be via a scalar field T, a cosmic time function, which has been proposed in various contexts. 3 ' 4 ' 5 ' 6 The gradient T a , if timelike, defines a preferred rest frame, and one can envision dynamics that would force it to be everywhere timelike. But while a scalar field is simplest, the norm of the gradient | T 0 | is "extra information", which has nothing to do with specifying a frame per se but rather specifies the rate of a particular cosmic clock. It may be that Nature provides such a clock; we just wish to point out that the clock rate is extra information. Constraining the gradient to have fixed norm is not a viable option since, as explained in section 3, this would lead inevitably to caustics where T;0b diverges. Another noteworthy feature of using a scalar is that, by construction, the 4-velocity of the preferred frame is necessarily hypersurface-orthogonal, i. e. orthogonal to the surfaces of constant T. Again, perhaps this is the way Nature works, but it is a presumption not inherent in the notion of a local preferred frame determined by microphysics. The alternative discussed in this paper is to describe the preferred frame by a vector field constrained kinematically to be timelike and of unit norm, which we call the (Ether field ua. Such a field is specified by three independent parameters at each point, and generally couples via covariant derivatives, so the theory is far more complicated than that of a scalar time function. It is instinctive to worry about ghost modes given a vector field without gauge invariance. However the unit constraint on the vector renders it an unfamiliar beast. All variations of the vector are spacelike, since they connect two points on the unit hyperboloid, so ghosts need not arise. There is a sparse history of studies of unit vector fields coupled to gravity. 7 ' 8 ' 9,10 ' 12 ' 13 ' 14 Here we focus on the particular approach and results in which we have been involved. 15,16 ' 17 ' 18 ' 19 ' 20 We begin with the action principle that defines the theory, and then discuss a Maxwell-like special case, linearized waves, PPN parameters, energy, stars and black holes, and cosmology. 2. Einstein-aether action principle In the spirit of effective field theory, we consider a derivative expansion of the action for the metric gab and aether ua. The most general action that is diffeomorphism-invariant and quadratic in derivatives is S = j Ref. [2],
^ fdAxV=g(R
+ KabmnVaumVbun
+ \(uaua
- 1)),
(1)
165
where Kabmn
= cigahgmn
+ c25am5bn + c35aJbm + c4uaubgmn.
(2)
The coefficients 0^2,3,4 are dimensionless constants, R is the Ricci scalar, and A is a Lagrange multiplier that enforces the unit constraint. The metric signature is (-) ), and units are chosen such that the speed of light defined by the metric gat, is unity. The constant G is related to the Newton constant G N by a c^-dependent rescaling to be discussed below. Other than the signature choice we use the conventions of Ref. [21]. The possible term RabUaub is proportional to the difference of the C2 and C3 terms via integration by parts, hence has been omitted. We have also omitted any matter coupling since we are interested here in the dynamics of the metricaether sector in vacuum. Note that since the covariant derivative of ua involves the Levi-Civita connection, which involves first derivatives of the metric, the aether part of the action in effect contributes also to the metric kinetic terms. We call the theory with this action Einstein-tether theory, and abbreviate using "jE-theory". Another way to express the theory is using a tetrad eaA rather than the metric, where A is a Lorentz index. Then the aether can be specified as ua — uAeaA, with a unit Lorentz 4-vector uA satisfying the constraint VABUAUB = 1, where T}AB is the fixed Minkowski metric. This decouples the normalization condition on uA from the dynamical metric. The Lagrange density is then of the form K^DauAD\>uB, where Da is the Lorentzcovariant derivative involving the spin-connection u^0, and K^ is a linear combination of the four terms gabr)AB, eAebB, eBebA, and ucuDeacehDJ]ABThis theory has a local Lorentz invariance, which can be used to set the components of uA to (1,0,0,0). That produces the form of the theory as presented by Gasperini. 7 One can also use a Palatini formalism, in which the spin connection is treated as an independent variable to be determined via its field equation. In this case the spin connection has torsion, because of the coupling to uA. If the solution u^D(e,u) is substituted back into the action, one returns to the tetrad form, but with different coefficients for each of the four terms in K^. The relation between these coefficients and the original ones has not yet been worked out. Given a metric and a unit vector field, there is a one parameter family of metrics that can be constructed (aside from simple rescalings). When expressed in terms of a different metric in this family, the action changes, but only insofar as the values of the Cj in (2) are concerned. More precisely,
166
consider a field redefinition of the form 9'ab = 9ab + {B~
l)uaUb,
U'a = Cua,
(3)
with C = [1 + (B - l ) u 2 ] - 1 / 2 (where ua := gamum and u2 = gmnumun). Lorentz signature of both gab and g'ab requires B > 0. The coefficient C is chosen such that g'abu'au'b — gabuaub, so the unit constraint is unchanged, hence in the action we can put simply C — J B - 1 / 2 . The action (1) for (g'ab, u,a) takes the same form as a functional of (gab, ua), but with different values of the constants c*. The general relation between the c; and the c^ has recently been worked out by Foster. 11 His results reveal, for example, that one can arrange for C\ + C3 = 0 by choosing B = 1 — c\ — c 3 (provided (c[ + c3) < 1 for Lorentz signature). A special case previously worked out by Barbero and Villasenor10 shows that the as-theory is equivalent via field redefinition to GR when the parameters satisfyb C1+C4 = 0, C1+C2+C3 = 0, and C3 = ±y/ci(ci — 2). Lorentz signature implies c\ < 0. Note that if one first makes a field redefinition such that c\ + c 3 = 0, then the BarberoVillasenor result reduces to the statement that the theory is equivalent to GR only if all coefficients vanish. Hence with the c\ + C3 = 0 description we ensure that non-zero coefncients always represent true deviations from GR. The field equation from varying the aether in the action (1) takes the form V 0 Jam - c 4 u a V m w a = Xum,
(4)
where Jam := KabmnVbun,
(5)
and un = ub\7bUn. The field equation from varying the metric in the action (1) together with a matter action takes the form Gab = T™ + 87rGT a r t t e r ,
(6)
b This corrects our earlier statement of the equivalence in Ref. [18]. We thank B.Z. Foster for pointing out this error.
167
where the aether stress tensor is given by 18 T%] = V m ( J{amub)
- Jmiaub) m
+C1 [(^mUa)(W Ub)
-
J(ab)um) {VaUm){VbUm)]
+C4 UaUb
+ [«n(V m Jmn)
- Ciil2} UaUb
--^9abLu.
(7)
The constraint has been used in (7) to eliminate the term that arises from varying yj^g in the constraint term in (1), and A has been eliminated by solving for it via the contraction of the aether field equation (4) with ua. The notation Lu = —KahmnVaumVbun is the aether lagrangian. Our goal is to determine the theoretical and observational constraints on the parameters Cj, and to identify phenomena whose observation could reveal the existence of the aether field. For such phenomena one can look at post-Newtonian effects, gravitational and aether waves, and cosmology. 3. Maxwell-like simplified theory Before considering the general, rather complicated, theory it makes good sense to ask if there is a simplification that might serve at least as a decent starting point. A great simplification occurs with the choice c\ + C3 = 0 and C2 = c\ = 0, so that the connections drop out of the aether terms in (1). The aether part of the Lagrange density then reduces to 2clU[a,b]u^
+ \{u2 - 1).
(8)
This theory was studied long ago by Nambu 22 in a fiat space context. It is almost equivalent to Einstein-Maxwell theory in a gauge with u2 = 1. The difference is that one equation is missing, since the action need only be stationary under those variations of u that preserve u2 = 1. The missing equation is an initial value constraint equation, Gauss' law. If the current to which ua is coupled is conserved, then Gauss' law holds at all times if it holds at one time. This theory coupled to dynamical gravity was first examined in Ref. [8], and further studied extensively in Ref. [15] and Ref. [9]. In Ref. [15] it was shown to be equivalent to Einstein-Maxwell theory coupled to a charged dust, restricted to the sector in which there exists a gauge such that the vector potential is proportional to the 4-velocity of the dust, i.e. the aether
168
field ua. The charge-to-mass ratio of the dust is given byc (8-KG/CI)1/2. The extremal value corresponds to c\ = 2. A number of results were established concerning static, spherically symmetric solutions, black holes, and linearized solutions. This case is appealing due to its simplicity, however a serious flaw was noticed: solutions can have "shocks" or caustics beyond which the evolution of the aether cannot be extended. In particular, 15 consider the aether configurations that can be written as the gradient of a scalar ua = T a , so that the Maxwell-like "field strength" tensor «[0,6] vanishes and ua is orthogonal to the surfaces of constant T. Then the field equations reduce to the vacuum Einstein equation, together with the vanishing of the Lagrange multiplier A and the statement that the gradient of T is a unit vector, T
= u a V 6 u a = 0.
(9)
The first equality holds since Ub is a gradient, and the second holds since it is a unit vector. If we launch geodesies orthogonal to an initial surface of constant T, they will generically cross after some finite proper time. Where they cross there is no well-defined value of ua, and the derivative Va«(, is singular. These are the shock discontinuities. A different demonstration of the existence of shocks appears in Ref. [9]. We note in passing that the preceding demonstration of shocks applies in a very different context, namely the version of fc-essence4 recently called "ghost condensation" d . 5,6 This is the theory of a scalar field
T h e convention with Maxwell Lagrangian given by —F2/16ir is adopted here. For a discussion of caustics in a more general fc-essence scenario see Ref. [23]. e T h e argument in Ref. [9] has a minor flaw, but the conclusion is correct. The negative
169
energy configurations in this case is related to the fact that the Lagrange multiplier A can be negative, so in the charged dust interpretation the mass density is negative. Before returning to the general class of Lagrangians we note that one might also consider the theory where the restriction on the norm of ua is enforced not rigidly by a constraint but rather by a potential energy term V(uaua) in the action. This approach was discussed by Kostelecky and Samuel,8 and more recently explored by Bjorken,24 Moffat,25 and Gripaios. 26 It has an additional, massive, mode, which should be checked for a possible wrong sign of the kinetic energy. 4. Waves The spectrum of linearized waves is important for several reasons. First, it can be used to constrain the theory a priori, by rejecting values of the parameters c* for which waves carry negative energy or for which there are exponentially growing modes. Second, wave phenomena can be used to place observational constraints on the parameters, using radiation from compact objects such as the binary pulsar, as well as cosmological perturbations. The spectrum of linearized waves around a flat spacetime background was worked out for the general theory defined by the action (1) in Ref. [19]f. The wave modes in a de Sitter background were found in Ref. [14] (for C4 = 0), which also studied the metric perturbations in inflation interacting with the vector as well as a scalar inflaton. Here we summarize the results for the modes around flat space. Since the aether has three degrees of freedom, the total number of coupled metricaether modes is five. There are two purely gravitational (spin-2) modes, two transverse aether (spin-1) modes in which the aether vector wiggles perpendicular to the propagation direction, and one longitudinal or "trace" (spin-0) mode. The waves all have a frequency that is proportional to the wave vector. Hence they are "massless" and have fixed speeds. The speeds for the different types of modes are all different, and each mode has a energy configuration described there is a time-independent pure gradient Ui = di<j>{S). This initial data with vanishing time derivative u^t = 0 indeed has negative energy, however the equation of motion implies that the time derivative does not remain zero (unless Ui = 0 ) . f T h e Maxwell-like special case was previously treated in Ref. [15], and the case with only c\ non-zero was treated in Ref. [16]. In Ref. [14] the modes were found in the small c, limit where the aether decouples from the metric (cf. section 6.2.2).
170 Table 1.
Wave mode speeds and polarizations.
squared speed 1/(1-C13)
polarization /112, / i n = -/122
(ci - \c\ + i c § ) / c X 4 ( l - C13)
/1/3 = [ c i 3 / s ( l - ci3)]uj
Cl23(2 - Cl 4 )/Ci 4 (l - Cl 3 )(2 + C13 + 3c 2 )
/lOO = -2t)0 /ill = /122 = — C14V0 /l33 = (2C14/C123)(1 + C2)VQ
particular polarization type. Table 1 gives the speeds and polarizations for the spin-2, spin-1, and spin-0 modes, in that order. The metric and aether have been expanded as gab = r]ab + hab and ua = ua + va, where r\ab is the Minkowski metric and ua is the constant background value that has components (1,0,0,0) in the coordinate system adopted. The gauge conditions hoi = 0 and Viti — 0 are imposed, where i stands for the spatial components. The propagation direction corresponds to i = 3, and 7 = 1,2 labels the transverse directions. The notation C123 stands for C\ + c? + C3, etc, and s is the wave speed. 4.1.
Stability
The squared speed refers to the squared ratio of frequency to wave-vector, so if it is negative for real wave-vectors the frequency is imaginary, implying the existence of exponentially growing modes 8 . The requirement that no such modes exist restricts the parameters of the theory h . For ct small compared to unity this requirement reduces to the conditions c\/c\n > 0 for the transverse vector-metric modes and C123/C14 > 0 for the trace mode. Lim argued 14 that one should additionally demand that the modes propagate subluminally (relative to the metric gab)- Although there is nothing wrong with local superluminal propagation in a Lorentz-violating theory, he pointed out that the vector field (in an inhomogeneous background) might tilt in such a way as to allow energy on such locally superluminal paths to flow around a closed spacetime curve. It is not clear to us that it is really necessary to impose this extra demand, since even in general relativity the classical field equations do not forbid the formation of closed timelike curves, around which relativistic fields could propagate. In any s
T h e factor s in the polarization /1/3 of the transverse aether mode implies that when s2 < 0 there is a 7r/2 phase shift of the metric perturbation relative to that of the aether. h T h i s is not yet an observational constraint since it has not been shown that the growing modes are not stabilized before their effects become apparent.
171
case, if we do make this demand, then in the case of small Q it implies C13 < 0, ci/cu < 1, and cus/cu < 1. On the other hand, if gravitational waves propagate subluminally relative to the "speed of light" for matter, then matter can emit gravitational Cerenkov radiation. Using this phenomenon, a very tight constraint on the difference between the maximum speed of high energy cosmic rays and that of gravitational waves was derived by Moore and Nelson.27 For cosmic rays of galactic origin, the constraint is Ac/c < 2 x 10~ 15 , while for extragalactic cosmic rays it is Ac/c < 2 x 10~ 19 . Cerenkov radiation in the additional, aether-metric modes has not been examined. Constraints could conceivably eventually be obtained using these processes. The requirement that all the modes propagate on the light cone of gab is satisfied if and only if C4 = 0, C3 = —Ci, and ci = c i / ( l — 2ci) . 4.2. Astrophysical
radiation
Since there are three additional modes, as well as a modified speed for the usual gravitational waves, one expects that the energy loss rate for orbiting compact binaries will be affected. Not only are there additional modes to carry energy, but also lower multipole moments may act as sources, for example there may well be a monopole moment generating the spin-0, "trace" mode. These phenomena have not yet been studied. Agreement with observations of binary pulsars should yield constraints on the c* parameters. Another potential source of constraints is the primordial perturbation spectrum in cosmology. Lim 14 has begun a study of this. 5. Newtonian limit and P P N parameters Carroll and Lim 13 have examined the Newtonian limit. They adopt the ansatz of a static metric, with the aether vector parallel to the timelike Killing vector. Restricting to the linearized field equations they recover the Poisson equation V2t/jv = inGwpm for the gravitational potential UN, where pm is the usual matter energy density and GN
= 1 G, / 9 ' 1 - C14/2
(10)
where G is the parameter appearing in the action (1). Actually c\ was set to zero in Ref. [13], but it can be restored without calculation by using the fact 18 that in spherical symmetry the effect of the a term can be generated by the replacements c\ —> C14 and C3 —+ C3 — C4.
172
The PPN formalism 31,32 can be applied to jE-theory since it is a metric theory, at least in the approximation (which is observationally known to be very accurate) that the matter is minimally coupled in the usual way to the metric1. In the general setting there are ten PPN parameters, but five of them vanish automatically in any theory that is based on an invariant action principle, so we need consider only the five remaining parameters, The two PPN parameters (3 and 7, known as the Eddington-RobertsonSchiff (ERS) parameters, are defined by the PPN expansion for the metric coefficients, goo = 1 - 2UN + 2pUN + • • • , gij = (l + 2~,UN + ---)5ij,
(11) (12)
where UN is the Newtonian gravitational potential. Thus (3 controls the non-linearity and 7 the space curvature due to gravity. In general relativity j3 = 7 = 1. The field equations are apparently too complicated to solve analytically, even assuming static, spherical symmetry. By numerical solution of the weak field equations in a 1/r expansion it was found18 that the metric takes the form (11,12), with UN is proportional to 1/r, consistent with the Newtonian limit. Moreover, the ERS parameters take precisely the same values as in general relativity in the generic case C123 7^ 0. (For the special cases with C123 = 0 see Ref. [18].) Hence the theory is indistinguishable from GR at the static, post-Newtonian level. To expose the post-Newtonian differences from GR it is necessary to examine the remaining PPN parameters, but these have not yet been computed for the jE-theory. The parameter £ is related to preferred location effects, and likely vanishes in iE-theory. The parameters a l i 2 are related to preferred frame effects and almost surely do not vanish. They will presumably arise when motion of the gravitating system relative to the asymptotic aether frame is allowed for*. One can hazard a guess based on the PPN parameters that were calculated for the vector-tensor theory without the unit constraint. 31 In that case, (3 and 7 are also unity in the case that corresponds most closely to JEtheory, namely u> = 0 in the vector-tensor parameters of Ref. [31] together 'Strictly speaking, if the matter couples also to the aether, producing local Lorentz violating effects in the matter sector, then the P P N formalism must be modified to allow for dependence of the P P N parameters on the composition of the matter sources. JA recent preprint of Sudarsky and Zloshchastiev 33 addresses this point.
173
with C4 = 0 in our parameters. 18 If this agreement persists, one would have in this case £ = ot\ = 0, but a2 7^ 0 generically. However, in the special case when C13 = 0, for which the spin-2 waves propagate on the light cone of gab, the vector-tensor parameters satisfy r = 77, and the result of Ref. [31] yields a2 = 0. Thus perhaps in this special case all the PPN parameters of Einstein-jEther theory agree with those of GR. Current limits on a^ and a2 are of order 1 0 - 4 and 1 0 - 7 respectively, so constraints of this order on the parameters of iE-theory might be expected, at least for generic parameters. 6. Energy 6.1. Total
energy
As discussed in section PPN, the Newtonian potential satisfies the Poisson equation with source term 87rGNPm- This implies that in terms of the metric coefficient goo ~ 1 — ro/r, the source mass is given by m = ro/2G^. Assuming the source couples minimally to the metric gab, this mass corresponds to an energy m (since the speed of light defined by gab is one in our units). Accordingly, one can infer that the energy of any isolated gravitating system is given by the same formula. This differs from the ADM mass ro/2G that one would have inferred from the action (1). The reason is that the aether stress tensor (7) adds \/r terms to the Newtonian field equation. Another path to the same conclusion is to examine the energy via its definition as the value of the Hamiltonian that generates asymptotic time translations. Using Wald's Noether charge formalism, Foster 28 showed that when one takes into account the falloff behavior of the fields at spatial infinity, the asther contribution to the energy flux integral takes the form ^(aether)
=
__i_
/ tf g
Rta^
y ^
( 1 3 )
where Kmnab is the tensor defined in (2). An equivalent result was found by Eling 29 using the Einstein pseudotensor. It was shown in Ref. [18] that for a spherically symmetric, static solution the line element has the asymptotic form ds2 = (1
_ r± r
+
_ _ _} dt2 _
(1 +
r_o + _ _ _} dj,2 _ r2dn^ r
(H)
and the asther has a ^-component of the form
„« = ( ! - £ ) + ....
(15)
174
The r-component of the aether starts at 0(1/r2), hence does not contribute to (13). Using this form in (13) one finds for the total energy
£=£(l-f).
(16)
Using the relation (10) between G and G N , this can be re-expressed as T-O/2GN- Thus the total energy is related to the ADM mass MADM = TO/2G by a constant rescaling.
6.2.
Positivity
For coefficients Cj such that 2 — CH is positive, positivity of the energy is thus equivalent to positivity of the ADM energy. The usual positive energy theorem 30 for GR assumes the dominant energy condition holds for the matter stress tensor, and proves that the total energy-momentum 4vector of the spacetime is future timelike. The aether stress tensor (7) does not appear to satisfy the dominant energy condition (for any choice of the Cj), so the proof does not go through as usual. Nevertheless, as discussed below, the energy of the linearized theory is positive for certain ranges of Cj. Perhaps a total divergence term that leaves the energy unchanged must be removed before positivity can be seen. Also, since the asymptotic value of the unit vector selects a preferred frame, it might be that the energy is always positive only in that particular frame. We can make no definite statement about the non-linear energy at this time, based on general formal grounds.
6.2.1. Linearized wave energy It is useful to examine the linearized theory to begin with. The energy density of the various wave modes has been found20 using the Einstein or Weinberg pseudotensors, averaging over oscillations to arrive at a constant average energy density for each mode. The energy density for the transverse traceless metric mode is always positive. For the vector modes it is positive provided (2c\ — c\ + c§)/(l - ci 3 ) > 0, and for the trace mode it is positive provided CH(2 — cu) > 0. For small c» these energy positivity conditions reduce to c\ > 0 and C\± > 0, respectively. In the Maxwell-like case c\z = C2 = c\ = 0 the linearized energy positivity requirement reduces to 0 < c\ < 2. The negative energy configurations discussed in section 3 do not show up in the linearized limit. Their energy density is proportional to — (Vu 0 ) 2 , which is quartic in the perturbation ui.
175
6.2.2. Gravitational decoupling limit The linearized waves are coupled metric-aether modes. A simpler limit to consider is a decoupling limit in which gravity is turned off. To access this limit formally we can let G and Q tend to zero, while the ratios c^ = Cj/G are held fixed. If the metric is expanded as g = rj + \/Gh in the action (1), and the limit G, Ci —> 0 is taken, then one is left with just the action for linear gravitons and a decoupled aether action where all metrics are replaced by rj and all covariant derivatives are replaced by ordinary partial derivatives. This limit was studied by Lim 14 in the case c\ — 0. Restoring the Ci dependence one finds perfect agreement between his results and the decoupling limit of the coupled linearized waves. As mentioned in section 3, Clayton showed that the energy can be negative in the Maxwell-like special case. He also claimed that this remains true for more general choices of the coefficients Ci. However, the case where only c\ is non-zero corresponds in the decoupling limit to a nonlinear sigma model (NLSM) on the unit hyperboloid which, like all NLSM's, has a stress tensor satisfying the dominant energy condition. 6.3. Summary
of constraints
on the
parameters
So far we have discussed constraints from requirements in the linearized theory of positive energy, stability (no exponentially growing modes), and subluminal propagation (not necessarily required). Taken together, the constraints of positive energy and stability in the linearized regime imply C\ > 0, C14 > 0, and C123 > 0 (for small parameters). These are likely necessary for a viable theory. The requirement of no superluminal propagation (which we do not feel is necessary) would additionally imply C13 < 0, c i < C14, a n d C123 < C14.
There is plenty of parameter space in which all the linearized constraints one might think of imposing are satisfied. It remains an important open question to determine whether energy positivity can be ensured beyond the linearized limit. But one thing is already notable, namely that this provides examples of a theory of a vector field which has no standard gauge symmetry and yet which has only stable, positive energy modes. The key factor making this possible is the constraint on the norm of the vector. 7. Stars and black holes? In static, spherical symmetry the aether vector must be a linear combination of the time-translation Killing vector and the radial vector. Solutions were
176
found in Ref. [18] for which the aether field has a radial component that falls off as 1/r2. The question arises as to what happens to this vector in the near field region of a star. Symmetry and regularity imply that the radial component vanishes at the origin of spherical symmetry, so if indeed a regular solution exists, ua must be parallel to the Killing vector at the origin and at infinity but not in between. Such solutions have recently been studied perturbatively by Sudarsky and Zloshchastiev.33 We have recently shown numerically that solutions with no radial component of the aether also exist. 34 In the case of a black hole there is no regular origin of spherical symmetry, but the question arises as to what happens to ua on the horizon. The vector ua cannot exist at a bifurcation surface (where the Killing vector vanishes, like the 2-sphere at the origin U = V = 0 of Kruskal coordinates). The reason is that the Killing flow acts there as a Lorentz boost in the tangent space of any point on the bifurcation surface, hence would act non-trivially on ua. Thus ua could not be invariant under the Killing flow. Since ua is constrained to be a unit vector it cannot vanish, hence we infer that it must blow up as the bifurcation surface is approached. One might think that the impossibility of an invariant asther at the bifurcation surface implies there is no regular black hole solution in this theory, since regularity on the future horizon is somehow connected to regularity at the bifurcation surface. It seems this is not necessarily the case. A result of Racz and Wald 35 establishes, independent of any field equations, conditions under which a stationary spacetime with a regular Killing horizon can be globally extended to a spacetime with a regular bifurcation surface, and conditions under which matter fields invariant under the Killing symmetry can also be so extended. In spherical symmetry the conditions on the metric are met for a compact Killing horizon with constant, non-vanishing surface gravity, so the result of Ref. [35] indicates that an extension to a regular bifurcation surface must exist. However, one of the conditions on the matter {i.e. aether) field is not met, namely, it is not invariant under the time reflection isometry. This is because the timelike vector ua obviously breaks the local time reflection symmetry. Thus the aether vector need not be regular at the bifurcation surface (although all invariants must be regular and, given the Einstein equations, the aether stress-tensor must remain regular in the limit that the bifurcation surface is approached). Hence, as far as we know, there is no argument forbidding regular black hole solutions. In fact, we have expanded the field equations about a regular, static fu-
177
ture event horizon and shown34 that locally regular solutions exist. Using a shooting method, we have shown that the free parameters can be chosen so that the solutions extend to asymptotically flat metrics at spatial infinity. Alternatively, it seems an attractive idea to numerically study the spherically symmetric time-dependent collapse scenario. The collapsing matter could be a scalar field, but more simply it could just be a spherical aether wave. 8. Cosmology Finally we turn to the role of the aether in cosmological models. Assuming Robertson-Walker (RW) symmetry, ua necessarily coincides with the 4velocity of the isotropic observers, so it is entirely fixed by the metric. The aether field equation has but one non-trivial component, which simply determines the Lagrange multiplier field A. Therefore the entire aether stress tensor is also determined by the metric. Like any matter field, when the asther satisfies its equation of motion, its stress tensor is automatically conserved. Hence, in RW symmetry, the aether stress tensor must be a conserved tensor constructed entirely from the spacetime geometry. One such tensor is the Einstein tensor itself. Another is the stress tensor of a perfect fluid with equation of state p = —\p, whose energy density varies with the scale factor a as does the spatial curvature, i.e. as 1/a2. The aether stress tensor is just a certain combination of these two conserved tensors, 16 ' 13 namely T$her
= - ^ 2 + ^ \Gab - ^R(gab + 2uaub)}. (17) z L b -l This is written using the conventions of Refs. [18,19] in which the field equations take the form (6). The effect of the cosmological aether is thus to renormalize the gravitational constant and to add a perfect fluid that renormalizes the spatial curvature contribution to the field equations. The renormalized, cosmological gravitational constant is given by 13 Q Gcosmo
=
l + (c13 + 3c2)/2-
(18)
Carroll and Lim 13 note that, since this is not the same as G N , the expansion rate of the universe differs from what would have been expected in GR with the same matter content. The ratio is constrained by the observed primordial 4 He abundance to satisfy \Gcosmo/G^ — 1| < 1/8. They assume
178
the positive energy, stability, and subluminality constraints discussed above, which imply GCOsmo < G < G N , so the universe would have been expanding more slowly t h a n in GR. In our notation, the resulting helium abundance constraint can be written as 15ci + 21c2 + 7c3 + 8C4 < 2, where we have included the c\ dependence omitted in Ref. [13].
Dedication This paper is dedicated by T J to Stanley Deser, with admiration and gratitude for his friendship, support, and guidance in gravitational exploration.
Acknowledgments We are grateful for helpful discussions during various stages of this research with S. Carroll, P. Chrusciel, T. Damour, J. Dell, S. Deser, G. EspositoFarese, A. Kostelecky, K. Kuchaf, E. Lim, C. Misner, I. Racz, M. Ryan, A. Strominger, M. Volkov, R. Wald, C. Will, R. Woodard, and probably others to whom we apologize to for having omitted t h e m from this list. This research was supported in part by the N S F under grants P H Y 9800967 and PHY-0300710 at the University of Maryland, by the D O E under grant DE-F603-91ER40674 at UC Davis, and by the CNRS at the Institut d'Astrophysique de Paris.
References 1. See for example the following and references therein: V.A. Kostelecky, ed., Proceedings of the Second Meeting on CPT and Lorentz Symmetry, Bloomington, USA, 15-18 August 2001 (World Scientific, Singapore, 2002); G. Amelino-Camelia, The three perspectives on the quantum-gravity problem and their implications for the fate of Lorentz symmetry, gr-qc/0309054; N.E. Mavromatos, CPT violation and decoherence in quantum gravity, grqc/0407005; T. Jacobson, S. Liberati and D. Mattingly, Astrophysical bounds on Planck suppressed Lorentz violation, hep-ph/0407370. 2. V A . Kostelecky, Phys. Rev. D 6 9 , 105009 (2004) [hep-th/0312310]. 3. R. Jackiw and S.Y. Pi, Phys. Rev. D68, 104012 (2003) [gr-qc/0308071]. 4. C. Armendariz-Picon, T. Damour and V. Mukhanov, Phys. Lett. B458, 209 (1999) [hep-th/9904075]. 5. N. Arkani-Hamed, H.C. Cheng, M.A. Luty and S. Mukohyama, JHEP 0405, 074 (2004) [hep-th/0312099]. 6. N. Arkani-Hamed, H.C. Cheng, M. Luty and J. Thaler, Universal dynamics of spontaneous Lorentz violation and a new spin-dependent inverse-square law force, hep-ph/0407034.
179 7. M. Gasperini, Class. Quantum Grav. 4, 485 (1987); Gen. Rel. Grav. 30, 1703 (1998); and references therein. 8. V.A. Kostelecky and S. Samuel, Phys. Rev. D40, 1886 (1989). 9. M.A. Clayton, Causality, shocks and instabilities in vector Held models of Lorentz symmetry breaking, gr-qc/0104103. 10. J.F. Barbero G. and E.J. Villasefior, Phys. Rev. D68, 087501 (2003) [grqc/0307066]. 11. B.Z. Foster, to be published. 12. J.D. Bekenstein, Relativistic gravitation theory for the MOND paradigm, astro-ph/0403694. 13. S.M. Carroll and E.A. Lim, Phys. Rev. D70, 123525 (2004) [hep-th/0407149]. 14. E.A. Lim, Can We See Lorentz-Violating Vector Fields in the CMB?, astroph/0407437. 15. T. Jacobson and D. Mattingly, Phys. Rev. D64, 024028 (2001). 16. D. Mattingly and T. Jacobson, Relativistic gravity with a dynamical preferred frame, in CPT and Lorentz Symmetry II, ed. V.A. Kostelecky (World ScientiOc, Singapore, 2002) [gr-qc/0112012]. 17. T. Jacobson and D. Mattingly, Phys. Rev. D 6 3 , 041502 (2001) [hepth/0009052]. 18. C. Eling and T. Jacobson, Phys. Rev. D69, 064005 (2004) [gr-qc/0310044]. 19. T. Jacobson and D. Mattingly, Phys. Rev. D 7 0 , 024003 (2004) [grqc/0402005]. 20. C. Eling and T. Jacobson, Einstein-JEther energy, in preparation. 21. R. M. Wald, General Relativity (University of Chicago Press, 1984). 22. Y. Nambu, Prog. Theor. Phys., Suppl., 190 (1968). 23. G.N. Felder, L. Kofman and A. Starobinsky, JHEP 0209, 026 (2002) [hepth/0208019]. 24. J. Bjorken, Emergent gauge bosons, hep-th/0111196. 25. J.W. Moffat, Int. J. Mod. Phys. D12, 1279 (2003) [hep-th/0211167]. 26. B.M. Gripaios, Modified gravity via spontaneous symmetry breaking, hepth/0408127. 27. G.D. Moore and A.E. Nelson, JHEP 0109, 023 (2001) [hep-ph/0106220]. 28. B.Z. Foster, unpublished. 29. C.T. Eling, unpublished. 30. E. Witten, Commun. Math. Phys. 80, 381 (1981). 31. C M . Will, Theory and Experiment in Gravitational Physics (Cambridge Univ. Press, Cambridge, 1993). 32. C M . Will, Living Rev. Rel. 4, 4 (2001) [gr-qc/0103036], 33. D. Sudarsky and K.G. Zloshchastiev, The coplanarity of planetary orbits places bounds on dynamical aether, gr-qc/0412086. 34. C. Eling and T. Jacobson, unpublished. 35. I. Racz and R.M. Wald, Class. Quant. Grav. 13, 539 (1996) [gr-qc/9507055].
HYPERBOLOIDAL SLICES A N D ARTIFICIAL COSMOLOGY FOR N U M E R I C A L RELATIVITY
CHARLES W. MISNER Department of Physics, University of Maryland, College Park, MD 20742-4111, USA E-mail: [email protected]
This preliminary report proposes integrating the Maxwell equations in Minkowski spacetime using coordinates where the spacelike surfaces are hyperboloids asymptotic to null cones at spatial infinity. The space coordinates are chosen so that Scri+ occurs at a finite coordinate and a smooth extension beyond Scri-f is obtained. The question addressed is whether a Cauchy evolution numerical integration program can be easily modified to compute this evolution. In the spirit of the von Neumann and Richtmyer artificial viscosity which thickens a shock by many orders of magnitude to facilitate numerical simulation, I propose artificial cosmology to thicken null infinity Scri-I- to approximate it by a de Sitter cosmological horizon where, in conformally compactified presentation, it provides a shell of purely outgoing null cones where asymptotic waves can be read off as data on a spacelike pure outflow outer boundary. This should be simpler than finding Scri-f- as an isolated null boundary or imposing outgoing wave conditions at a timelike boundary at finite radius.
It is a pleasure to dedicate this work to my friend and colleague Stanley Deser as we celebrate his ancient works and his continuing contributions to a broad variety of problems in physics. 1. Introduction As LIGO 1 approaches its first stage design sensitivity and other 2 gravitational wave observatories progress, theoretical descriptions of the expected waveforms from the inspiral of binary black holes and neutron stars remain less detailed than one might hope. The largest efforts are devoted to solving Einstein's equations by discrete numerical methods based on a space+time split of the equations and intend to impose as a boundary condition an asymptotic Minkowski metric. Initial formulations were given the ADM name although (probably inconsequentially) none uses the conjugate sets of fields gij and 7rlJ' favored by ADM. Variations on these formulations have 180
181
shown practical improvements, 3,4 ' 5 and others 6 with less developed implementations hold the theoretical suggestion of better numerical behavior. Because the (expected) observable gravitational waves will be seen far from their sources, some efforts have asked that the data set representing the state of the gravitational field at any computational step be not that on an asymptotically flat spacelike hypersurface, but instead that on an outgoing light cone. Thus the Pittsburgh group 7 has focused on slicing spacetime along null cones. By extracting a conformal factor as in Penrose diagrams, null infinity becomes a finite boundary in the computational grid. But these outgoing null cones do not have a useful behavior near the wave sources, leading to a study matching them to flatter spacelike slices there. Friedrich8 formulated the Einstein equations in a way where the conformal factor could be one of the dynamical variables, and also considered the case where the time slices were spacelike but asymptotically null, as for hyperboloids in Minkowski spacetime. A large program to implement this approach is being pursued by Sascha Husa 9,10 and others at AEI-Potsdam. This paper aims to formulate within the simpler problem of solving Maxwell's equations some aspects of these methods which could be attacked with smaller computational resources. In particular we consider the question of whether fairly straightforward evolution algorithms designed for spacelike slicings of spacetime will encounter special problems when those slices are asymptotically null, and the question of imposing boundary conditions at null infinity (Scri+). Work with Mark Scheel and Lee Lindblom 11 considered the first of these questions, but employed the scalar wave equation. Those preliminary results were presented at the KITP Workshop "Gravity03" in June 2003. Vince Moncrief12 then suggested that since the coordinate choice which brought Scri+ to a finite radius displayed a conformal metric which was nonsingular at Scri+, it would be better to study instead the Maxwell equations whose properties under conformal transformations are simpler that those of the scalar wave equation. A new suggestion is given below that the boundary conditions at Scri+ might be easier to handle if Scri+ were approximated by a de Sitter cosmological horizon with a cosmological constant chosen for numerical convenience rather than for physical interest.
182
2. Maxwell's Equations In formulating a 3+1 statement of the Maxwell equations I, of course, revert to lessons from Arnowitt and Deser.13 Following their Schwinger tradition I use a first order variational principle 51 = 0 with
I = J g^i-F^daAp
+ \F^Faf3)d4x
(1)
- ff" V ° )
(2)
where
are components of an operator that maps covariant anti-symmetric tensors onto contravariant anti-symmetric tensor densities. This mapping has an inverse whose components are 9livaP=
2i9na9vl3—9ii.f}9va)/y/-9
•
(3)
Note that these two symbols are not related merely by the usual lowering of indices, as one map raises the density while the other lowers it. Their product
9l>vpc
.
(4)
gives the identity mapping of antisymmetric tensors onto themselves. In this variational principle the fields A^ and F^ are varied independently and yield the equations d„ru
=0
(5)
and F^ = d»Av - dvA^ u
Here V
= V^F^
.
(6)
arises as ^=g^^Fa0
.
(7)
Again following the given tradition I seek a proto-Hamiltonian form pq — H for this Lagrangian and rewrite it by replacing the FM„ field in I by its equivalent ^ v field, giving
1 = J{-rvd»Au
+ \rvT0g^ap)dAx
.
(8)
The only term here containing time derivatives is —$°xdoAi so that S)i = £ 0 i
(9)
183
and Ai are the only fields for which one obtains evolution equations. The scalar potential t/> = -Ao
(10)
and the 3 ^ are therefore Lagrange multipliers which enforce the constraints diST = 0
(11a)
and B^ = 9ij,aP^
= diAj-
djAi
.
(lib)
The evolution equations can then be written as d0Ai = -Ei - dtxp
(12a)
do& = djff*
(12b)
and
where Ei=giO,a0dal3
and
& = Tj
•
(13)
There is no evolution equation for ip = — Ao but one can be supplied as a gauge condition. There is also no evolution equation directly from the variational principle for B^ but one can be deduced by taking the time derivative of equation (lib) and evaluating the time derivatives on the right hand side using equation (12a). The result is -doBij = dtEj - djEi
•
(14)
In a numerical example mentioned below, the fields were taken to be Aj and 35l using the evolution equations (12a) and (12b). Equation (lib) was treated not as a constraint but as merely making B^ an abbreviation for the right member of that equation, thus introducing second (spatial) derivatives into the system of equations. For that example of a wave packet the gauge chosen was Coulomb gauge with ip = 0 for all time. The remaining constraint (11a) is easily seen to be preserved in time as a consequence of the evolution equation (12b). Another possible system of equations would be to use (12b) and (14) as evolution equations. Since the vector potential does not appear in these equations, it can be ignored and not evolved. The constraint (lib) would then have to be replaced by its integrability condition d[kBij} = 0
(15)
184
treated as a constraint equation. It is easily seen from (14) that this constraint is preserved in time (differentiably) as a consequence of (14). The Gauss constraint (11a) remains part of this system and is preserved in time by exact solutions. These equations (12b) and (14) are exactly what one writes in classical electromagnetism in a material medium (e.g., Section 16-2 of [14]); only the constitutive equations (17) are new. With the usual 3+1 decomposition of the metric ds2 = -a2dt2
+ jij (dxi + pidu)(dxj
+ (3jdu)
(16)
these are Ei = (a/VTbyJD'' + B ^
(17a)
and $ « = a^ikijlBkl
+ W
- 2)^'
(17b)
which are more complex than the E = D/e and H = B/jtx of simple isotropic media. The Einstein aether is, in coordinate terms, anisotropic not only in space but also in spacetime, but conveniently linear. 3. Conformal Invariance This exploration of numerical evolution with the Maxwell equations was provoked by Vince Moncrief s reminder that the Maxwell and Yang-Mills equations (in four spacetime dimensions) have the simplest possible conformal structure. A conformal transformation g^, \—> CPg^ changes the metric geometry to a different one which shares the same light cones. Under this change the metric dependent factor g^v af3 in the variational integral / does not change since g^v i-» £l~2giiu while ^f^g i-> Q 4 ^ / ^ . As nothing in the variational integral changes under a conformal transformation, it follows that if the fields AM and F^ or ^ u satisfy the field equations in one metric, so do they also in any conformally related metric. Although nothing in section 2 depends on the metric having any special properties (other than Lorentz signature) such as flatness or satisfying the Einstein equations, these conformal properties in four spacetime dimension are particularly useful since the metric (21) we intend to use is singular only in a conformal factor s2/q2 which we now see will appear nowhere in our field equations. A similar simplification occurs when the metric of interest is the Schwarzschild metric in the hyperboloidal slicings used in [11]. In the 3+1 decomposition of the field equations we note that Ai,'JDi,Ei,ip,Bij,F)'l:> and /?* are conformally invariant, while 7^ H-»
185
9,2jij,a i-> fia, y/7 K-> fi3-^. Thus the metric structures which appear in the constitutive equations (17), (a/-y/7)7y and 0^/77^7-'' as well as /? l , are conformally invariant. The constraint equations (11) and (15) have an even stronger invariance: they are metric invariant. No metric quantities appear in these constraint equations when they are written in terms of our choice of fields. Thus if two hypersurfaces are diffeomorphic then initial conditions for the Maxwell equations which satisfy the constraints in one Lorentzian manifold can be imported to the other where they will again satisfy the constraints. If the two manifolds are not conformally equivalent, however, the subsequent evolutions of these initial conditions will generally be inequivalent. Gauge conditions present greater difficulties. In the absence of sources the "Coulomb" gauge ^ = 0 is conformally invariant, but only 3dimensionally coordinate invariant. On the other hand the simplest Lorentz gauge d^y/^gg^Av) = 0 is coordinate invariant but not conformally invariant. The set of field equations which evolve 2)' and B^ without the use of the vector potential are conformally invariant but do not require a gauge condition.
4. Background Spacetime The background metric for this study will be the de Sitter spacetime ds2 = -dT2 + dX2 + dY2 + dZ2 + {R2/L2){dT-dR)2
.
(18)
which is here presented in Kerr-Schild form as flat spacetime plus the square of a null form. The cosmological constant here is A = +3/L2. Since dT—dR is null in both the underlying flat spacetime and after the Kerr-Schild modification, the hyperboloidal slices we used in flat spacetime should remain asymptotically null also in this modified metric. Therefore we introduce the same coordinate change as in our earlier scalar work 11 T
+
-r2
7=» r^F and
<19)
186
where R2 = X2 + Y2 + Z2 = XiXi leads to the metric ds2 =
and r2 = x2 +y2 + z2 = x"x"
(s2/q2)[-a2du2
+
i
[
s
i
+
This (21a)
r
L2(1+lr)4j
(dr + (3du)2 + r W ]
where 1 - i r1~2
a2 = (1 + \r2
(21b)
L2(\ + \rY
(21c)
and /3
r + L2{l + \r)2_
1+
s2r2
2
i (l + ir)4
(21d)
I have verified that this metric (21) actually is the de Sitter metric by two tests: I have run a GRtensor II 15 Maple 8 worksheet to calculate the Einstein equations for it and find Gt = -(3/L2)SS
(22)
and I have also had that worksheet calculate the Riemann tensor which is Wap
= (1/L2)(5^
- 5ffi)
(23)
and shows that this is a spacetime of constant curvature, therefore de Sitter. Richard Woodard 16 has confirmed this by showing that the metric (18) can be reduced to the familiar static deSitter metric by the coordinate change T = T + f{R) where df/dR = -R2/(L2 - R2). 5. Causal Structure This de Sitter metric is presented in equations (21) with an apparent singularity at r = 2 where q = 0. This is probably a coordinate singularity since the curvature does not go bad there, and also because the de Sitter spacetime in notorious for being presented in many different guises corresponding to different coordinate systems, many of which cover only a patch of the full manifold. But for present purposes we only need this coordinate patch, and the singularity at R = oo will be removed by a conformal transformation later.
187
By dropping the conformal factor in the metric (21) we arrive at a nonsingular metric ds2 = -a2du2
(24) 2 2
+ 1 + L2{l
sr + \rY
{dr + pdu)2 + r2dQ.2
using the same abbreviations as in equations (21). Note, however, that a and (5 are now the lapse and shift of this regulated metric. This metric no longer satisfies the Einstein equations (with A term) but is related to the original (de Sitter) metric in a known way. From this point on we use only this metric, and the previous metric will be called ds2 = Q2ds2 following the practice in Wald's discussion (Appendix D of [17]) of conformal transformations, with Q, = s/q in our case. The causal structure of the regulated metric (24) is simpler than that of the original metric (18). The hypersurfaces of constant u are everywhere spacelike since the regulated metric is positive definite when du = 0. The hypersurfaces of constant r are spacelike only in an interval around r = 2 as seen from their normal vector dr being timelike there:
/L)2 ( v , f = ^ " - | r f r £2 2
•
<*>
(l + \r ) This norm is plotted in Figure 1 over a broad range and again in Figure 2 giving the interval around r = 2 in more detail. 6. Analysis What can we learn using the tools described above? A few lessons already appeared in the work [11] with Scheel and Lindblom. Suppose only the time coordinate were changed to parameterize a family of hyperboloids, here [T - s{u - l)] 2 - R2 = s 2
(26)
in the original coordinates of equation (18). Then the outgoing coordinate speed of light dR/du would become unboundedly large as R —> oo. The Courant condition usually met in numerical implementations would then make the allowable time step impractically small. This can be cured by "Analytic Mesh Refinement" (AnMR), which is just a coordinate transformation such as that from R to r in equation (20), since outgoing waves get redshifted as the u = const hypersurfaces tend toward outgoing null cones. Because this redshift gives long wavelengths, high resolution is not needed
188
Figure 1. The squared norm ( V r ) 2 = grr of the normal Vr to hypersurfaces of constant r in the conformally regulated metric (24) is plotted over a range from the origin to beyond the location r = 2 of the surface which becomes Scri+ in the high resolution limit L —• oo. This example uses L/s = 10.
Figure 2. The squared norm ( V r ) 2 = gTT of the normal V r to hypersurfaces of constant r in the conformally regulated metric (24) is plotted over a narrow range around r = 2 where r = const are spacelike pure-outflow hypersurfaces on which the entire forward light cone is oriented toward increasing r when grr < 0. This example uses L/s = 10.
and the large R domain can be compressed to a finite range. Thus we find that the outgoing coordinate speed of light dr/du becomes
(i + H 2
(27)
which remains less than 4 out to r = 2 or R = oo. The ingoing radial light speed c;n is similarly bounded in magnitude but becomes positive in
189
the region where (Vr) 2 is negative rather than the simpler value c in = — (1 — ^ r ) 2 in the Minkowski (L = oo) case where it merely touches zero on Scri+. Thus the encouraging results from [11] are unchanged by introducing the cosmological term. 7. Exact Solutions There is in principle no difficulty in finding exact solutions to these equations in Minkowski spacetime. One merely takes solutions from any textbook and performs the coordinate transformation to our uxyz compactified hyperboloidal slicing coordinates. In detail is it a great convenience to have GRTensor15 available to perform some of the algebra and much essential checking. Exact solutions play two roles. The first is that they provide initial data for numerical integrations whose physical interpretation is known. Secondly, though comparison, they allow testing the accuracy of the numerical solution. Indeed, the whole point of carrying out these numerical evolutions is to explore in a simpler context than full general relativity different approaches for possible adaption to Einstein's equations. Thus one may explore and compare different integration schemes either to understand them better as in the work of Baumgarte, 18 or to test possible new schemes as in [19]. Questions which may be treated in this simplified context include the stability of different formulations, the control of constraints, the treatment of boundary conditions, the extraction of asymptotic wave amplitudes and waveforms, etc. Here we address the use of hyperboloidal time slices and the possibilities for carrying the integration out to and beyond Scri+ using a finite grid. We find that a sufficient variety of examples can be found by assuming that the vector potential A = A^dx^ has the form A = / sin 2 6 dcp - ip dT
.
(28)
Three cases are (1) a uniform magnetic field with ip = 0, / = BzR2/2 (2) a static dipole electric field with / = 0,i/> = Z/R? (3) the Baumgarte 18 choice of a wave packet, which is our principle test case, with ip = 0 and / = ( 1 - 2XU) exp(-At/ 2 ) - ( i + 2XV) exp(-XV2)
(29)
where U and V were defined by U = T-R
= s(u
~ j
(30a)
190
and V = T + R = s(u+
-^r-)
•
(30b)
In spite of first impressions one finds that / = 0(R2) at small R or small r, but numerical implementations must avoid evaluating this / at r — 0 as subtle cancellations occur. To convert equation (28) to uxyz coordinates requires
-dT = du+
X
[
dx*
(31)
s (l-±r2)2 % % 2 and, from X = sx /(l — \r ) and the standard rectangular to spherical coordinate transformation, sin2 6 d
191
1510-
^y 5 *~~-—""•"""""
0"
2
2.5
-5-
-10-
^ \
/
/
\ / \/
Ey @ u = - 2 Ey @ u = 0 10Ey @ u = 1.4
Figure 3. For the Baumgarte choice of EM wave packet from equations (28) and (29) we plot the conformally invariant electric field 33y along the x-axis at three different hyperboloidal times u = —2, 0, and 1.4 . Note that the field values are finite and generally nonzero at Scri+ which is x = 2. Those at the latest time have been here magnified by a factor of 10.
3. T. W. Baumgarte and S. L. Shapiro, Phys. Rev. D59, 024007 (1999) [grqc/9810065]. 4. M. Shibata and T. Nakamura, Phys. Rev. D 5 2 , 5428 (1995). 5. M. Alcubierre, B. Briigmann, T. Dramlitsch, J. A. Font, P. Papadopoulos, E. Seidel, N. Stergioulas and R. Takahashi, Phys. Rev. D62, 044034 (2000) [gr-qc/0003071]. 6. L. Lindblom and M. A. Scheel, Phys. Rev. D67, 124005 (2003) [grqc/0301120], h t t p : / / l i n k . a p s . o r g / a b s t r a c t / P R D / v 6 7 / e l 2 4 0 0 5 . 7. J. Winicour, Living Reviews Relativity 4 (2001) [Online article], http://www.livingreviews.org/Articles/Volume4/2001-3winicour/. 8. H. Friedrich, in H. Friedrich and J. Frauendiener, eds., Lecture Notes in Physics 604, pp. 1-50 (Springer, 2002) [gr-qc/0209018]. 9. S. Husa, in H. Friedrich and J. Frauendiener, eds., Lecture Notes in Physics 604, pp. 239-260 (Springer, 2002) [gr-qc/0204043]. 10. Y. Zlochower, R. Gomez, S. Husa, L. Lehner and J. Winicour, Phys. Rev. D68, 084014 (2003) [gr-qc/0306098], http://link.aps.org/abstract/PRD/v68/e084014.
192
0.8
/
0.6-
Ey
/
0.4-
'
-1 ^ v .
u
\
/
0.2-
-2
\
/
\
1
\
2
/^~
3
"~4
-0.2"
Figure 4. Waveform extraction: for the same analytic solution as in Figure 3 we plot 5)v as function of u at Scri+ (x = 2). Note that at constant r one has T = su + const so this waveform is not distorted from its (T, R) presentation as is the pulse shape (Figure 3) when plotted as function of r. 11. C. W. Misner, M. Scheel and L. Lindblom, Wave propagation with hyper boloidal slicings (2003), http://online.kitp.ucsb.edu/online/gravity03/misner. 12. V. Moncrief, private communication (2003). 13. R. Arnowitt and S. Deser, Phys. Rev. 113, 745 (1959), http://link.aps.org/abstract/PR/vll3/p745. 14. J. R. Reitz, F. J. Milford and R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Massachusetts, 1979). 15. P. Musgrave, D. Pollney and K. Lake, GRTensor II software (2001), http://www.grtensor.org. 16. R. Woodard, private communication (2004). 17. R. M. Wald, General Relativity (University of Chicago Press, 1984). 18. A. M. Knapp, E. J. Walker and T. W. Baumgarte, Phys. Rev. D 6 5 , 064031 (2002) [gr-qc/0201051], h t t p : / / l i n k . a p s . o r g / a b s t r a c t / P R D / v 6 5 / e 0 6 4 0 3 1 . 19. D. R. Fiske, Phys. Rev. D69, 047501 (2004) [gr-qc/0304024], http://link.aps.org/abstract/PRD/v69/e047501.
SOME APPLICATIONS OF T H E A D M FORMALISM
J.E. NELSON* Dipartimento and Istituto
di Fisica Teorica, Universita degli Studi di Torino Nazionale di Fisica Nucleare, Sezione di Torino via Pietro Giuria 1, 10125 Torino, Italy E-mail: [email protected]
The ADM Formalism is discussed in the context of 2 + 1-dimensional gravity, uniting two areas of relativity theory in which Stanley Deser has been particularly active. For spacetimes with topology IRxT 2 the partially reduced and fully reduced ADM formalism are related and quantized, and the role of "large diffeomorphisms" (the modular group) in the quantum theory is illustrated.
1. Introduction Over forty years ago Arnowitt, Deser and Misner (ADM) studied the 3 + 1decomposition of general relativity, its initial value problem, the dynamical structure of the field equations and calculated the Hamiltonian.* This extraordinary piece of work has become a fundamental ingredient of modern relativity theory. It is now regularly taught as an integral part of relativity courses, and usually occupies at least a chapter in relativity textbooks. The problem had actually been considered previously by Dirac 2 who applied his theory of constrained systems 3 to the gravitational field. But Dirac's treatment was incomplete and in a particular gauge. In this Section I briefly summarise the ADM results, and in Section 2 discuss the main differences between the 3 and 4 dimensional theories. In Section 3 the second-order, partially reduced, ADM formalism, for spacetimes of topology R x T 2 is reviewed, and I show how in principle the system can be quantized. In Section 4 the first-order fully reduced holonomy approach is presented. In Section 5 the two approaches are related, both classically and quantum mechanically, using the action of the modular *Work supported by the Istituto Nazionale di Fisica Nucleare (INFN) of Italy and the Italian Ministero dell'Universita e della Ricerca Scientifica e Tecnologica (MIUR). 193
194
group, or "large diffeomorphisms — those that remain after ADM reduction. The 3 + 1-decomposition of the Einstein-Hilbert action calculated by ADM is IBin = fd4xy/^
(4)
fl = fdt
f d3x(irijgij
- N'Hi - NH),
(1)
where spacetime is of the form R x E , and time runs along M. In (1) the metric has been decomposed as ds2 = N2dt2 - gij(dxi + Nidt)(dxj
+ Njdt),
i,j = 1,2,3,
(2)
and 7ru = ^fg {Kli — g^K), where K%:* is the extrinsic curvature of the surface E labeled by t = const. a In (2) the lapse Nl and shift N functions are related to the non-dynamical components of g^ and their variation in (1) leads to the supermomentum and super-Hamiltonian constraints on gij
and 7rkl Hi = -2Vj^i
H = -y= gijgkfa***1 - \^^kl)
= 0,
-VdR = 0, (3)
where Vj is the covariant derivative for the connection compatible with gij, and indices are now raised and lowered with g^. The H,Hi in (3) are non-polynomial in g^ and irkl and involve (gij)"1. They are directly proportional to the components G°^ of the Einstein tensor defined by Qtlv
=
SlEin og^xV
=
Rtlv
_ \Rg»^ i
(4)
so finding a solution to (3) would correspond to finding a general solution of Einstein's equations (the other components of (4) are zero by the Bianchi identities G»v\v = 0). The constraints (3) generate, through the Poisson brackets obtained from (1) {9ij(x),
nkl(y)}
= \{8k5lj + 5k5\)5\x
- y),
(5)
three-dimensional diffeomorphisms in E, and the time development of the variables gij(x),Trkl(y). a T h i s is standard ADM notation: gij and R refer to the induced metric and scalar curvature of a time slice, while the spacetime metric and curvature are denoted ' 4 'g^ l / and (4)R.
195
One can ask what effect the constraints (3) would have when applied on wave functions ip(g). If the brackets (5) are represented by letting the momenta 7rIJ act by differentiation
and the metric components gij(x) by multiplication, the supermomentum constraint Hiip(g) = 0 is easy to interpret, since
/ SxiN'H^ig)
= 5V(g) = ^L^Nj
+V^),
(7)
and implies that one should identify wave functions of metrics gij and ijij when they differ as gij^gij+ViNj+VjNi.
(8)
But (8) is a Lie derivative, or coordinate transformation, in the spatial surface S, so the supermomentum constraint reflects the freedom to choose the 3 spatial coordinates on S. The space of metrics (8) with g^ identified with g^ was named superspace in 1963 by Wheeler.4 The super-Hamiltonian constraint Hip(g) = 0 (also known as the Wheeler DeWitt equation 5 ) is much harder to interpret, and alone does not generate the dynamics, or time reparametrization invariance, of wave functions ip(g). Instead one needs to use the full Hamiltonian, namely the combination
/ dzx{Nini + Nn).
(9)
The gravitational field in 4 spacetime dimensions has correctly (for a massless field) 2 independent degrees of freedom per spacetime point. This is most easily seen by noting that the induced metric of a time slice g^ has 6 independent components, and there are the 4 constraints (3). 2. 2 + 1-Dimensional A D M Decomposition In 2 + 1 spacetime dimensions the description of Section (1) is essentially identical, apart from a factor of \ in the super-Hamiltonian (3). The counting of degrees of freedom is, however, quite different. There are in fact zero degrees of freedom, and this can be seen in several ways. The simplest is perhaps to note that now the induced metric gij,i,j = 1,2 has only 3 independent components, but there are 3 constraints Ji = 0, Jii = 0 analogous to (3). Alternatively, since the Weyl tensor vanishes in 3 dimensions
196
(but not in 4, see [6]), it follows that the full Riemann curvature tensor RaPnv c a n he decomposed uniquely in terms of only the Ricci tensor R^ the scalar curvature R and the metric tensor g^, itself.
R\nvk = g\vR^k - g^vRxk - gxkR^u + g^kR\v +-^R{9nug\k-g\u9iik)2
(io)
2
In fact in d dimensions Ra/3)iu has d (d — 1)/12 independent degrees of freedom and R^ has d(d+ l ) / 2 . These coincide when d = 3. In terms of the Einstein tensor Ga/3 [Eq. (4)] the decomposition (10) is R\pivk
=
£A^/3 Zvkot Ga
,
(11)
a/3
so that when Einstein's vacuum equations G = 0 are satisfied, the full curvature tensor (all components) are zero, i.e. Rx^k = 0 and spacetime is flat. Thus vacuum solutions of Einstein's equations correspond to flat spacetimes, and there are no local degrees of freedom. It is possible, however, to solve the field equations and introduce some dynamics, in several ways. The first — developed extensively by Deser et al.r and others, is to add sources, or matter, thus creating local degrees of freedom. When Einstein's equations read
where Ta$ is the stress-energy tensor of the sources, the curvature (11) is no longer zero, but is proportional, from (12), to Ta/3. The second creates propagating massive gravitational modes by adding a topological term to the action, always possible in an odd number of dimensions.6 For gravity in 3 dimensions, this is the Chern-Simons form J(LJabAdwab
+ ^ujacA^Aujda),
(13)
where the components of the spin connection u>^b are to be considered as functional of the triads e^ by solving the torsion equation. Ra = dea - ujab Aeb = 0 , with e^e^ijab = g^. Variation of (13) with respect to the metric tensor g^ gives the Cotton tensor C^=g-i
e^Dx(R^-^R\,
which is symmetric, traceless, conserved, and vanishes if the theory is conformally invariant. Therefore, adding the Chern-Simons term (13) to the
197
three-dimensional scalar curvature action (1) with a constant factor l//x leads to the field equations
which can be transformed into ( • + M2) R»* = terms in (R^
.
(14)
In the linearized limit the R.H.S. of (14) vanishes and it is shown in [6] that the solutions of (14) correspond to massive, spin ±2, particles. A way to introduce global rather than local degrees of freedom in flat spacetime is to consider non trivial topologies. Recall that curvature is defined by commutators of covariant derivatives, or, by parallel transport around non-collapsible curves i.e. curves which are not homotopic to the identity. The change effected by parallel transport around closed curves of this type is often called holonomy — and is used to characterise flat spacetimes. A simple example is when the spatial surfaces are tori, i.e. S = T2 — then the meridian and parallel are clearly non-collapsible. This will be discussed explicitly in Sections (3) and (4). 3. Second-Order, Partially Reduced A D M Formalism Here I summarise work by Moncrief8 and Hosoya and Nakao, 9 adding a cosmological constant A. It is known that any two-metric gij on T,g, where S 3 is a Riemann surface of genus g, is conformal (up to a diffeomorphism) to a finite-dimensional family of constant curvature metrics gij(ma), 9ij=e2Xgij(ma), labeled by a set of moduli ma,a
= 1
R(g) =
1, 0, -1,
(15)
63 — 6 (see Abikoff10), and 9 = 0, g = l, g>\.
(16)
A similar decomposition of the momenta 7ry gives nij = e~2X^§ (pV + ^ ' T T / V ^ + V V + W
- gijVkYk)
,
(17)
where Vj is the covariant derivative for g^, indices are now raised and lowered with gtj, and pli — the momentum conjugate to g^ — is transverse traceless with respect to Vj, i.e., ViP J J = 0.
198
This decomposition uses York time, 11 the mean (extrinsic) curvature K = "7'\JH = T, shown to be a good global coordinate choice in [8]. The supermomentum constraints now imply that Yl = 0, while the super-Hamiltonian constraint, n = -\VSe2X(T2
- 4A) + VSe-2XPijPij
+ 2^3 AX-'-R
= 0,
(18)
reduces to a differential equation for the conformal factor A as a function of gij,plj and T. For g > 1 a solution of (18) always exists and the threedimensional Einstein-Hilbert action is IEin = JdT^pa^-H(m,p,T)y (19) where pa are the momenta conjugate to the moduli ma defined by
pa= d2xnij
(2o)
L ^'
and H(m,p,T)
is an effective, or reduced, ADM Hamiltonian H{m,p,T)
= J ^gd2x
= f e2X(m>p>T^d2x,
(21)
which represents the surface area at time T, with A(m,p, T) determined by (18). The reduced ADM Hamiltonian (21) generates the T = K or time development of ma, pP through the Poisson brackets {may}
= 6*.
(22)
For 5 = 1 the modulus is the complex number m = mi+im2 (with rti2 > 0), with momenta p = p1 + ip2 satisfying the Poisson brackets {m,p} = {fh,p} = 2,
{m,p} = {fh,p} = 0,
(23)
and da2 =rri21 \dx + mdy\2
(24)
is the spatial metric for a given m where x and y each have period 1. The surface curvature (16) is zero and (18) is explicitly solved. The reduced ADM Hamiltonian (21) becomes H(m,p,T)
= (T2 - 4 A ) " 1 / 2 [m22pp]l/\
(25)
One can recognise in (25) the square of the momentum with respect to the Poincare (constant negative curvature) metric on the torus moduli space m2~2dmdfh.
(26)
199
Hamilton's equations for the motion of m, p on the hyperbolic upper half plane (Teichmuller space) using the reduced Hamiltonian (25) can be solved exactly 12 ' 13 and correspond to motion on a semicircle, a geodesic with respect to the metric (26). This reduced phase space can, in principle, be quantized by replacing the Poisson brackets (22) with commutators,
[ma,f]=ihC
(27)
representing the momenta as derivatives,
and imposing the Schrodinger equation itj^H=H^m,T),
(29)
where the Hamiltonian H is obtained from (25) by some suitable operator ordering. With the ordering of (25), the Hamiltonian is H=
,
h
Ai/2,
(30)
where Ao is the scalar Laplacian for the constant negative curvature moduli space with metric (26). Other orderings exist, but all consist of replacing Ao in (30) by A„, the weight n Maass Laplacian (see e.g. Carlip 14 ). This approach also depends on the arbitrary, albeit good, choice of K = Tf/y/9 = T as a time variable. It is not at all clear that a different choice would lead to the same quantum theory. 4. First-Order Fully Reduced, A D M Formalism The first-order, connection approach to (2+l)-dimensional gravity, in which the triad one-form ea = ealldxil and the spin connection u>°6 = uabMcbM are treated as independent variables was inspired by Witten 15 (see also [16]) and developed by Nelson, Regge and Zertuche. 17 ' 18 ' 19 ' 20 The three dimensional Einstein-Hilbert action is IEin = f(dwab - cvad Au d b + ^ea A eb) A e c e a b c ,
a,b,c = 0,1,2. (31)
For A < 0 this action can be written (up to a total derivative) as b / c s = -J j(d VAB - l^lAE A QEB) A b
For A > 0 see e.g. the discussion in [21].
SlCDeABCD,
(32)
200
where A,B,C... = 0,1,2,3, eabc3 = — eabc, the tangent space metric is VAB = (—1,1,1, —1) and the (anti-)de Sitter 50(2,2) spin connection Q.AB is
(33)
"^(^"(f)' with A = —a - 2 . The canonical 2 + 1-decomposition of (32) is
iEm = Jd3x(niABnfDeABCD
- n0ABRABijyi.
(34)
In (34) the curvature two-form RAB = d flAB — ClAC AQ,cB has components aS = (l/a)Ra Rab + A e a A eb [proportional to the constraints (3)] and R (proportional to a rotation constraint Ja(, on the triads), where Rab = duab - ujac A ucb,
Ra = dea - toab A eb
(35)
are the (2+l)-dimensional curvature and torsion. The field equations (constraints) derived from the action (34) are simply RAB = 0, and imply that the 50(2,2) connection Q,AB is flat, or, equivalently, from (35) that the torsion vanishes everywhere and that the curvature Rab is constant. They generate, through the Poisson brackets {fi/B(x), n^iy)}
= ±eijeABCDS2(x
- y),
(36)
infinitesimal gauge and coordinate transformations 5QAB = DuAB on the connections ClAB. Since the connection Q,AB is flat, it can be written locally in terms of an SO(2, 2)-valued zero-form ij)AB as dGAB = Q.AC GCB• This sets to zero all the constraints RAB = 0 and is therefore a fully reduced ADM formalism in which the Hamiltonian is identically zero. However, some global degrees of freedom remain, as can be seen by now taking into account the nontrivial topology of the Riemann surface. For each path a on S define the holonomy (Wilson loop) AB
GAB = e x p P f/ n ft
,
(37)
Ja
where P denotes path-ordered, and Ga depends on the base (starting) point and the homotopy class {a} of a, and satisfies Gap = GaGp. Integrating the brackets (36) along paths p, a with non-zero intersection gives {Gp,Ga}^0, and this is the starting point for holonomy quantization.
(38)
201
It is actually more convenient to use the spinor groups SL(2,R) ® SL(2,R) for SO(2,2) (see [18] for details). For each path a we have GABlB
= S-\a)1AS{a\
(39)
where ^A are the Dirac matrices and S — S+
= ±s(-S±(pfaS±(ari
{S+(pfa,S~(a);}
= 0,
a,f3,...
2S±(p3aiyaS±(a3p1)^,
+ = l,2,
(40)
where s is the intersection number (now set to 1) and o,1,p1 (resp. <J3,p3) are the segments of paths before (resp. after) the intersection. The gauge invariance can be implemented by taking traces since, if 5 is an open path ff^ff) = tr S±(a) = tvS±(S-1aS)
= ^(tViS),
(41)
where now 5~icr8 is closed. For g = 1 it is enough to have just six traces Ftf,i = 1,2,3, corresponding to the three paths 71,72,73 = 7i • 72- From (40) they satisfy the non-linear cyclical Poisson bracket algebra 18 {Rf,Rf}
= TyE±(eijkR±-RfRf),
e123 = 1,
(42)
= 0.
(43)
and the cubic Casimir 1 - (Rf)2 - (Rt)2 - (Rf)2 + 2RfRfRt
The traces (holonomies) of (42) can be represented classically as Rf = coshrf,
Rf = coshr^,
Rf = cosh(r^ +rf),
(44)
where rf2 are real, global, time-independent (but undetermined) parameters which, from (42) satisfy the Poisson brackets0 {rt,rt)
= =F^,
K 2 , r r , 2 } = 0.
(45)
The above fully reduced system can be easily quantized either by replacing the Poisson brackets (45) by the commutators
P±,f±]=T**f*
[^2,^=0,
(46)
c T h e parameters r * 2 used here have been scaled by a factor of 1/2 with respect t o previous articles.
202
or by directly quantizing the algebra (42). This gives for the (+) algebra d q^R+Rt ~ q'^RtRi = (lk ~ Q-^Rt,
(47)
where q = exp2id and tan0 = -(hy/^A)/8. The algebra (47) is related 20 to the Lie algebra of the quantum group SU(2)q, and can be represented [up to rescalings of 0(h)] by e.g. Ri = |(Aj + A^1), i = 1,2,3 where the Ai satisfy AXA2 = qA2Ai,
AxAiAz
= q%.
(48)
The first of (48) is called either a ^-commutator, or a quantum plane relation, or it is said that Ai, A2 form a Weyl pair. Relations (48) can be satisfied by the assignments A\ = eri,A2 = er2,A3 = e~^1+fi2^ with fi,f2 satisfying (46). 5. Classical and Quantum Equivalence 5.1. Classical
Equivalence
The classical solution of Section (3) can be related to the parameters rf of Section (4) as follows.21 The ADM reduced actions (19) and (34) are related by
/ = jdt [d2XKij9ij = fd3x =
2
niabClfDeABCD
/ -(pdm + pdfh) - HdT - d(p1mi + p2m2)
/ •
and show that with the time coordinate t determined by T = — (2/a) cot(2i/a) the parameters rx 2 are related to the complex modulus m and momentum p through a (time-dependent) canonical transformation. Explicitly, with ra(t) = r~e^ + r + e - ^ , a = 1,2. and the rf satisfying (45), then m = r2-1(t)r1(t)
and
p= - i
4A 4A
f 2 2 (*)
(50)
will satisfy the Poisson brackets (23). The Hamiltonian (25) is now
H=
VW^K{rTrt~rtr^
d
T h e (—) algebra is the same as (47) but uses q
l
rather than q.
(51)
203
and generates the development of the modulus and momentum (50) as functions of the parameters ra± and time T through dp r Tr1 ~ = {p,H}, 5.2. Large
dm , „.. — = {m,H}.
, x (52)
Diffeomorphisms
The reduction to the modulus m and momenta p means there are no more "small diffeomorphisms" — coordinate transformations (the constraints which generate them are all identically zero). But there remain "large diffeomorphisms" due to the topology. These are transformations that are not connected to the identity, cannot be built up from infinitesimal transformations and are generated by "Dehn twists", i.e. by the operation of cutting open a handle, twisting one end by 2n, and regluing the cut edges. For g > 1 the set of equivalence classes of such large diffeomorphisms (modulo diffeomorphisms that can be deformed to the identity) is known as the mapping class group. For g = 1 it is also called the modular group, and the Dehn twists of the two independent circumferences 71 and 72 (which have intersection number +1) act by
T : 71 -> 71 • 72,
72 -> 72-
(53)
These transformations induce the modular transformations 5 : m —> —m _1 ,
p —> fh2p,
T : m—nn + 1,
p —»p,
(54)
which preserve the Poincare metric (26), the Hamiltonian (25) and the Poisson brackets (23). Figure 1 illustrates this group action on the modulus configuration space, with the invariant semicircle representing the geodesic motion of the modulus m. Classically, one could ask that observables be invariant under all spacetime diffeomorphisms, including those in the modular group. Since equation (54) shows that the modular group is well behaved on configuration space, invariant functions of m exist (see [14]). So the reduced ADM approach of Section 3 looks like a standard "Schrodinger picture" quantum theory, with time-dependent states ip(m,T) whose evolution is determined by the Hamiltonian operator (30).
204
K Figure 1.
Modular transformations S and T.
On the traces of holonomies the transformations (53) induce the following
S:Rf
#2 i
R2
Rl ,
R3
2RX R2 — R3 ,
T: Rf
#3
Rn
R2
R3
2R3 R2
i
!
Ri ,
(55)
which preserve the algebra (42). The corresponding transformations on the holonomy parameters preserve their Poisson brackets (45) S
r
•2 )
T
+ r.2
'
2
1 )
'2 •
(56)
In this approach the modular group action (56) on the parameters is not well behaved since it mixes r\ and r^, and quantization normally requires a polarization. So the quantum theory of Section 4 resembles a "Heisenberg picture" quantum theory, with time-independent states ip(r), and, for some ordering, time-dependent operators (50). 5.3. The Quantum
Modular
Group
Here I present work in collaboration with Carlip. 22 It is useful to note that the modular transformations can also be implemented quantummechanically, by conjugation with the unitary operators 19 ' 22
{>+**>}.
T = T+T-
= exp
S = S+S
=expi ^
[2(pf + p ) + m t ( m t p + pm t ) + (mj3t +$m)m]
I.
The S transformation for p differs from its classical version (54) b : p —>m'-—(m'p + pm1),
(57)
205
by terms of order ft. In terms of the holonomy parameters these are
ft±=6XP ±
{ ^ 9 ± ) 2 } ' ^ ± = e x p { ± ^ [ ^ ± ) 2 + ^ ± ) 2 ]}- (58)
Using the above construction the two representations, classically equivalent as shown in Section 5.1 can be related as follows. Start by diagonalizing the commuting moduli operators m and m\ considered as functions of time and initial data rf2 through (50). Now if the r2(i) are "coordinates" u(t) and r\(t) their "momenta" then m and p act as m ~ w_1 — , p~u2. (59) au The normalized eigenstates of m with eigenvalues m (and m for fh)) are T_,
_
,.
ami
_.
Klm,m,t\u,u)
_
f
a
a 2
_ _2~\
.„„.
=
uexp <—-—mu +—mu >. (60) z7rft l 4ft 4ft J So candidates for "Schrodinger picture" wave functions are the superpositions •tp(m,m,t) = duidu2K*(m,fh,t\u,u)^(u,u) (61) of the "Heisenberg picture" wave functions ip(u,u). Inverting (61) gives f d2m _ ~ ip(u,u) = / ^K{m,m,t\u,u)%p(m,fh,t), (62) Jr m 2 where T is a fundamental region for the modular group. Now apply the T transformation (54) to (62) - f d2m _ ~ m 2 IT 2'
I
r
QTVCi
m 2
2
~
K(m + l,m + l,t\u,u)ip(m+
l , m + l,t)
= / — ^ K ( m , m , t \ u , u)ip(m, m,t), (63) JT-irm22 where T~XT is the new fundamental region obtained from T by a T"1 transformation, and in (63) ip(m,m,t) and the integration measure are modular invariant. A similar argument holds for the S transformation, and shows that there are no invariant "Heisenberg picture" wave functions i>(u, u), since the integration regions in (62) and (63) are disjoint except on a set of measure zero. Further, ip(u, u) and Tip(u, u) are orthogonal since r
<^|TV>)= /
JT-ij7
r
if-™
^ o mi
/
Jjr
r/2™/
m2
^7Yrn'2262(m-m')iP(m,m,t)i>*{m',fh',t)=0: (64)
206
and similarly for S. Equation (64) shows t h a t the modular group splits the Hilbert space of square-integrable functions of (u 1,1*2) into an infinite set of orthonormal fundamental subspaces consisting of wave functions of the form (62) for a fixed fundamental region T. It is shown in [22] t h a t they are physically equivalent, because matrix elements of invariant operators can be computed in any of these subspaces, and each one is equivalent to the ADM Hilbert space. References 1. Ft. Arnowitt, S. Deser and C. W. Misner, The Dynamics of General Relativity, Ch. 7, pp. 227-264, in Gravitation: an introduction to current research, L. Witten ed. (Wiley, New York 1962) and gr-qc/0405109. 2. P.A.M. Dirac, Proc. Roy. Soc. Lond. A246, 333 (1958). 3. P.A.M. Dirac, Can. J. Math. 2, 129 (1950). 4. J. A. Wheeler in Relativity Groups and Topology, 1963 Les Houches Lectures, C. DeWitt and B. S. DeWitt eds. (Gordon and Breach Science Publishers Inc., New York 1964). 5. B. S. DeWitt, Phys. Rev. 160, 1113 (1967). 6. S. Deser, R. Jackiw and S. Templeton, Ann.Phys. 140, 372 (1982). 7. S. Deser, R. Jackiw and G. 'tHooft, Ann. Phys. 152, 220 (1984). 8. V. Moncrief, J. Math. Phys. 30, 2907 (1989). 9. A. Hosoya and K. Nakao, Class. Quant. Grav. 7, 163 (1990); Prog. Theor. Phys. 84, 739 (1990). 10. W. Abikoff, The Real Analytic Theory of Teichmiiller Space, Lecture Notes in Mathematics 820 (Springer-Verlag, Berlin, 1980). 11. J. W. York, Phys. Rev. Lett. 28, 1082 (1972). 12. K. Ezawa, Phys. Rev. D49, 5211 (1994); Reduced Phase Space of the First Order Einstein Gravity on R x T 2 , Osaka preprint OU-HET-185 (1993). 13. Y. Pujiwara and J. Soda, Prog. Theor. Phys. 83, 733 (1990). 14. S. Carlip, Phys. Rev. D47, 4520 (1993). 15. E. Witten, Nucl. Phys. B311, 46 (1988). 16. A. Achiicarro and P. K. Townsend, Phys. Lett. B180, 89 (1986). 17. J. E. Nelson and T. Regge, Nucl. Phys. B328, 190 (1989). 18. J. E. Nelson, T. Regge and F. Zertuche, Nucl. Phys. B339, 516 (1990). 19. J. E. Nelson and T. Regge, Phys. Lett. B272, 213 (1991). 20. J. E. Nelson and T. Regge, in Integrable Systems and Quantum Groups, Pavia 1990, M. Carfora, M. Martellini, and A. Marzuoli eds. (World Scientific, Singapore, 1992). 21. S. Carlip and J. E. Nelson, Phys. Lett. B324, 299 (1994); Phys. Rev. D 5 1 , 5643 (1995). 22. S. Carlip and J. E. Nelson, Phys. Rev. D59, 024012 (1999).
INTEGRABILITY + SUPERSYMMETRY + BOUNDARY: LIFE O N T H E E D G E IS N O T SO DULL A F T E R ALL!
R A F A E L I. N E P O M E C H I E * Physics
Department, P.O. Box 248046, University of Coral Gables, FL 33124, USA E-mail: nepomechieSphysics.miami. edu
Miami
After a brief review of integrability, first in the absence and then in the presence of a boundary, I outline the construction of actions for the N = 1 and N = 2 boundary sine-Gordon models. The key point is to introduce Fermionic boundary degrees of freedom in the boundary actions.
1. Introduction Quantum field theories (QFTs) with enhanced spacetime symmetries, such as integrability or supersymmetry, are attractive to theorists both as candidate models of real physical systems, and as toy models which can be probed more deeply than would otherwise be possible by exploiting their symmetries. Introducing a spatial boundary in such theories, whose effects can be physically important, poses a particular challenge to theorists, since boundary conditions generically break bulk spacetime symmetries. Hence the fundamental question: to what extent can bulk spacetime symmetries be maintained in the presence of a spatial boundary? One expects that the "more" bulk symmetries there are, the harder it is to maintain such symmetries when a boundary is introduced. In particular, it was believed for some years that it is essentially impossible to maintain both integrability and supersymmetry in the presence of a boundary. My main message here is that this belief is wrong: there do exist nontrivial integrable supersymmetric boundary QFTs. Although I address this question in the specific context of the sine-Gordon model, I expect that corresponding results can also be found for other models. This result may have *Work supported in part by the National Science Foundation under Grants PHY-0098088 and PHY-0244261. 207
208
applications in various areas, including condensed matter physics (in connection with impurity problems) and superstring theory. However, I have been motivated not so much by any particular application, but rather, by the two general convictions that systems with enhanced spacetime symmetries can be very interesting, and that boundary effects can be very important. The remainder of this article is organized as follows. Sec. 2 briefly reviews some general features of integrability in the absence of boundaries, and considers as an example the sine-Gordon model. 1 Sec. 3 briefly reviews integrability in the presence of a boundary, focusing on the boundary sineGordon model. 2 Sees. 4 and 5 outline the construction of actions for the N = 1 and N = 2 boundary sine-Gordon models, respectively.3'4 The key point is to introduce Fermionic boundary degrees of freedom in the boundary actions. Sec. 6 lists some interesting open problems, and points out related recent work on superconformal boundary Liouville models. 2. Integrability In this Section, I very briefly review some general features of integrability in the absence of boundaries, and consider as an example the sine-Gordon model. See Zamolodchikov and Zamolodchikov1 for a much more detailed
2.1.
Generalities
A QFT is integrable if it has an infinite set of mutually commuting local integrals of motion (IMs). According to the Coleman-Mandula theorem, 5 an integrable, Lorentz-invariant QFT in D spatial dimensions has a trivial S matrix, unless D = 1. Therefore, I henceforth restrict to 1 spatial dimension, with coordinate x. In this Section, I assume that there is no spatial boundary; i.e., the theory is defined on the line —oo < x < oo. A trivial example of an integrable QFT is the theory of a free massive scalar field (j){x,t), with Lagrangian density
£=^0^0-|mV.
(1)
Consider the following integrals of local densities 6 ' 7
I
hn = /
dx
\(dtdW + i(ax"+V)2 + \m2(d24>)2
/•OO
dx
'2n+l -oo
:dt
n = 0,1,2,... .
(2)
209
Using the equation of motion df
2.2. The sine-Gordon
model
As an example, let us consider the so-called sine-Gordon model, which is among the first known and most-studied integrable QFTs. It is convenient to go to Euclidean space (x, y), with z = x + iy, z = x + iy. The Lagrangian density is 777
CQ = 2dz(fidgip - -p cos{(3ip),
(3)
where
d-zTs+1 = a,e s _!,
dzTs+1 = a 2 e s _!,
s
= i,3,...,
(4)
210
starting with
T 2 - {dz<$>)2,
60 = —2 cos
T4 = (d24>)2 -\{dA)\ -
e 2 = (dz
i>,
(5)
and the corresponding barred quantities are obtained by complex conjugation. It follows from (4) that the local charges /"OO
OO
/ are conserved-00
dx {Ts+l + e s _ i ) ,
Ps=
dx {Ts+1 + es-i),
(6)
J—00
Aps=0=-fps,
s = l,3,....
(7)
dy dy The energy and momentum are given by Pi + Pi and Pi— Pi, respectively. The charges with s > 3 are nontrivial — their existence proves the classical integrability of the model. The classical spectrum includes solitons and antisolitons. Indeed, the classical potential V(
f°°
= — /
0
dx dxf = — Mx
= ° ° . y) - fix
= - ° ° . y)\ -
(8)
the solutions with T = +1 and T = — 1 are called solitons and antisolitons, respectively. There are also solutions with T = 0 called breathers. The quantum sine-Gordon model has particle-like states corresponding to these classical solutions, for 0 < 01 < 87r. The exact two-particle S matrix is given in [1]. 3. Integrability in the presence of a boundary Following [2], let us now consider what happens when a spatial boundary is introduced. That is, I consider an integrable QFT on the half line —00 < x < 0, which evidently has a boundary at x = 0. The first problem to be addressed is to determine the boundary conditions which preserve integrability. Another important problem is to determine the so-called boundary S matrix, which describes scattering off the boundary. Integrability implies that particles reflect elastically from the boundary, and that the
211
boundary S matrix obeys a boundary generalization10 of the Yang-Baxter Eq. Turning again to the example of the sine-Gordon model, let us consider the Lagrangian 2
f
dx {2dz(f>dz4> - 4cos) + B(
.
(9)
x=0
,
J —c
The Lagrangian density is essentially (3) with the coupling constant scaled away and with m = 2. The boundary potential B{4>) does not change the bulk equation of motion, but does affect the boundary condition, which follows from the variational principle, dB d<j)
= 0.
(10)
x=0
The question is: which B{<j>) (if any) preserves integrability? Clearly, the corresponding boundary conditions must be compatible with some nontrivial IMs. Since the boundary breaks translational invariance, momentumlike quantities ~ Ps—P3 cannot be conserved. The only hope is for energylike quantities ~ Ps + Ps to be conserved. Hence, consider the quantity
Ha= I dx [(rs+1 + es_i) + (Ts+1 + es_1)] .
(11)
J — oo
Computing theo "time" derivative, ^o / dy'
dx dv[
]= I
dx idx (Ta+i - 0 s _ i - T s + i + 0 s _ i )
= i(T s+1 -e s _ 1 -T s+1 + es_i) •oo
=i-r^s,
J—oo
x=o
(12)
uy
where the second equality on the first line follows from current conservation (4). Hence, if there exists a quantity S s obeying (12), then Hs - iT,s is an IM. That is, the boundary potential B((p) should be chosen so that (considering the first nontrivial case, s = 3)
( T 4 - e 2 - T 4 + e2) v
=4-S 3 .
(13)
' x=o dy Ghoshal and Zamolodchikov2 solved the constraint (13) for the boundary potential, and obtained the result fl(0) = 2 a c o s ( i ( t f - 0 „ ) ) ,
(14)
where a and 4>o are arbitrary boundary parameters. The model (9), (14) is known as the boundary sine-Gordon model. The corresponding boundary
212
condition (10), which reads {dx<j) — asin(|((/> —
(15)
where ip and -tp are components of a Majorana Fermion field. (Again, the dimensionless bulk coupling constant has been scaled away, and the mass parameter has been fixed to m = 2.) Indeed, this model has conserved supercurrents d-zTz = dzQ_i
,
dzTz2 = dzQ_i
,
(16)
where T1=dz<j>i), 2
6_i =-2Visin^; 2
(17)
2
and corresponding conserved supersymmetry charges /»OC
OO
/
dx(Ti+G_l2), Pi= dx(Tl+Q_i). (18) Moreover, this model has an infinite set of local integrals of motion 15 d— d „ s = l,3,..., (19) TPS == 0 = — Ps, dy dy (the corresponding currents for s = 1,3 are generalizations of (5) with additional terms involving the Fermion field) and is therefore integrable. Bulk S matrices have been proposed for the solitons 16 and breathers. 8 ' 17 One finally arrives at the question: are there boundary interactions which preserve both integrability and supersymmetry? This question was first addressed by Inami, Odake and Zhang, who proposed the Lagrangian 18
L=
f ./-oo
dx £ 0 + #(,V>,V0
, x=0
(20)
213
where CQ is given by (15), and B((j), ip, xjj) is a boundary potential. Imposing both integrability (as in Eq. (13)) and supersymmetry, they found that the boundary potential is fixed up to a sign,
£(,V,V) = ± ( 8 c o s J + V V ] •
( 21 )
That is, unlike the nonsupersymmetric (N = 0) boundary sine-Gordon model (9), (14), this model has no boundary parameters. This no-go result (namely, that the combined constraints of integrability and supersymmetry do not allow any free parameters in the boundary action) seemed to me aesthetically unsatisfactory and even paradoxical. 19 Hence, I decided to revisit this problem. 3 My main idea was to introduce a Fermionic boundary degree of freedom in the boundary action. Indeed, the N = 1 solitons have an Ising-type RSOS degree of freedom,16 and Ghoshal and Zamolodchikov2 introduced such a Fermionic boundary degree of freedom to describe the Ising model in a boundary magnetic field. Thus, instead of (20), I proposed the Lagrangian dx
JCQ
±iprl> + ia-^a - 2 / ( $ a ( > T i>) + B{cf>)
(22)
J —c
where Co is given by (15), a is a Hermitian Fermionic boundary degree of freedom which anticommutes with both ip and i/i, and B((f>), /() are boundary potentials. Imposing both integrability (as in Eq. (13)) and supersymmetry (~ Pi ± P i ) , one finds that the boundary potentials are given by 3 1 B(0) = 2 a c o s ( - ( ^ - 0 o ) ) ,
fc* 1 /(0) = ^ - B i n ( - ( 0 - D ) ) ,
(23)
where C, D are known functions of a,
214
generalization 21 £o = ^ ( -dz
+tp~dzijj+ + ^ ~ 9 2 V + + tp+dztp~
+ tp+dzip~) + gcos(p+ip~ip~ + gcos(p~ip+ip+ + g2 sin
dzT% = dzQt i ,
2
2
(25)
2
where Tf = 5 ^ ^ ,
eti=
2
9^
sin ip± ;
(26)
2
and corresponding conserved supersymmetry charges
dx(??-Q*i). 5
•00
5
-Pi=/ 2
di(rf-eix).
i-00
2
(27)
2
Moreover, this model has an infinite set of local integrals of motion -^-Ps=0=-^Ps, a = l,3,..., (28) dy dy and is therefore integrable. The bulk S matrix was proposed in [22]. This model can be formulated on the half-line most simply when the bulk mass vanishes (g = 0), in which case a suitable Lagrangian is 4 (see also [23]) L =J
dx Co + [ - fo+il>- + 4>~ip+) - \aT Ya+
- B&+
, V")
+\f+(
4
2
2
1 1 B(^+ ,
- tpo)), (30)
215
Hence, there are three (!) boundary parameters a, <^J. For the bulk massive case ( 5 / O ) , the boundary action has more terms, and I have performed an analysis only up to first order in g. The boundary S matrix for this model has been discussed by Baseilhac and Koizumi. 24 (See also [23,25].) 6. Outlook Already for the the nonsupersymmetric (N = 0) boundary sine-Gordon model, there are many interesting questions that remain unanswered, such as its relation to i3-perturbed boundary minimal conformal field theories (CFTs). In the bulk case, it is known 26 that the S matrices of the perturbed minimal models are restrictions of the sine-Gordon S matrix. One would like to know if something similar happens in the boundary case. For the minimal models, the possible conformal boundary conditions (CBCs) have been classified by Cardy. 27 (A CBC is characterized in part by its boundary entropy, 28 similar to the way that a bulk CFT is characterized by its central charge.) The boundary S matrix of a perturbed CFT describes the boundary "flow" from one CBC to another. Such issues (and more!) can eventually also be addressed for the N = 1 and N = 2 boundary sine-Gordon models which have been discussed here. This work has also recently led to progress in formulating superconformal boundary Liouville models. As is well known, the Liouville model is closely related to the sine-Gordon model. It is conformal invariant, not just integrable. For the N = I and N = 2 boundary Liouville models, the same Ansatze (22), (29) give one-parameter families of boundary actions which are N = 1 29 ' 30 and N = 2 al superconformal invariant, respectively. Dedicated to Stanley Deser, with gratitude for his guidance and support, and with best wishes. References 1. 2. 3. 4. 5. 6.
A.B. Zamolodchikov and ALB. Zamolodchikov, Ann. Phys. 120, 253 (1979). S. Ghoshal and A.B. Zamolodchikov, Int. J. Mod. Phys. A9, 3841 (1994). R.I. Nepomechie, Phys. Lett. B509, 183 (2001). R.I. Nepomechie, Phys. Lett. B516, 376 (2001). S.R. Coleman and J. Mandula, Phys. Rev. 159, 1251 (1967). P.P. Kulish, Conservation laws for the sine-Gordon equation, Serpukhov preprint (1974); Theor. Math. Phys. 26, 132 (1976). 7. V.E. Korepin and L.D. Faddeev, Theor. Math. Phys. 25, 1039 (1975). 8. R. Shankar and E. Witten, Phys. Rev. D17, 2134 (1978).
216
9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19.
20. 21. 22. 23. 24. 25. 26.
27. 28. 29. 30. 31.
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A N EFFECTIVE FIELD THEORY D E S C R I P T I O N FOR KALUZA-KLEIN S U P E R G R A V I T Y ON A D S 3 x S 3
H. NICOLAI Max-Planck-Institut fur Gravitationsphysik Albert-Einstein-Institut Muhlenberg 1, D-1^476 Potsdam, Germany E-mail: nicolaiiSaei.mpg.de H. S A M T L E B E N II. Institut fur Theoretische Physik der Universitat Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany E-mail: henning. samtlebenSdesy. de
We review the recent construction of a three-dimensional effective field theory that describes the interactions of the infinitely many TV = 8 supermultiplets contained in the spin-1 Kaluza-Klein towers t h a t arise in the compactification of six-dimensional N = (2,0) supergravity on AdS3 x S 3 . The theory forms a gauged N = 8 supergravity with gauge group SO(4) x Too over the infinite dimensional coset space SO(8, oo)/ (SO(8) x SO(oo)), where Too is an infinite dimensional translation subgroup of SO(8, oo).
1. Introduction Gravity and supergravity in three space-time dimensions have been a long standing interest of Stanley Deser's (for a selection, see Refs. [1-9]). The seminal contributions that Stanley and his co-workers have made and continue to make in this area foreshadowed many recent developments. In this contribution, we would like to review yet another recent result concerning half maximal gauged supergravity in three dimensions, 10 ' 11 for which Refs. [1-3] and the Chern Simons formulation of D = 3 gravity 12,6 and supergravity 13 are essential prerequisites (see also Ref. [9]). More specifically, we will discuss a model of Kaluza Klein supergravity in six dimensions 14 compactified on AdS3 x S3, for which the effective theory describing the non-linear couplings of the massless theory and two infinite towers of massive states can be described in terms of a certain D = 3, N = 8 217
218
gauged supergravity coupled to infinitely many N = 8 supermultiplets. This model is remarkable insofar as it appears to be the first and only example of a Kaluza Klein theory for which not only the massless sector, but also part of the massive sector can be completely and consistently treated within the effective theory approach 3 . However, we should emphasize that there is still a 'missing link' because the present construction does not include the remaining tower of massive 'spin-2' multiplets; the extension of our results to this remaining Kaluza Klein tower and hence to the full Kaluza Klein theory, remains an open problem. Our construction exploits the special properties of gauged supergravities in three dimensions, 18 ' 19 ' 20,21,22 which have no analog in dimensions D > 4. The AdS3 x S3 background which we consider here is half maximally supersymmetric, implying TV = 8 supersymmetry in three dimensions. The effective D = 3 theory describes a massless N = 8 supergravity multiplet and two entire infinite spin-1 towers and their interactions in terms of a gauged supergravity over a single irreducible coset space. The information about the effective D = 3 theory is thus encoded in the infinite dimensional coset space G/H — SO(8,oo)/(SO(8) x S0(oo)), which can be 'regularized' to any finite number M of multiplets M
M
fc2 n l2
S
8 x S0
G/H = SO (8, J2 + Y, ) /( °( ) fc>2
i>l
M
n
M
(51 + 5Z /2 )) - (!)
fc>2
fc2
(>1
The significance of the numbers k and I will be explained below. The parameter n denotes the number of tensor multiplets in the six-dimensional theory. 14 When the theory is gauged, a subgroup Go of the global symmetry G is promoted to a local symmetry, with associated minimal couplings. In contrast to higher dimensional gauged supergravities, the associated vector gauge fields appear with a Chern Simons term; in the case at hand, the Chern Simons gauge group is the semi-direct product Go = SO(4)
K
Tx c SO(8, oo),
(2)
where X^ is an infinite dimensional translation group. At this point we make use of the results of Ref. [11] establishing the link between YangMills and Chern-Simons type gauged supergravities in three dimensions: the Chern Simons type theory with gauge group (2) is on-shell equivalent a We note, however, that a similar idea was invoked in Ref. [16] to describe the effective low energy dynamics of a string compactified to four space-time dimensions. For a general introduction to Kaluza Klein supergravity, see Ref. [17].
219
to a Yang-Mills type gauged supergravity with Yang Mills gauge group GYM = S0(4), which is just the isometry group of the three sphere S3. The complete Lagrangian with an arbitrary number of multiplets from the spin-1 Kaluza Klein towers then assumes the simple form e-'C = -IR
+ Ig^TrP^
- e ^ c s -W
+ Cferm ,
(3)
with all the complexity of the nonlinear Kaluza Klein theory encoded in the coset space structure of (1) and the precise form of the embedding tensor which describes the minimal coupling of vector and scalar fields. The full fermionic Lagrangian as well as the supersymmetry transformation rules are obtained from Ref. [10] upon using the explicit form of this embedding tensor, given explicitly in (20), (43) below. In particular, the scalar potential W given in (23) is a function on the coset manifold (1), and yields the Kaluza Klein scalar and vector masses by virtue of a three-dimensional variant of the Brout-Englert-Higgs mechanism on the infinite-dimensional space (1). The fact that this agreement extends to the complete self-interactions induced by the Kaluza Klein compactification is a consequence of the uniqueness of the effective locally N = 8 supersymmetric theory. The present results may also shed some new light on the issue of consistent truncations in Kaluza Klein theories. 23 Among the maximally supersymmetric theories, complete consistency of the truncation was established for the AdS4 x S7 truncation of D = 11 supergravity in Ref. [24]b, and for the AdS7 x S4 truncation in Ref. [25], while the issue is still open for the AdSs x S 5 truncation of type II supergravity. The consistency of the AdS3 x S3 theory was discussed via a perturbative approach in Ref. [26]. The new aspect of the present work is that it shows the existence of infinitely many consistent truncations to certain subsectors of the Kaluza Klein theory, where any finite number of spin-1 multiplets may be retained. This is not to be expected to be the case for the spin-2 tower, in accordance with the known fact that there exists no consistent truncation with a finite number of massive spin-2 states in Kaluza Klein theory. 2. Supergravity spectrum on AdS3 X S3 The mass spectrum of AT = (2,0), D = 6 supergravity compactified on AdS3 x S3 has been computed in Ref. [27] by linearizing the equations of b
Whereas recent work emphasizes the consistency of the truncated equations motion, this proof proceeds via an analysis of the supersymmetry variations, whose consistency implies the consistency of the truncated equations of motion, see section 6 of Ref. [24].
220
motion around the AdS background, and in Ref. [28] by group theoretical arguments in terms of unitary irreducible representations of the supergroup SU(2|1,1) L x SU(2|1,1) R . The field content of the six-dimensional theory 14 comprises the supergravity multiplet with graviton, gravitini and five selfdual tensor fields, and n tensor multiplets, each containing an anti-selfdual tensor field, four fermions and five scalars. The scalars in six dimensions belong to a coset space u-model SO(5,n)/(SO(5) x SO(n)). The AdS3 x S3 background endows one of the five tensor fields of the supergravity multiplet with a vacuum expectation value such that the .R-symmetry group is broken from SO(5) down to SO(4). Together with the SO(n) rotating the tensor multiplets, this group survives as a global symmetry of the D — 3 effective theory. The spectrum of the three-dimensional theory is hence organized under the AdS3 supergroup SU(2|1,1) L x SU(2|l,l) f l whose bosonic extension SO(3) L x SO(3) fi = SO(4) corresponds to the isometry group of the three-sphere S 3 , and a global SO (4) x SO(n). It consists of three KaluzaKlein (KK) towers: a spin-2 tower, and two spin-1 towers transforming as vector and singlet under SO(n), respectively. Let us describe briefly the lowest levels of these three Kaluza Klein towers, 27 which we have displayed in Table 1. The SO(4) representations are labeled by their spins [j'1,.7'2] under S O ( 3 ) L X S O ( 3 ) H while the numbers (A, so) label the representations of the AdS group SO(2,2). The spin2 tower starts from the massless supergravity multiplet, which in three dimensions does not carry propagating degrees of freedom. It comprises the metric, gravitinos and pure gauge modes of the SO(4) vectors, see below. The lowest level of the spin-1 SO(n)-vector tower is occupied by the degenerate short (spin-i) multiplet (2,2)s that contains 8 scalars and 8 fermions, all transforming in the vector representation of SO(n). By contrast, the spin-1 SO(n)-singlet tower starts from the generic multiplet (3,3)s whose bosonic content is given by 26 scalars and 6 propagating vector fields.
3. Effective theory of the lowest mass multiplets The next task is the explicit construction of the three-dimensional theory that describes the coupling of the lowest level multiplets of the three Kaluza Klein towers, collected in Table 1. For that we need the result of Ref. [11] establishing the equivalence of Yang Mills gaugings with Chern Simons gaugings in three dimensions, in the following sense: a Yang Mills gauged
Table 1. A 2 2
SQ
Lowest mass spectrum on AdS 3 x S 3 .
SO(4)gau8e
SO(4) K i o b
SO(n)glob
# dof
Nonpropagating gravity multiplet ( 3 , 1 ) 5 + ( 1 , 3 ) s 2 [0,0] [0,0] 1 0 -2
[0,0]
3 2 3 2
3 2 3 2
Ml
1 1
1 -1
[0,1] [1,0]
[*.o]
[0,0] [0,1]
1 1
0 0
[1,0] [0,0] [0,0]
1 1 1
0 0 0
Spin 1 hypermultiplet ( 2 , 2 ) s L2' 2J
[0,0]
4n
[io] [o,|]
4n
[1 11 L 2 ' 2J Spin-] multiplet ( 3 , 3 ) s [0,0] 1 [1,1]
[0,0]
4n
[1.*]
[0.*]
1
til]
&°] rLi2 ' i2iJ [0,0] [0,0]
1 1 1 1
9 12 12
[*.o] [o.i]
1 1
[0,0]
1
2
0
5 2 5 2
l 2 1 2
3 3 3
0 1 -1
I2' 2J [1,0] [0,1]
7 2 7 2
1 2
[*.°]
4
0
2
4n
[0.5] [0,0]
16 3 3 4 4 1
supergravity with gauge group Go is equivalent on shell to a Chern Simons gauged supergravity with additional scalar fields and gauge group Go « T, where T is a translation group containing a subgroup transforming in the adjoint of Go- In particular, all gauged supergravities in three dimensions may be obtained as deformations of that ungauged theory in which the entire bosonic matter content has been dualized into the scalar sector. The supergravity multiplet ( 3 , l ) s + ( l , 3 ) s in Table 1 contains the non-propagating fields in three dimensions. The corresponding topological supergravity theory has been given in Ref. [13] as a Chern-Simons theory based on the supergroup SU(2|1,1) L x SU(2|1,1) H . The coupling of this theory to propagating matter in the n spin-^ hypermultiplets (2,2)s has been constructed in Ref. [10]. It comes as an SO(8,n)/(SO(8) x SO(n)) coset space model with SO (4) gauge group and non-propagating Chern Simons vector fields. We shall now describe the extension of this construction to the spin-
222
1, SO(n) singlet, supermultiplet (3,3)s which contains the SO(4) YangMills gauge vectors, in order to describe the full lowest mass spectrum of supergravity on AdS3 x S 3 . Counting of degrees of freedom shows that after dualizing all degrees of freedom into the scalar sector, the spectrum of Table 1 consists of 32 + 8n bosonic and the same number of fermionic degrees of freedom. The theory describing this field content should thus be obtainable as a Chern Simons gauging of the N = 8 theory with coset space G/H = SO(8,4 + n)/(SO(8) x SO(4 + n)) .
(4)
According to the above discussion, the relevant gauge group must be of the type Go = SO(4) gauge « T C SO(8,4 + n) ,
(5)
where SO(4) gauge is to be identified with the YM gauge group and T = T6 denotes an abelian group of six translations that transform in the adjoint representation of SO(4)gaUge. In order to identify the group (5) within G, we first embed the SO(4) subgroup into the compact subgroup H c G according to SO(4) gauge c C
SO(4) + x SO(4)_ x SO(4) 2 x SO(n) SO(8)
x
SO(4 + n)
,
(6)
where SO(4) gauge = diag (S0(4) + x SO(4) 2 )
(7)
denotes the diagonal subgroup of SO(4) + and SO(4) 2 . Under this SO(4) gauge, the 32 + 8n scalars that parametrize the coset space (4) transforming as a bivector (8„, 4 + n) under H decompose into ( [ i , i ] + 4 . [ 0 , 0 ] ) x ( [ | , I ] + n-[0,0])
(8)
= n • [|, \] + An • [0,0] + [0,0] + 4 • [I, \] + [0,1] + [1,0] + [1,1] , in accordance with Table 1. Likewise, the fermions transforming as (8C,4 + n) under H decompose as (2.[i,0]+2-[0,I])x([i,I]+n-[0,0])
(9)
= 2.[l,i]+2.[0,i]+2.[i,l] + 2.[i,0]+2n.[i,0]+2n-[0,i].
223
Moreover, the factors SO(4)_ and SO(n) in (6) commute with SO(4) gauge and thus represent global symmetries of the gauged theory. With the further identification SO(4) glob = SO(4)_ ,
SO(n)giob = SO(n) ,
(10)
the decomposition according to (6) precisely reproduces the desired spectrum of representations. Note however, that the vector degrees of freedom of Table 1 appear among the scalars in (8). These are the Goldstone bosons which give mass to the associated Chern Simons gauge vectors. Accordingly, the gauge group (7) is enlarged to (5) by the essentially unique set of six nilpotent abelian translations T c G transforming in the adjoint representation of SO(4)gaUge. This part of the gauge group is broken at the groundstate in order to account for the massive vectors. We have thus identified the proper gauge group (5) within G. It remains to verify that this gauge group is compatible with the algebraic constraints that supersymmetry imposes on its embedding tensor, 10 ' 21 see (14) below. The Lagrangian and supersymmetry variations of the most general D = 3, N = 8 gauged supergravity have been given in Refs. [10,21]. As shown there, the theory is completely specified by the choice of coset space G/H and the symmetric embedding tensor QMM, which characterizes the gauge group Go C G and encodes the minimal coupling of vector fields to scalars according to D^S=(d„ + eM^B^t^)S.
(11)
The matrix <S G G = SO(8,4 + n) here contains the scalar fields of the theory; by tM we denote the generators of g = Lie G acting by left multiplication, with the curly indices M, N referring to the adjoint representation of g. The number of vector fields involved in (11) is equal to the rank of @A4Af; a n d the gauge algebra is generated by (QM/stN). Because the embedding tensor @Mx characterizes the theory completely, it remains to identify the tensor which correctly reproduces the gauge group (5), and at the same time is compatible with the algebraic constraints imposed by supersymmetry. It turns out that there is a unique QMJ^ that fits all the requirements. To construct it explicitly, we need to write out generators of £) more explicitly. We denote by indices I, J,... and indices r,s,... the vector representations of SO(8) and SO(4 + n), respectively. The generators {tM} of g split into the compact generators {X^J\X^T^}, and the noncompact generators {YIr} with standard commutation rela-
224
tions [XIJ, XKL] = 2 (*'[* XLV - 6J^K XLV) , [Xra,Xrtv]=2(5r^Xv^-Sa[uX^T),
[XIJ,YKr]
[Xrs, YKu] = -2Su^r YKs^ ,
[YIr, YJs]
-2SK^Y^r,
=
= dIJ Xrs + 8rs XIJ . (12)
In particular, the current (11) decomposes into = }-QlJXu
S-'D^S
+ \Qr;Xrs
+ VIfirYIr .
(13)
In this basis, the algebraic constraints imposed by supersymmetry on the embedding tensor 6 ^ ^ from (11) read 10 ®IJ,KL = ®[IJ,KL] ,
&IJ,rs
=
®Ir,Js ,
&IJ,Kr = Q[IJ,K]r ,
®Kr,su
=
QK[T,SU] •
@rs,uv =
&[rs,uv] i (14)
In order to describe the embedding (6), we further need to split these indices into / = (i, i) and r = (i, r) with I,J,...:
i,j,---£
r,s,...:
{1,2,3,4},
t , j , - . - e {1,2,3,4}, M
The generators {t
t, J, ••• G {5,6,7,8} , f,s, ••• e { 1 , 2 , . . . ,n} . (15)
} accordingly decompose into
Q = (x^ij\Xi\X^,X^],Xif,X^\
© {Yij,Yif,Y~%j,Y~lf}
. (16)
Prom these we may explicitly build the generators of so(4) g a u g e and the six abelian nilpotent translations t as so(4) g a u g e = | J [ « l = Xl«l + XM} , t = ( T ' ^ I = X^rt — X^fi + Y^ — Y^l\ .
(17)
It is straightforward to verify that the J^ close into the SO(4) algebra (7) while the mutually commuting generators T ^ transform in the adjoint representation under J^, This is the Lie algebra underlying (5). Similarly defining vector fields C[ij]
=
B[ij]
+
B[ij\
^
A[ij]
_
B[ij]
_
B[i}] +
Bij
_
Bji
we start from the following ansatz for the embedding tensor QM^
+\d2 (Afii]T+W
- ,4-WlT-W]) ,
^g)
j
n
(19)
225
with real constants g\, g2, and where A^1^ denote the selfdual and antiselfdual part of A^lj\ respectively, etc. Translating (19) back into the basis (12), (16), this embedding tensor takes the form ®ij,kl = (32 + 2#i) eijkl , Qiikl = -g2 eijki , 0 ij)fcf = (gx + g2) eijkl , e
i3,ki = (92 ~ 2gi) eijkl , Qikji
= -g2 eyfcJ , 9 ~ fef = (gi - g2) eijkl . (20)
The choice of a relative minus sign between selfdual and anti-selfdual components in (19), or equivalently the relative coupling constant (—1) between the two SO (3) factors in SO (4) is necessary to ensure that terms proportional to 8!?J drop out in (20), such that the supersymmetry constraints (14) are satisfied for any choice of gi and g2. That is, at this stage we still have a class of physically distinct theories for different choices of g\, g2. We further emphasize that these constraints harmonize beautifully with the particular non-semisimple type of gauge group (5). Indeed, coupling the diagonal SO(4) of (7) requires a nonvanishing contribution in 0 / j , r s . By means of (14) this induces a nonvanishing Qir,js which in turn precisely corresponds to coupling the nilpotent contributions of (17). In other words, the diagonal SO(4)gaUge from (7) alone is not a consistent Chern Simons gauge group; supersymmetry requires its non-semisimple extension to (5). We may now state the complete bosonic Lagrangian of the threedimensional theory that describes the coupling of the lowest matter multiplets. It is given as a gravity coupled Chern Simons gauged coset space CT-model e-lC = -\R
+ \g»v V'JV1; - e ^ c s
- W ,
(21)
with G/H = SO(8,4 + n)/(SO(8) x SO(4 + n)). The kinetic scalar term is obtained from putting together (11), (13), and (20), while the Chern Simons term is
£cs = \^vpK
0
^ (dvB? + | SNvc QVK B*B^J ,
(22)
with QMN from (20) and the SO(8,4 + n) structure constants from (12). The potential W is given as a function of the scalar fields as W = ~
+ \ eIJKLMNpQ
(T{U,KL]T{IJ,KL\
Tjj,KLTMN,PQ
) ,
(23)
in terms of the so-called T-tensor Tu,KL
= VMIJVKL
QMN
,
Tu,Kr
=
VMJjVMKr
& MM ,
(24)
226
where V defines the group matrix 5 in the adjoint representation: S~hMS
= ^ VMu XIJ +X-2VMra fs + VMir YIr
.
(25)
Hence, like all other terms in (21), the scalar potential W depends crucially on the precise form of the embedding tensor (20). For the fermionic contributions and full supersymmetry transformations we refer to Ref. [10]. Here, we just quote the variations of the gravitinos ipfi and fermion fields \Ar (neglecting cubic spinor terms)
^
= ( ^
?" - ^ T
u
, K r )
eA ,
(26)
with SO (8) T-matrices Tf,. These variations are likewise expressed in terms of the T-tensor from (24) and may serve as BPS equations for bosonic solutions. In particular, they show that an AdS ground state preserving all supersymmetries requires TjjtKr = 0. Recall that we seek that theory whose groundstate at the origin 5 = 1 precisely corresponds to the sixdimensional AdS3 x 5 3 background with full N = (4,4) supersymmetry. Since T at this point reduces to the embedding tensor ©, together this imposes a nontrivial relation between the constants g\, g^ in (20) &U,Kr = 0
=»
6.,.
fcf
= 0
=»
52 = -51 •
(27)
That is, existence of a maximally supersymmetric AdS groundstate eventually fixes the ratio <7i/
227
4. Effective theory of massive Kaluza Klein s t a t e s We now wish to extend the results described in the foregoing section to the two infinite towers of spin-1 multiplets ^spin-1 = / ,(k
+ l,k + l ) s + n-5^(1 + 1,1 + 1) 5 ,
fc>2
(28)
/>1
that appear in the KK spectrum, transforming in the singlet and the vector representation of SO(n), respectively. In this way, we are able to construct an effective D = 3 theory that comprises the entire spectrum (28) of spin-1 multiplets, and which is obtained by gauging the TV = 8 theory with the coset space given in (1) above. The generic spin-1 multiplet is given in Table 2. It contains 16fc2 degrees of freedom, and following Ref. [28] we will designate it by (k + 1, k + 1)5. Indeed, its number of bosonic degrees of freedom reproduces the infinite dimensional coset (1), i.e. gives agreement for each value of k and I separately. It remains to determine the corresponding Chern Simons gauge group and its embedding. According to Ref. [11], this group, which we denote by SO(4)gauge * (Too ® Te), is obtained from exponentiating an extension of the non-semisimple algebra (17) described above, by a set of additional nilpotent generators too corresponding to the massive vector fields appearing in (28). For transparency, we will now describe in detail the theory which couples the distinguished lowest spin-1 multiplet (3, 3)5 to a single additional higher spin-1 multiplet, say, (k + l , k + l ) s from the SO(n) singlet tower. The extension to an arbitrary number of these multiplets and multiplets from the SO(n) vector tower is straightforward. To this aim we consider the coset space G/H = SO(8, 4 + k2)/ (SO(8) x SO(4 + k2)) .
(29)
Similar to (6) the gauge group SO(4) gauge is embedded via SO(4) gauge c
SO(4) + x SO(4)_ x SO(4) 2 x SO(4)fc
C
S0(4) + x SO(4)_ x SO(4) 2 x SO(fc2)
C
SO(8)
x
SO(4 + A;2)
,
(30)
with SO(4) gauge = diag(sO(4) + x SO(4) 2 x SO(4)fc) ,
(31)
228 Table 2. Spin-1 multiplet ( k + l , k + l ) s . A
so
SO(4)gauge
SO(4) K l o b
#dof
fc
0
[0,0]
(fc + 1) 2
fc + ±
[o.« [i°]
2fc(fc + l)
k+\
l 2 1 2
[fc k] [2< 2 j [fc k-l]
fc + 1
0
[il]
4fc2
fc + 1
1
fc + 1
-1
fc+f fc+f fc + 2
[2'
2 J
r k - i fci [ 2 ' 2j [fc-1 fc-l]
[ 2 ' 2 J [fc fc-2] [2' 2 J [fc-2 k]
[ 2 '2J
1 2 1 2
[fc-1
fc-2]
[fc-2
0
[fe-2
2fc(fc + 1)
[0,0]
fc2-l
[0,0]
fc2-l 2fc(fc - 1)
fc-l]
[|.o] [o,|]
fc-2]
[0,0]
(fc-1) 2
[ 2 - 2 J [ 2 - 2 J L 2 • 2 J
2fc(fc - 1)
and an embedding SO(4)fc C SO(fc2) :
fc2^[^fi,^i]
.
Working out the products of the relevant representations, viz. (l-[|,i]+4.[0,0])x [ ^ W ]
(bosons),
(2 • [1,0] + 2 • [0, i]) x [4=1, i = i ]
(fermions) ,
it is then straightforward to verify that under these subgroups the fields reproduce the correct representation content as given by Eq. (28) together with Table 2. Again, the vector degrees of freedom show up through the associated Goldstone scalars, transforming in the same representations as the massive vector fields [A=a,$]
+
[H,i=2].
(32)
In line with our general arguments above we now seek a theory with Chern Simons gauge group SO(4) gauge x ( f W ® T 6 ) C SO(8,4 + k2) ,
(33)
with SO(4) gauge from (31), extending (5) by 2 ( f c 2 - l ) generators transforming as (32) under SO(4) gauge and closing into T 6 , in order to correctly describe a theory with SO(4) Yang Mills gauging and 2(A;2 - 1 ) massive vector fields.
229
To this end, we introduce the index split / = (i,i) and r = (?, ab) with a, b, • • • = 1 , . . . , k, according to (29) and extending (15). The generators {tM} of S0(8,4 + k2) accordingly decompose as 0 = ix[ii\Xi3,X^,X^,Xil
© {Yi3,Yitab,Y*s,Yl'ab}
. (34)
The commutation relations can be read off from (12). For the SO(k2) subgroup, we thus have tj£ab,cd
j{ef,gh-i
__ cd,ef j^ab,gh
_
cd,gh^ab,ef
_
ab,ef j£cd,gh
,
ab,g/i-v-cc^e/
(35) where the metric rjab,cd = VacVbd serves to lower and raise indices, such that Vub,efVef'cd = 5cjt ,
(36)
and the tensor r)at, is the quadratic invariant of the fc-dimensional representation of SO(3); it is symmetric for bosonic representations, i.e. odd k, and skew-symmetric for fermionic representations, i.e. even k. The generators of so(4) gaug e are defined as the extension of the previous SO(4) to the new diagonal SO(4) subgroup (31) viz. *o(4) gauge = { J M = xM+xM
+ % Ct/alvbdX^+^Q^V*^}
•
(37) By Qj , we here denote the generators of SO (4) in the spin [^r^O] an<^ spin [0, ^p-] representation, respectively. Accordingly, ^ . a 6 is symmetric in ab for fermionic (even k) and skew-symmetric for bosonic (odd k) representations. Explicit expressions for these generators can be constructed in terms of Clebsch-Gordan coefficients. The SO(3) commutation relations in this representation are ' >±(fc) >±(fc) 1 _ 9 (r t±{k) ^ij '^mn J ~ ^ \ui[m>>n]j
_ s- r±(k)\ u j[m^n]i J '
,±(k) ">ij
_
—
, i jjmn ^2C
tr(c{ffc)CJffc))=fc(fc2-l),
>±(fc) Wn '
(38)
in obvious matrix notation. Let us also record the relation Cimac Cmj cb =
4 (k
~ ^ijVab
(39)
+ Cjj ab '
which follows from (38) and the (anti-) selfduality of the £'s. The generators of the translation subgroup (T^ ® T6) of (33) are given by t = { r ^ 1 = X[ij] - X^
+ Yi3 - Yji\
,
(40)
230
which is the same as before, and by t( fc ) :
= {( P [^l •2k] ) 1/iabj cd ©{(Pf k
(Xjcd
_
yj cd\ 1
V
(X^cd — Y3cd)
k-l] )
(41)
ab,j cd
where (Fri-aii) . \
.
cd I 2 ' 2 J / iab,j ab,j cd
=^
»?„c»7M «« + ^ k-12 2k
A-W _
Cy « Vbd 1 r +(fc) > - W
ac ^'/ac ^ij7' hd bd ~~ Jc* t? M ^im ac ^mj S>j m ac ^>n bd '
( P [4,^2]) ia6j . cd = T P 1 ^ ^ <*« - W C^ac Vbd ,fc+1 _a c r - W _ J_ r+(k) ,-{k) 2 2 '
2/c "
"sij 6d
fc
>>im ac ^mj bd '
are the projectors onto the [^y^, f ] and [|, ^ ^ ] representations, respectively, in the tensor product [ I , I ] x [ ^ , ^ ] of SO(4) gauge . It is straightforward to show that indeed [tW,tW]ct,
[t<*>,t]=0
[t(fc),t]=0.
for k^l,
(42)
The embedding tensor (20) acquires the additional components (fc) Ky
_
0 (fc)
_o(*0 ~ ^iabjcd
0
ij,abcd~
^iabjcd
_
n
( f c )
^i^abcd
_
0
~
^ij,abcd
( f c )
_ _o(fe) ~~ ^iabjcd
_
~
> + (*!)
_
9lSijacVbd
A-(fc)
9l Uj bd Vac ,
_ 0,f+(k)„u_flir('!)tl C4^ ~ Ul^ijac'lbd ill ^ij bd 'lac ' \*°)
which are obviously compatible with the algebraic constraints (14) imposed by supersymmetry (with antisymmetry under interchange of the indices i, j and the pairs ab and cd, each pair being regarded as a single SO(fe2) index). Moreover, they do not obstruct the existence of an N = (4,4) supersymmetric AdS groundstate (27) if we keep g\ = —g-i as we did in (20). The components in the first line of (43) can be read off directly from (37) (keeping in mind the relative factor (—1) between the two SO(3) factors in SO(4)), and give rise to the generalization of J^ from (37). The remaining components, i.e. the second line in (43) lead to the additional contribution in (19) e£l
B^tM = ef a } 6 J c d (Biab - Biab)
(X3cd - Y*cd) .
(44)
These components can be determined in two a priori different ways. On the one hand, they are related to the first line of (43) by supersymmetry (14), i.e. complete antisymmetry in the SO(8,4 +fc2)indices, implying
231
e.g. &\ab,jcd = ®ij,abcd- ®n t n e ° t n e r hand, their values are proportional to the difference between the two projectors from (41)
©faLcd = ^ ( p [ * , ^ ] - p [ ^ , i ] ) i a f e , c d '
w
again featuring the relative factor of (—1) between the different 'chiralities' that we saw already in (19). This remarkable coincidence guarantees that (44) picks out precisely 2{k2 — 1) vector fields from the a priori 4fc2 fields (Biah - Biab). Likewise, the combinations
Ti<* = e\kXiai(X3cd-YJcd)>
(46)
appearing in (44) correspond to a projection of the 4fc2 nilpotent generators (X? c d - Yjcd) onto a subset of 2(k2 - 1) generators, which span (41). Had supersymmetry (14) imposed another value for S ; a b , c d , the minimal coupling (44) would have involved too many vector fields and generators. The Chern Simons gauge group identified by (43) precisely realizes the desired algebra (33). The Lagrangian of the theory coupling the higher spin-1 multiplet (k + 1, k + 1)5 is then given by (21)-(25) with the coset space G/H from (29) and the embedding tensor QMAr by (20), (43). Its fermionic part and supersymmetry transformation rules are obtained from Ref. [10]. Finally, the theory describing the entire spectrum (28) is straightforwardly constructed starting from the coset (1) and summing over the additional contributions (43) of the embedding tensor 0 for the different k. Note that all Q^ in (43) act in different sectors; consequently, there are no divergent infinite sums of any kind in the limit of infinitely many multiplets. Similar comments apply to the multiplets from the spin-1 tower in the vector representation of SO(n). Acknowledgments H. N. would like the organizers of the Deserfest, M, Duff, K. Stelle, J. Liu and R. Woodard, for the invitation to a very enjoyable meeting. References 1. S. Deser, R. Jackiw and S. Templeton, Ann. Phys. 140, 372 (1982), Erratumibid. 185, 406 (1988). 2. S. Deser, R. Jackiw and G. 't Hooft, Ann, Phys. 152, 220 (1984). 3. S. Deser and R. Jackiw, Ann, Phys. 153, 405 (1984). 4. S. Deser and R. Jackiw, Ann. Phys. 192, 352 (1989).
232
5. S. Deser and Z. Yang, Mod. Phys. Lett. A4, 2123 (1989). 6. S. Deser, Gravity And Gauge Theories In Three-Dimensions, Invited lecture given at 1st Int. Symp. on Particles, Strings and Cosmology (PASCOS-90), Boston, MA, Mar 27-31, 1990. 7. S. Deser and J. McCarthy, Phys. Lett. B246, 441 (1990), Addendum-ibid. B248, 473 (1990). 8. S. Deser, J.G. McCarthy and A.R. Steif, Nucl. Phys. B 4 1 2 , 305 (1994) [hepth/9307092]. 9. S. Deser and J.H. Kay, Phys. Lett. B120, 97 (1983). 10. H. Nicolai and H. Samtleben, Phys. Lett. B514, 165 (2001) [hep-th/0106153]. 11. H. Nicolai and H. Samtleben, Nucl. Phys. B668, 167 (2003) [hepth/0303213]. 12. E. Witten, Nucl. Phys. B311, 46 (1988). 13. A. Achiicarro and P. K. Townsend, Phys. Lett. B180, 89 (1986). 14. L. J. Romans, Nucl. Phys. B276, 71 (1986). 15. H. Nicolai and H. Samtleben, JHEP 0309, 036 (2003) [hep-th/0306202]. 16. A. Giveon and M. Porrati, Phys. Lett. B246, 54 (1990); Nucl. Phys. B355, 422 (1991). 17. M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rep. 130, 1 (1986). 18. H. Nicolai and H. Samtleben, Phys. Rev. Lett. 86, 1686 (2001) [hepth/0010076]; JHEP 0104, 022 (2001) [hep-th/0103032], 19. H. Lu, C. N. Pope, and E. Sezgin, SU(2) reduction of six-dimensional (1,0) supergravity, hep-th/0212323; Yang-Mills-Chern-Simons supergravity, hepth/0305242. 20. T. Fischbacher, H. Nicolai, and H. Samtleben, Class. Quant. Grav. 19, 5297 (2002) [hep-th/0207206]. 21. B. de Wit, I. Herger, and H. Samtleben, Nucl. Phys. B 6 7 1 , 175 (2003) [hepth/0307006]. 22. T. Fischbacher, H. Nicolai, and H. Samtleben, Commun. Math. Phys. 249, 475 (2004) [hep-th/0306276]. 23. G.W. Gibbons, talk at this conference. 24. B. de Wit and H. Nicolai, Nucl. Phys. B 2 8 1 , 211 (1987). 25. H. Nastase, D. Vaman, and P. van Nieuwenhuizen, Phys. Lett. B469, 96 (1999) [hep-th/9905075]. 26. G. Arutyunov, A. Pankiewicz, and S. Theisen, Phys. Rev. D 6 3 , 044024 (2001) [hep-th/0007061]. 27. S. Deger, A. Kaya, E. Sezgin, and P. Sundell, Nucl. Phys. B536, 110 (1998) [hep-th/9804166]. 28. J. de Boer, Nucl. Phys. B548, 139 (1999) [hep-th/9806104].
M A S S I V E GRAVITY IN A D S A N D M I N K O W S K I BACKGROUNDS
M. PORRATI Department of Physics, New York University 4 Washington Place New York NY 10003, USA E-mail: massimo.porratiSnyu.edu I review some interesting features of massive gravity in two maximally symmetric backgrounds: Anti de Sitter space and Minkowski space. While massive gravity in AdS can be seen as a spontaneously broken, UV safe theory, no such interpretation exists yet in the flat-space case. Here, I point out the problems encountered in trying to find such completion, and possible mechanisms to overcome them.
1. Introduction While doing theoretical research in gravity, both classical and quantum, standard or "super," at some stage we inevitably encounter a major contribution due to Stanley Deser. My experience is no exception: in [1], Boulware and Deser presented a comprehensive and in some way definitive study of massive gravity in four dimensions. Their analysis pointed out the true reason underlying the problems faced by a quantum theory of massive gravity. It has nothing to do with the incompleteness of Einstein gravity at high energy. Rather, it is a truly infrared problem. The problem is best explained using the ADM formalism,2 another major contribution to gravity due to Stanley. In ADM, the physical, propagating degrees of freedom of massless gravity are the space metric,
234
them appear nonlinearly in the action of massive gravity. So, their equations of motion do not produce any new constraint, and one ends up with 6 propagating degrees of freedom. One of them is always either a ghost or a tachyon. The only exception to this conclusion obtains when Einstein's action is modified by adding a Pauli-Fierz 3 mass term that, at quadratic order in the metric fluctuation h^ — g^ — r)mnv, reads M2 SM
= SEinstein
+ ^
^
f J d ^ h ^ h ^
- h2).
(1)
In this action, the lapse (better hoo ~ N2 — 1) appears linearly, so that it still acts as a multiplier, and does eliminate the unwanted sixth degree of freedom. On the other hand, Boulware and Deser showed that this property of the Pauli-Fierz mass term holds only in the quadratic approximation. In any Lorentz-invariant mass term, N does enter nonlinearly in the complete, interacting Lagrangian. Correspondingly, the full nonlinear theory propagates 6 degrees of freedom (and the Hamiltonian is unbounded below). The results of Boulware and Deser mean that massive gravity cannot be a consistent quantum theory. The recent revival of interest in massive gravity, or, more generally, in long-distance modifications to gravity, does not contradict that. The question posed in recent years is not whether a quantum theory of massive gravity exists, that makes sense up to a very high (Planckian) energy. The question is instead whether massive gravity makes sense as a low-energy effective field theory, up to the shortest scale at which we have experimentally tested gravity. From this point of view, the UV cutoff of the theory is not Mpi but the (somewhat smaller!) scale ~ (100/um) - 1 ~ 1 0 - 3 eV. Surprisingly, finding an effective theory that works up to such a small cutoff and is compatible with experiment is nevertheless difficult. Let us review the problems, starting with the case that is better understood theoretically, namely, massive gravity in AdS space. 2. Massive Gravity in AdS Space This section is based on [4,5]. Neither in flat space, nor in anti de Sitter space is the long-distance behavior of Einstein's gravity changed by coupling it to massive particles. The effect of massive particles is always encoded in local operators that do not give a mass term to the graviton, because of general covariance. On the other hand, in AdS space, the effect of massless particles is subtler. Let us take for instance a very simple case: a conformally coupled free scalar.
235 Free means here that the scalar interacts only with gravity. By integrating out the scalar field, one gets a nonlocal action for the graviton, that can be written schematically as S=^gJd4x^(R-2A)+WcFT[g}-
(2)
A is the (negative) cosmological constant of the background, and W C F T M denotes the generating functional of the connected correlators of the CFT. Denote by
+ Ztfhcp
= 0.
(3)
Here, L^£ is the standard Einstein kinetic operator in AdS, and x2ixr S2WcFT
ya(3 ""
Sg^Sga0
= (T^T^)CFT.
(4)
g=h
So, E is the two-point function of the stress-energy tensor in the CFT. Implicit in this notation is the fact that while the CFT is integrated out exactly, gravity is treated classically, i.e. all graviton loops are being ignored. This approximation makes sense for computing infrared quantities on a weakly curved background. By a wise choice of counterterms, £ can be made transverse traceless with respect to the background metric. By construction, the equations of motion are covariant. So, one can decompose the metric fluctuation into transverse-traceless (TT), longitudinal, and trace components, and choose the gauge h^ — /i^J + . In this gauge the equations of motion split into 1
32?rG
(A - 2A) + S(A)
Kt = 0,
(3A - 4A)c?!> = 0.
(5)
Here, A is the Lichnerowicz operator, 6 a curved-space generalization of the Laplacian, that commutes with the covariant divergence and trace defined in terms of the background metric. The scalar operator S(A), computed at the pole of the propagator, gives the graviton square mass. In the Einstein theory, the pole is at A = 2A, and E(2A) = 0. When the mass is smaller than 2|A|, this prescription gives M 2 « 32TTG£(2A).
(6)
236
Now, even before doing any explicit computation, it is clear that this mass is parametrically smaller than the curvature radius of AdS, L = y^|A|/3. Indeed, £ is computed by a correlator in the CFT, that depends on L but not on the Newton constant G. So, £ can be at most 0(cL~4), where c is the central charge of the CFT; thus, M ~ a^Lpianck/L2
< 1/L.
(7)
Here a is a number of order one. Of course, it could still be zero. The analysis performed in Ref. [4] shows that a indeed vanishes when the conformally coupled scalar is given standard (reflecting) boundary conditions at the boundary of AdS. When the field is given more general boundary conditions, that allow for an energy flow into and out of the AdS space, then a is nonzero. More precisely, the representation theory of the isometry group of AdS^ SO(2,3), shows that a conformally coupled scalar can belong to two representations, called .D(1,0) and Z?(2,0), respectively (see [4] or [7] for notations and further results). When reflecting boundary conditions are given, then the scalar belongs to either D(1,0) or D(2,0).8 In general, the scalar field may be a linear combination of modes belonging to both representations. In the case that the linear combination is the same for all modes, then the scalar propagator is A{x,y) = aA1{x,y)+0A2(x,y),
a + 0 = l.
(8)
Here AE(x, y) is the propagator for modes in the D{E, 0) irrep. Transparent boundary conditions, first proposed in [9], and natural from the point of view of the holographic AdS/CFT duality, 10 ' 4 are a = /3 = 1/2. The result of [4,5] is S
^ =
a / ?
5 ^ '
(9)
So, the graviton mass vanishes for standard boundary conditions, but it can be nonzero for nonstandard ones. Physically, the Higgs field (a vector belonging to .0(4, l), 4 in this case) is a composite field, a "bound state" of size L. In conclusion, in AdS one can give a (tiny) mass to the graviton by a Higgs mechanism. The Higgs is a composite vector of size L, and 1/L is the cutoff for the effective theory of massive gravity. Above that energy, the correct description is in terms of ordinary gravity plus all the degrees of freedom of the CFT.
237
3. Minkowski Space No analog of the Higgs-like mechanism described in the previous section is available in Minkowski space. The best one can do is to introduce the appropriate Goldstone fields needed to make massive gravity explicitly covariant under general coordinate transformations. This can be done at the full nonlinear level.11 In the Goldstone boson language, the disease first noticed by Boulware and Deser, that the lapse starts propagating at nonlinear level, manifests itself in a different guise: massive gravity becomes strongly interacting at an extremely small energy scale: E~{MPlM4)V5,
E ~ (MPiM2)1/3.
or
(10)
The first scale holds for the standard PF mass term, while the second is the best that can be achieved by judiciously improving the action by adding appropriately chosen higher-order operators. 11 For a graviton of Compton wavelength Mv1 ~ 10 28 cm, the cutoff length corresponding to the highest energy cutoff in Eq. (10) is 0(1000km)\ This is the scale below which massive gravity becomes strongly interacting and essentially uncontrollable within perturbation theory. The existence of such very low scale is not confined to the theory of a single massive graviton. A similar bound also exists in the DGP model. 12 The DGP model is the first ghost-free example of a mechanism in which gravity can be localized on a 4d brane in a space of infinite transverse volume. It describes a theory where 4d general covariance is unbroken, but the graviton is a metastable state. Its main property is that, on the 4d brane, gravity looks 4d at short distance, while it weakens at large distance. Interesting cosmological applications of this scenario have been proposed, for instance in [13]. The model can be described by the action
Zx'-iikL***1™ 1
JdM
-Kh) + T^R(7)
8TTG5
16TTG
(11)
where A4 is a 5d manifold with boundary dM, g is the 5d metric, 7 is the 4d induced metric on the boundary, and K is the extrinsic curvature. The DGP model is closely related to massive gravity. In fact, the brane-to-brane graviton propagator can be written as D^AP)
= D^;r(P, \P\/L),
(12)
238
where •D™f\s've(p, m 2 ) is the propagator for 4d massive gravity, and the ratio of the two Newton constants, L = G5/G defines the transition length from standard 4d gravity to the 5d behavior. The analysis of [14] (see also [15]) shows that the DGP model becomes perturbatively strongly coupled at a scale E = ( M p ; L _ 2 ) 1 / 3 . This is what one would get by naively substituting the "running mass" \p\/L into the first of the bounds in Eq. (10). Ref. [14] also shows that this problem cannot be cured by adding local counterterms to the action Eq. (11). So, strong coupling at an unacceptably low scale seems an ubiquitous problem plaguing IR-modified gravity. It could be resolved, perhaps, by a nonperturbative re-summation of Feynman diagrams, or by introducing other degrees of freedom that change the theory before it becomes strongly coupled. In the rest of this paper, I will briefly discuss the second possibility: the only one that would give us full computational control of the theory. For simplicity, I will discuss massive gravity a . The strong coupling problem stems from the breakdown of the linearized approximation. This happens at a lower than expected scale because the linearized graviton fluctuation generated by a conserved source with stress energy tensor TM„ is (in momentum space): Kuip) =
hnv(p)+P»PvV(p), 8-ITG
z
•""
p2
+
M2
T^{p) -
\T^(J>)
*(p) =
-
8nG
T"(p). (13)
The first term, h^ is well-behaved in the limit M —> 0; the factor —1/3 in its trace component is the origin of the famous van Dam-Veltman-Zakharov discontinuity,18 and it also ensures that the only propagating degrees of freedom are the 5 physical polarizations of a massive spin-2 field. On the other hand, \& diverges in the massless limit. At liner order, this infrared divergence is harmless, since ^ is a gauge mode: it vanishes in the onegraviton scattering amplitude, when h^v{p) is contracted with a conserved source. At the next order, though, it contributes an amplitude that may dominate over the linear term. For a point-like source of mass M, inspection of Eq. (13) shows that this happens at distances r = 0(GM/MA). A 1/5 The length scale {M Mpi)~ is attained for M = MPX. Can we modify massive gravity in such a way as to keep the same onegraviton amplitude in between conserved sources as given by Eq. (13)? The a
Refs. [16,17] are a first attempt to studying these issues in the DGP model.
239
answer is: yes, by modifying the graviton's kinetic term. Let us add to the linearized action SM in Eq. (1) the term
SA = 3 ^ Jd4xh(d^h^
- ah),
(14)
where A is an arbitrary constant. Decompose next the metric into its transverse-traceless, longitudinal, and trace components: hlw = h^
+ rilt^ + dlldv^ + a^Al),
0 M j = O.
(15)
By computing the double divergence of the equations of motion (S(SM + SAl/Shnv - TMl/ = 0) we get (M 2 D + AD 2 )$ = d^duT^.
(16)
So, even when A ^ 0, <& still obeys a homogeneous equation when TM„ is conserved, and so it can be set to zero when computing the field generated by a localized source. This is the key property that guarantees that only 5 physical polarizations propagate in the one-graviton scattering amplitude. The unphysical gauge mode \& changes. It becomes
For all p2 ^ 0, this mode can be made arbitrarily small in the limit A —> oo b . Formally, the limit A —> oo eliminates the dangerous mode that triggers the breakdown of the linear approximation. Moreover, when expanding the metric as in Eq. (15), the change in the kinetic term can be easily interpreted as giving a very large kinetic term to the unwanted modes $ and ^ , that, therefore, decouple. This argument is of course still hand-waving and it would be interesting to make it sounder. One objection that can be raised against it is that it makes h^v propagate 7 degrees of freedom, two of which are ghosts, instead of the physical 5. So, for any finite A, the theory is unstable unless the ghosts decouple from all physical amplitudes, as the do at linear order. The existence of two extra degrees of freedom is another simple application of the methods devised by Boulware and Deser.1 By writing the action Static sources and quantum loop computations both require to use Euclidean momenta, so this condition is generic.
240
in terms of the 3d metric, linearized lapse hoo, and shift hoi, we see t h a t SA
is A SA = jr-g
f / d*x(hu
- h00)didohi0
H
.
(18)
So, dihio and hu — hQQ become a new pair of (propagating) canonical variables. Now the total number of degrees of freedom is thus 6 (coming from hij) plus 1. T h e new degree of freedom is a boson with first-order action in the time derivative, so its energy is unbounded below. W h e t h e r massive gravity — or D G P , where a more sophisticated version of this mechanism may be at work, according to the analysis of [16] — can be made perturbatively stable and calculable is yet to be proved, many years after the groundbreaking investigations of Boulware and Deser. T h e problem is still tantalizing, so much so t h a t it may be fit t o conclude this review by mentioning t h a t another intriguing route t o solve the strong coupling problems of gravity has been recently opened: by explicitly breaking Lorentz invariance, gravity can be made finite-range, consistent with existing d a t a , and weakly coupled down to distances 0 ( 1 0 0 / x m ) . 1 9 ' 2 0 ' 2 1 Acknow
ledgments
Work supported in part by N S F grant PHY-0245068. References 1. D.G. Boulware and S. Deser, Phys. Rev. D6, 3368 (1972). 2. R. Arnowitt, S. Deser and C. Misner, in Gravitation: An Introduction to Current Research, ed. L. Witten (New York, Wiley, 1962). 3. M. Fierz, Helv. Phys. Acta 12, 3 (1939); M. Fierz and W. Pauli, Proc. Roy. Soc. 173, 211 (1939). 4. M. Porrati, JEEP 0204, 058 (2002) [hep-th/0112166]. 5. M. Porrati, Mod. Phys. Lett. A18, 1793 (2003) [hep-th/0306253]. 6. A. Lichnerowicz, Propagateurs et commutateurs en relativite generate, Institut des Hautes Etudes Scientifiques, Publications Mathematiques, 10 (1961). 7. W. Heidereich, J. Math. Phys. 22, 1566 (1981). 8. P. Breitenlohner and D.Z. Freedman, Ann. Phys. 144, 249 (1982). 9. S.J. Avis, C.J. Isham and D. Storey, Phys. Rev. D18, 3565 (1978). 10. R. Bousso and L. Randall, JHEP 0204, 057 (2002) [hep-th/0112080]. 11. N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Annals Phys. 305, 96 (2003) [hep-th/0210184]. 12. G.R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B485, 208 (2000) [hep-th/0005016], 13. C. Deffayet, G.R. Dvali and G. Gabadadze, Phys. Rev. D65, 044023 (2002) [astro-ph/0105068].
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14. M.A. Luty, M. Porrati and R. Rattazzi, JEEP 0309, 029 (2003) [hepth/0303116]. 15. C. Deffayet, G.R. Dvali, G. Gabadadze and A.I. Vainshtein, Phys. Rev. D65, 044026 (2002) [hep-th/0106001]. 16. M. Porrati and J.W. Rombouts, Phys. Rev. D69, 122003 (2004) [hepth/0401211]. 17. G. Gabadadze and M. Shifman, Phys. Rev. D69, 124032 (2004) [hepth/0312289]; G. Gabadadze, hep-th/0403161. 18. H. van Dam and M.J. Veltman, Nud. Phys. B22, 397 (1970); V.I. Zakharov, JETP Lett. 12, 312 (1970). 19. N. Arkani-Hamed, H.C. Cheng, M.A. Luty and S. Mukohyama, JHEP 0405, 074 (2004) [hep-th/0312099]. 20. V. Rubakov, hep-th/0407104. 21. S.L. Dubovsky, hep-th/0409124.
M A N Y ROADS LEAD TO M = 2 S E I B E R G - W I T T E N THEORY
S T E P H E N G. N A C U L I C H * Department of Physics Bowdoin College Brunswick, ME 04011, USA H O W A R D J. S C H N I T Z E R t Martin
Fisher School of Physics Brandeis University Waltham, MA 02454, USA
The Seiberg-Witten solution plays a central role in the study of J\f = 2 supersymmetric gauge theories. As such, it provides a proving ground for a wide variety of techniques to treat such problems. In this review we concentrate on the role of IIA string theory/M theory and the Dijkgraaf-Vafa matrix model, though integrable models and microscopic instanton calculations are also of considerable importance in this subject.
Outline 1. 2. 3. 4. 5.
Introduction Review of M = 2 Seiberg-Witten theory The SW curve from IIA string theory/M theory The Dijkgraaf-Vafa matrix model approach Concluding remarks
1. I n t r o d u c t i o n Neither of the authors has had the privilege of collaborating with Stanley Deser, much to our regret. Since Stanley is still in the full bloom of his career, there is still a great deal of time to remedy this. Stanley has been a wonderful colleague, with whom we have enjoyed talking about physics 'Research supported in part by the NSF under grant PHY-0140281. tResearch supported in part by the DOE under grant DE-FG02-92ER40706. 242
243
Microscopic
Integrable systems
Nekrasov's approach
instanton calculations / M = 2 SW theory
M-theory/ IIA string theory
Dijkgraaf-Vafa matrix model
Statistical matrix models
Figure 1.
Roads to Seiberg-Witten theory
and many other topics for many, many years. We hope this situation will continue well into the future. We dedicate this article to a celebration of Stanley's achievements. The theme of our discussion is to argue that M = 2 Seiberg-Witten (SW) theory 1 lies at the cross-roads of many different aspects of gauge theories. The various paths that intersect SW theory are quantum field theory, string theory, integrable models, and most recently the DijkgraafVafa (DV) matrix model.2 Because of time limitations, we will not survey all these roads, but concentrate on just two because 1) they are of current interest and 2) they seem to involve drastically different concepts, which nevertheless lead to a better understanding of SW theory. These different approaches to a common problem enhance our confidence in the various methods, and lends support to applying these techniques to new problems. In our own endeavors on SW theory we have been fortunate to have had a number of excellent collaborators: Isabel Ennes, Marta Gomez-Reino, Carlos Lozano, Henric Rhedin, and Niclas Wyllard.
244
2. Review of J\f = 2 Seiberg-Witten theory Af = 2 supersymmetric (susy) Yang-Mills theory with gauge group Q and hypermultiplets in representation R can be described in the low-energy region exactly by the SW effective theory. The underlying microscopic Lagrangian is rpa
Ag2
""
jrifiva
rpa 327T 2
TpfAisa
""
(1) plus fermion and hypermultiplet terms. In Eq. (1), F£v and <j>a denote the gauge field strength and the bosonic components of the N — 1 chiral superfield belonging to the AT = 2 vector multiplet respectively, both in the adjoint representation. The vacuum is defined by (2)
which implies that <j> may be diagonalized to give H) •
(3)
Generically this breaks Q to U(l) r a n k g. If only <j> acquires a vacuum expectation value (vev), this defines the Coulomb branch. If only the scalar fields in the matter hypermultiplet have vevs, this defines the Higgs branch. We focus on the Coulomb branch. The Seiberg-Witten program for AT = 2 susy gauge theories centers on the low-energy effective Lagrangian 1
Ass = ~r
Im
47T
d2F{A) *a dT{A) -Ai \ Id2e~„ ; . . ^ 6 A dA* ' 2 7 " " " dAidA* / '
wiwj
(4)
where T{A) is the holomorphic prepotential and A1 are TV = 1 chiral superadds. On the Coulomb branch one may write 1 -eff
47T
J Imin^F^ + Imd^(a^y
where
F""i + \ Reir^F^ 9 M (ao)j + fermions
F^ (5)
245
a, = order parameters, (O-D)J
= —~
(6a)
= dual order parameters,
(6b)
(JCbj
T
u = -z.—o— = coupling matrix or period matrix. (6c) v oaiddj ' One requires Im Ty > 0 for a positive kinetic energy. As a result of non-renormalization theorems, the holomorphic prepotential only receives perturbative corrections at one-loop, but there is an infinite series of instanton contributions. Thus oo •'{.•"•)
==
•'classical T •» 1 —loop T / J ^ d=l
•>d—instanton i
\> )
where A = the quantum scale (Wilson cutoff), b — the coefficient of the /? function,
(8)
with b positive for asymptotically-free theories. For example, for M = 2 SXJ(N) theory with Nf hypermultiplets in the fundamental representation •
N
Fi-loop(a) = ^ Y^ z ^ K ' ~ a^2
lo
/ _ § f^~A~
i=i & N
Nf
^ ]T(a*+ m,)2 log ( Oj + m/ ^ ^i=l^ 1=1 ; ' ° v^ A )
(9)
yielding Ti:j(a) ~ log(fli - a,j) H
(i ^ j)
(10)
at large a. Significantly r^ is not single-valued, and a Riemann surface emerges for which r^ is the period matrix. In general, the Seiberg-Witten data needed to find the prepotential are: 1) A Riemann surface or algebraic curve specific to the gauge group and matter content, dependent on moduli m, which are related to the order parameters a»; 2) a preferred meromorpbic 1-form A = the SW differential; 3) a canonical basis of homology cycles on the surface (A^, Bk)-
246
Microscopic Lagrangian for M = 2 susy YM
M-theory or geometric engineering
•
Nekrasov approach
Instanton calculus
Instanton expansion of Af = 2 prepotential from microscopic Lagrangian
SW curve and differential for Coulomb branch of N = 2 susy YM
DV matrix model
Instanton expansion of Af = 2 prepotential from SW theory
Instanton expansion of N = 2 prepotential fromDV matrix model
i
test of SW theory Figure 2.
test of matrix model
Computing the instanton expansion
Given this data, the program to solve for the prepotential is: 1) Compute the period integrals of the SW differential 2ni a,k =
2-rri (ao )k=i
X.
(11)
JB,
2) Integrate ao to find !F(a). 3) Test this against explicit results from £miCro where possible. 4) Test this against the predictions of the DV matrix model.
3. The S W curve from IIA string t h e o r y / M theory Klemm, et. a/.,3 demonstrated that SW theory could be derived from string theory, using a technique called geometric engineering. Subsequently, Witten 4 gave a systematic method to find SW curves by lifting 10-dimensional IIA string theory to 11-d M-theory. In IIA language, this involve configurations of two (or more) parallel NS5 branes spanned by a number of D4 branes between each NS5 neighboring pair. For pure Af = 2
247
(x4 +ix5)
*
XQ
Figure 3. IIA brane configuration for A/" = 2 SU(JV) gauge theory SU(./V) gauge theory (with no hypermultiplets), one has two parallel NS5 branes, connected by N D4 branes, as shown in Fig. 3. In the lift to 11-d one can visualize the IIA brane picture as a Riemann surface, with (fattened) lines for the D4 branes becoming branch-cuts which connect two Riemann sheets, i.e., the two NS5 branes. This gives the hyperelliptic curve
y2 = Y[(x-eif-4A2N
.
(12)
More interesting are the theories yielding noH-hyperelliptic curves, particularly since, for a long time, string theory was the only method available to obtain such curves. For example, the curve for STJ(Ni) x SU(iV2) gauge theory, with hypermultiplets in bifundamental representations (Wi, N2) and (Ni,N
(13)
where the polynomials A(x), B(x), and C(x) depend on the details of the theory. Even more interesting is the case when an orientifold 06 plane is placed on the middle NS5 brane, 5 as shown in Fig. 4. The presence of the orientifold plane implies a reflection symmetry on the brane-picture, and an involution imposed on a curve of type (13). An 0 6 + plane corresponds to SU(A0 + OH gauge theory, while an 0 6 ~ plane yields SU(AT) + H.
248
G$06
:
]
x6 Figure 4.
IIA brane configuration for M = 2 SU(JV) + f T l / H
One of the significant contributions of the Brandeis group is the development of methods for finding the instanton expansion for non-hyperelliptic curves such as (13), and others. 6 In order to compute the instanton expansion for the prepotential one needs to compute
a-i ~
/
-
y
(aD)%
f xdy_
(14)
in some approximation scheme since an exact solution seems inaccessible. The method developed by our group was hyperelliptic perturbation theory, 6 i.e., a systematic expansion of the non-hyperelliptic curve about a fiducial hyperelliptic curve. The treatment of SU(iV) + m and SU(JV) + g gauge theories typify the method, but a number of other examples including other groups were also considered. Explicit calculations enabled us to obtain ^l-inst from our methods. More recent calculations 7 ' 8 involving the renormalization group (RG) have simplified the method, and allow an easier access to two or more instanton contributions. In every case, the results from our hyperelliptic expansion agree with those of microscopic calculations, when available. The most systematic of these are by Nekrasov and others, 9 and recently by Marifio and Wyllard. 10 This agreement supports the M-theory approach to SW theory, as well as the validity of our approximation methods.
249
4. The Dijkgraaf—Vafa matrix model approach A major advance in our understanding of susy gauge theory was provided by Dijkgraaf, Vafa, and collaborators. 2 They showed that a suitably formulated matrix model will describe the low-energy physics of N = 1 or N = 2 susy gauge theories. This is proved by showing that, in the computation of the effective superpotential Weg and the period matrix T^ of the gauge theory, the space-time part of Feynman integrals cancels, leaving a zerodimensional theory, i.e., a matrix model. 11 An alternate proof12 shows that the generalized Konishi anomaly equations of the gauge theory have a form identical to those of the resolvent operator of the matrix model. We will describe the application of the DV model to SW theory. To be specific, consider the matrix model appropriate to Af = 2 U(7V) theory with Nf hypermultiplets in the fundamental ( • ) representation. Consider M x M matrices $ together with M-vectors Q1, Qi (I = 1 to Nf). One wishes to compute the partition function2 ~ volG
f d $ dQ1 AQj exp
--W(^,Q,Q) 9s
(15)
with G the unbroken matrix gauge group. The choice of W(§, Q, Q) determines the physics. For our example W = W0($) + W ma tter($, Q, Q),
(16)
W&x) = a f[(x - ei )
(17)
where
breaks Af = 2 susy toAf= 1. One takes a —> 0 at the end of the calculation to recover Af = 2 physics from the a-independent quantities Weg and T ^ . The matter interaction in (16) is taken to be of the same form as the analogous superpotential in the underlying gauge theory. For the theory we are considering, 13 ' 14 M
Nf
^matter = ^ £ Q « / * ° 6 QM • a,b 1=1
(18)
To evaluate the partition function (15), one expands the action about a stationary point 2 $ = $o ,
Q = Q = 0,
(19)
with fluctuations $ = $„ + * = d i a g ( e i l M i ) + (tfij),
(20)
250
where ^ij is an Mi x M, matrix, and N
N
Y,Mi = M,
G = H U(M{) .
(21)
One makes the gauge choice ^ij = 0 for i ^ j , which requires the introduction of ghost matrix fields.18 One obtains the Feynman rules from the expansion of the action, as well as for the ghost contributions, from which one can do a perturbative evaluation of the partition function. In the limit Mi ^> 1 for all i, with Si =gsMi=
finite,
(22)
one may express the partition integral in a topological expansion,2 Z = exp F(e, S) = exp £
- ^ Fx (e, S),
(23)
where F(e, S) is the free-energy and x = 2 — 2g — /i is the Euler number for a two-dimensional surface with g handles and h holes. The leading terms are F(e, 5) = ^ F s p h e r e ( e , 5) + - F d i s k ( e , S) + • • • , (24) 9s 9s where the sphere contribution has g = h = 0, while the disk contribution has g = 0, h — 1, the latter arising from diagrams with Q or Q running along the boundary. (The h > 2 diagrams are suppressed relative to x — 1 for the limit gs —> 0, M, —> oo, with ^ M j = 5, fixed.) On the gauge theory side, the gauge symmetry is broken to U(N)->YlU(Ni), i
^Ni
= N.
(25)
i
It is important to emphasize that Mi —> oo on the matrix-model side, while Ni remains finite on the gauge theory side. In order to study a generic point on the Coulomb branch of the M — 2 SW theory, one takes Ni = 1 for all i, that is, U(N) -> [U(1)]N .
(26)
The effective superpotential of the gauge theory is then given by 2,13 ' 14 WeS(e, S) = - J2 i=l
9Fsph
™{e'S) '
- F d i s k (e, S) + 2vrz r 0 £ i=l
Su
(27)
251
where r 0 = r(A 0 ) is the gauge coupling of XJ(N) at scale A0. To proceed, one finds the extremum cWeff
(28)
= 0,
dSi
(Sj)
which defines (Si). The period matrix is then obtained from only the sphere contribution to the free-energy T
ij(e)
d2Fspheie(e,S) dSidSj
1 2iri
= TT7
(29) (Si)
One lets a —• 0 to obtain N — 2 results. (Actually, however, Eq. (29) is independent of a.) While Eq. (29) gives the matrix of 11(1)^ couplings as a function of the {ej}, the period matrix (6c) from SW theory, d2T(a) Tij(a)
=
(30)
daida-i
is expressed in terms of different parameters on moduli space. The {aj} are physical order parameters, while {e{\ are "bare" order parameters, so one needs a relation between Oj and e^. One can show that the desired relation is
13 N ei
d
^2 « c " 9s(tT ^ s p h e r e + ( t r dS 3=1 th
(31)
$u) disk (S)
where tr is the trace over the i diagonal block in * only. The computation of (31) involves the calculation of tadpole diagrams with external ^a legs in the matrix model. Using Eq. (31), one may re-express (29) as a function of a,i. Finally, integration of Tjj(a) yields the M — 2 prepotential T(a). The procedure just outlined for calculating T(a) is shown schematically in Fig. 5. In practice, one computes the matrix-model quantities to a certain power in A, which increases with the number of loops in the matrix model, but corresponds to the instanton expansion in gauge theory. That is, nonperturbative information in the gauge theory is obtained from perturbative calculations in the matrix model! In these calculations (n + l)-loop perturbation theory in the matrix model corresponds to the n-instanton term of the prepotential of the gauge theory. 13 Recent work8 using the renormalization group has improved the situation, so that only n-loops in the matrix
252
DV Matrix model
Topological expansion of free energy F(e, S) = g~2Fsph + g^F^ +•
Tadpole calculation relates a» to e;
(a)
~
..( P\ _ ij\e) —
T T
WeS(e,S) from F(e, 5)
1 a 2 F B P. h(e,s) .
2wi
d S
d S
(Si)
ddidaj
Ho)
Weff((5i»
(Af = 2 prepotential)
(Af = 1 superpotential)
compare TV = 2 prepotential from SW theory Figure 5.
Matrix model flow chart
model theory are required to obtain n-instanton accuracy in the gauge theory. Another improvement 15 allows one to obtain the one-instanton prepotential for U(JV) + NfD using only the contribution from inhere• In every case, our matrix model calculations agree with results from "conventional" SW theory, and microscopic calculations. Prom the matrix model approach, one can also directly derive 18 ' 13 the SW curve and differential from matrix-model resolvent equations. Particularly noteworthy is the treatment of no?vhyperelliptic curves within the context of matrix models. 16 It is possible to derive the correct SW curve for U(7V) + CD and U(Ar) + [] gauge theories, as well as the one-instanton
253
contribution to the prepotential. There are subtle points 1 6 in constructing nj for these two theories, and additional subtle issues 1 6 ' 1 7 in choosing the correct matrix-model vacuum state for XJ(N) + [ ] . W h e n these issues are dealt with correctly, one obtains agreement with our previous results from hyperelliptic perturbation theory, as well as microscopic calculations of Nekrosov and others, 9 and Marino and Wyllard. 1 0
5. C o n c l u d i n g r e m a r k s Very often a single problem plays a central role in testing a wide variety of methods. Seiberg-Witten theory may not be the hydrogen a t o m of strongly interacting gauge theories, but it does seem to have a privileged position in regard to M = 2 susy gauge theories. As our initial road-map indicated, SW theory can be treated by rather diverse techniques, among t h e m IIA string t h e o r y / M theory, the DV matrix model, microscopic instanton calculations, and integrable systems. 1 9 ' 2 0 We have surveyed just two of these approaches due to time limitations, but the subject is vast and still developing in the other areas as well. H J S wishes t o t h a n k the organizers of the Deserfest for the opportunity to honor Stanley with this survey of topics which have occupied us for a long time.
References 1. N. Seiberg and E. Witten, Nucl. Phys. B426, 19 (1994) [hep-th/9407087]; Nucl. Phys. B 4 3 1 , 484 (1994) [hep-th/9408099]. 2. R. Dijkgraaf and C. Vafa, Nucl. Phys. B644, 3 (2002) [hep-th/0206255]; Nucl. Phys. B644, 21 (2002) [hep-th/0207106]; hep-th/0208048; R. Dijkgraaf, et. al., Phys. Lett. B573, 138 (2003) [hep-th/0211017]. 3. A. Klemm, W. Lerche, P. Mayr, C. Vafa, and N. Warner, Nucl. Phys. B477, 746 (1996) [hep-th/9604034]. 4. E. Witten, Nucl. Phys. B500, 3 (1997) [hep-th/9703166]. 5. K. Landsteiner and E. Lopez, Nucl. Phys. B516, 273 (1998) [hepth/9708118]; K. Landsteiner, E. Lopez, and D. Lowe, JHEP9807, 011 (1998) [hep-th/9805158]. 6. S. Naculich, H. Rhedin, and H. Schnitzer, Nucl. Phys. B 5 3 3 , 275 (1998) [hepth/9804105]; I. Ennes, S. Naculich, H. Rhedin, and H. Schnitzer, Int. J. Mod. Phys. A14, 301 (1999) [hep-th/9804151]; Nucl. Phys. B536, 245 (1998) [hepth/9806144]; Phys. Lett. B 4 5 2 , 260 (1999) [hep-th/9901124]; Nucl. Phys. B558, 41 (1999) [hep-th/9904078]; I. Ennes, C. Lozano, S. Naculich, and H. Schnitzer, hep-th/9912133; for a review, see I. Ennes, S. Naculich, H. Rhedin, and H. Schnitzer, hep-th/9912011.
254 7. M. Matone, Phys. Lett. B357, 342 (1995) [hep-th/9506102]; G. Bonelli and M. Matone, Phys. Rev. Lett. 76, 4107 (1996) [hep-th/9602174]; Phys. Rev. Lett. 77, 4712 (1996) [hep-th/9605090]; T. Eguchi and S.-K. Yang, Mod. Phys. Lett. A l l , 131 (1996), [hep-th/9510183]; J. Sonnenschein, S. Theisen, and S. Yankielowicz, Phys. Lett. B367, 145 (1996) [hepth/9510129]; E. D'Hoker, I.M. Krichever, and D.H. Phong, Nucl. Phys. B494, 89 (1997) [hep-th/9610156]; G. Chan and E. D'Hoker, Nucl. Phys. B564, 503 (2000) [hep-th/9906193]; E. D'Hoker, I.M. Krichever, and D.H. Phong, hep-th/0212313; M. Gomez-Reino, JHEP 0303, 043 (2003) [hepth/0301105]. 8. M. Gomez-Reino, S. Naculich, and H.J. Schnitzer, JHEP 0404, 033 (2004) [hep-th/0403129]. 9. N.A. Nekrasov, hep-th/0206161; hep-th/0306211; H. Nakajima and K. Yoshioka, math.AG/0306198; N. Nekrasov and S. Shadchin, hep-th/0404225. 10. M. Marino and N. Wyllard, hep-th/0404125. 11. R. Dijkgraaf, M. Grisaru, H. Ooguri, C. Vafa, and D. Zanon, JHEP 0404, 028 (2004) [hep-th/0310061]. 12. F. Cachazo, M. Douglas, N. Seiberg, and E. Witten, JHEP 0212, 071 (2002) [hep-th/0211170]; N. Seiberg, JHEP 0301, 061 (2003) [hep-th/0212225]. 13. S.G. Naculich, H.J. Schnitzer, and N. Wyllard, Nucl. Phys. B 6 5 1 , 106 (2003) [hep-th/0211123]; JHEP 0301, 015 (2003) [hep-th/0211254v3]; JHEP 0308, 021 (2003) [hep-th/0303268]; Nucl. Phys. B674, 37 (2003) [hep-th/0305263]. 14. R. Argurio, V.L. Campos, G. Ferretti, and R. Heise, Phys. Rev. D67, 065005 (2003) [hep-th/0210291]; I. Bena and R. Roiban, Phys. Lett. B555, 117 (2003) [hep-th/0211075]; H. Ita, H. Nieder, and Y. Oz, JHEP 0301, 018 (2003) [hep-th/0211261]; R.A. Janik and N.A. Obers, Phys. Lett. B553, 309 (2003) [hep-th/0212069]; S.K. Ashok, et al, Phys. Rev. D67, 086004 (2003) [hep-th/0211291]. 15. M. Gomez-Reino, hep-th/0405242. 16. S.G. Naculich, H.J. Schnitzer, and N. Wyllard, JHEP 0308, 021 (2003) [hepth/0303268]; Nucl. Phys. B674, 37 (2003) [hep-th/0305263]. 17. M. Aganagic, K. Intriligator, C. Vafa, and N. Warner, hep-th/0304271; K. Intriligator, P. Kraus, A.V. Ryzhov, M. Shigemori, and C. Vafa, hepth/0311181; A. Klemm, K. Landsteiner, C.I. Lazaroiu, and I. Rankel, hepth/0303032. 18. R. Dijkgraaf, S. Gukov, V.A. Kazakov, and C. Vafa, hep-th/0210238. 19. J.D. Edelstein, M. Marino and J. Mas, Nucl. Phys. B 5 4 1 , 671 (1999) [hepth/9805172]; J.D. Edelstein, M. Gomez-Reino, and J. Mas, Nucl. Phys. B561, 273 (1999) [hep-th/9904087]. 20. A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, and A. Morozov, Phys. Lett. B355, 466 (1995) [hep-th/9505035]; E.J. Martinec and N.P. Warner, Nucl. Phys. B459, 97 (1996) [hep-th/9509161]; R. Donagi and E. Witten, Nucl. Phys. B460, 299 (1996) [hep-th/9510101]; E. D'Hoker and D.H. Phong, hep-th/9912271; A. Gorsky and A. Mironov, hep-th/0011197.
NON-COMMUTATIVE TOPOLOGICALLY MASSIVE GAUGE THEORY
NICOLA CAPORASO t , LUC A GRIGUOLO*, SARA PASQUETTI* and DOMENICO SEMINARA 1 Dipartimento di Fisica, Polo Scientifico, Universitd di Firenze; INFN Sezione di Firenze Via G. Sansone 1, 50019 Sesto Fiorentino, Italy * Dipartimento di Fisica, Universitd di Parma; INFN-Gruppo Collegato di Parma Parco Area delle Scienze 7/A, 4-3100 Parma, Italy We investigate the perturbative dynamics of noncommutative topologically massive gauge theories with softly broken supersymmetry. The deformed dispersion relations induced by noncommutativity are derived and their implications on the quantum consistency of the theory are discussed.
1. I n t r o d u c t i o n Non-commutative q u a n t u m field theory is a fascinating theoretical laboratory where highly non-trivial deformations of space-time structures induce novel and unexpected dynamical effects at q u a n t u m level. Recently they have attracted a lot of attention, mainly due to the discovery of their relation to s t r i n g / M theory. 1 ' 2 In particular, Seiberg and W i t t e n 2 realized t h a t a certain class of q u a n t u m field theories on non-commutative Minkowski space-times can b e obtained as a particular low energy limit of open strings in t h e presence of a constant NS-NS JB-field. Prom a purely field theoretical point of view they appear as a peculiar non-local deformation of conventional q u a n t u m field theory, presenting a large variety of new phenomena not completely understood, even at perturbative level. Four dimensional non-commutative gauge theories are in fact afflicted by the infamous U V / I R mixing 3 t h a t complicates the renormalization program and it may produce tachyonic instabilities. 4 We will not t r y to address these problems in D •= 4: 5 our investigations will be instead concentrated on the three dimensional, non-commutative, topologically massive electrodynamics for a number of reasons. First of all the presence of a single physical polarization and of an explicit gauge-invariant mass for the photon should simplify the analysis of the U V / I R mixing and elucidate t h e n a t u r e of t h e tachyonic 255
256
instabilities. Secondly, planar non-commutative gauge theories with ChernSimons terms have been proposed as effective description of the Fractional Hall Effect.6'7 Last but not least, two of us a share with Stanley an insane passion for the ubiquitous Chern-Simons term and its unusual dynamical properties: we hope he will enjoy our non-commutative exercises on topologically massive gauge theory, of which he is a Master. 2. Non-commutative U(l)
Yang-Mills-Chern-Simons
2.1. Pure gauge model, and its
symmetries
b
Non-commutative topologically massive U(l) gauge theory in three dimensions is governed by the Lagrangian S = - i fd3x FM„ * F^-
mgfd3x
ex>lu (\AX * d»Av + f Ax * AM * Av) , (1)
where the field-strength is given by FM„ = d^Av - d„AM - ig[A^, Av]+
(2)
and the * stands for the usual Moyal product,
(f*g)(x) = / ^ / S /( ^' V)9{t ' z)e-^0ii(v-xY(z-x)i•
(3)
Differently from the commutative case, which simply describes the propagation of a free massive boson, the new non-commutative incarnation is an interacting theory, resembling more a non-abelian model than an abelian one c . This richer structure at the level of interactions is however paid when considering the global symmetries. The constant tensor d^ present in the definition of the Moyal product does waste the original Lorentz invariance.[In three dimensions, there is no Lorentz-invariant constant antisymmetric twotensor.] In the case of space-like non-commutativity (tf'"'#M„ < 0), the residual symmetry can be identified with the spatial rotation 5 0 ( 2 ) and the translations. For time-like non-commutativity(i?A"/i?M1/ > 0), SO(2) is replaced by 50(1,1), but the theory is not unitary. 8 Finally, dealing with a
L.G. and D.S. plead guilty for leading astray the innocent souls of N.C. and S.P. with this project. b I n what follows we shall only consider t h e case of space-like and therefore the constant tensor i)*"' is chosen to be #e°'" / , where the index 0 denotes the time component. c This is not surprising, because the actual gauge group of (1) can be identified with a particular realization of C/(oo).
257
light-like non-commutativity (^V'&IU, = 0) is trickier, but one can show that the residual group is "R". The discrete symmetries (C,P,T) instead follow closely the known path of the commutative case: C is conserved, while P and T are again broken. The equations of motion derived from the action (1), but not the action itself, are obviously gauge invariant against the *—gauge transformation A^(x) = u{x) * A^x)
* u\x)
dfj,u(x) * u\x),
(4)
generated by *—unitary functions u(x) *v)(x) = u(x)^ * u(x) — 1. The presence of the Chern-Simons term in (1) produces indeed a non-vanishing variation, SS
=
? f / d 3 z e V i > f * 9AU *UU d^u * n* * dvu) -
_ ^ £ / d3xeXfiUd\(u^
* d^u* Av) = 4ir2 ( ^
(5)
J w(u) + total divergence
where w(u) is the non-commutative version of the usual winding index. An example of transformation u, for which w(u) is not zero, is given by u, = [l-P(i)y)] + e ^ P ( I l ! , ) , 2
(6)
2
where F(x,y) = 2 e x p ( - ( z + y )/|i?|) is a •-projector ( P * P = P) and b(t) is any function such that b(t)\_ = 27r. In this particular case, we find w(u) — 1. Thus, as occurs for non-abelian topologically massive gauge theory, 9 the consistency of the quantum theory requires that the mass mg is quantized according to the relation 4* 3 ( ^ )
2.2. The J\f = 1 supersymmetric
= 2nk.
(7)
extension
At the perturbative level one of the most puzzling feature of noncommutative field theory is the phenomenon of the ultraviolet-infrared (UV/IR) mixing. The non-local nature of the interaction, while softening the behavior at large momenta, moves the UV divergences into the IR region. This effect generically endangers the stability of the perturbative vacuum, the unitarity and the infrared finiteness of the theory. An elegant way to have under control these potential problems is to consider the supersymmetric extension of the model. Supersymmetry improving the ultraviolet behavior of a theory will also act, via (UV/IR) mixing, as an
258
infrared regulator. In fact, if the number of supersymmetries is sufficiently large, all the undesired divergences will disappear from the infrared region. In three dimensions, for the case of the Yang-Mills Chern-Simons (YMCS) system, it is enough to consider the Af = 1 extension of the model, whose Lagrangian is obtained by minimally coupling a Majorana fermion to the action (1),
SS-YMCS = - \ J d3xF^ irF^ -\mgsx^
jd3xAx
+ ^J
d3x\ * (ip -mf)*X
* d„Av + l-gmg f d3xex^Ax
* A^* Av.
+ (8)
In Eq. (8) we have also softly broken the supersymmetry to M = 0 by choosing different masses for the gauge field and the Majorana fermion. This will not jeopardize the cancellation of the infrared singularities because, in this case, they are related just to the leading ultraviolet divergences. Besides this breaking will provide us with a much richer and interesting model: by taking, in fact, different limits for the masses, we can focus our attention, for example, either on the pure bosonic theory (m/ —> oo) or on the usual supersymmetric gauge theory {mj = 0, mg = 0) or on the Chern-Simons theory (mg - t o o ) . Finally a remark is in order. Naively one may expect that there is no problem with the UV/IR mixing for the YMCS system. In fact topologically massive commutative gauge theories are super-renormalizable models, that actually result UV-finite in perturbation theory. Thus, apparently, there is no UV divergence to be moved in the IR region. However their finiteness originates partly from their symmetries: the simultaneous presence of Lorentz and gauge invariance forbids the potential linear divergences. In the non-commutative set-up Lorentz invariance is lost and the linear divergences will reappear as infrared divergences via (UV/IR) mixing. However the theory is still UV-finite.
3. The one-loop two-point function The simplest way to address the question of vacuum stability and unitarity is to analyze the one-loop one-particle irreducible two-point function for the gauge boson. At the tree level, this function coincides with the commutative one since the *—product is irrelevant in the quadratic part of the action
259
(l) d . Its tree level form in the Landau gauge is in fact r
Ji™e(p) = VnvP2 ~ VnVv - imge^xPx-
(9)
In the commutative case, when computing the one-loop correction, the only effect of the radiative corrections is to properly renormalize the two transverse structures in (9). In fact they can be recast in the general form I W / > ) = ne(p)(7?MI/^2 - VnVu) - imgU.0{p)etxl/xP\-
(10)
The two functions II e and II 0 , computed in [9,10], govern the commutative wave-function and the mass renormalization respectively. This simple setting cannot be promoted to the non-commutative case as it originates from the simultaneous presence of gauge and Poincare invariance which is now broken. Once the Lorentz invariance is lost, we cannot expect just one wave-function (Ze = 1 — II e ) and mass (Zm = 1 — Il 0 ) renormalization, since different components of the gauge field may renormalize in different ways. More importantly, even the transversality of the one-loop correction to the T ^ e may be endangered. This possibility, for example, takes place in the non abelian gauge theory at finite temperature, 11 ' 12 where the space-time symmetries are destroyed by the choice of a preferred reference system, the thermal bath. Therefore, before proceeding, we must carefully reexamine the Ward Identity that controls the longitudinal part of the II^j,. A tedious exercise, with the non-commutative version of the BRST transformation, shows that in any covariant £ gauge (and thus also in the Landau gauge) the following Ward identity holds p A UXa(p) = 5 r „ ( p ) {fva
- P2^a
- imge»a/)pp - iIT a (p)) ,
(11)
where r „ is defined through the following vacuum expectation {c(x)[Au(y),c(y)]i,)0 = ifd3zQ(x - y - z)T„(z), with Q the exact ghost propagator. In the commutative case F^ is compelled by Lorentz invariance to be proportional to pv and the above identity entails transversality. In the non-commutative model, there are two new possible vectors that can appear in the expansion of T„, jT = Wp,,
x»
= e^Ppcpp,
(12)
and the above argument seems to break. However a detailed one-loop analysis shows that Tv has surprisingly no component along p^ and x^' • Therefore d
I t holds the following property: / f + g = / fg-
260
the transversality is preserved at one-loop. At higher loops, the situation is less clear, but there are indications that this property is preserved. Once we have convinced ourselves that the transversality is kept, we can write the most general form for the nM1/, which is also compatible with the bosonic symmetry
nM„ = n\P2^
+ ir 2 ^fV - n°imge^xPx + n^p^+p.x^-
(is)
Actually the last tensor structure will not appear at any order in perturbation theory because of the accidental invariance $ —> —•& that IIM„ possesses. This, combined with the Bose symmetry, implies that ft° must be even in •d and odd in p but such a scalar cannot be built. We are left with
r v = wlP2^
+ n e 2 M v - n°imge^xP\
(u)
x y At the end of the day the only effect of non-commutativity is to produce two different wave-function renormalization: one for the component (p • A) and one for the component {\- A). The commutative case (10) is recovered when ITf = Ilf, because 77^ -p^Pv/p2 = XtiXv/x2 +P M p„/p 2 . Summing the general form (14) of the radiative correction to the tree level contribution (9) and inverting the total result, we obtain the renormalized propagator nR
1\
1
( fZiXuXv
.
[Z^PuPv
. . R p*
where 21 = 1-11?,
22 = 1-111
Zm = l-IL°,
(m*) 2 =
272 miZ, ^ ^ .
In next section, by looking at different features of G^u(p) at one-loop, we shall illustrate how the non-commutativity affects the spectrum of the theory, its unitarity and its vacuum stability. But for accomplishing that, we need the explicit form of scalar functions IIj, n | and I P , whose evaluation is lengthy and tedious. In the following we shall not report on the details of the computations, which will appear in [13]. The final result is given for completeness in appendix A. Here we shall limit ourselves to some general comments on their properties. Each function displays two contributions, which originates respectively from the "planar" and "non-planar" diagrams. The former is identical to the commutative (non abelian) case, while the latter carries the effects of the non-commutativity. They are both finite and they cancel each other when •& goes to zero. This decoupling occurs
261
since the softly broken supersymmetric model smoothes the effect of the UV/IR mixing. Finally both contributions possess a physical threshold at p2 = 4m 2 , but two unphysical threshold at p2 = 0 and p 2 = m 2 . The last feature will complicate our future analysis. 4. Dispersion relation and the stability of the vacuum The spectrum of the non-commutative Yang-Mills Chern-Simons system is entirely encoded in the poles of the above propagator e . Firstly it contains an unphysical pole at p2 = 0, which describes the longitudinal degree of freedom still propagating in any covariant gauge. Secondly, it contains the relevant physical pole at
p2 = (rn*f(P,p)=™2°Z^f\v
(15)
9
Zi{p,p)Z2{p,p) which represents the effect of the radiative corrections on the tree level pole at p2 — m2. Since the Lorentz invariance is broken, Eq. (15) does not depend only on p2 but also on the new invariant p 2 , which is simply proportional to the euclidean norm of the spatial momentum for the case of space-like non-commutativity. Therefore the pole condition (15) should not be thought as an equation for evaluating the radiative corrected mass, but rather as an equation that determines the energy of the excitation in terms of its momentum, namely the dispersion relation. In a relativistic theory, this question is pointless because the functional form of the dispersion relation is fixed by the Poincare symmetry. The simplest way to solve Eq. (15) is to proceed perturbatively. At the lowest order in (g2/mg) we have: „2
& zi ™2 E* =p +m
l~)(2n o ( P ) p)-nf(p,p)-n|(p,p)
(16)
771,
where we have factored out the dependence of the one-loop lis on the coupling constant. In order that Eq. (16) provide a reasonable dispersion relation for a stable physical excitation, two criteria must be met: (a) it has to be gauge invariant; (b) it has to be real. These two requirements are far from being manifest, since the explicit form of the lis is plagued by many complex contributions (see appendix A) coming from the unphysical thresholds at p 2 = 0 and p 2 = m 2 and moreover our perturbative computation has been performed in the Landau gauge. e
Similar investigations has been performed in [7].
262
The first point can be easily clarified by evaluating the combination 211° (p,p) — Ilf (p,p) — Tl^iPift) a t t n e threshold p2 = m2 A series of unexpected cancellations occur and the final result is completely real. This apparent miracle is just a signal that unitarity is preserved. The above combination can be in fact reinterpreted as the S-matrix element describing the transition from one particle state to one particle state. 10 Thus, if unitarity is not violated, this element must be free from unphysical cuts. The interpretation of the combination 2Tl°(p,p) - H\(p,p) - Yl%{p,p) for p2 = m2 as an element of the S-matrix also solves the second puzzle. In fact we know that S-matrix elements are gauge invariant. An alternative proof can be also given by means of the Nielsen identity. For space-like non-commutativity, the explicit form of the gauge boson dispersion relation (16) reads
*{j*^-*H^-*±i&*)}(17> where we have introduces the dimensionless variables £ = mgp and // = mf/nig for convenience. In this dispersion relation we can distinguish essentially three terms: the first bracket contains the fermion contribution, the second parenthesis collects instead the gauge contribution, while the last piece is the remnant of the UV/IR mixing. This expression in fact finite in the infrared region £ —» 0 for finite [i. In the mere supersymmetric case fx = 1 Eq. (17) dramatically simplifies and we are left just with the bosonic contribution, but with a different coefficient: —18 instead of —27. If /J, —> oo, i.e. if we approach the pure Yang-Mills Chern-Simons system, the UV/IR mixing will rise again. In fact the last bracket will produce an infrared divergent term of the form — 4e~ 5 /£. The rising of this negative divergent contribution at small £ for sufficiently large \i will always make the square of the energy negative in a certain region of the spatial momenta (see Fig. 1). In other words, the massive excitation becomes a tachyon and the perturbative vacuum is no longer stable. Varying the other two parameters g2/mg and m2,d will not affect the picture: it will only change the specific value of ^ at which the tachyon will appear. Thus, when we reach the critical value of fj,, we must to resort with non-perturbative techniques to select the new vacuum. At the moment, the nature of this new vacuum is only matter of speculation. 7 One may conjecture that the transition tuned
263
p/m
Figure 1. The dispersion relation of the boson for small spatial momenta, when g2/mg = 0 . 1 ml-d = 1 and n = 100 - • 300.
by the tachyonic mode will lead the system to a sort of stripe phase analogous to that proposed by Gubser and Sondhi 14 for
264
Appendix A. Analytic expression of the different l i s If we introduce the following basic integrals (k = —1,0,1) 0fc + l
/ _ 1 \fc + l e - ? ^ / x ( x - l ) T ) 2 + X M H ( 1 - : E ) / i l
/ /•!
^ ^ j y - ^
I, (A.l)
y/x{x - l)r)2 + X\i\ + (1 - X)H2
and the dimensionless variable rf = p2 /ml, the non-planar contributions for the different lis are given by Gluon Sector: TTe — li l,np —
(g^7ly)(i,i)-(4-^+^^>(i,i))
8nma
/ 5 ( l - t | 3 ) a ( n pP i) )(l,0) - i^!)To(np)(l,0)V^T^(0,0) -^T0^(0,0))
2
r/9(4-772)
5
n2,np" ~8^m~g[{
(
+
9
-
4
(np)
^
2
5
-
4
^
(A.2)
(4-5^ + ^)
1 (Mj+
4
- (
-
^
\
(np)
A (M)J
( l - » 7 2 ) ^ nwp ) / n n ^ , ( 5 - V ) - ( 9 - 4 7 , 2 ) e - « -T|' (0,0) + ^^
n:n p
^
167rm 9
4
- 2^
2 l^np).
r i r (0,0)
-
3^ - ^
4
(A.3)
n
i( P)/ + 12^7(1,1)
( 1 - 2\2 ^(1+3^)^)^^^(1-2^)1^
v2Z Fermion TT e
l.np"
n:2 , n p ~
(A.4)
Sector:
^(V>W>-M^W>),n; B =-^7V,), ,2Tr(np)
2
2wmgrj I
^T™>{^n)
1, (np)/ + -Tr>^,n) £
•
The planar contribution are identical to that of Pisarski and Rao. 10
(A.5)
265
References 1. A. Connes, M. R. Douglas and A. Schwarz, JHEP 9802, 003 (1998); M. R. Douglas and C. M. Hull, JHEP 9802, 008 (1998). 2. N. Seiberg and E. Witten, JHEP 9909, 032 (1999). 3. S. Minwalla, M. Van Raamsdonk and N. Seiberg, JHEP 0002, 020 (2000); A. Matusis, L. Susskind and N. Toumbas, JHEP 0012, 002 (2000). 4. M. Van Raamsdonk, JHEP 0111, 006 (2001); A. Armoni and E. Lopez, Nud. Phys. B632, 240 (2002). 5. L. Alvarez-Gaume and M. A. Vazquez-Mozo, Nud. Phys. B668, 293 (2003). 6. L. Susskind, hep-th/0101029; A. P. Polychronakos, JHEP 0104, 011 (2001). 7. J. L. F. Barbon and A. Paredes, Int. J. Mod. Phys. A17, 3589 (2002). 8. J. Gomis and T. Mehen, Nud. Phys. B591, 265 (2000). 9. S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48, 975 (1982); Annals Phys. 140, 372 (1982) [Erratum-ibid. 185, 406 (1988); 281, 409 (2000)]. 10. R. D. Pisarski and S. Rao, Phys. Rev. D 3 2 , 2081 (1985). 11. D. J. Gross, R. D. Pisarski and L. G. Yaffe, Rev. Mod. Phys. 53, 43 (1981). 12. H. A. Weldon, Annals Phys. 271, 141 (1999). 13. N. Caporaso, L. Griguolo, S. Pasquetti and D. Seminara, in preparation. 14. S. S. Gubser and S. L. Sondhi, Nud. Phys. B605, 395 (2001); P. Castorina and D. Zappala, Phys. Rev. D68, 065008 (2003). 15. J. M. Cornwall, Phys. Rev. D 5 4 , 1814 (1996). 16. J. Ambjorn, Y. M. Makeenko, J. Nishimura and R. J. Szabo, JHEP 0005, 023 (2000).
T H E M A I N POSTULATES A N D RESULTS OF LOOP Q U A N T U M GRAVITY
LEE SMOLIN Perimeter Institute for Theoretical Physics, 35 King Street North, Waterloo, Ontario N2J 2W9, Canada, and Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada E-mail: IsmolinlSperimeterinstitute. ca
1. Introduction It is an honor and a pleasure to report back to Stanley some of the results of a life spent working on the problem of quantum gravity. I was a student of Stanley's, in all the meanings of the term, and he was and remains an inspiration-as a person no less than as a scientist. Perhaps the following can be read as a very late term paper for the course on quantum gravity I took from him my first year of graduate school. Because I took the assignment seriously, I have followed Stanley's example and found myself working on all approaches to quantum gravity. But more and more of my effort-along with a rapidly growing number of colleagues and friends-has been spent on loop quantum gravity. This is an approach to quantum gravity based on a simple and conservative premise: Construct a quantum theory of spacetime based only on the experimentally well confirmed principles of general relativity and quantum mechanics. Most approaches to quantum gravity have concluded or assumed that this could not be done. However, in the mid 1980's the situation changed because it was understood that general relativity could be most simply formulated as a kind of gauge theory. 1 ' 2 Given the tremendous advances in our understanding of gauge theories that had occurred in the decade previous, this made it possible to take a fresh approach to quantum gravity based on asking a simple question: can a consistent and sensible physical theory be constructed with no other assumptions than the principles of quantum theory and general relativity? Remarkably, after 17 years of work by a community of about a hundred 266
267
physicists and mathematicians, a great deal of evidence has accumulated that the answer to this question is yes. The result is a language and dynamical framework for studying the physics of quantum spacetimes, which is completely consistent with the principles of both general relativity and quantum field theory. The picture of quantum spacetime geometry which emerges is to many compelling, independently of the fact that it has been derived from a rigorous quantization of general relativity. The basic structure that emerges is of a new class of quantum gauge field theories, which are background independent, in that no fixed spacetime metric is needed to describe their quantum dynamics. Instead, the geometry of space and time is coded in the degrees of freedom of a gauge field4"15. While a number of technical issues remain open, enough is known about the physics of loop quantum gravity that a number of applications to problems of physical interest are under development. There is by now a well understood detailed description of the quantum physics of black hole and cosmological horizons 16 ' 17 ' 18 that reproduces the Bekenstein-Hawking results on the relationship between area and entropy. 19 There is under development an approach to quantum cosmology that shows that cosmological singularities are eliminated 20 and has already led to some predictions for effects observable in CMB spectra. 20,21 And there are published predictions for observable Planck scale deviations from energy momentum relations 22 ' 23 that imply predictions for experiments in progress such as AUGER and GLAST. For those whose interest is more towards formal speculations concerning supersymmetry and higher dimensions than experiment, there are also results that show how the methods of loop quantum gravity may be extended to give background independent descriptions of quantum gravity in the higher and super realms. 31 " 35 Like any mature field, work in loop quantum gravity proceeds through a variety of methods and levels of rigor. The first results were found using direct extensions of methods from ordinary gauge theories, with modifications in regularization methods appropriate to theories without background metrics. More recently, all the key results have been confirmed using mathematically rigorous methods. 7 ' 8 In this review I employ a nonrigorous style, grounded on intuitions about the behavior of gauge fields that particle physicists and condensed matter physicists will find familiar. But the reader should keep in mind that, as will be noted, many of the key results listed are confirmed by completely rigorous methods. This is good, as we have so far no experimental checks on the behavior of gauge fields in this context, it is essential to be able to confirm our intuitions with rigorous
268
results. This brief introduction proceeds by discussion the four basic physical observations that underlie the successful results of loop quantum gravity. This is followed by two sections that describe the key methods and results. After this I list the 35 main results, so far found for this theory. For those interested in a more detailed introduction to loop quantum gravity, there are several textbooks, monographs and review papers now available. Early monographs are [4,5]. A recent textbook is by Rovelli.6 Comprehensive introductions to the rigorous side of the subject are given in [7], also soon to appear from CUP, as well as the very recent review.8 Review papers on different aspects of loop quantum gravity include [8-15]. For reasons of space, I have left out of this volume most of the references cited, but these are available in an expanded online version, hepth/0408048, available at http://arxiv.org/abs/hep-th/0408048. 2. The four basic observations While the principles assumed are only those of general relativity and quantum mechanics, there are four key observations that make the success of loop quantum gravity possible. These are I. Classical general relativity is a background independent theory, hence any theory which is to have general relativity as a low energy limit must be background independent. A background independent theory is one whose formulation does not assume or require the existence of any single preferred spacetime metric or connection. Instead, all the fields that define the geometry of spacetime are fully dynamical, none are fixeda. The principle of background independence does not imply that the fundamental theory must be based on fields living on a manifold. But it does imply that whenever manifolds appear, whether fundamentally or as part of the low energy effective theory, the dynamics of the fields will be invariant under active diffeomorphisms of the manifold. The reason for this was worked out first in detail by Dirac, and his book 24 is still a good source for understanding the connection between background independence and diffeomorphism invariance b . We note that what is necessarily assumed from general relativity is only the principles of background independence and diffeomorphism invariance. a T h e reader should be aware that the word "background independence" has other connotations besides that given here. More about this as an answer to a FAQ. b T h i s is also described in detail in [6] and in [25-29].
269
While many papers in LQG are concerned with the quantization of the Einstein action, we can equally well study other actions, including supergravity and terms of higher powers in curvature. Loop quantum gravity is perfectly compatible with the expectation that the Einstein equations are just the low energy limit of a more fundamental theory. II. Duality and diffeomorphism invariance may be consistently combined in a quantum theory. By duality we mean here the conjecture that the dynamics of a quantum gauge field can be described equivalently in terms of the dynamics of one dimensional extended objects. In the context of YangMills theory this conjecture was explored in detail by Polyakov, Migdal, Mandelstam, Nielson and others, where the one dimensional objects were Wilson loops. More recently, in the background dependent context, the idea of duality between gauge fields and extended objects has been central string theory, for example in the AdS/CFT correspondence. In the background independent context, the idea is central to loop quantum gravity. As we will see, the key point is a method for constructing a diffeomorphism invariant quantization of any diffeomorphism invariant gauge field theory. This cannot be done using Fock states, as the inner product on Fock space depends on a background metric, whose presence breaks diffeomorphism invariance. But, as I will describe, it can be done if one works in a space of states created by the action of Wilson loops. Indeed, from a technical point of view, the main achievement of loop quantum gravity is the discovery of a new kind of quantum gauge field theory, which is exactly invariant under diffeomorphisms of a manifold. This is understood in a great deal of detail, and the Hilbert space, inner product, states, observables and path integral representation are all understood in closed form. That understanding is robust, and has been achieved by several regularization procedures. Significantly, there is a completely rigorous formulation of these diffeomorphism invariant gauge theories. A recent uniqueness theorem 30 greatly limits the possibilities for quantum descriptions of diffeomorphism invariant gauge theories apart from the one studied in LQG. III. General relativity and all related theories, such as supergravity, can be formulated as gauge theories. This means that the configuration variable is a gauge field, and the metric information is contained in the conjugate momenta to the gauge field. This extends to coupling with all the known kinds of matter fields. I V . Further, general relativity, supergravity and related theories can be put in a special form in which they are constrained topological field
270
theories. These are theories whose actions differ from the action of a topological field theory 0 by the imposition of a non-derivative, quadratic constraint equation. That constraint diminishes the number of gauge invariances, leading to the emergence of local physical degrees of freedom, while maintaining the diffeomorphism invariance of the theory. This is true of general relativity in all dimensions and is also known to be true of supergravity, in d = 4, at least up to N = 2, 31 " 33 and for d — l l . 3 4 As a result, the techniques which give us consistent diffeomorphism invariant gauge theories can be applied to give consistent quantizations of general relativity and supergravity. Many results follow, which will be described in more detail below. Most of the key results, have been confirmed as theorems in the rigorous formulation of diffeomorphism invariant quantum field theory. The fundamental result which follows from these four observations is that quantum geometry is discreteA. Operators which measure the areas and volumes of diffeomorphism invariantly defined surfaces and regions may be constructed. They are finite, after an appropriate regularization, respecting diffeomorphism invariance, and they have discrete, computable spectra. Hence, the theory predicts minimal physical areas and volumes. By diagonalizing these observables we find an orthonormal basis of diffeomorphism invariant states, which are certain labeled graphs called spin networks. Their evolution can be described in closed form in either a Hamiltonian or path integral language. Furthermore, the dynamics of the matter fields can be constructed and studied. Because of the existence of minimal quanta of volume, all divergences of ordinary quantum field theories are eliminated. This is because there simply are no degrees of freedom in the exact theory that correspond to gravitons or other quanta with wavelength shorter than the Planck length. There is evidence as well that singularities of general relativity are eliminated. It should be apparent from this summary that loop quantum gravity, as a research program, is rather different from other research programs such as string theory, that are based on new hypotheses about nature, such as supersymmetry, the existence of higher dimensions, and the unification of all elementary particles and forces by means of strings. These are interesting hypotheses, but it is proper to characterize them as speculative, as they C
A topological field theory is a field theory whose equations of motion are all trivial, so that there are no local degrees of freedom. The solutions are parameterized instead entirely by topological and boundary information. d Precise statements and references for all these results are given in section 4.
271
have no direct support from experiment. Nor are there results that show that any of these assumptions are necessary consequences of a consistent unification of quantum theory with gravity and spacetime. It is fair to say that approaches such as string theory have so far failed to lead to a complete and well denned theory 6 . Nor have they led to many results that concern the behavior of quantum spacetime at the Planck scale. There is a widely held view that, to give a deeper picture of quantum spacetime, string (or perhaps M) theory require a background independent formulation. It is natural to try to construct such a theory using the methods of loop quantum gravity, some preliminary results in this direction are given in [34,35]. There are several different approaches to a background independent quantum theory of gravity. Besides loop quantum gravity, others include dynamical triangulations, 37 ' 38 causal sets 36 and the Gambini-Pullin discrete quantization approach to quantum gravity.39 Although these are independently motivated research programs, some of their results are relevant for loop quantum gravity, because they concern models which can be understood as arising from spin foam models by simplifications which eliminate certain structures. It is also the case that while most results concern quantum general relativity, the methods of loop quantum gravity can be applied to construct a large number of different background independent quantum theories, and it can easily incorporate speculative hypotheses about the dimension of spacetime, the form of the action or the presence of additional symmetries such as supersymmetry. All of these theories incorporate the basic principles of background independence and diffeomorphism invariance, so that the spacetime geometry is completely dynamical. At the same time, by showing that a consistent diffeomorphism invariant quantum field theory can be constructed that represents general relativity, coupled to arbitrary matter fields, in the 3+1 dimensional world we observe, the success of loop quantum gravity undermines the claim that supersymmetry, strings or higher dimensions are necessary for a consistent quantum theory of gravity. 3. W h y loops? The kinematics of diffeomorphism invariant quantum gauge theories Many of the key results of LQG concern the construction of a new kind of quantum gauge theory which is diffeomorphism invariant. This construce
A detailed list of results from string theory is contained in [12].
272
tion has been successfully applied to a wide variety of theories, including topological field theories, general relativity and supergravity. It works in any dimension and all the standard kinds of matter fields can be included. The basic ideas behind this construction are very simple, from a physics point of view, although the math needed to realize them rigorously is a bit more involved. We begin by considering a theory of a connection f , Ala, valued in a Lie algebra (or superalgebra) G on a d + 1 dimensional spacetime manifold M. The manifold has no fixed metric defined on it, fixed or dynamical, all that is fixed is the topology and differential structure of M- To quantize a spacetime theory using Hamiltonian methods, we assume that M = £ x R, where £ proscribes the topology and differential structure of what we will call "space". We begin with the Hamiltonian theory. The initial configuration space C will consist of the possible configurations of the gauge field Ala on S. To this must be added the conjugate momenta, which are represented by the electric field E?, which is a vector density on S, valued in the lie algebra G. Together, they coordinatize the phase space, on which is defined the Poisson bracket relations, {Ai(x),E$(y)}=63(x,y)5ba$ij.
(1)
We note that because Ef is a density this is well defined in the absence of a background metric. The physical configuration space consists of equivalence classes of C under the action of the gauge symmetries of the theory. These include ordinary gauge transformations, with gauge group G(T) and the diffeomorphisms of S, denoted, Diff{T). The physical configuration space we are interested in is then, rdiffeo =
G- connections on S
The problem we want to solve is how to write a corresponding Hilbert space of gauge and diffeomorphism invariant states. We proceed in three steps. 1) We construct a Hilbert space of gauge invariant states, called Hkin, on which the diffeomorphisms of £ act unitarily and without anomalies. 2) We mod out by the unitary action of the diffeomorphisms, to find the f All fields described here are forms, the form indices are a = 1 , . . . , d, while the indices i, j ... are valued in the Lie algebra, G
273
subspace ^diffeo
c
^kin
(3)
of diffeomorphism invariant states. 3) We endow 7^dlffeo with an inner product, making it a Hilbert space. For the first step, we cannot take the Fock space, which is the usual starting point for quantization of field theories. The reason is that Fock space depends on a background metric, and this prevents the construction of a unitary, anomaly free realization of the diffeomorphism group. The unique 6 approach that does work is to take Wilson loops as elements of the normalizable basis. Given a loop 7 £ £ the Wilson loop is defined as the path ordered exponential T[y,A}=TrPe^A.
(4)
The physical idea behind this is the same as the old ideas that motivated the idea of duality in Yang-Mills theories: that the vacuum is a dual superconductor, so that physical excitations are defined by Wilson loops acting on a vacuum. Thus, duality follows from the requirement that the space of ordinary gauge invariant states carries a unitary and anomaly free representation of the diffeomorphism group!! We then have, %[A] = f[y,A}\0)=T[f,A].
(5)
For the reason just mentioned, the vacuum cannot be the Fock vacuum, as that depends on a background metric. Instead we take simply * o [ 4 = (A\0) = 1.
(6)
Given a set of loops 7*, i = 1 , . . . M, for M finite, we can build up complicated states
%t[A] = n f [ ^ ] l ° > = \[ThuA). i
(7)
i
Such states are not normalizable in Fock space. But this is of little concern for us, as there can be no Fock space in the absence of a background metric. It is also the case that there are no operators in the theory that represents Ala(x) or their field strengths F^b(x). The gauge field is represented only by the non-local operators T[y, A], but the limits for very small loops, which classically would give the field strengths, do not exist. g
The strongest uniqueness theorem is [30].
274
In a certain sense, this representation is closer to the Hilbert spaces of lattice gauge theory than to Fock space. Of course, a lattice is a background structure, and we don't use a fixed lattice here, we simply consider all such states. The new representation can be thought of roughly as the direct sum of the Hilbert spaces of all lattice gauge theories, with all possible lattices. This indeed is how the construction is done in the rigorous approach. One might object that this will lead to too many states, and this intuition can be made precise. However, we are not interested in these states themselves, as they are not invariant under diffeomorphisms. Instead we will find we can construct exactly states corresponding to their spatial diffeomorphism invariant classes. It turns out that there are exactly enough of these, so that we end up with a separable basis of normalizable diffeomorphism invariant states h . The key idea is that quantum geometries are built up from such states. There is a translations between gauge fields and gravity, which follows from observations III and IV above. In 3 + 1 dimensions the correspondence gives G = SU(2) or 5 0 ( 3 ) ' which we will assume for the following. This gives us correspondences between flux of the electric field Ef and geometric quantities. Given a surface S, it turns out that Area of S = HG Electric field flux through S.
(8)
In the dual superconductor picture, electric field flux is quantized. This is realized on states of the form (5) which are eigenstates of the operators that measure electric flux. Hence they are eigenstatesJ of the operators that measure the areas of surfaces S. So it follows that the dual superconductor picture leads to a quantization of areas. Note the HG in (8). This is necessary because area and electric flux have different dimensions. When the correspondence is worked out in detail, they are there because h and G are parameters of a quantum theory of gravity. Because of them, flux quanta turn into quanta of area, with a minimal quanta of area given by the Planck area KG. Further, given a region R G E the volume of R can be expressed (in
h For technical reasons, this requires that the diffeomorphisms be piecewise smooth, see [43]. 'For N = 1 supergravity, G = Osp{l, 2). J This is modulo some subtleties concerning simultaneous intersections of loops and surfaces.
275
d = 3) as Volume of R = f y/\det(E?)\.
(9)
It is then of interest to construct simultaneous eigenstates of the operators that measure the volumes of all regions flgE, and the areas of all surfaces that separate the regions. This is done by combining loops into graphs. This is necessary because it can be shown that after suitable regularization, the operator corresponding to (9) annihilates states of the form of (7) unless there are points where at least two loops intersect. Evidently, volume is a property associated with intersections of loops. The eigenstates turn out to be spin network states. To give the definition of spin network states we return to general G. A spin network T is a graph whose edges are labeled with irreducible representations of G. The nodes are also labeled by invariants (or intertwiners). If a node n has edges incident on it with labels i,j, k, I, than the node has to be labeled by an invariant, defined by a map, li:i®j®k®l^Id.
(10)
For each set of labels on the incident edges there is a finite dimensional space of such invariants. We require that there be a non-trivial such invariant at each node. Given such a spin network V we can define a spin network state, V&r [A], which is a generalization of a Wilson loop. It is gotten by writing the parallel transports of the gauge field A for each edge, in the representation labeling that edge, and then tracing them together, using the invariants labeling each node, to get a gauge invariant functional of the connection associated to the whole graph. The inner product of W kin is then chosen so that the spin network states comprise an orthonormal basis (r|r')=5rr'.
(11)
This is of course natural, as such states form the orthonormal basis of gauge invariant states of any lattice gauge theory. Because there is no fixed lattice, there are too many states. Because one can change any state depending on a graph F to an orthogonal state by the smallest deformation of the embedding of the graph in E, there is no countable basis. Thus, the space of states 7ikm is not separable. This is remedied by now implementing the second step, which is to define a unitary realization of the diffeomorphism group, and mod out by it. This
276
of course is a step that could not be carried out in a theory on a fixed metric background. We define a unitary action of the diffeomorphisms DiffiT,) by £>(0)o|r> = | 0 - 1 o r > ,
M4>&Diff{Y,).
(12)
It is trivial to check the unitarity. This follows from the fact that for any spin network V and any diffeomorphism
*(T) = Jdii(A)T\T,A]*(A)
(13)
is known precisely, the measure required is called the AshtekarLewandowski measure. Heuristically this is analogous to the Fourier transform as (A\T) = T[T,A] gives a basis of states that the Hamiltonian acts simply on. Indeed, the key discovery that makes the quantization of general relativity possible is that the Hamiltonian and diffeomorphism constraints, which make up the hamiltonian, take any state T[T, A] to a state of the same form, but with a different T. This is analogous to the fact that the Fourier transform is useful in ordinary quantum mechanics because the hamiltonian acts algebraically on momentum eigenstates. One then defines a subspace of diffeomorphism invariant states HdiSeo C W k i n
(14)
containing all states for which *[ o T] = * [ r ] , for all <j> £ Diff{H). It is easy to construct such states. For example, let K. be any knot or graph invariant, then
* K [r] = /c[r]
(15)
is in HdlSeo. Similarly, let {r} be the diffeomorphism equivalence class of the network T. Then the characteristic state defined by
M>{r}[r'] = i i f r ' e { r } ,
(16)
and zero otherwise is in 7idlSeo. In fact, it can be shown that these provide an orthonormal basis of 7idlSeo.
277
Thus, the diffeomorphism invariant states perfectly combine the principles of duality and diffeomorphism invariance. Each state gives an amplitude to diffeomorphism classes of collections of Wilson loops. These states have physical meaning, given to them by the fact that they diagonalize diffeomorphism invariant observables that measure the geometry of the spatial slice. Examples of these which we mentioned above are the volume of the universe and the area of its boundary. To complete the theory one has to translate the dynamical evolution equations to act on these diffeomorphism invariant states. This can be done in particular examples such as general relativity coupled to arbitrary matter fields in 3 + 1 dimensions and supergravity. Exact expressions for the dynamical evolution of these states is known in both hamiltonian and path integral form. The action changes the graphs by local rules. A typical action is to act at a trivalent node, converting it to a triangle. From the quantum Einstein equations we have exact closed form expressions for the amplitudes for these processes. We now turn to the methods by which these were constructed. 4. Dynamics of constrained topological field theories Observation I V means in details that all gravitational theories of interest, including general relativity and supergravity in 4 dimensions, can be described by an action, which is generically of the formk, a _ ^topological I ^constraints . omatter
/ i yi
To describe the detailed form, its simplest first to fix the dimension to be four, in which case G = SU(2). The first term describes a topological theory called BF theory. It depends on a 2 form Bl and the field strength Fl of a connection, A1, all valued in a Lie algebra of SU(2). Thus, i = 1,2,3. The action is, ^topological
=
j JM
(£i
A
p. _ ^ B i
A B.y
^
2
k This kind of formulation of general relativity was first discovered by Plebanksi 4 4 and later independently by [45,46]. The corresponding simplification of the Hamiltonian theory was independently discovered by Sen 1 and formalized by Ashtekar. 2 By now several different connections are used in loop quantum gravity. These include the selfdual part of the spacetime connection, 1 , 2 and a real SU{2) connection introduced by Barbero 4 7 and exploited by Thiemann. 4 8 There are also alternate formulations that use both the left and right handed parts of the spacetime connection. 3 1 ' 4 9
278
The field equations which follow are Fi = KB\
(19)
P A B * = 0,
(20)
where V is the SU{2) gauge covariant derivative. The second term contains a quadratic function of Bl, which can be expressed as ^constraint
=
f (f)..BiABj, JM
(21)
where 4>ij is a symmetric, traceless matrix of scalar fields. Variation of the independent components of <j> produces a quadratic equation in Bl whose solution turns the theory into general relativity. These are Bl ABj = ^-5ijBk ABk.
(22)
The solutions to this are all of the following form: there exists a frame field e'a for / = 0 , . . . ,3 = 0, i, such that the Bl are the self-dual two forms of the metric associated to ela. That is, B i = e 0 A e i + (t)e« fc e J -Ae fc)
(23)
where the % is there for the Lorentzian case and not for the Euclidean case. Thus, the metric is not fundamental, instead the frame field appears when we solve the equations, (22), that constrain the degrees of freedom of the topological field theory. The third term contains coupling to matter fields, such as spinors, scalars and Yang-Mills fields. Interestingly enough, these can be written in terms of the fields involved in the other two terms. 46 It turns out that the same trick works in all dimensions, where Bl is now a r f - 2 form. For d = 2 + 1, the BF theory is equivalent to general relativity. 50 For spatial dimension d > 3 the extension has been given in [51]. It is also known how to express supergravity in d = 3 + 1 32 and d = 10+1 3 4 as constrained topological field theories. Now we can return to the canonical quantization, and discover the structure we assumed above. Given a choice of time coordinate, t, which represents E as constant time slicings1, we can find the canonical momenta to 'However, among the gauge symmetries of the theory are diffeomorphisms of M that take any "spatial" slice representing £ to any other slice. If the theory implements the quantum version of the constraint that generates this gauge symmetry, it will not depend on the choice of slice used.
279
A\. We see from the form of the action that the only time derivatives are in the first term, ^topological
=
[ dt J {Bl f\Ai
+ ---).
(24)
Hence the canonical momenta to the gauge field are contained in conjugate electric fields, which can be expressed in terms of the pull back of the two forms of our theory to the spatial manifold S. Ef = eabcBbci,
(25)
abc
1
where the e is defined on the spatial manifold E, making E a vector density on E. When the quadratic constraints from gc°nstraints a r e s o i v e c j ) all the metric information is contained in the Bl,s, and hence is represented by the electric field Ef conjugate to the gauge field A1. The gauge field A\ will turn out to code components of the spacetime connection. This is a reversal of the older ADM way of understanding the dynamics of the gravitational field, but it turns out to be deeper, and much more progress can be made with it. We can easily see how the restriction from the topological BF theory to general relativity by a quadratic equation works in the hamiltonian formulation. As the field equations of the BF theory, (19,20) are expressed as spacetime forms, they pull back to equations in the three surface E. These must hold the canonical theory. The covariant conservation of Bl given by (20) pulls back, given (25), to Gauss' law g* = VaEai = 0.
(26)
As in any Yang-Mills gauge theory, these are first class constraints that generate the ordinary gauge transformations. The field equation of the topological field theory (19) pulls back to Fab = F*b + AeabcEci
= 0.
(27)
These are constraints which tell us that the curvature is entirely determined by the Ef. Hence, the connection has no local independent degrees of freedom. To give local dynamics to the connection we want to impose conditions on the fields, which will restrict the number of independent constraints. As mentioned above, to get general relativity in any dimension, it suffices to impose quadratic conditions on the Bx,s. In fact, it is easy to see how this is realized in the Hamiltonian form of the theory, for the case of four spacetime dimensions.
280
The required quadratic constraints are equations of motion, hence they should be generated by the hamiltonian. This suggests that a hamiltonian at most cubic in fields should suffice. However in spacetime diffeomorphism invariant theories the hamiltonian must be a linear combination of constraints, so as to avoid any preferred time coordinates. On general grounds we expect there must be four more constraints, to generate the four dimensional diffeomorphisms. These constraints should come from restricting the constraints (27) that give the topological field theory. The simplest way to do this turns out to lead to general relativity"1. First, we can trace (27) with one power of the momenta, to find, Da = FahE\ = FlhE\ = 0.
(28)
It is easy to show that these generate the diffeomorphisms of E. The next simplest thing to do is to trace (27) with two powers of the momenta. This yields the desired cubic constraint, H = eijkEa^Ebkrab
= eijkE^Ebk
[^ b + AeabcEci] = 0.
(29)
This is the Hamiltonian constraint. Remarkably, unlike older approaches, it is polynomial in the fields. This makes it possible to translate it into a quantum operator using non-perturbative methods". Can one stop here? One can if the constraints form a closed algebra. It turns out that the system of 7 constraints, consisting of (26), (28) and (29) do form a closed first class algebra. Hence it is consistent to impose only these four of the nine equations in (27). The result must be a theory with local degrees of freedom0 It is not hard to show that it is general relativity. It is also straightforward to show that for d = 4 the constraints (26,28,29) do follows from the action (17).
m
T h e same reasoning in the supersymmetric case leads to supergravity. " T h e form (29) holds for the Ashtekar-Sen variables, in which, for the Lorentzian case, Ala is a complex variable. One can also work with the real Barbero variable, at a cost of an additional term added to the Hamiltonian constraint in the Lorentzian theory. However, it is still possible to convert the Hamiltonian constraint to a finite, well defined operator in this case. °To count degrees of freedom, note that in the topological field theory Gauss' law (26) follows from (27), but this is not true of the restricted set (28,29). Hence there is a net of two (9 — 7) fewer equations, so there are two physical degrees of freedom. These are the polarizations of gravitational waves.
281
4.1. Horizons,
black holes and
boundaries
Loop quantum gravity has led to a number of important results about black holes and cosmological horizons. In most of these results a horizon is modeled as a boundary of spacetime. Conditions are imposed at the boundary which capture the physical feature that at a horizon nulls rays neither diverge nor converge. As a result, only the degrees of freedom on the horizon and its exterior are quantized. This method certainly limits the kinds of questions that can be asked about black holes but, within this limitation, there is a complete description of the states associated with the black hole horizon. These results are made possible by the relationship with topological field theory. The reason is easy to describe. Let us now assume that the spatial manifold S has a boundary d £ = B which is a closed, oriented two dimensional surface. For the classical and quantum dynamics to be well defined we must fix boundary conditions at B and we must also add a boundary term to the action of the theory. There is a very natural class of boundary conditions which follow from the relation to topological field theory. The reason is that the only derivatives of the action of general relativity are shared with the topological field theory. We can make use of natural relationships between topological field theories of different dimension, called the ladder of dimensions15. This works in any dimension, but there is an especially nice situation in 3 + 1 dimension because the natural boundary theory associated with a BF theory is a Chern-Simons theory on the three dimensional boundary of spacetime. To realize this we add a boundary term to (17), given by ^boundary
=
* 47r
f
YcS{A),
(30)
JBXR
where Yes (A) is the Chern-Simons three form of the connection pulled back to the spacetime boundary. One can then show that the classical and quantum dynamics is well satisfied, so long as a boundary condition is satisfied on B, which is 16
B'ls = ^ V
(31)
It turns out that this condition precisely characterizes black hole horizons, with the constant k related to the surface gravity at the horizon. 18 p
Whose relevance for quantum gravity was first emphasized by Louis Crane.
282
It applies also to cosmological horizons16 and to timelike boundary conditions in the case of a non-zero cosmological constant, of either sign.49 In this case the consistency of the boundary condition (31) with the constraint (29) implies that,
Since Chern-Simons theory is precisely understood, this leads to a detailed understanding of the physics at these boundaries, including horizons.16"49 4.2. The path integral formulation:
spin
foam
Path integral formulations of loop quantum gravity are known as spin foam models. There are several different constructions of them, and up to technical issues q , they yield the same class of theories. Each construction yields a spacetime history, or spin foam, T which evolves an initial spin network state |rj n ) into a final state | r o u t ) . A spin foam history !F, is itself a combinatorial structure, whose boundary is the initial and final state dT = r o u t - r,„.
(33)
Here are several equivalent characterizations of a history, T. • A spin foam as a causal history. A causal spin foam is a succession of local moves rrii that transform the initial state Tmt to the final state r o u t . 5 2 " 5 6 The moves rrii are considered the discrete analogues of the events of a continuous spacetime. They have a partial order, which give the history a discrete analogue of the causal structure of a Lorentzian spacetime. Consequently, each causal spin foam history is endowed with discrete analogues of structures such as light cones and horizons. • A spin foam as a one dimension higher analogue of a Feynman diagram in which spin networks play the role normally played by incoming and outgoing particle states. 57 Propagators for edges are two dimensional spacetime surfaces, they meet along edges which are propagators for nodes. Each surface is labeled by a representation, as the edge it propagates. Similarly, each edge of the spin foam is labeled by an invariant corresponding to the node of the spin network it propagates. Interactions occur at vertices in For the interested reader these are described below.
283
the foam, where several edges meet. Any cross section of such a spin foam is a spin network. The interactions are events where the spin network changes by a local move. • Spin foams as Feynman diagrams of a matrix model. The analogy to Feynman diagrams is made precise by showing that there for every spin foam model there is a matrix model (more precisely a dynamical system on a group manifold) such that the spin foams are the Feynman diagrams of that matrix model. 58 Each foam is then a Feynman diagram, and like in ordinary quantum field theory, there are sums over intermediate states on internal propagators. In the spin foam case these are sums over representation labels on two surfaces and sums over invariant labels on edges. But unlike ordinary QFT, in a large class of physically relevant models, the sums over intermediate states are finite. • A spin foam as a triangulation of spacetime For simplicity, consider the case where the initial and final spin network are four valent graphs. They then are each dual to a three dimensional triangulation (technically, a pseudo-manifold), in which each node is dual to tetrahedra, whose four faces are dual to the four edges incident on that node. 59,60 Each face of the dual triangulation is now labeled by a representation, while each tetrahedron is labeled by an invariant. There are then spin foams constructed from a four dimensional triangulation, whose boundary is the three triangulation dual to the initial and final spin network. Each four simplex is dual to an event in the previous characterizations of a spin foam. Following the rules of quantum theory, the dynamics of the theory is specified when we have assigned an amplitude to each history. The amplitude of each foam AlF] is given by a product A[F}=Af-1
Yl
A[event]
(34)
events
of factors for each event (or vertex or four-simplex), where each amplitude is a function of the labels of surfaces and edges incident at that event. N is a normalization factor which depends on the labels on lower dimensional structures. Several models have been studied as candidates for quantum general relativity. The best studied is the Barrett-Crane model, which is derived by following a strategy which is naturally indicated by the realization that general relativity is a constrained topological field theory. The first step is to define a spin foam model which exactly realizes the quantization
284
of the BF theory given in (18). The second step is to implement the quadratic constraint, (22), on the sums over labels in the path integral representation of the topological field theory. It turns out there is a very beautiful and natural way to do this. The result is that the path integral measure for quantum general relativity is exactly the same as the measure of the corresponding topological field theory. The only difference is that in quantum general relativity the sum over labels is restricted to a subset of representations and invariants. This restriction implements the quadratic constraint and, by destroying the topological invariance, leads to a theory with local degrees of freedom. This simple construction is known to work for quantum general relativity in all dimensions, for both Lorentzian and Euclidean theories. 51 Following the conventional rules of quantum theory, amplitudes of physical interest are to be constructed from summing the corresponding amplitudes over all spin foam histories. Thus, we have
(rout|rin>=
Y,
A
(35)
^-
Fs.t.dF=rout-rin
A spin foam model then provides a precise realization of the sum over spacetime histories, conjectured and described formally in early work by Hawking, Hartle and others. Given such a model, one can use it to construct different amplitudes of interest including projection operators onto the space of physical states, annihilated by all the constraints. One can construct as well physical evolution amplitudes, such as the amplitude to evolve from the initial state to the final state through histories with a fixed spacetime volume. 5. The main results of loop quantum gravity We can now turn to listing some of the main results which have been obtained concerning loop quantum gravity and spin foam models. 5.1. The fundamental
results of the canonical
theory
The first set of results concern the construction of a Hilbert space of states of a gauge field valued in a Lie algebra or superalgebra G, invariant under local gauge transformations and diffeomorphisms of E. The setting for the results that follow is a bare manifold S, with no metric structure, on which is defined the phase space of a gauge theory, with gauge group G. The only non-dynamical structure that is fixed is a
285
three manifold E, with a given topology and differential structure. There are no fixed classical fields such as metrics, connections or matter fields on E. The only exception is in modeling the quantization of spacetime regions with boundary, as in the asymptotically flat or AdS context, or in the presence of a black hole or cosmological horizon. In these cases fields may be fixed on the boundary <9E to represent physical conditions held fixed there. The dynamics is formulated in terms of a diffeomorphism invariant action which is a functional only of that gauge field and its derivatives. 1. The states of the theory are known precisely. 67 ' 68 ' 69,70 ' 7 ' 6 The kinematical Hilbert space W kin has been rigorously constructed and has all the properties described in the last section. The Hilbert space TidlSeo of spatially diffeomorphism invariant and gauge invariant states of a gauge field on a manifold E, has an orthonormal basis, |{r}) whose elements are in one to one correspondence with the diffeomorphism equivalence classes of embeddings of spin networks 61 into E. 62 The inner product is given by
({r}|{r'}> = £ { r } { r , } .
(36)
Here {r} refers to the equivalence class of graphs under piecewise smooth diffeomorphisms of S. In the case of pure general relativity in 3 + 1 dimensions, with vanishing cosmological constant, the gauge group is SU(2). In this case the labels on the edges are given by ordinary SU(2) spins. For details see [61,62]. 2. Certain spatially diffeomorphism invariant observables have been constructed. After a suitable regularization procedure these turn out to be represented by finite operators on Hd'Seo, the space of spin network states. 10 ' 63 ' 64 In the case of general relativity and supergravity, these include the volume of the universe, the area of the boundary of the universe, or of any surface defined by the values of matter fields. These operators all preserve the diffeomorphism invariance of the states. 66 Other operators also have been constructed, for example an operator that measures angles in the quantum geometry 71 and the lengths of curves in £. 7 2 3. The area, volume and length operators have discrete, finite spectra, valued in terms of the Planck length. 63 * 64,65,72 There is hence a smallest possible volume, a smallest possible area, and a smallest possible length, each of Planck scale. The spectra have been computed in closed form. 4. The area and volume operators can be promoted to genuine physical observables, by gauge fixing the time gauge so that at least locally time is measured by a physical field.73'66 The discrete spectra remain for
286
such physical observables, hence the spectra of area and volume constitute genuine physical predictions of the quantum theory of gravity. 5. Among the operators that have been constructed and found to be finite on JiAlReo is the Hamiltonian constraint (or, as it is often called, the Wheeler de Witt equation 74 " 76 ). Not only can the Wheeler deWitt equation be precisely defined, it can be solved exactly. Several infinite sets of solutions have been constructed, as certain superpositions of the spin network basis states, for all values of the cosmological constant. 68 ' 48 These are exact, physical states of quantum general relativity. 6 If one fixes a physical time coordinate, in terms of the values of some physical fields, one can also define the Hamiltonian for evolution in that physical time coordinate 73 and it is also given by a finite operator on a suitable extension of 7Ydlffeo including matter fields. 7. Coupling to all the standard forms of matter fields are understood, including gauge fields, spinors, scalars and higher p-form gauge fields1".6'7 It is known how to extend the definition of the spatially diffeomorphism invariant states to include all the standard kinds of matter fields, and the corresponding terms for the Hamiltonian constraint are known in closed form, and are finite on the space of diffeomorphism invariant states. These states are also invariant under ordinary Yang-Mills and p-form gauge transformations3. Inclusion of matter fields does not affect the finiteness and discreteness of the area and volume observables. 8. There is a rigorous formulation of the Hamiltonian quantization of general relativity in which all the preceding results are reproduced as theorems. 70,48 ' 7 In the context of this rigorous approach, there is a fundamental uniqueness theorem, 30 which essentially says the following': T o r supergravity, see result 31, below. s To my knowledge whether loop quantum gravity suffers from the fermion doubling problem is an open question. ' A more precise statement of the theorem follows 77 . Let A(e) be the holonomy of a classical connection along a piecewise analytic path e, let Ef(S) = fs T r ( / * E) be the electric flux of the classical two-form *E through the piecewise analytic surface S where / is a Lie algebra valued smooth scalar field. Let D be the group of piecewise analytic spatial diffeomorphisms of the hypersurface and G the group of local gauge transformations. Let W be the Weyl algebra generated by [A{e),A{e')\ = 0 and Wf(S)A{e)Wf(S)-1 = exp(ihCXE ( s ) ) • A(e), where we have, A{e)* = A{e~l)T (T=transposition of matrices) Wf{S)* = W _ / ( S ) . Finally consider group of automorphisms of W labeled by the semidirect product GxD given by ag(A(e)) — g(b{e))A(e)g(j'(e))-1 (g G G; 6(e), / ( e ) beginning and end point of e) ad(A{e)) = A(d(e)) and similar for Wf{S). We are looking for representations n of the *algebra W on a Hilbert space H which
287
Consider an approach to a quantum gauge theory on a d > 2 dimensional manifold without metric. Assume that the Wilson loop operator and area (or generally electric flux) operator are well defined on a kinematical hilbert space, Hkm. Assume also that 7ikm carries a unitary anomaly free representation of Diff(E,) so that the space 'HdsS of diffeomorphism invariant states may be constructed (formally as a subspace of a dual space of 7Ykln. Then the Hilbert space of the theory is isomorphic to that just described,
5.2. Results
on path integrals
and spin
foams
9. The dynamics of the spin network states can be expressed in a path integral formalism, called spin foams (for the most recent review see [15]). 13-15,52-60 rp n e m s t ; 0 r j e s by which spin network states evolve to other spin network states, called spin foam histories, are explicitly known. A spin foam history is a labeled combinatorial structures, which can be described as a branched labeled two complex. Spin foam models have been derived in several different ways, and the results agree as to the general form of a spin foam amplitude. These include: 1) by exponentiation of the Hamiltonian constraint, 2) directly from a discrete approximation to the classical spacetime theory, 3) by constraining the summations in a finite state sum formulation of a four dimensional topological invariant, 4) from a matrix model on the space of fields over the group, 5) by postulating spacetime events are local moves in spin networks. 10. Evolution amplitudes corresponding to the quantization of the Einstein equations in 3 + 1 dimensions, are known precisely 60,15 for vanishing 1. contains a cyclic vector Q, i.e. the states 7r(a)fi span H as we let a vary through W. 2. implements G\xD unitarily by U(g,d)n(a)Q = 7r(a9i<j(a))f2 for all a E W. 3. is weakly cont. wrt t —• •n{Wtj{S)) for all / , S for real parameter t, and that fi is in the common dense domain of all the iz(Ef(S). That means that matrix elements of n(Wtf(S) become matrix elements of the unit operator as t —> 0 which by Stone's theorem means that the flux operators ir(Ef(S)) exist. This technical assumption is necessary also in Stone von Neumann's theorem without which the uniqueness of the Schroedinger rep of QM is wrong. A different way of saying this is that {-K, f2, H) are the GNS data for a state (pos. lin. functional) u> on W satisfying the invariance condition us o a 3 ) ( j = w and the continuity assumption. Then the THEOREM is; There is only one solution, the AILRS Hilbert space. Moreover, the representation TTAILRS is irreducible (that is, not only one vector is cyclic but every vector is cyclic) which excludes the existence of spurious superselection sectors.
288
and non-vanishing values of the cosmological constant, and for both the Euclidean and Lorentzian theories. 11. The sum over spin foams has two parts, a sum over graphs representing histories of spin networks, and, on each, a sum over the labels. The sums over labels are known from both analytic and numerical results to be convergent 78 ' 79 for some spin foam models, including some corresponding to the quantization of the Einstein equations in 2 + 1 and 3 + 1 dimensions. 12. For some spin foam model in 2 + 1 dimensions, it has been shown that the sum over spin foam histories is Borel summable. 80 13. The physical inner product, which is the inner product on solutions to all the constraints, has an exact expression, given in terms of a summation over spin foam amplitudes. 57 14. The spin foam models have been extended to include gauge and spinor degrees of freedom. Recently spin foam models with matter have been extensively studied in 2 + 1 dimensions. 81 15. Spin foam models appropriate for Lorentzian quantum gravity, called causal spin foams, have quantum analogues of all the basic features of general relativistic spacetimes". These include dynamically generated causal structure, light cones and a discrete analogue of multifingered time, which is the freedom to slice the spacetime many different ways into sequences of spatial slices.52 The spatial slices are spin networks, which are quantum analogues of spatial geometries v .
5.3. Results
on black holes and
horizons
16. Several kinds of boundaries may be incorporated in the theory including timelike boundaries, in the presence of both positive and negative cosmological constant, and null boundaries such as black hole and cosmological horizons. 1 6 - 1 8 ' 3 1 ' 4 9 In all these cases the boundary states and observables are understood in terms of structures derived from Chern-Simons theory.
u
For more details on these models and the resulting physical picture, see [53]. I t should be mentioned also that there are very impressive results relevant to the spin foam program from a research program called causal dynamical triangulations. This work shows that Lorentzian and Euclidean path integrals for quantum gravity fall into different universality classes, with the Lorentzian path integrals being more convergent. 38 " 42 For the first time there are results that suggest that a dynamical 3 + 1 spacetime may emerge from critical behavior of a path integral over combinatorially defined spacetimes. 4 2 v
289
17. The boundary theory provides a detailed microscopic description of the physics of horizons and other boundaries. The horizon entropy (37) is completely explained in terms of the statistical mechanics of the state spaces associated with the degrees of freedom on the horizon. This has been found to work for a large class of black holes, including Schwarzschild black holes. 17 ' 18 LQG also gives the correct results for dilatonic black holes. 85 The boundary Hilbert spaces decompose into eigenspaces, one for each eigenvalue of the operator that measures the area of the boundary. 16 For each area eigenvalue, the Hilbert space is finite dimensional. The entropy may be computed and it agrees precisely with the Bekenstein-Hawking semiclassical result, <±«UNewton
The calculation of the entropy involves a parameter, which is called the Imirzi parameter. This can be understood either as a free parameter that labels a one parameter family of spin network representations, or as the (finite) ratio of the bare to renormalized Newton's constant. The Imirzi parameter is fixed precisely by an argument invented by Dreyer, involving quasi normal modes of black holes.82 Dreyer's argument depends on a remarkable and precise coincidence between an asymptotic value of the quasi normal mode frequency and a number which appears in the loop quantum gravity description of horizons. The value of the asymptotic quasi normal mode frequency was at first known only numerically, but it has been very recently derived analytically by Motl. 83 Once Dreyer's argument fixes the Imirzi parameter, the Bekenstein-Hawking relation (37) is predicted exactly for black hole and cosmological horizons w . 18. Corrections to the Bekenstein entropy have been calculated and found to be logarithmic. 86 19. Suitable approximate calculations reproduce the Hawking radiation. They further predict a discrete fine structure in the Hawking " T w o caveats should be mentioned. 1) Dreyer's calculation leads also to the conclusion that transitions where punctures, i.e. ends of spin networks, are added or subtracted to the boundary, must be dominated by creation and annihilation of spin 1 punctures. 2) The match between the quasi normal mode frequencies and the black hole entropy appears to require that the boundary Hilbert space corresponding to a Schwarzschild black hole is defined to have the labels of all the edges puncturing the horizons equal. This may be justified by noting that then the horizon quantum geometry has maximal symmetry, as does the classical geometry. If one includes states where the quantum geometry is arbitrarily asymmetric, the match does not appear to work, 8 4 but in that case the quasi normal mode spectrum should be altered as well.
290
spectrum. 87 ' 88 At the same time, the spectrum fills in and becomes continuous in the limit of infinite black hole mass. This fine structure stands as another definitive physical prediction of the theory. Thus, to summarize, loop quantum gravity leads to a detailed microscopic picture of the quantum geometry of a black hole or cosmological horizon. 18 This picture reproduces completely and explains the results on the thermodynamic and quantum properties of horizons from the work of Bekenstein, 89 Hawking90 and Unruh. 91 This picture is completely general and applies to all black hole and cosmological horizons. Finally, there is a first result indicating that black hole singularities are eliminated by the same mechanism that eliminates cosmological singularities. 92 5.4. Results
on the low energy
behavior
20. A large class of putative ground states can be constructed in 7{km which have course grained descriptions which reproduce the geometry of flat space, or any slowly varying metric. 93 " 97 Linearizing the quantum theory around these states yields linearized quantum gravity, for gravitons with wavelength long compared to the Planck length. 96 21. Excitations of matter fields on these states reproduce a cut off version of the matter quantum field theories, but with a physical, Planck scale cutoff. As a result of the discreteness of area and volume, the ultraviolet divergences of ordinary QFT are not present. 7 22. It is understood rigorously how to construct coherent states in Hkl" which are peaked around classical configurations.97 23. Formulations of the renormalization group for spin foam models have been given in [98,99]. As a byproduct of this work it is shown that while the Wilsonian renormalization group is not a group, it does have a natural algebraic setting, as a Hopf algebra. 24. For the case of non-vanishing cosmological constant, of either sign, there is an exact physical state, called the Kodama state, which is an exact solution to all of the quantum constraint equations, whose semiclassical limit exists x . 100 That limit describes deSitter or anti-deSitter spacetime. Solutions obtained by perturbing around this state, in both gravitational 11 and matter fields,101 reproduce, at long wavelength, quantum field theory X
A related state in the context of Yang-Mills theory was previously discussed by Jackiw. 1 0 5 Some properties of Jackiw's state were discussed by Witten. 1 0 6 More about this in a FAQ.
291
in curved spacetime and the quantum theory of long wave length, free gravitational waves on deSitter or anti-deSitter spacetime y . It is not yet known whether or not the Kodama state can be understood in a rigorous setting, as the state is not measurable in the AshtekarLewandowski measure which is an essential element of the rigorous approach to the canonical quantization. At the same tine, the integrals involved are understood rigorously in terms of conformal field theory. It is thus not known if the state is normalizable in the physical inner product of the exact theory. One can ask whether the projection into the Hilbert space of linearized gravitons on deSitter spacetime gotten by truncating the Kodama state to quadratic order is normalizable or not. The answer is it is not in the Lorentzian case, and it is delta functional normalizable in the Euclidean case. 107 25. The inverse cosmological constant turns out to be quantized in integral units, so that k = 6TT/GA
(38)
16
is an integer. This is related to a basic result, which is that in 3 + 1 dimensions, a non-zero cosmological constant implies a quantum deformation of the gauge group whose representations label the edges of the spin networks, where the level which parameterizes the quantum deformation is given for the Euclidean case by Eq. (38) (and for the Lorentzian case by ik). 26. The thermal nature of quantum field theory in a deSitter spacetime is explained in terms of a periodicity in the configuration space of the exact quantum theory of general relativity. 101 ' 11 27. In both flat space and around deSitter spacetime, one may extend the calculations that reproduce quantum theory for long wavelength gravitons and matter fields to higher energies. These calculations reveal the presence of corrections to the energy-momentum relations of the form of E2 = p2 + M2 + ahiE3
+ (3l2PlE4 + ••• .
(39)
Given a candidate for the ground state, the parameters a and /3 are computable constants, that depend on the ground state wavefunctional. 22,23,11 Thus, given a choice for the ground state the theory yields predictions for modifications of the energy momentum relations. 28. In the 2 + 1 dimensional theory, modifications of the energy momentum relations (39) are present. They are understood as indicating, y
For a possible A = 0 analogue of the Kodama state, see [104].
292
not a breaking of Poincare invariance, but a deformation of it. 108 It is conjectured, 11 ' 109 but not shown, that the same will be true of quantum gravity in 3 + 1 dimensions. This will be discussed further in the next section. To summarize, the situation with regard to the low energy limit is very much like that of condensed matter systems. It is possible to invent and study candidates for the ground state, which have reasonable physical properties and reproduce the geometry of flat or deSitter spacetime. By studying excitations of these states one reproduces conventional quantum field theory, as well corrections to it which may be compared with experiment. There is not so far a first principles calculation that provides an exact or unique form of the ground state wavefunction. This is also the case in most condensed matter physics examples. It remains an open question whether one can derive first principles or model independent physical predictions for the energy momentum relations (39) from the theory. This will be discussed further below.
5.5. Results
concerning
cosmology
29. An approach to quantum cosmology has been developed from applying the methods of loop quantum gravity to the spatially homogeneous case. 20 There is a physical (but so far not rigorous) argument that the states studied in this approach correspond to spatially homogeneous states in the Hilbert spaces, HdlS of the full theory. The space of states of the homogeneous theory preserves the property of the full states that the operator that measures the volume of the universe has a discrete spectrum. The evolution of these states has been studied in detail and it has been found generically that the usual FRW cosmology is reproduced when the universe is very large in Planck units. At the same time, the cosmological singularities are removed, and replaced by bounces where the universe re-expands (or pre-contracts). The replacement of cosmological singularities by bounces has been confirmed in other non-perturbative studies of quantum cosmological models. 39 - 110 When couplings to a scalar field are included, there is a natural mechanism which generates Planck scale inflation as well as a graceful exit from it. 20 Using this formulation of quantum cosmology it has recently it has been argued that loop quantum gravity effects lead naturally to a version of chaotic inflation which may also explain the lack of power on long wave
293
lengths in the CMB spectrum. 21 30. Another approach to inflation within loop quantum gravity is given in [111]. Exact homogeneous quantum states can be constructed for general relativity coupled to a scalar field with an arbitrary potential. This may be used to construct exact quantum states corresponding to inflating universes, in the homogeneous approximation. 3 1 . It has recently been shown that loop quantum gravity effects eliminate the chaotic behavior of Bianchi IX models near singularities. 112 5.6. Results
concerning
supergravity
and other
dimensions
32. Many of these results extend to quantum supergravity for N = 1 and several have been studied also for N — 2. 3 2 ' 3 3 , 3 1 33. The same methods can also be used to solve quantum gravity in 2-f 1 dimensions50 and in some 1 + 1 dimensional reductions of the theory. 113 They also work to solve a large class of topological field theories, 114 ' 115 giving results equivalent to those achieved by other methods. Further, loop methods applied to lattice gauge theories yields results equivalent to those achieved by other methods. 116 34. Spin foam models are known in closed form for quantum general relativity in an arbitrary dimension d > 4. 51 5.7. Other extensions
of the
theory
35. The mathematical language of spin networks and spin foams can be used to construct a very large class of background independent quantum theories of gravity. These may be called generalized loop quantum gravity models. The states of such a quantum theory of gravity are given by abstract spin networks 2 . The graphs on which the spin networks are based are defined combinatorially, so that the need to specify the topology and dimension of the spatial manifold is eliminated. 52 ' 55 In such a theory the dimension and topology are dynamical, and different states may exist whose coarse grained descriptions reveal manifolds of different dimensions and topology. The histories of the theory are given by spin foams labeled by the same representations. The dynamics of the theory is specified by evolution amz An abstract spin network is a combinatorial graph whose edges are labeled by representations of A and whose nodes are represented by invariants of A. associated with the representation theory of a given Hopf algebra or superalgebra, A.
294
plitudes assigned to the nodes of the spin foams (or equivalently to local moves by which the spin networks evolve). Such generalized loop q u a n t u m gravity models have been proposed to serve as background independent formulations of string and M theory. 1 1 7 ' 5 5 T h e y also offer new possibilities for the unification of physics, because the topology of spacetime is not assumed a priori, and hence must be emerge dynamically in the low energy limit. There is then the possibility of new physics coming from obstructions to recovering trivial topology at low energies. It has been proposed t h a t these may account for t h e existence of m a t t e r degrees of freedom, 1 1 8 and perhaps even q u a n t u m theory itself. 119 5.8. Comments
on the
results
O n t h e basis of these results, it can be claimed t h a t loop q u a n t u m gravity is b o t h a consistent quantization of general relativity and a physically plausible candidate for the q u a n t u m theory of gravity. T h e failure of q u a n t u m general relativity in perturbation theory is explained by the fact t h a t there are, in this quantization of general relativity, no degrees of freedom t h a t correspond t o gravitons or other perturbative q u a n t a with wavelength shorter t h a n the Planck scale. T h e ultraviolet divergences are eliminated because a correct quantization, t h a t exactly realizes spatial diffeomorphism invariance, t u r n s out t o impose an ultraviolet cutoff on the physical spectrum of the theory. T h e assumptions, made by other approaches, t h a t spacetime is smooth and lorentz invariant at arbitrarily short scales, are not used in the quantization procedure, and in fact t u r n s out to be contradicted by the results. References 1. 2. 3. 4.
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C O U N T E R T E R M S , HOLONOMY A N D S U P E R S Y M M E T R Y
K.S. STELLE* Theoretical Physics Group, Imperial College London, Prince Consort Road, London SW7 2AZ, UK E-mail: k. stelleQimperial. ac.uk
The divergence structure of supergravity has long been a topic of concern because of the theory's non-renormalizability. In the context of string theory, where perturbative finiteness should be achieved, the supergravity counterterm structures remain nonetheless of importance because they still occur, albeit with finite coefficients. The leading nonvanishing supergravity counterterms have a particularly rich structure that has a bearing on the preservation of supersymmetry in string vacua in the presence of perturbative string corrections. Although the holonomy of such manifolds is deformed by the corrections, a Killing spinor structure nevertheless can persist. The integrability conditions for the existence of such Killing spinors remarkably remain consistent with the perturbed effective field equations.
1. Supergravity Counterterms The ultraviolet divergences of quantized general relativity and its various matter couplings have posed a key problem for the reconciliation of quantum mechanics and relativity. The potential for ultraviolet trouble with gravity was apparent already since the 1930's from rudimentary power counting, in consequence of Newton's constant having dimensionality [length]2. When detailed calculations of gravitational Feynman diagrams became possible in the 1970's,1 this became a reality with the first calculations of divergence structures that are not present in the original second-order action. As ever, in the key issues involving gravity and its quantization, Stanley Deser played a major role in this development. 2,3 As disastrous as the ultraviolet problem was for quantized field theories containing gravity, there was nonetheless some hope that a clever combina* Research supported in part by the EC under RTN contract HPRN-CT-2000-00131 and by PPARC under rolling grant PPA/G/O/2002/00474. 303
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tion of fields might save the day by arranging for the divergences to marvelously cancel. The prime candidate for an organizing principle that might engineer this was supersymmetry, and when supergravity came forth,4 there was palpable hope that it might enable the construction of some jewel-like theory that could resolve (maybe uniquely) the ultraviolet problems. This hope was encouraged by the development of non-renormalization theorems for chiral supermatter 5 and by initial calculations showing that supergravity also had better-than-generic ultraviolet behavior. For one-loop Feynman diagrams, the divergences cancel in pure N = 1 supersymmetry, as one can see by summing the contributions of the different field species occurring in the loop. A range of differing arguments was advanced on formal grounds to why these cancellations occur and why they could be expected to persist at the two-loop level (despite the prohibitive difficulty of actually performing such calculations). One approach 6 that has much current resonance focused on helicity conservation properties. In many of these early developments, the lively scientific atmosphere at Brandeis yielded important understanding of these ultraviolet problems. For me, as a graduate student there at the time, it was a marvelous training ground for learning the way physics should really be done, but one with a decidedly European flavor. Stan Deser was without doubt the leader in these matters, and it summons pleasant recollections to think back to how these fundamental issues were grappled with. Given Stan's status as doyen of the canonical formalism, another natural development we got into at the time was the canonical formulation of supergravity.7 Although not directly related to the issue of infinities, this revealed a number of essential duality properties of the theory and it also provides, via the duality-related form of the constraints, a link to the Ashtekar variable program for quantum gravity. The clearest reason for ultraviolet cancellations was the requirement that the counterterms preserve local supersymmetry. This was given a clear expression in the detailed analysis of N = 1 supergravity counterterms that we performed together with Stanley and John Kay in Ref. [8] The result was not ultimately encouraging for the prospects of finiteness, but it was intriguing nonetheless. The first relevant N = 1 supergravity counterterm occurs at the three-loop level, at which order power counting leads one to expect an expression quartic in curvatures, since at one loop the leading logarithmic divergences are of fourth order in derivatives and each loop adds two more to this count. However, at one and two loops, the possible counterterm structures happen to vanish subject to the classical
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field equations, so they can be eliminated by field redefinitions renormalizations. This is in striking contrast to the situation obtaining in pure General Relativity, where there is an available f d4xRlivp(TRpc"3'13Ra^" counterterm which moreover has a definite nonzero infinite coefficient, as found in the heroic calculations of Ref. [9] The most intriguing aspect of the three-loop supergravity counterterm was its geometrical structure: the purely gravitational part is the contracted square of the Bel-Robinson tensor 10 T^^p = —RxaPfj,R\pPu + *RXa%,*R\0pv Subject to the Einstein field equations, this tensor is covariantly divergence-free on any index, totally symmetric and totally traceless. Thus, it is a higher-order analogue of the stress tensor, whose contracted square occurs in the the (nonrenormalizable) one-loop divergences of the gravity plus Yang-Mills system. 3 Similarly, in N = 1 supergravity plus super Yang-Mills, one encounters the stress-tensor supermultiplet {T^u, JuaiCp), where J M a is the matter supersymmetry current and CM is the matter axial current. These come together in the counterterm In extended supergravities, the gravitational and lower-spin contributions give expressions that manage to vanish subject to the classical field equations at the one- and two-loop levels, as Stanley and John Kay found already in the N = 2 case. 11 In the early days, it was hoped that this situation might continue on to higher orders, but the added constraints of local supersymmetry (and this for all degrees of extension) prove to be exhausted at the next, three-loop, order. The corresponding N = 1 counterterm, whose structure continues to figure importantly in quantum gravity discussions in the string era, is a natural generalization of the one-loop matter divergence structure:
A37 = J ^ ( ( T ^ + H""^)(rMI/a/j + H^ap) +iJ* ,a V0p.W - ic^uc^p),
(i)
where H^ap = -^Padndu+'Tud^fpx, Jmp = \RxaPTGpTlnf\[i, C^p = A -^/ a757/i//3A hi which fap = daipp - dpipa is the Rarita-Schwinger field strength for the gravitino field. H^ap plays the role of the 'matter' contribution to the Bel-Robinson 'stress', while JMa/3 and C^ap are similarly higher-order analogues of the supersymmetry current JM and axial current CM. Although direct Feynman diagram calculations of the divergent coefficients of such higher-loop counterterms remain out of reach, other tech-
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niques for evaluating such divergences have progressed immensely since the late 1970's. Clever use of unitarity cutting rules plus dimensional regularization 12 have yielded the result that analogues of the D = 4 threeloop counterterm do indeed occur with an infinite coefficient for all N < 6 extended supergravities, but that the N = 8 theory (which is the same as N = 7) manages to remain finite until it too succumbs at five loops (by which time the ordinary Feynman diagram approach would involve something like 1030 terms . . . ) . Similar considerations also apply to supergravity divergences in higher dimensions, where the corresponding divergences occur at lower loop orders, e.g. in D = 11 one has divergences already at the two loop level. In the nonlocal parts of the four-graviton amplitude in such cases one again finds analogues of the (Bel — Robinson) 2 counterterm. 13 The special circumstance of the N = 8 theory remains highly intriguing. It is likely to be analogous to that which obtains for maximal {i.e. 16-supercharge, corresponding to N = 4 in D = 4) super Yang-Mills theory in 5 dimensions, which also becomes nonrenormalizable, but later than previously expected. The previous SYM expectation was based on power counting in ordinary superspace 14 ' 15 ' 16 leading to an anticipated divergence at the four-loop level in D = 5 or at three loops in D = 6. This was in agreement with explicit two-loop calculations 17 in D = 6 showing finiteness at that level. Using the unitarity cutting rule techniques, it is now known, 12 however, that the onset of divergences is in fact delayed until 6 loops in D = 5 although it does occur at the previously expected 3 loops in D = 6. This late onset of the D = 5 SYM divergences can be understood 18 using superspace power counting together with a more powerful 12-supercharge harmonic superspace formalism.19 Although the story remains incomplete, it may be anticipated that something similar is going on in maximal supergravity, perhaps with an N = 6 harmonic superspace formalism. Since these field-theory divergences, sooner or later, ineluctably arrive, the SYM situation does not, unfortunately, revive the hope for finiteness of supergravity itself, but it does have a bearing on the sorts of finite radiative corrections to be expected in superstring theories, to which we now turn.
2. String Corrections String theory may be viewed as a 'physical' regulator for the divergent supergravity theories. Instead of a Feynman integral cutoff, one has the string length yfa'. Counterterms that would have occurred with divergent coefficients in a supergravity field theory now occur with finite a'-dependent
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coefficients in quantum string corrections. In particular, the (curvature) 4 counterterms of D = 4, 3-loop supergravity now are present at the a ' 3 level in superstring theories. Generally, these string corrections have been calculated using the Neveu-Schwarz-Ramond (NSR) formalism in string light-cone gauge. As in the analogous field theory case, the first corrections not vanishing subject to the classical supergravity field equations (which are now removable by nonlinear redefinitions of the background fields that the string propagates on) occur at the a'3 level. One may write the string tree-level correction in the general form A7 = £ a ' 3 / dwxy/=jj e^* Y,
(2)
where the dependence on the dilaton <j> is appropriate to string tree level. A similar form is obtained at one string loop, but without any e^ factor, as is appropriate for one-loop order in string perturbation theory. Prom the string light-cone gauge calculations, the integrand Y in (2) can be written in terms of a Berezin integral over an anticommuting spinor field ipL,R- The (curvature) 4 correction thus takes the form 20 ' 21
Y = J d^Ld*i>R exp(i,LTi^LRijkl^Rrkl^R),
(3)
where i, j = 1 , . . . 8 are light-cone transverse indices, P-7 = ^ ( r T ^ — P T l ) are SO(8) Gamma matrices and ipi a n d ^R a r e left- and right-handed SO (8) chiral spinors. From the fact that Berezin integration gives zero except when a linear expression in each spinor field is integrated over, one sees immediately that (3) produces exclusively (curvature) 4 corrections. Letting a and a be 8-valued R, L spinor indices, one has, up to a proportionality constant, y
_
,Q!lO!2"-0!8,/8l/32.../38p*l*2
..
p»7i8
pjlj2
. . . pj7J8
Xiii1i2ji;;2rti3i4:/'3j4-Ki5i6j5j6JXi7j8j7j8
•
y*-)
Working this out in more detail, one finds Y = Y0 — Y"2, where Vn
— _L/-*i---*8yJi"-is p . . . . D. . . . D. . . . / ? . . . . 64 -"nil2.Jl.j2 W473J4 1516J5J6 *7*8J7J8I 1 ri\ ...is r .Ji---.78 P D f? f> X 2 — 256 *1 J 2 j l J 2 - n ' » 3 * 4 J 3 J 4 - n ' i 5 i 6 J 5 J 6 J X J 7 « 8 J 7 J 8 I 0
_
•y
in which t11-^
(x.\ \°)
is defined by
t' 1 -* 8 M i l i a . . . Mi7is = 2 4 M ^ M / M f c W
- 6(M^M/)2.
(6)
In making a light-cone gauge choice for the string variables in order to derive the form of these (curvature) 4 corrections, one has in fact to restrict
308
the background curvature to the transverse 8 coordinates i i . . . is, so in fact the term Y% contributes a total derivative here, since it becomes the Euler density in D = 8. Since the string tree-level correction (2) multiplies this by e~2<^, there still is a contribution to the Einstein equation, but this becomes proportional to d
(7)
in which the correction Xij arises from the connection variations, giving a correction of the form X^ = WhWeXikje,
(8)
where Xn-jt is an expression cubic in curvatures with symmetries similar to those of the curvature tensor: [ik], \jl] antisymmetric but [ik] «-> \j£] symmetric under pair interchange. Tracing the corrected gravitational equation and combining it with the dilaton equation one obtains to this order 2 D 0 - r a ' 3 X = O,
(9)
where the X = g^X^ correction arises purely from the gravitational equation trace, since the dilaton equation itself does not have order a! 3 corrections for initially supersymmetric spaces, as we have seen above. Moreover, for the special holonomy manifolds in question, one finds 9l3Xikjt — guZ .
(10)
309
Thus, X = DZ and consequently one can solve explicitly for the dilaton correction: for <j> = const+i, where fa is the correction to the initially constant dilaton. One finds fa = —^a'3Z so the corrected Einstein equation becomes Rij = a' 3 (ViVj-Z + S7kVlXikie)
.
(11)
3. Special Holonomy To see how the corrected form (11) of the Einstein equation influences the background field solutions that initially have special holonomy, consider first the case of spaces with structure M§ = K x K7 where Kr is, at order a ' 0 , a 7-manifold with holonomy Gi- Similar conclusions are obtained for 8-manifolds of Spin 7 holonomy.25 To study the G 2 case, pick the following basis for the SO (8) Dirac V matrices: r- = a2 ®Tl
f8 = -ax ® 1 8 ,
i=l,...,7;
(12)
where the P- are antisymmetric imaginary 8 x 8 SO(7) T-matrices; signs are chosen such that iT-- ••T-= tg- Chiral SO (8) spinors are eigenspinors of f9 = f1- • -f^- = a3 ® 1 8 , so * = (
+
j where * + and * _ are real
8-component SO(7) spinors. Consequently, for manifolds of G2 C SO(7) holonomy, the 8± representations decompose as 8± —> 7 © 1. Accordingly, the (curvature) 4 correction Y oc J d V + ^ V - exp [ ( ^ + T% V»+) (^_ TkJ i>-) Rijkt]
(13)
satisfies the requirements for vanishing of 'explicit' metric variations in (3), and the resulting corrections to the Einstein equations arising solely from the connection variations are of the form (11). The value of Y in (13) is zero for manifolds of initial G2 holonomy {i.e. before the effects of a' corrections are included), owing to the presence of the holonomy singlets in both the ^±. decompositions, together with the rules of Berezin integration, which give a vanishing result for f d6 integrals without a corresponding 9 in the integrand. This accounts for the absence of direct a'3 corrections to the dilaton equation, as we have noted. The vanishing of Y for such spaces does not, however, imply the vanishing of its full variation. This has to be performed without restriction to spaces of any particular holonomy, although the initial holonomy is subsequently used in evaluating the result after variation. The only surviving
310
terms in the variation of (13) are those where the singlets in the 8± decompositions go onto the 'front' and 'back' of the same varied curvature, since the only way one can get a nonvanishing result is to keep the singlet products from contracting with unvaried curvatures. This observation gives a way to write the variation in a nice fashion (where now i, j = 1 , . . . , 7): frY oc fmil '"ie fn-»»",J8 P • • •
/?••••
Ft • • • rij
rkt
X7Vi.Sn„
(14)
where c^fc = ifjTijk f] is the covariantly constant 3-form that characterizes a Gi holonomy manifold. The variation (14) thus takes the general form (11) with Xijkl. ~ CikmCjin Z
,
(15)
where 7">n =
lfmii-i
6
nji-jj
n
p.
. . .
Zmn
7, — n
(1P>\
i.e. the corrected Einstein equation is now Rij = ca,3\yiVjZ
+ cikmcilnVkVtZmn]
.
(17)
The corrected Einstein equation (17) modifies the curvature at order a ' 3 so as to give an apparently generic SO(7) holonomy, i.e. the initial G 2 special holonomy is lost as a result of the a' corrections. To see this, note that the integrability condition for the existence of an ordinary Killing spinor r\ satisfying V,?? = 0 is RijkecHmn = 2Rijmn where CHU
=
iei}ktmnPCmnp
=
^
T
. .
u v
is
t h e
H o d g e
d u a l
o f
c..k
i n
D
=
7
T a k
.
ing the trace of this integrability condition, one finds that G
Z .
(18)
Going over to a Darboux complex coordinate basis i, j = 1 , . . . , 6 —> a, a = 1, 2, 3, one then has the standard Calabi-Yau result i? a 5 = c o / 3 V a V 5 Z ,
(19)
311
which is a cohomologically trivial (1,1) form, but which does not destroy the Kahler structure (which depends on the vanishing of the Nijenhuis tensor, which is not disturbed). 4. Corrected Killing Equations Despite the fact that the string a! corrections perturb manifolds of initially special holonomy into manifolds of generic Riemannian holonomy, another remarkable property of the Bel-Robinson-descendant string corrections is that a manifold's initial supersymmetry can nonetheless be preserved. This can happen because the Killing spinor equation can itself be modified in such a way that its corrected integrability condition reproduces precisely the corrected Einstein equations. To see how this can happen, seek a condition Vj?7 = 0, where V* = V, + a'3Qi. We need to choose Qi such that the integrability condition [V,, Vj] 77 = 0 yields the corrected Einstein equation. One has directly the integrability condition \RijkeTMr]
+ ca'3QijV
= 0,
(20)
where Qij = Vi Qj - Vj Qt .
(21)
In the case of a manifold of initial G2 holonomy, one can use the Fierz identity Ti TJ fj Ti + 77 fj = t to find Rijkt ckem + 4ca' 3 i fj T m Qtj TJ = 0,
(22)
where fjQij 77 = 0. Multiplying by I \ and using the Fierz identity, one obtains a supersymmetry integrability condition involving the corrected Ricci tensor Rij=2ca'3fjT{jkQi)kri
.
(23)
The condition (23) must then be consistent with the corrected Einstein equations for some choice of the Killing spinor correction Qi. In principal, one should be able to find the Killing spinor correction Qi by an exhaustive study of the supersymmetry properties of the (curvature) 4 counterterm. This would require first determining the structure of the superpartners to the pure (curvature) 4 part by varying it subject to the original a'0 supersymmetry transformations but subject to the a'0 field equations, then relaxing the latter and calculating the required corrections to the gravitino supersymmetry transformation. This is a long process which has not been carried out for the maximal D = 11 and D = 10
312
supergravities. However, the requirements for Qi nonetheless allow one to find out its structure. The answer, i.e. the solution to (20) is Qi = -icijkVjZkeTe.
(24)
The integrability condition for the modified Killing spinor condition then reproduces precisely the corrected Einstein equation (17). The Killing spinor correction (24) seemingly depends on special properties of the order a' ° manifold, since it involves the G% manifold's covariantly constant 3-form Cijk- However, another remarkable structural feature emerges here. The Killing spinor correction (24) can be rewritten in a form that does not make use of any special tensors on the manifold: (25) Moreover, this is precisely the same expression as one finds from the study of corrections to D = 6 Kahler manifolds,22 so there is a strong argument for the universality of the result (25). 5. Conclusion The quantum field theoretic approach to quantum gravity, of which Stan Deser is a key pioneer continues to yield important insights into a theory of which we still have only glimpses. The main approach to quantum gravity has changed from the canonical formulation to supergravity and on to superstrings, but there is considerable continuity in certain central elements of the story. The ultraviolet problem for gravity, which gave rise to much soul-searching about the nature of the entire perturbative quantum gravity program, has now more or less been solved. Accordingly, one can now begin to actually look at the perturbatively finite theory that lies behind. Despite the evolution in dynamical details, the analytical approach of focusing on symmetries and their consequences remains an important strategy. In the examples that we have looked at, ultraviolet counterterms that spelled the end of supergravity as a fundamental theory in its own right remain nonetheless of keen interest as finite local contributions to the supergravity effective action for superstrings or M-theory. They have a set of 'miraculous' properties that appear to make them precisely tailored to preserving the integrity of the underlying, still incompletely known, string or M-theory. In particular, they may lead to important insights into the structure of M-theory, for which we still have no full microscopic formulation.
313
An example of this is the link between the C[3] Ai? 4 coupling in M-theory and the RA t e r m s t h a t arise in type IIA string theory at the one loop level, which are in t u r n related t o M-theory (curvature) 4 terms by dimensional oxidation. These two types of terms are related 2 7 by on-shell supersymmetry; the relation is also crucial 2 5 for the way in which t h e supersymmetry of an initially SU(5) holonomy Kahler manifold can be preserved despite the fact t h a t the (curvature) 4 corrections in this case destroy the Kahler structure, yielding a general complex D = 10 manifold. Via a sequence of 'miracles' analogous to those we have sketched here for the string tree level C?2 case, t h e initial supersymmetry of such a background t u r n s out to be preserved t h a n k s to interrelated corrections t o the Einstein and 4-form field equations. T h e Cp] Ai? 4 t e r m s play a key role 2 8 in this mechanism, because they force the turning on of a necessary amount of 4-form flux. T h e same t e r m s are also crucial for t h e elimination of t h e sigma-model anomalies of the M5 b r a n e 2 9 and for duality between M2 and M5 branes. T h e quartic curvature corrections thus are deeply related t o the internal consistency of our best chance for a fundamental theory of q u a n t u m gravity. References 1. 2. 3. 4.
5.
6.
7. 8. 9.
G. 't Hooft and M. Veltman, Ann. Inst. Henri Poincare 20, 69 (1974). S. Deser and P. van Nieuwenhuizen, Phys. Rev. D10, 401 (1974); 411 (1974). S. Deser, H.-S. Tsao and P. van Nieuwenhuizen, Phys. Rev. D 1 0 , 3337 (1974). D.Z. Freedman, S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. D 1 3 , 3214 (1976); S. Deser and B. Zumino, Phys. Lett. B 6 2 , 335 (1976). D. Capper and G. Leibbrandt, Nucl. Phys. B85, 492 (1975) ; K. Fujikawa and W. Lang, Nucl. Phys. B88, 61 (1975); R. Delbourgo, Nuovo Cirnento 25A, 646 (1975); S. Ferrara and O. Piguet, Nucl. Phys. B 9 3 , 261 (1975); P.C. West, Nucl. Phys. B106, 219 (1976); M.T. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B159, 42 (1979). M.T. Grisaru, P. van Nieuwenhuizen and J.A.M. Vermaseren, Phys. Rev. Lett. 37, 1662 (1976); M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Phys. Rev. D15, 996 (1977); M.T. Grisaru and H.N. Pendleton, Nucl. Phys. B124, 81 (1977); S.M. Christensen, S. Deser, M.J. Duff and M.J. Grisaru, Phys. Lett. B 8 4 , 411 (1979). S. Deser, J.H. Kay and K.S. Stelle, Phys. Rev. D16, 2448 (1977). S. Deser, J.H. Kay and K.S. Stelle, Phys. Rev. Lett. 38, 527 (1977). M.H. Goroff and A. Sagnotti, Nucl. Phys. B266, 709 (1986); A.E.M. van de Ven, Nucl. Phys. B378, 309 (1992).
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10. I. Robinson, unpublished; L. Bel, C. R. Acad. Sci. 247, 1094 (1958). 11. S. Deser and J.H. Kay, Phys. Lett. B76, 400 (1978). 12. Z. Bern, L. Dixon, D. Dunbar, M. Perelstein and J.S. Rosowsky, Nucl. Phys. B530, 401 (1998); Class. Quantum Grav. 17, 979 (2000); Z. Bern, L. Dixon, D. Dunbar, A.K. Grant, M. Perelstein and J.S. Rosowsky, Nucl. Phys. Proc. Suppl. 88, 194 (2000). 13. S. Deser and D. Seminara, Phys. Rev. D62, 084010 (2000) [hep-th/0002241], 14. M.T. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B159, 429 (1979). 15. P.S. Howe, K.S. Stelle and P.K. Townsend, Nucl. Phys. B236, 125 (1984). 16. P.S. Howe and K.S. Stelle, Phys. Lett. B137, 175 (1984). 17. N. Marcus and A. Sagnotti, Phys. Lett. B135, 85 (1984); Nucl. Phys. B256, 77 (1985). 18. P.S. Howe and K.S. Stelle, Phys. Lett. B554, 190 (2003) [hep-th/0211279]. 19. A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E. Sokatchev, Class. Quantum Grav. 2,155 (1985). 20. D.J. Gross and E. Witten, Nucl. Phys. B277, 1 (1986). 21. M.D. Freeman and C.N. Pope, Phys. Lett. B174, 48 (1986). 22. P. Candelas, M.D. Freeman, C.N. Pope, M.F. Sohnius and K.S. Stelle, Phys. Lett. B177, 341 (1986). 23. H. Lii, C.N. Pope and K.S. Stelle, JHEP 0407, 072 (2004) [hep-th/0311018]. 24. H. Lii, C.N. Pope, K.S. Stelle and P.K. Townsend, JHEP 0410, 019 (2004) [hep-th/0312002]. 25. H. Lii, C.N. Pope, K.S. Stelle and P.K. Townsend, hep-th/0410176. 26. D. Constantin, Nucl. Phys. B 706, 221 (2005) [hep-th/0410157]. 27. M.B. Green and M. Gutperle, Nucl. Phys. B498, 195 (1997) [hepth/9701093; M.B. Green, M. Gutperle and P. Vanhove, Phys. Lett. B409, 177 (1997) [hep-th/9706175], 28. K. Becker and M. Becker, Nucl. Phys. B477, 155 (1996) [hep-th/9605053]; K. Becker and M. Becker, JHEP 0107, 038 (2001) [hep-th/0107044]. 29. M.J. Duff, J.T. Liu and R. Minasian, Nucl. Phys. B452, 261 (1995).
MILNE A N D T O R U S U N I V E R S E S M E E T
A. WALDRON Department
of Mathematics, University of California, Davis, CA 95616, USA E-mail: wallyiSmath.ucdavis.edu
Three dimensional quantum gravity with torus universe, T 2 x R, topology is reformulated as the motion of a relativistic point particle moving in an 5/(2, Z) orbifold of flat Minkowski spacetime. The latter is precisely the three dimensional Milne Universe studied recently by Russo as a background for Strings. We comment briefly on the dynamics and quantization of the model.
Stanley has made many fundamental contributions to our understanding of life in various dimensions. Notable amongst these is dimension three. Therefore, on the occasion of his seventy third birthday, I present him a reformulation of three dimensional gravity in terms of a point particle moving in a flat three dimensional spacetime. 1. Introduction The limited tractability of quantum gravity means that minisuperspace reductions to quantum mechanical degrees of freedom are an important calculational tool. Moreover in three dimensions, where gravitons carry no field theoretic degrees of freedom, a minisuperspace reduction might be expected to faithfully represent the full theory. Indeed, at least for toroidal spatial topologies'1, this is the case:1 on any given toroidal spatial slice, a Weyl transformation brings the 2-metric to a flat one. The space of conformally fiat metrics on the torus is parameterized by the coset Sl(2, M.)/SO(2). Furthermore, a peculiarity of three dimensions is that the space dependence of this Weyl transformation can be gauged away. Therefore, as observed long ago by Martinec, 1 all that remains is the quantum mechanical evolution of a
Modulo questions of inequivalent quantizations discussed by Carlip. 2 315
316
metric moduli in the space R+ x 5/(2, R)/SO(2). In our previous work3 we made the additional observation that this system was in fact a relativistic particle with mass proportional to the cosmological constant. In addition we showed that this model enjoys a conformal (spectrum generating) symmetry and falls into a large class of novel conformal quantum mechanical models. In this short note we show that for the case of three dimensional spatially toroidal gravity, the metric moduli space is an 5/(2, Z) orbifold of flat three dimensional Minkowski spacetime. The derivation of this model is given in Section 2. Dynamics and solutions are discussed in Section 4 while quantization and a discussion of possible physical computations may be found in Section 5. 2. M i n i s u p e r s p a c e Our model is obtained by rewriting three dimensional cosmological gravity in the limit where all spatial derivatives are discarded, so we parameterize the metric as ds2 = -N(t)2dt2
+ hij(t)dxidxj
,
(1)
where i,j = 1,2. We study a toroidal topology T 2 x R with 0 < xz < 1. The extrinsic curvature is simply
J_
K
h » = -^Tr 2N H '
which implies K = hljKij Hilbert action now reads S=
(2)
— - j ^ ^ log h with h = det hij. The Einstein
fdt d2xVh N
-K2
+ KijRv
- 2A .
(3)
Extracting the spatial volume hij = h1'2
h^ ,
(4)
the unit volume metric hij is parameterized by the coset Sl(2,R)/SO(2). Requiring, in addition, that spatial sections are toroidal, means that we must identify metrics related by the left action of 5/(2, Z). The resulting 5/(2, Z)\Sl(2, E ) / 5 0 ( 2 ) orbifold is just the usual upper half plane modded out by modular transformations. Introducing coset coordinates UM (M = 1,2) and invariant metric UMGMNUN
= - i hij hij
(5)
317
as well as the field redefinition ri = RN,
R = h1/2,
(6)
yields S = ^
fdt{-[-R2
+ R2UMGMNUN]
- 4A77} .
(7)
This is the action of a relativistic particle in three dimensions, (mass) 2 = 4A (tachyonic in AdS!) with metric ds2m = -dR2 + R2 dUMGMNdUN
.
(8)
Hence, the time evolution of the metric moduli UM and R in this minisuperspace truncation is described by the dynamics of a fictitious particle moving in a three dimensional Lorentzian spacetime which we will denote Wl. 3. Geometry of the Metric Moduli Space To understand the space 9DT better, write the unit volume spatial metric in terms of a zweibein liij = (ee')y
(9)
where the Sl(2) valued zweibein e has Iwasawa decomposition
so that
ds2m = -dR2 + R2 * 1 + M .
(11)
This space is the three dimensional Milne Universe. Here — 00 < T\ < 00, 0 < T2 and 0 < R. Now change coordinates
R = VZ2~X*-Y*,
Y
Tl
= Y-x,
R
^ = Y^X-
(12)
The metric becomes the flat three dimensional Minkowski one ds2m = -dZ2 + dX2 + dY2
(13)
and our fictitious particle action is simply S
\2 fdt{ - [-Z 2 + X 2 +y 2 ]-4Ar?}.
(14)
318
Figure 1. Lobacevskii's non-Euclidean plane realized as a hyperboloid in the forward lightcone.
To complete the discussion, we still need to identify the topology of the space 971. Firstly note that the inverse of the above coordinate transformation is 2 V r2
T2 /
r2
2 V r2
r2 /
(The second solution with an overall — is ruled out by positivity of R and T2 which requires Z > X.) Surfaces of constant R are hyperboloids in the forward lightcone (since Z > X) isomorphic to the upper half plane H. This is simply Lobacevskii's ur non-Euclidean geometry depicted in Figure 1. The boundary dM at r 2 = 0 maps to the boundary of each hyperboloid while the cusp at r = ioo maps to (X, Y, Z) = (oo,0, oo). The unit circle |r| = 1 corresponds to the line {X = 0}n{i? 2 = Z2 -X2 -Y2 | Z > 0}. Therefore, before modding out by modular transformations, the metric moduli space is simply the interior of the forward lightcone in three dimensions with usual flat Minkowski metric. Now we must mod out the upper half plane PSl(2,R)/SO{2) =M M valued torus moduli U by the left action PSl(2,Z) which acts independently on each hyperboloid. This group is a discrete subgroup of the three
319
dimensional Lorentz group generated by isometries T : r i-+ r + 1 and S : r H-> —1/r. In terms of the new variables (X,Y,Z) these isometries leave R invariant and act as T : (X, Y, Z) ^ I ( x + 2 F + Z, - 2 [ Z - F - Z], - X + 2Y + 3 ^ ) ,
S:(x,y,z)^(-x,-y,z).
(16)
It is easy to verify that an infinitesimal transformation r —> r + i is generated by — = Ydx - Xdy + Ydz + Zdy = MYX + MYZ ,
(17)
the sum of a rotation in the (X, F)-plane and an (Y, Z)-boost (or better, a lightcone boost). The resulting orbifold is precisely the three dimensional Milne universe studied by Russo4 in a String theoretic context. Finally, note that the generators of the S7(2, R) isometry subgroup are e+ = dTl = MYZ + MYX ,
h = 2ndTl + 2r2dT2 = -2MXz
e_ = - n (ndTl + 2r2dT2) + r 2 2 9 Tl = MYZ ~ MYx •
, (18)
Hence, we may also view this subgroup as the 50(2,1) Lorentz group in the natural way. 4. Dynamics The time coordinate t in the relativistic particle model (14) plays no preferred role, since the the "einbein" 77 ensures reparameterization invariance. Instead classically, we may only predict trajectories in the space 9Jt. The beauty of this model is that for a free relativistic particle these are simply straight lines with slopes subject to Einstein's relativistic dispersion relation. Explicitly, in the gauge 77 = 1, straight line geodesies are Z=Z0
+ Pzt,
X = X0 + Pxt,
Y = Y0 + PYt,
(19)
subject to a mass-shell condition --Pi + ^ l + -Py = 4A.
(20)
To convert these to metric solutions, note that the two-metric takes the compact form 'Z + X (hij)
=
Y
Some fundamental solutions include:
Y
z-xr
<21>
320
Kasner Setting Y(t) = 0 and A = 0 the mass shell constraint becomes (Z + X)(Z-X) = 0.
(22)
The Kasner solution to this constraint is Z = t+^,X=t~^ which yields the metric ds2 = — ^dt2 + 2t(dx1)2 + (dx2)2. The standard Kasner metric is obtained by changing time coordinates to "cosmological time" T = R = y/Z2 - X2 so that ds2 = -dr2 + (rdx1)2 + (dx2)2 ,
(23)
which amounts to the gauge rj = r in the relativistic particle model. de Sitter Reintroducing the cosmological constant alias the relativistic mass 2\/A we solve the mass shell constraint via Z = 2\fht, X = Y = 0 yielding metric ds 2 = -j^dt2+2yfh.t [(dx1)2 + (dx2)2}. Changing time coordinates r = r(t) to the gauge rj = exp ( 2 - \ / A T J yields the steady state de Sitter metric ds2 = -dr2 + e 2 V X r [ ( & 1 ) 2 + (dx 2 ) 2 ].
(24)
Anti de Sitter We can also consider tachyonic trajectories corresponding to negative cosmological constant, X = 2y/\A\t, Y = 0 and Z = ZQ (say). This yields a novel Anti de Sitter metric dt2 2 ds = - Z Q 2 + 4 J U 2 + (Z0 + 2^\A\t)(dx1)2 + (Z0 - 2^\A\t)(dx2)2 . (25) This metric becomes singular at t = ± Z o / ( 2 y |A|) at which points the volume of spatial slices R = 0. Therefore, it represents only a coordinate patch of Anti de Sitter space. It would be interesting to study the compatibility of the 1? torus orbifold on the spatial coordinates (re1, a;2) with the geodesic completion of the above metric. 5. Quantization and Conclusions We have presented a simple three dimensional relativistic particle model that describes toroidal gravity in three dimensions. The model would be
321
trivially solvable if it were not for the 5/(2, Z) orbifold of the flat particle background necessary to identify equivalent gravity metrics. The Hilbert space of physical states in this model is dictated by the Hamiltonian constraint { - 3 z + 3 x + # + 4 A } * = 0.
(26)
Here we have a chosen a particular quantization corresponding to the natural operator ordering stated. An initial investigation of this Klein-Gordon equation has been conducted in [4], the key difficulty being to automorphize with respect to 5/(2, Z). A more detailed study will appear. 5 Finally, an old observation6 is that Hamiltonian constraints taking this Klein-Gordon form naturally imply a second quantization of the theory, jocularly dubbed "third quantization". A natural candidate for interactions would be
D I Q U A R K S AS INSPIRATION A N D AS OBJECTS
FRANK WILCZEK Attraction between quarks is a fundamental aspect of QCD. It is plausible that several of the most profound aspects of low-energy QCD dynamics are connected to diquark correlations, including: paucity of exotics (which is the foundation of the quark model and of traditional nuclear physics), similarity of mesons and baryons, color superconductivity at high density, hyperfine splittings, A J = 1/2 rule, and some striking features of structure and fragmentation functions. After a brief overview of these issues, I discuss how diquarks can be studied in isolation, both phenomenologically and numerically, and present approximate mass differences for diquarks with different quantum numbers. The mass-loaded generalization of the Chew-Prautschi formula provides an essential tool.
/ had hoped to present something really worthy for my friend Stanley Deser's birthday, like my solution of the cosmological term problem — but a few details of that work still need clarification. What you find here instead is a report of some recent work on the strong interaction. We have what appears to be the fundamental theory of that interaction (ratified by the 2004 Nobel Prize), but one message I'd like to emphasize today is the phenomenology exhibits some striking regularities that are very poorly understood, including regularities that play a central role in determining the structure of ordinary matter as we observe it. Superficially Stanley's brain has not been concerned so much with why there are protons and neutrons and nuclei, but since that very brain is made from those very things, perhaps it's not entirely inappropriate to discuss those issues on this occasion.
1. Diquarks as Inspiration 1.1. Diquarks
in Microscopic
QCD
In electrodynamics the basic interaction between like-charged particles is repulsive. In QCD, however, the primary interaction between two quarks can be attractive. At the most heuristic level, this comes about as follows. Each quark is in the 3 representation, so that the two-quark color state 322
323
3 ® 3 can be either the symmetric 6 or the antisymmetric 3. Antisymmetry, of course, is not possible with just 1 color! Two widely separated quarks each generate the color flux associated with the fundamental representation; if they are brought together in the 3, they will generate the flux associated with a single anti-fundamental, which is just half as much. Thus by bringing the quarks together we lower the gluon field energy: there is attraction in the 3 channel. We might expect this attraction to be roughly half as powerful as the quark-antiquark 3 ® 3 —» 1. Since quark-antiquark attraction drives the energy in the attractive channel below zero, triggering condensation (qq) ± 0 of qq pairs and chiral symmetry breaking, an attraction even half as powerful would appear to be potentially quite important for understanding low-energy QCD dynamics. One can calculate the quark-quark interaction due to single gluon exchange, and of course one does find that the color 3 channel for quarks is attractive. Going a step further, one can consider magnetic forces, and distinguish the favored spin configuration. One finds that the favorable spin configuration is likewise the antisymmetric one, i.e. \ ® \ —> 0. With antisymmetry in color and spin, and a common spatial configuration, Fermi statistics requires that the favorable diquark configuration is also antisymmetric in flavor. For non-strange diquarks, this means isosinglet, in the context of flavor SU(3) it means flavor 3. We shall denote the favorable diquark configuration as [qq'}, and speak of "good" diquark. We shall also have occasion to consider the spin triplet flavor symmetric configuration (still color 3!), which we will denote this as (qq1) and speak of the "bad" diquark. Since the spin-spin interaction is a relativistic effect, we might expect it to be strongest for the lightest quarks; that is, we expect the splitting (ud) — [ud] > (us) — [us] > (uc) — [uc] « 0. One can also calculate forces between quarks due to instantons. The same antisymmetric channel emerges as the most favorable, with attraction. At asymptotically high densities in QCD one can justify the use of weak coupling to analyze quark interactions near the Fermi surface. The attractive quark-quark interaction in the good diquark channel is responsible for color superconductivity, and more particularly color-flavor locking. In that context it triggers condensation of diquarks, with color symmetry breaking. This leads to a rich theory, including calculable — weak coupling, but nonperturbative — mechanisms for confinement and chiral symmetry breaking. In vacuum we do not have color breaking, of course, or (therefore) diquark condensation; but the dominant role of good diquarks at high density is definitely another motivation for studying their properties in general. As
324
a practical matter, it might help us understand the parameters governing the approach to asymptopia, which is important for constructing models of the internal structure of neutron (—»center quark) stars. As a corollary to the fact that quark attraction that favors good diquark formation, we might expect repulsion between good diquarks. Indeed, when two good diquarks overlap the cross-channels, involving one quark from each diquark, will have unfavorable correlations. The repulsion might be manifested in the form of a force or, in response to attempts at fusion, re-arrangement into baryon plus single quark. 1.2. Phenomenological
Indications
These heuristic, perturbative, and quasi-perturbative considerations suggest several applications of diquark ideas within strong interaction phenomenology. Since the relevant calculations are not performed in a wellcontrolled approximation, we should regard this as an exploratory activity. To the extent that we discover interesting things in this way — and we do! — it poses the challenge of making firmer, more quantitative connections to fundamental theory. A classic manifestation of energetics that depends on diquark correlations is the S — A mass difference. The A is isosinglet, so it features the good diquark \ud); while E, being isotriplet, features the bad diquark (ud). The £ is indeed heavier, by about 80 MeV. Of course, this comparison of diquarks is not ideal, since the spectator s quark also has significant spin-dependent interactions. A cleaner comparison involves the charm analogues, where E c —Ac = 215 MeV. (Actually this comparison is not so clean either, as we'll discuss later. One sign of uncleanliness is that there either E c (2520)§ or E c (2455)| might be used for comparison; here I've taken the weighted average.) One of the oldest observations in deep inelastic scattering is that the ratio of neutron to proton structure functions approaches 1/4 in the limit x->l lim
i.) { -> - .
(1)
In terms of the twist-two operator matrix elements used in the formal analysis of deep inelastic scattering, this translates into the statement lim
(2)
325
where spin averaging of forward matrix elements, symmetrization over the fis, and removal of traces is implicit, and a common tensorial form is factored out, together with similar equations where operators with strange quarks, gluons, etc. appear in the numerator. Equation (2) states that in the valence regime x —* 1, where the struck parton carries all the longitudinal momentum of the proton, that struck parton must be a u quark. It implies, by isospin symmetry, the corresponding relation for the neutron, namely that in the valence regime within a neutron the parton must be a d quark. Then the ratio of neutron to proton matrix elements will be governed by the ratio of the squares of quark charges, namely ( - | ) 2 / ( | ) 2 = 1/4. Any (isosinglet) contamination from other sources will contribute equally to numerator and denominator, thereby increasing this ratio. Equation (2) is, from the point of view of symmetry, a peculiar relation: it requires an emergent conspiracy between isosinglet and isotriplet operators. It is also, from a general physical point of view, quite remarkable: it is one of the most direct manifestations of the fractional charge on quarks; and it is a sort of hadron = quark identity, closely related to the quark-hadron continuity conjectured to arise in high density QCD. It is an interesting challenge to derive (2) from microscopic QCD, and to estimate the rate of approach to 0. A more adventurous application is to fragmentation. One might guess that the formation of baryons in fragmentation of an energetic quark or gluon jet could proceed stepwise, through the formation of diquarks which then fuse with quarks. To the extent this is a tunneling-type process, analogous to pair creation in an electric field, induced by the decay of color flux tubes, one might expect that the good diquark would be significantly more likely to be produced than the bad diquark. This would reflect itself in a large A/E ratio. And indeed, data from LEP indicates that the value of this ratio is about 10 for leading particles (that is, at large z). In the Particle Data Book one also finds an encouraging ratio for total multiplicities in e + e~ annihilation: Ac : S c = .100 ± .03 : .014 ± .007; in this case the c quarks are produced by the initiating current, and we have a pure measure of diquarks. There are also several indications that diquark correlations have other important dynamical implications. The AI = 1/2 rule in strangenesschanging nonleptonic decays has also been ascribed to attraction in the diquark channel. The basic operator ^7^(1 — 75)^7^(1 — 75)^ arising from W boson exchange can be analyzed into [us][ud], (us)(ud), and related color-6 diquark types. Diquark attraction in [us][ud] means that there is
326
a larger chance for quarks in this channel to tap into short-distance components of hadronic wavefunctions. This effect is reflected in enhancement of this component of the basic operator as it is renormalized toward small momenta. Such an enhancement is well-known to occur at one-loop order (one gluon exchange). Stech and Neubert have advanced this line of thought significantly.1
1.3. Correlations
and the Main Problem
of
Exotics
Our present understanding of the strong interaction is disturbingly schizophrenic. On the one hand we have an algorithmically definite and very tight relativistic quantum field theory, quantum chromodynamics (QCD), which we can use to do accurate quantitative calculations in special circumstances. Many hard (i.e., large momentum-transfer) processes and processes involving heavy quarks can be treated using the techniques of perturbative QCD. The spectroscopy of low-lying states, and a few interesting matrix elements of operators (currents, twist-two operators, weak Hamiltonian matrix elements) can be calculated by direct numerical solution of the fundamental equations, using the techniques of lattice gauge theory. These quantitative calculations are famously successful, with accuracies approaching 1% in favorable cases, and amply justify faith in the theory. The basic degrees of freedom in QCD include massless gluons and almost-massless u, d quarks, and the interaction strength, though it "runs" to small coupling at large momentum transfer, is not uniformly small. We might therefore anticipate, heuristically, that low-energy gluons and quarkantiquark pairs are omnipresent, and in particular that the eigenstates of the Hamiltonian — hadrons — will be complicated composites, containing an indefinite number of particles. And indeed, according to the strictest experimental measure of internal structure available, the structure functions of deep inelastic scattering, nucleons do contain an infinite number of soft gluons and quark-antiquark pairs (parton distributions ~ dx/x as x —> 0). On the other hand, the phenomenological quark model has been used with considerable success to organize a lush jungle of observations that would otherwise appear bewildering. This model is built upon degrees of freedom whose properties are closely modeled on those of the fundamental theory; nevertheless, its success raises challenging conceptual questions. For the main working assumption of the quark model is that hadrons are constructed according to two body plans: mesons, consisting of a quark and an antiquark; and baryons, consisting of three quarks. This paucity of
327
body plans seems out of step with the heuristic expectations we mentioned earlier. And, lest we forget, the most developed and useful model in strong interaction physics is traditional nuclear physics, based on nucleons as degrees of freedom. In this model the effective residual interactions are feeble compared to the interactions responsible for constructing the nucleons from massless ingredients in the first place; this allows us to employ essentially non-relativistic dynamics, and we don't consider particle production. Furthermore, and not unrelated: the nuclear forces have a "hard core" repulsion, and saturate. The puzzles posed by the success of the quark model and traditional nuclear physics come into sharp focus in the question of exotics. Are there additional body plans in the hadron spectrum, beyond qqq baryons and qq mesons (and loose composites thereof)? If not, why not; if so, where are they? As a special case: why don't multi-nucleons merge into single bags, e.g. qqqqqq — or can they? The tension between a priori expectations of complex bound states and successful use of simple models, defines the main problem of exotics: Why aren't there more of them? A heuristic explanation can begin along the following lines. Low-energy quark-antiquark pairs are indeed abundant inside hadrons, as are low-energy gluons, but they have (almost) vacuum quantum numbers: they are arranged in flavor and spin singlets. (The "almost" refers to chiral symmetry breaking.) Deviations from the "good" quark-antiquark or gluon-gluon channels, which are color and spin singlets, cost significant energy. States which contain such excitations, above the minimum consistent with their quantum numbers, will tend to be highly unstable. They might be hard to observe as resonances, or become unbound altogether. The next-best way for extraneous quarks to organize themselves appears, according to the preceding considerations, to be in "good" diquark pairs. Thus a threatening — or promising — strategy for constructing low-energy exotics apparently could be based on using these objects as building-blocks. There are two reasons "good" diquark correlations help explain the paucity of exotics: because of their antisymmetry, they lock up spin and flavor; and because of their repulsion, they forbid mergers. These two aspects are exemplified in the next two paragraphs. Tetraquarks play an important role in modeling the observed low-lying nonet of scalar 0 + mesons including / 0 (600) = a, K ( 9 0 0 ) , / O ( 9 8 0 ) , O O ( 9 8 0 ) . It appears perverse to model these as conventional qq mesons, since the
328
isotriplet ao(980) is the heaviest component, but would (on this assignment) contain no strange quarks. A serious and extensive case has been made that an adequate model of these mesons must include a major admixture of qqqq. Then both /o(980),ao(980) are accommodated as [is][/s], with I = u or d. For our purposes, the most important observation is that if the quarks (antiquarks) are correlated into good diquarks (antidiquarks), as we expect they will be for the lowest-lying states, then the non-exotic flavor structure of the nonet is explained; indeed, for the flavor one obtains 3 ® 3 = 8 © 1 with the same charges as for qq. For this reason they are called cryptoexotics. qqqq can organize alternatively into two color singlet qq mesons, of course, and sophisticated modeling includes both channels (with diquarks dominating at short distances, mesons at larger distances). The non-existence of low-lying dibaryons is related to the (or at least, a) foundational problem of nuclear physics: Why do protons and neutrons in close contact retain their integrity? Essentially the same question arises in a sharp form for the H particle studied by Jaffe.2 It has the configuration uuddss. In the bag model it appears that a single bag containing these quarks supports a spin-0 state that is quite favorable energetically. A calculation based on quasi-free quarks residing in a common bag, allowing for one-gluon exchange, indicates that H might well be near or even below A A threshold, and thus strongly stable; or perhaps even below An threshold, and therefore stable even against lowest-order weak interactions. These possibilities appear to be ruled out both experimentally and by numerical solution of QCD (though possibly neither case is completely airtight). Good diquark correlations, together with repulsion between diquarks, suggests a reason why the almost-independent-particle approach fails in this case. Note that for this mechanism to work requires that essentially nonperturbative quark interaction effects, beyond one gluon exchange, must be coming into play.
2. Diquarks as Objects From all this it appears that diquarks may be very useful degrees of freedom to use in understanding QCD. If we're going to do that, the first step should be to study them in a pure and isolated form, and determine their parameters. This is not straightforward, due to confinement, since the diquarks are colored. But I believe there are attractive ways to do something approaching isolating them, both physically and numerically. Of course, the same problem arises for quarks. Our considerations will
329
apply to them in a non-trivial way, as well. In rapidly spinning baryons centrifugal forces lead to a geometry where a quark at one end of a line of color flux is joined to two quarks at the other. The two-quark end then makes a little laboratory where one can compare good and bad diquark configurations with each other, assess the effects of strangeness, and (comparing with mesons) normalize them relative to single quarks. Famously, the Chew-Frautschi formula M2 = a + aL
(3)
organizes trajectories of resonances (Chew-Frautschi formula) with the same internal quantum numbers but different values of Jp; here a is a universal constant ~ 1.1 Gev2 while a depends on the quantum numbers, and L is an orbital angular momentum, quantized in integers. Recently Alex Selem and I have used this formula, together with some refinements and extensions, to do extensive and I think quite successful hadron systematics. My main point below, extracted from that work, will take off from one such refinement. The formula M2 = <JL arises from solving the equations for a spinning relativistic string with tension a/(2n), terminated by the boundary condition that both ends move transversely at the speed of light. We might expect it to hold asymptotically for large L in QCD, when an elongated flux tube appears string-like, the rotation is rapid, quark masses are negligible, and semiclassical quantization of its rotation becomes appropriate. The primeval CF formula M 2 = a + aL, with simple non-zero values of a (e.g., a = \a) can result from quantization of an elementary non-interacting string, including zero-point energy for string vibrations. In the following section we (that is, Alex and I) generalize the classical formula to the form appropriate for string termination on massive objects. There will be corrections that depend on the masses of the objects at the end. Using these corrected formulas, we are able to identify (overdetermined) values of the masses of various kinds of quasi-isolated quarks and diquarks, directly from spectroscopic data. 2.1. Generalization
of the Chew-Frautschi
Formula
We can generalize the Chew-Frautschi formula by considering two masses mi, ?Ti2 connected by a relativistic string with constant tension, T, rotating with angular momentum L. Our general solution naturally arises in a parameterized form in which the energy, E, and L are both expressed in
330
terms of the angular velocity, u>, of the rotating system. In the limit that mi, rri2 —* 0, the usual Chew-Prautschi relationship E2 oc L appears. Considering masses mi and ra-i at distances r\ and r^ away from the center of rotation respectively. The whole system spins with angular velocity u>. It is also useful to define: H =
,
h
(4)
v2»
where the subscript i can be 1 or 2 (for the mentioned masses). It is straightforward to write the energy of the system: rp
ru>ri
-1
rp
pUJT2
1
E = mi71 + m272 + - / , du + - / —===du. (5) w w 7o v 1 - w2 Jo V1 - u2 The last two terms are associated with the energy of the string. Similarly, the angular momentum can be written as: T f^ri u2 T fur2 u2 L = m 1 wri7i+m 2 wr^7 2 + ^ / du+^ / rfu. W Jo v 1 — u w Jo V1 — u Carrying out the integrals gives: T E = mi7i + m272 H (arcsinfwri] + arcsin[u;r2]), L = miwri7i + m 2 wr 2 72 rj-i
(6)
(7a) (7b)
-t
- I — ^ - { - u r i y / l - (wri) 2 +arcsin[tjr 1 ]-a;r2\/l - (wr2)2+arcsin[a;r2]J . Furthermore the following relationship between the tension and angular acceleration holds for each mass: T miW2ri = —. (8) We can use this to eliminate the distances r\ and r
From Eq. (8) we also know that uri is just T/(mijfw). We are now in a position to replace these terms in Eq. (7) and write E and L in terms of the parameter o> and other quantities assumed known, namely the masses and the string tension T. The resulting expressions are
331
a bit opaque, but we can make good use of them either by plotting E2 vs L parametrically, or by making appropriate expansions, for the cases of either very light or very heavy masses, to obtain analytic expressions for E2 vs L. The terms associated with each mass decouple from one another, so we may construct expansions for each separately. We adopt the convention that the contribution from one mass is preceded with a 5, as in SE. It is useful to define another variable Xi = rriiu/T, If we expand in xt, then, we find the contribution to the energy, 5E, and angular momentum, 5L, due to one light mass is SEHght = g TTT
+\ n ^
x
r +± m W
l m i 1/2
+ O (m\»x*'*) ,
3 mi 3/2 nfrrii
5/2\
(10a)
, , „ , .
to order x/ . For a system with two light and equal masses, we would of course just multiply the right hand side of these expressions by two to obtain the total energy and angular momentum. Note that for a very light mass it appears that u> —» oo as L —> 0, so this is a singular limit. If we let both masses go to zero, and therefore take only the first term for each mass from the light-mass expansion (Eq. (10)), then we recover the familiar Chew-Frautschi relationship for the string with massless ends: E2 = (2TTT)L.
For the first corrections at small m\,rai algebra,
(11)
(and L ^ 0) we find, after some
EKVaL + KL'ini,
(12)
with
«44,
d3)
and 3
3
3
H? = m{ + m l .
(14)
This is a useful expression, since it allows us to extract expressions for quark and diquark mass differences from the observed values of baryon and meson mass differences. Numerically, K W 1.15 G e V " for a « 1.1 GeV 2 . For heavy-light systems the corresponding formula is E-M
= J?—+2*KL-*II%,
(15)
332
where M is the heavy quark mass and fi is the light quark mass. Note that the usual correction due to a zero-point vibrations, i.e. a classic intercept of the type E2 = a + (2nT)L, yields corrections of the form E —* VCTL + a)'(2\faL). It becomes subdominant to mass corrections at large L.
2.2. Nucleon-Delta
Complex
As a small taste of the much more extensive analysis presented in [3], our fit to the bulk of non-strange light baryons is presented in Table 1. The entries contain central values of masses as quoted in the Particle Data Tables, together with spin-parity assignments. By definition nucleons have isospin | , deltas have isospin | . We have included only resonances rated 2* or better. The first series assumes maximal alignment between orbital and spin angular momentum. For L = 0 there is a unique nucleon state, since (assuming spatial symmetry) spin symmetry and color antisymmetry imply flavor symmetry. For larger values of L there is both a good diquark and a bad diquark nucleon state. The latter is made by assembling the 1 = 1 bad diquark with the I = \ quark to make 1
333 Table 1. Fit to nucleon and delta resonances, based on the standard baryon body plan. I. M a x i m a l spin alignment for "good" and "bad" diquarks Angular Momentum 0 1 2 3 4 5 6 II.
A. (L)
[ud]—1
B.
T
(ud)—1
fr—T
iV(939) 1/2+ AT(1520) 3 / 2 " AT(1680) 5/2+ Af(2220) 9/2+ iV(2600) 1 1 / 2 iV(2700) 13/2+
A(1232) 3/2+ A(1950) A(2400) A(2420) A(2750) A(2950)
7/2+ 9/211/2+ 13/215/2+
AT(1675) 5 / 2 7V(1990) 7/2+ AT(2250) 1 1 / 2 "
"Bad" diquark w i t h net spin 1 anti-aligned and "good" diquark w i t h net spin 1 anti-aligned
Angular Momentum (L) 1 2
A.
B.
(ud)—1
fr—-i or < » — T
AT(1535) 1/27^(1720) 3/2+
3 4 III.
[ud]—1
— 4
A(1700) 3 / 2 A(1905) 5/2+ A(2000) 5/2+
JV(1700) 3/2~ N(2000) 5/2+ JV(2190) 7/2+
A(2300) 9/2+ "Bad" diquark w i t h net spin 2 anti-aligned
Angular Momentum (L) 1 2 3
A . (ud)—1 4-—T or «•—i A(1620) 1/2" iV(1650) l / 2 ~ A(1920) 3/2+ ./V(1900) 3/2+ iV(2200) 5 / 2 "
IV. "Bad" diquark w i t h net spin 3 anti-aligned Angular Momentum (L) 2 3
A.
(ud)—1
A(1910) 1/2+ AT(2080) 3 / 2 -
odd L. We will discuss a possible microphysical origin for this separation momentarily below, in a separate subsection. We will give less textual detail in describing the remaining series, since most of the required explanation is so similar. The second series includes cases where the spin and orbital angular momenta sum up to one less than the maximum possible J. It starts at L = 1. There is a unique good diquark nucleon series, corresponding to the second term in L®\
= {L+\)®{L~\),
(16)
334
Regge Trajectory for even-L Nucleons (series IA).
All Nucleons of series IA
S- .-
y' /
\
•
y
/ •
•
Nucleons Fitted line (E2=1.07*L + .781)
Angular Momentum (L)
Ajigular momentum (L)
(a)
(b)
Regge Trajectory for even-L Deltas (series IB).
r
Regge Trajectory for Lambdas (with [ud]—s).
J'''' y \
i
•
:
\
y
;
X
X
X
c
I
j y:
"m'
|
y'"''
y '
\.y/
|
!
•
•
\
i
Deltas Fitted line (E2=1.18*L + 1.429)
Angular Momentum (L)
(c)
I
Flttadhr.a(ea.1.08-L+1.2ll)|
Angular Momentum (L)
(d)
Figure 1. Various E2 vs L plots, (a) is a plot of all nucleons of series IA, showing "even-odd effect". (b-d) are plots of prominent Regge trajectories.
but two bad diquark series, corresponding to the second and third terms in
L®l®\
= {L + \)®{L + \)®{L + \)®{L-\)®(L-\)®{L-\)
(17)
(with of course the understanding that negative values are to be dropped, and that for L — 0 L + \ occurs only once). For L = 0 there is no clean separation of two ends, and hence no effective approximate isospin conservation to stabilize the bad diquark; so the absence of those states is not surprising. The only case where a doubling is apparent is for the | + A(1905), A(2000); we predict that there are many additional doublets yet
335
to be resolved. (In our fit to the meson sector, several more doublets of this kind do appear.) Beginning with the third series we should not, and do not, find a gooddiquark nucleon column. The fourth series is very poorly represented; this is not wholly unexpected, since it is predicted to start at L = 2. The surprising feebleness of spin-orbit forces manifests itself most abundantly for L = 2. We find two nearly degenerate good-diquark nucleons iV(1680),iV(1720) with Jp = | + , | + ; and a host of nearly degenerate bad-diquark nucleons and deltas: N(1990)l+, TV(2000)§+, N(1900)%+, + + + + A(1950)| , A(1905)§ , A(2000)§ , A(1920)| , A(1910)± + !
2.2.1. Even-Odd Effect and Tunneling We have mentioned that the even and odd L members of a sequence representing different rotational states of a bone with given internal quarkstructure can lie on different trajectories. A possible microphysical explanation for this is connected with the possibility of quark tunneling from one bone-end to the other. Imagining the bone in a fixed position, such tunneling produces the same effect as rotation through ir. We should construct internal spatial wave-functions which are symmetric or antisymmetric under this interchange. The former will be nodeless, and lower in energy than the latter, which have a node. The symmetric states will allow only even L, the antisymmetric states will allow only odd L. Thus if tunneling of this kind is significant we should expect an even-odd splitting the trajectory, with the odd component elevated. This is what we observe in the nucleon and delta trajectories. (For this and the subsequent related assertions, see Fig. 1). A larger effect might be expected for the trajectories with bad diquarks, since the ends won't be sticky. This too is what is is observed. In the A trajectory the dominant quark configuration has [ud] on one end and s on the other. It requires triple tunneling to mimic the effect of a 7r rotation. Thus we do not expect an even-odd effect here, and none is evident in the data, which has entries for L = 0 through 5.
2.3. Results
and
Conclusion
By comparing good nucleons with the corresponding bad nucleons and deltas, using Eqs. (12, 13, 14) we can get a more quantitative handle on the diquark mass differences. They begin as equations for differences between
336
the three-halves power of the masses. Prom the mass-difference between AT(1680) and A(1950) we find (udf2
oi/4
- [udf2 = — ( 1 . 9 5 0 - 1.680) = .28 GeV 3 / 2 .
(18)
K
From this, we see that (ud) — [ud] itself ranges from 360 to 240 MeV as [ud] ranges from 100 to 500 MeV. This constitutes a powerful indication of the importance of these diquark correlations, since such energies are quite large in the context of hadron physics. A similar comparison among hyperons involves £(2030) and £(1915) and leads to (us)3/2 - [usf2 = — ( 2 . 0 3 0 - 1.915) = .12 GeV 3 / 2 .
(19)
From this, we see that (us) — [us] itself ranges from 150 to 100 MeV as [us] ranges from 200 to 600 MeV. This is smaller than (ud) — [ud], as expected, but still a very significant energy. A more adventurous comparison is to mesons. Since the same sort of picture, with flux tubes joining weakly coupled ends and feeble spin-orbit forces, works very well for them too, we are encouraged by the data to compare diquark-quark to antiquark-quark configurations. (By the way, this baryon-meson parallelism poses a challenge for Skyrme model or large N approaches to modeling hadrons, since these approaches treat mesons and baryons on vastly different footings.) To be concrete, let us continue to consider orbital angular momentum L = 2 states with maximal spin and orbital alignment. They are as follows: • • • • • • • •
[ud] - u : JV(1680) (ud)-u: A(1950),iV(1990) [ud] - s : A(1820) [us] - u : £(1915) (us) - u : £(2030) s-u : #3(1780) J-u:/9(1690),w(1670) s-s: 0(1850)
Now a remarkable thing that appears here, upon comparing the first line with the seventh, or the third with the sixth, is that the mass of the good diquark [ud] is roughly the same as that of u itself! This comparison is somewhat contaminated by tunneling and mixing effects (e.g., tunneling induces mixing between [ud] — s and [us] — d), but it's a striking — and
337
by no means isolated — phenomenon that at large L, there is a marked convergence between mesons and baryons. Another interesting qualitative pattern is (ud) > [us] > s > [ud]. The near-equality between effective [ud] and u effective masses, inferred in this way, contrasts with what appears at low L, even for heavy quark systems, e.g. Ac(2625) versus Z?(2460) at L = 1 are split substantially. On the other hand, this difference of 165 MeV is far less than the conventional "constituent quark" mass ~ 300 MeV, and also far less than the 275 MeV difference between Ac(2285) and £>*(2010) at L = 0. (Note that heavyquark hadrons are only half as stretched as their light-quark analogues, for the same L, so L = 1 is ultra-minimal.) Part of the reason, I suspect, is that the stretched flux tubes we encounter at larger L can be terminated more smoothly on diquarks, which are extended objects, than on single quarks; this gives the diquarks an additional energetic advantage. Another part is simply that the c spin somewhat interferes with the [ud\ correlation, and spatial separation lessens this effect. Altogether, the concept of diquarks as objects appears to emerge quite naturally and inescapably as an organizing principle for hadron spectroscopy. As we examine it more carefully, we find that the energies in play are very significant quantitatively, and that several qualitative refinements with interesting physical interpretations suggest themselves. It would be wonderful to illuminate these effects further by numerical experiments in lattice gauge theory. The simplest way to see diquark dynamics is just to look at two quarks coupled to a static color source, and in this way to compare the energy of different spin configurations. It would be desirable to verify the strong dependence of the splitting on the quarks' masses. One could also study the diquark repulsion, by bringing together the static sources of two such source-diquark systems. Although it seems very difficult to simulate spinning systems using known techniques of lattice gauge theory, one could study quark and diquark systems "in isolation" (attached to a flux tube) by artificially introducing a position-dependent mass for the light quarks, that becomes large outside a pocket wherein it vanishes. This would, by pushing the quarks away from the source, mimic the effect of a centrifugal force. With insight gained from such studies, we would be empowered not only to connect the spectroscopic regularities to foundational QCD, but also to do better justice to the other fundamental dynamical questions that this circle of ideas wants to encompass.
338
Acknowledgments Whatever is new in this paper emerged from joint work with Alex Selem, of which a more complete description is in preparation. I have also greatly benefited from discussions and earlier collaborative work on related subjects with Robert Jaffe, and from ongoing conversations with Rich Brower and John Negele.
References 1. M. Neubert and B. Stech, Phys. Lett. B231, 477 (1989). 2. R. Jaffe, Phys. Rev. Lett. 38, 195 (1977); Erratum 38, 617 (1977). 3. A. Selem and F. Wilczek, Hadron Systematics, Diquark Correlations, and Exotics, in preparation. This paper includes many further details and references.
D E SITTER B R E A K I N G IN FIELD THEORY
R.P. WOODARD Department of Physics University of Florida Gainesville, FL 32611, USA E-mail:
woodardlSphys.ufl.edu
I argue against the widespread notion that manifest de Sitter invariance on the full de Sitter manifold is either useful or even attainable in gauge theories. Green's functions and propagators computed in a de Sitter invariant gauge are generally more complicated than in some noninvariant gauges. What is worse, solving the gauge-fixed field equations in a de Sitter invariant gauge generally leads to violations of the original, gauge invariant field equations. The most interesting free quantum field theories possess no normalizable, de Sitter invariant states. This precludes the existence of de Sitter invariant propagators. Even had such propagators existed, infrared divergent processes would still break de Sitter invariance.
1. Introduction Stanley Deser has been my mentor for over two decades. One of the many things he taught me is that theoretical physics is characterized by long periods of stagnation, punctuated by bursts of activity after some insight or technical advance makes progress possible. When this happens one has to push forward as far and as fast as possible because these opportunities don't arise often. Stanley's career has exemplified this, starting in the late 50's with the canonical formulation of gravity that his work with Arnowitt and Misner made possible.1"13 Another fine example is the way Stanley and various collaborators exploited the newly developed technology of dimensional regularization and the background field formalism in the mid 70's to analyze the one loop divergences of gravity combined with other theories. 14 " 17 Much of my recent work has dealt with exploiting a technical advance that has made it possible to get interesting results from quantum field theory during inflation. The advance is the development of relatively simple propagators for massless fields on a locally de Sitter background of arbitrary spacetime dimension. This has made it possible to use dimensional 339
340
regularization to go beyond the coincidence limits of one loop stress tensors — the technology for which had been codified before I graduated. 18 One can now get at the deeply nonlocal, ultraviolet finite parts of quantum processes during inflation. Section 2 of this article reviews what has been done. Section 3 explains why some of my methods offend the aesthetic prejudices of the mathematically minded. However much more attractive the formalism might seem their way, it would be neither practical, nor physically correct, nor would its most interesting predictions be free of the unaesthetic properties of my techniques. The various problems of practicality and of principle are described in section 4. Section 5 summarizes my conclusions. 2. Quantum Field Theory During Inflation I model inflation using a portion of the full de Sitter manifold known as the open conformal coordinate patch. If the D-dimensional cosmological constant is A = (D — 1)H2, the invariant element is, ds2 = a2 (-dr]2 + dx • dx J,
where
a(r]) = ——— .
(1)
The conformal time 77 runs from —00 to zero. The various propagators have simple expressions in terms of the following function of the invariant length t(x\x') between x*1 and x'M, y{x;x') = 4sm2(^He(x;x')')
= aa'H2(\\x-x'\\2-{\r]-r]'\-id)2')
.
(2)
One might expect that the inflationary expansion of this spacetime makes quantum effects stronger by allowing virtual particles to persist longer than in flat space. Indeed, it is simple to see that any sufficiently long wavelength (A > 1/H) virtual particle which is massless on the Hubble scale can exist forever.19 However, one must also consider the rate at which virtual particles emerge from the vacuum. Classical conformal invariance causes this rate to fall off exponentially, so any long wave length virtual particles which emerge become real, but very few emerge. 19 To get enhanced quantum effects during inflation requires quanta which are effectively massless and also not conformally invariant. Even one such particle can catalyze processes involving conformally invariant particles. It has long been known how to write the propagator for a massless, conformally coupled scalar in arbitrary dimension, 18
""^-(S^V 1 )® '
(3
>
341
Massless fermions are also conformally invariant in any dimension and their propagator is closely related, i&j
(x;x') = (aa') — id^j (aa')T-HAc!(x;x')
.
(4)
One can compute with these propagators but the results are not much different from flat space on account of conformal invariance. The advance that has made interesting quantum effects computable is explicit expressions for the propagators of particles which are massless and not conformally invariant. The first of these is the minimally coupled scalar, iAA(x;x') = iAcf(x;x') HD-2T(D-l)j
D T2(f) /4\f-2
HD-^fir(n+D-l)(yy ( 4 7 r ) f ^ \ n r ( n + f ) U)
/7T x , , , / |
1 r ( n + f + 1) m » - g + 2 ) n - f + 2 T(n+2) UJ J '
W
This might seem a daunting expression but it isn't so bad because the infinite sum on the final line vanishes in D = 4, and each term in the series goes like positive powers of y(x;x'). This means the infinite sum can only contribute when multiplied by a divergent term, and even then only a small number of terms can contribute. Fascinating physics has been revealed by endowing such a scalar with different sorts of interactions. When a quartic self-interaction is present one can compute the VEV of the stress tensor 20 ' 21 and the scalar selfmass-squared 22 at one and two loop orders. The resulting model shows a violation of the weak energy condition — on cosmological scales! — in which inflationary particle production drives the scalar up its potential and induces a curious sort of time-dependent mass. When a complex scalar of this type is coupled to electromagnetism is has been possible to compute the one loop vacuum polarization 23,24 and use the result to solve the quantum corrected Maxwell equations. 25 Although photon creation is suppressed during inflation, this model shows a vast enhancement of the 0-point energy of super-horizon photons which may serve to seed cosmological magnetic fields.26,27,28 Finally, when a real scalar of this type is Yukawa coupled to a massless Dirac fermion it has been possible to compute the one loop fermion self-energy and use it to solve the quantum corrected Dirac equation. 29 The resulting model shows explosive creation of fermions which should make inflation end with the super-horizon modes in a degenerate Fermi gas!
342
Electromagnetism is a special case, being conformally invariant in D — 4 but not generally. My favorite gauge fixing term is an analogue of the one introduced by Feynman in flat space, - (D-4)HaA0)2
CGF = -^-'(rTAw
.
(6)
Because space and time components are treated differently it is useful to have an expression for the purely spatial part of the Minkowski metric, %V=V^+^V.
(7)
In this gauge the photon propagator takes the form, i
MAyj
(x; x') = rj^aa'iABix;
x') - Sffiaa'iAcix;
x') .
(8)
The B-type and C-type propagators are,
r(n+2) u)
y
[
'
j A c ( I ; I ^, A r f ( « 0 + ^g{ ( n + 1 ) a^) ( f ) » As with the A-type propagator (5), the infinite sums in (9) and (10) vanish in D = 4. In fact the -B-type and C type propagators agree in D — 4, and the photon propagator is the same for D = 4 as it is in flat space! No results have been published using the photon propagator but L. D. Duffy and I are computing the one loop scalar self-mass-squared in scalar QED. We expect its secular growth to eventually choke off the inflationary particle production that so enhances the one loop vacuum polarization. 25 Of course this can't eliminate scalars which have already been ripped out of the vacuum, or the vacuum polarization they induce. A similar computation of the scalar self-mass-squared of the Yukawa scalar fails to show any secular growth at one loop order, 30 implying that the scalar-catalyzed production of super-horizon fermions goes to completion.29 Gravitons are also massless without conformal invariance. I define the graviton field ^^{x) as follows, 9ij,u(x)=a2(r]llI/
+ Kiplxl/(x)),
where
K2 = WirG.
(11)
343
My favorite gauge fixing term is an analogue of the de Donder term used in flat space, 31 CGF = ~ n
f l
-
2
f V , , Fp = vpa(^P,a
(D-2)Ha^X)(12) With these definitions the graviton propagator takes the form of a sum of three constant index factors times the three scalar propagators, pv^pa
~ \l>Pa,p +
(*;*')= £ [
V-v-1 per
iA](x;x')
(13)
I=A,B,C
The index factors are, iv1
rpA pa
P-V-L pa
IV-1- i
pa
=
2r
)p(pVa)v
D-3
Vp„Vp
-4fe(X) • (D-2)(D-3)
( I > - 3 ) ^ + V (D-3)5X + Vpa
(14) (15) .(16)
The full power of the dimensionally regulated graviton propagator has not so far been exploited in published work. However, the one loop graviton self-energy32 has been computed using a D = 4 cutoff. The expectation value of the invariant element has also been obtained at two loop order. 33 These results indicate that the back-reaction from graviton production slows inflation by an amount which eventually becomes nonperturbatively large. 34 N. C. Tsamis and I have used the dimensionally regulated formalism to compute the expectation value of the metric at one loop order. E. O. Kahya and I are also using it to compute the one loop scalar selfmass-squared induced by graviton exchange. This might have important consequences for models which inflate for a very large number of e-foldings. 3. What Bothers People Despite all the results that have been obtained, and the ones which are attainable, the response of the theoretical physics community has been — underwhelming. Different segments of the community have different reasons for ignoring my work. Many inflationary cosmologists feel that causality precludes interesting quantum field theoretic effects. Some of them even seem to have forgotten that the density perturbations which figure so prominently in recent observation 35 ' 36 are driven by precisely the same inflationary particle production 37,38 that underlies each of the effects reported in the previous section! String theorists are not much interested
344
in physics that doesn't make essential use of their candidate for a theory of everything. They also flirt with the notion that there are no observables in de Sitter, which requires them to disbelieve that quantum corrections to the field equations mean anything. Loop space gravity people have trouble achieving correspondence with most forms of perturbation theory, including mine. And phenomenologists seek to work out the consequences of popular theories, so the fact that few people pay attention to my work serves to justify continuing to ignore it! There isn't much I can do about this. But I could converse with one segment of the community if only it was possible to overcome the distaste its members have for the methods I use. I refer to the mathematical relativists. They are prepared to accept that quantum field theory might have interesting effects during inflation, and that these can be quantified in a reliable way. They are even willing to let me use Minkowski-signature perturbation theory starting with a prepared initial state! However, they are strongly attracted by the analogy between Minkowski space and de Sitter space, the maximally symmetric solutions of Einstein's equations for A = 0 and A > 0, respectively. They feel that manifest de Sitter invariance on the full de Sitter manifold should be as powerful an organizing principle for quantum field theory with A > 0 as Poincare invariance has been for A = 0. So it bothers them that my open conformal coordinate patch (1) does not cover the full de Sitter manifold and that the gauge fixing terms I use — (6) and (12) — are not de Sitter invariant.
4. You Can't Always Get What You Want Mick Jagger and Keith Richards are not my favorite authorities on much of anything, but one of their songs seems relevant here. I will argue that it isn't necessary, convenient or even possible to impose de Sitter invariant gauges and work on the full manifold. Nor would doing so lead to de Sitter invariant results for the most interesting processes if it were possible. Necessity is the simplest issue. Everyone understands that it isn't necessary to use de Sitter invariant gauges, just as it isn't necessary to use Poincare invariant gauges in flat space. Nor is there any logical problem with restricting physics to the open conformal coordinate patch (1), especially if one contemplates releasing a prepared state from a finite initial time. The condition 77 = constant defines a perfectly good Cauchy surface. Information from the rest of the full de Sitter manifold can only propagate to the future of such a surface by passing through it as part of the initial
345
condition. Indeed, the case for restricting to (1) can be put much more strongly if one imagines — as I do — the local de Sitter background as merely a model for the more complicated geometry of the inflating epoch of cosmology. I am not interested in quantum field theory on perfect de Sitter space but rather in potentially observable quantum phenomena from the epoch of primordial inflation. In that case the relevant symmetries are homogeneity and isotropy, not the full de Sitter group, and the conformal coordinate patch — with arbitrary a(rj) — is the coordinate system in which these symmetries are manifest. The reason people typically prefer to maintain manifest Poincare invariance in flat space is that it makes things simpler. That this is not true for de Sitter can be seen by comparing propagators in my gauges with those in the simplest de Sitter invariant gauges. It will sharpen the distinction if we take D = 4. In that case the photon propagator in my gauge (6) is the same function of conformal coordinates as it is in flat space, Au
(:r; a/)
"Qiiv
£>=4
(17)
47r 2 Ax 2
where Ax = ||x — x'|| 2 — (|7y — r\'\ — i5)2. It is worth pointing out that this expression applies to any homogeneous and isotropic geometry in conformal coordinates, not just the special case of de Sitter. The simplest de Sitter invariant photon propagator of which I know was obtained by Allen and Jacobson 39 with the gauge fixing term,
A„v = -\(
~aD-4(r,^A^~(D-2)HaA0f
. (18)
Their propagator takes the form, i
M A„j
where
(x; x')
y(x;x')
= a{y) yg„ (x; x') + p(y) [Mnj (x; x') n„] (x; x') , aa'H2A2
(19)
^gv (x;x') is the parallel transport matrix
M and L n x;x') and n„ (x;x') are the gradients with respect to x and w x of the geodesic length. In D = 4 the coefficient functions are,
a{y) =
ML 4TT2
{\+
(4-§i/) ln
4-2/ + ' (4-y) 2
(!)}•
(20)
346
The 3 + 1 decomposition of the parallel transport matrix is, ^gv = aa'
1 0 \ 0 SmnJ
2 / -(a+a1)2 4 - j / \-{a+a')aa'HAxm
(a+a')aa'HAxn\ a2a'2H2AxmAxn) (22)
The other tensor has the following 34-1 decomposition _1 /aa'y+2a2 + 2a'2 ~ ~y\ 2aa'2HAxm 4 y{A-y)
-2a2a'HAxn 0
/ (a+a')2 \(a+a')aa'HAxm
-{a+a')aa'HAxn a2a'2H2AxmAxn
However much one may admire manifest de Sitter invariance, I hope we can all agree that it doesn't simplify propagators. But suppose you are fanatical about de Sitter invariance and you prefer to compute on the full de Sitter manifold in a gauge which is manifestly de Sitter invariant, no matter how much harder it is. In that case you risk violating the invariant equations of motion! The problem arises from combining the causal properties of de Sitter with the constraint equations of any gauge theory. Before gauge fixing the constraint equations are elliptic, and they typically result in a nonzero response to sources throughout the de Sitter manifold, even in regions which are not future-related to the source. But gauge fixing in a de Sitter invariant manner results in hyperbolic equations for which the response to sources is zero for regions which are not future-related to the source. As far as I know this problem was first noted by Penrose 40 in 1963. Tsamis and I encountered it for gravity in 1994 31 and recent studies for electromagnetism have been conducted by Bicak and Krtous. 41 To better understand the problem let us adopt the standard closed coordinatization of the full de Sitter manifold, ds2 = -dt2+H~2
cosh2(Ht) (dX2+sin2(X)d62+sin2(x)
sin 2 (0)d0 2 ) . (24)
Consider the invariant Maxwell equations for a pair of oppositely charged point particles,
d^V^g^g^F^
=q J dr[zl(r)S4(x~z+(r))
-
Z»(T)64(X-Z-(T
(25) When the +q charge is stationary at x — 0 and t n e ~Q charge is stationary at x — n a perfectly good solution exists, Ap = 5lA0(t, x) = ^sech(Ht)
cot( X ) .
(26)
347
Suppose we try to find a gauge parameter 6{x) such that the transformed field, A' — Afj, — d^O obeys the de Sitter invariant condition •A'M;M = 0, dJ^Tt
= dJ^
= - ^ sinh(2tfi)^|^
s i n (0)
=5(x). (27)
You might think this is easy with a Green's function, dM (y=ggPvdvG(x\
x'j\ =54(x-
x')
9(x) = f d4x'G(x;
=>
x')S(x'). (28)
However, the retarded Green's function, CT*(x;x') = ^
A t )
[25{y(x;x'))+e(-y(x;x'))]
,
(29)
contains a ^-function tail term which is nonzero throughout the volume of the past light-cone. Because the source (27) actually grows as t —> — oo, the integral (28), and even its gradient, fail to converge. Note that the electric field of (26) points from the +q to the — q charge and is nonzero throughout the full de Sitter manifold manifold, F*° = ^ - s e c h 3 ( # t ) c s c 2 ( x )
(30)
47T
This isn't at all what one gets by integrating the photon retarded Green's function against the current density in a de Sitter invariant gauge,
if{x) = Jd'x'
Grel {x;x')Jv{x')
.
(31)
One can recover the retarded Green's function from the Allen-Jacobson propagator (19) by simply taking the imaginary part and multiplying by -26(t-t'), ret ,- G v
<^qM{ 47r M(s) (_ ~^"'_<^ ( 4 - y )9H/) } W(l;I . 2
H26(At)
/
f j/
^{{^yfd{~y)} H ^ H ^ -
(32)
The retarded Green's function is causal, so the response from it vanishes in the vast region of the full de Sitter manifold which is not future-related to either of the source world lines. It turns out that the Allen-Jacobson Green's function does give the correct response within the open conformal coordinate patch, so a de Sitter invariant gauge can at least be imposed locally in electromagnetism. (I
348
thank A. O. Barvinsky for correcting me about this, and I apologize to Allen and Jacobson for having said otherwise at the Deserfest.) The same does not seem to be true in gravity. Antoniadis and Mottola have shown that de Sitter invariant graviton propagators — which are also much more complicated than the one in my favorite gauge (12) — lead to local violations of the linearized Einstein equations! 42 These violations are not present when using my non-invariant propagator (13). 31 Note that the problem reconciling causality and the constraints is classical. I advance for your consideration the folly of working much harder to quantize a formalism that doesn't even correctly reflect classical physics. When confronted with the causality obstacle de Sitter fanatics sometimes respond that the problem arises from the constraints not having been imposed throughout the initial value surface. When this is done the full system can be evolved just fine. I don't dispute this but it misses the point. The issue is not whether physics can be done on the full de Sitter manifold. There was never any doubt about that: (26) is the instantaneous Coulomb potential of Coulomb gauge. The issue is rather whether or not physics can be done maintaining manifest de Sitter invariance. The answer is no. Imposing the constraints can always be subsumed into adding a surface gauge condition that breaks manifest de Sitter invariance. Moving from classical to quantum field theory, recall that the condition for getting enhanced quantum effects during inflation is massless particles which are not classically conformally invariant. There are two such particles: the massless, minimally coupled scalar and the graviton. Consideration of these particles is the only phenomenological justification for studying quantum field theory during inflation, so we cannot dismiss them if they happen to violate aesthetic prejudices. As it happens, the free quantum field theory of neither system possesses a normalizable, de Sitter invariant wave functional. This was proved long ago for the massless, minimally coupled scalar by Allen and Folacci.43 It can be seen for my graviton propagator (13) by simply performing a naive de Sitter transformation coupled with the compensating gauge transformation needed to restore my noninvariant gauge (12). 44 Contrary assertions for gravitons are always based upon using de Sitter invariant gauges on the full manifold, which I have just shown to be incorrect. The fact that the most interesting free quantum field theories have no de Sitter invariant states means that the propagators of these fields must break de Sitter invariance, not just through the gauge fixing function but in a fundamental way. One can see this in the factor of ln(aa') on the
349
second line of expression (5) for iA^(:r; x'). There is no sense complicating a marginally tractable formalism to respect a symmetry which is not there. My final point is that even if manifestly de Sitter invariant propagators had existed, the most interesting interactions would still break de Sitter invariance. I don't mean the interaction vertices would be noninvariant. They are manifestly invariant. What I mean instead is that higher order processes can involve integrals over interaction vertices. De Sitter invariant propagators would make the integrands invariant but would not guarantee that the integrals were invariant. Consider the invariant volume of the past light-cone from some observation point xM back to the initial value surface (IVS) on which the prepared state was released, V(x) = f dAx' V - f f ( ^ ) 0(At)0(-y(x; x')) . (33) v Jivs ' The integrand is manifestly de Sitter invariant — one inside the past lightcone and zero outside — but the integral grows as the observation point is taken later and later after the state was released. One can see from the scalar retarded Green's function (29) that this example is not artificial. Unsuppressed integrals over the volume of the past light-cone occur in many of these computations. 20 They give factors of ln(a) every bit as important as the explicit ones from the de Sitter breaking terms of iA^(x;a;'). It would not be far wrong to say that extracting these secular logarithms is the whole point of studying quantum field theory during inflation. 5. Conclusions Massless particles which are not conformally invariant can mediate interesting quantum effects during inflation. Even a single non-conformal massless particle can catalyze surprising processes which would otherwise not go. 19 It is now possible to study this by modeling inflation as the open coordinate patch of de Sitter space, and by exploiting simple gauge fixing terms. This bothers de Sitter fanatics, who would prefer to work on the entire de Sitter manifold and to employ only de Sitter invariant gauge conditions. That would not be easy because de Sitter invariant gauges complicate propagators. It is also incorrect physically because a de Sitter invariant gauge converts elliptic constraint equations into hyperbolic evolution equations. Whereas the former require a nonzero response throughout the manifold to a sufficiently distributed source, the later give zero response in the vast regions of de Sitter space which are not future-related to a source worldline. In some cases the use of a de Sitter invariant gauge even leads to violations
350
of the original, gauge invariant field equations within the region which is future-related to the a source worldline! 42 T h e free q u a n t u m field theories of the two massless particles which are not conformally invariant admit no de Sitter invariant states. This means t h a t the propagators of these fields cannot be de Sitter invariant. 4 3 , 4 4 Even had all propagators been de Sitter invariant, interactions would still break this invariance. T h e infrared logarithms which signal this breaking are at the heart of what makes q u a n t u m field theory during inflation potentially observable. I think we'd all rather have interesting q u a n t u m dynamics without symmetry t h a n sterile dynamics with a beautiful symmetry. As Mick and Keith put it: "You can't always get w h a t you want. B u t if you t r y sometime you find, you get what you need!" Acknowledgments It is a pleasure to acknowledge discussions and work on this problem with A. O. Barvinsky, T. Jacobson, E. O. Kahya, N. C. Tsamis and S. H. Yun. This work was supported by N S F Grant PHY-0244714 and by the Institute for Fundamental Theory at the University of Florida. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
R. Arnowitt and S. Deser, Phys. Rev. 113, 745 (1959). R. Arnowitt, S. Deser and C.W. Misner, Phys. Rev. 116, 1322 (1959). R. Arnowitt, S. Deser and C.W. Misner, Phys. Rev. 117, 1595 (I960). R. Arnowitt, S. Deser and C.W. Misner, Nuov. Cim. 15, 487 (1960). R. Arnowitt, S. Deser and C.W. Misner, Phys. Rev. Lett. 4, 375 (1960). R. Arnowitt, S. Deser and C.W. Misner, Phys. Rev. 118, 1100 (1960). R. Arnowitt, S. Deser and C.W. Misner, J. Math. Phys. 1, 434 (1960). R. Arnowitt, S. Deser and C.W. Misner, Phys. Rev. 120, 313 (1960). R. Arnowitt, S. Deser and C.W. Misner, Phys. Rev. 120, 321 (I960). R. Arnowitt, S. Deser and C.W. Misner, Ann. Phys. 11, 116 (1960). R. Arnowitt, S. Deser and C.W. Misner, Nuov. Cim. 19, 668 (1961). R. Arnowitt, S. Deser and C.W. Misner, Phys. Rev. 121, 1556 (1961). R. Arnowitt, S. Deser and C.W. Misner, Phys. Rev. 122, 997 (1961). S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 1 0 , 401 (1974). S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 1 0 , 411 (1974). S. Deser, H.-S. Tsao and P. van Nieuwenhuizen, Phys. Lett. B50, 491 (1974). S. Deser, H.-S. Tsao and P. van Nieuwenhuizen, Phys. Rev. D10, 3337 (1974). N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982). 19. R.P. Woodard,Quantum Effects during Inflation, astro-ph/0310757. 20. V.K. Onemli and R.P. Woodard, Class. Quant. Grav. 19, 4607 (2002) [grqc/0204065].
351 21. V.K. Onemli and R.P. Woodard, Quantum effects can render w < — 1 on cosmological scales, gr-qc/0406098. 22. T. Brunier, V.K. Onemli and R.P. Woodard, Two Loop Scalar Self-Mass during Inflation, University of Florida preprint UFIFT-QG-04-5, in preparation. 23. T. Prokopec, O. Tornkvist and R.P. Woodard, Phys. Rev. Lett. 89, 101301 (2002) [astro-ph/0205331]. 24. T. Prokopec, O. Tornkvist and R.P. Woodard, Ann. Phys. 303, 251 (2003) [gr-qc/0205130]. 25. T. Prokopec and R.P. Woodard, Ann. Phys. 312, 1 (2204 [astro-ph/0310757]. 26. A.C. Davis, K. Dimopoulos, T. Prokopec and O. Tornkvist, Phys. Lett. B501, 165 (2001) [astro-ph/0007214]. 27. K. Dimopoulos, T. Prokopec, O. Tornkvist and A.C. Davis, Phys. Rev. D 6 5 , 063505 (2002) [astro-ph/0108093]. 28. T. Prokopec and R.P. Woodard, Am. J. of Phys. 72, 60 (2004) [astroph/0303358]. 29. T. Prokopec and R.P. Woodard, JEEP 0310, 059 (2003) [astro-ph/0309593]. 30. L.D. Duffy and R.P. Woodard, Yukawa Scalar Self-Mass on a Conformally Flat Background, University of Florida preprint UFIFT-QG-04-3, in preparation. 31. N.C. Tsamis and R.P. Woodard, Commun. Math. Phys. 162, 217 (1994). 32. N.C. Tsamis and R.P. Woodard, Phys. Rev. D 5 4 2621 (1996) [hepph/9602317]. 33. N.C. Tsamis and R.P. Woodard, Ann. Phys. 253, 1 (1997) [hep-ph/9602316]. 34. N.C. Tsamis and R.P. Woodard, Nucl. Phys. B474, 235 (1996) [hepph/9602315]. 35. D.N. Spergel et al., Astrophys. J. Suppl. 148, 175 (2003) [astro-ph/0302209]. 36. H.V. Peiris et al., Astrophys. J. Suppl. 148, 213 (2003) [astro-ph/0302225]. 37. A.A. Starobinskii, JETP Lett. 30, 682 (1979). 38. V.F. Mukhanov and G.V. Chibisov, JETP Lett. 33, 532 (1981). 39. B. Allen and T. Jacobson, Commun. Math. Phys. 103, 669 (1986). 40. R. Penrose, in Relativity, Groups and Topology, Les Houches 1963, eds. C. DeWitt and B. DeWitt (Gordon and Breach, New York, 1964). 41. J. Bicak and P. Krtous, Phys. Rev. D64, 124020 (2001) [gr-qc/0107078]. 42. I. Antoniadis and E. Mottola, J. Math. Phys. 32, 1037 (1991). 43. B. Allen and A. Folacci, Phys. Rev. D35, 3771 (1987). 44. G. Kleppe, Phys. Lett. B317, 305 (1993).
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PUBLICATIONS OF STANLEY D E S E R
Publications 1. Relativistic Two-Body Interactions, Ph.D. Thesis (1953). 2. Covariant Meson-Nucleon Equation (with P.C. Martin), Phys. Rev. 90, 1075 (1953). 3. Solutions of Meson-Nucleon Equation in Adiabatic Limit (with R. Arnowitt), Phys. Rev. 92, 1061 (1953). 4. Electromagnetic Effects in Two-Nucleon Systems, Phys. Rev. 92, 1542 (1953). 5. Radiative Effect in Meson-Nucleon Scattering, Phys. Rev. 93, 612 (1954). 6. Virtuai States of Two-Meson Systems (with R. Arnowitt), Phys. Rev. 94, 798 (1954). 7. Low Energy Limits and Renormalization in Meson Theory (with M.L. Goldberger and W. Thirring), Phys. Rev. 94, 711 (1954). 8. Energy Level Displacement in u.-mesic Atoms (with M.L. Goldberger, W. Thirring, and Baumann), Phys. Rev. 96, 774 (1954). 9. High Energy Multiple Photon Production (with R. Arnowitt), Nuovo Cim. 2, 707 (1955). 10. Functional Integrals and Adiabatic Limits in Field Theory, Phys. Rev. 99, 325 (1955). 11. Renormalization of Derivative Coupling Theories (with R. Arnowitt), Phys. Rev. 100, 349 (1955). 12. General Relativity and the Divergence Problem in Quantum Field Theory, Rev. Mod. Phys. 29, 417 (1957). 13. Structure of Vertex Function (with W. Gilbert and G. Sudarshan), Phys. Rev. 115, 731 (1959). 14. Structure of Forward Scattering Amplitude (with W. Gilbert and G. Sudarshan), Phys. Rev. 117, 266 (1960). 15. Integral Representation of Two-Point Functions (with W. Gilbert and G. Sudarshan), Phys. Rev. 117, 273 (1960). 16. Quantum Theory of Gravitation I—Linearized Theory (with R. Arnowitt), Phys. Rev. 113, 745 (1959). 17. Dynamics of General Relativity, in Les Theories Relativistes de la Gravitation (CNRS, Paris, 1959). 18. Dynamical Structure and Definition of Energy in General Relativity (with R. Arnowitt and C.W. Misner), Phys. Rev. 116, 1322 (1959). 19. Consistency of Canonical Reduction of General Relativity (with R. Arnowitt and C.W. Misner), J. Math. Phys. 1, 434 (1960). 353
354
20. Gravitational-Electromagnetic Coupling and the Classical Self-Energy Problem (with R. Arnowitt and C.W. Misner), Phys. Rev. 120, 313 (1960). 21. Interior Schwarzchild Solutions and Interpretation of Sources in General Relativity (with R. Arnowitt and C.W. Misner), Phys. Rev. 120, 321 (1960). 22. JVote on Positive-Definiteness of Energy of the Gravitational Field (with R. Arnowitt and C.W. Misner), Ann. Phys. 11, 116 (1960). 23. Canonical Variables for General Relativity (with R. Arnowitt and C.W. Misner), Phys. Rev. 117, 1532 (1960). 24. Energy and the Criteria for Radiation in General Relativity (with R. Arnowitt and C.W. Misner), Phys. Rev. 118, 1100 (1960). 25. Canonical Variables, Energy and Criteria for Radiation in General Relativity (with R. Arnowitt and C.W. Misner), Nuovo Cim. 15, 487 (1960). 26. Finite Self-Energy of Classical Point Particles (with R. Arnowitt and C.W. Misner), Phys. Rev. Lett. 4, 525 (1960). 27. Gravitational-Electromagnetic Coupling and the Classical Self-Energy Problem (with R. Arnowitt and C.W. Misner), Phys. Rev. 120, 321 (1960). 28. The Wave Zone in General Relativity (with R. Arnowitt and C.W. Misner), Phys. Rev. 121, 1556 (1961). 29. Heisenberg Representation in Classical General Relativity (with R. Arnowitt and C.W. Misner), Nuovo Cim. 19, 668 (1961). 30. Coordinate Invariance and Energy Expressions in General Relativity (with R. Arnowitt and C.W. Misner), Phys. Rev. 122, 997 (1961). 31. Canonical Variables, Energy and Radiation in General Relativity (with R. Arnowitt and C.W. Misner), in General Relativity and Gravitational Waves (Interscience, 1961). 32. The Spin-Statistics Theorem (with R. Arnowitt), J. Math. Phys. 3, 637 (1962). 33. The Dynamics of General Relativity (with R. Arnowitt and C.W. Misner) in Gravitation: An Introduction to Current Research, ed. L. Witten (Wiley, NY, 1962). 34. On Moller's Gravitational Stress-Tensor, Phys. Lett. 7, 42 (1963). 35. Proprietes Asymptotiques du Champ Gravitationel, Cahiers de Phys. 157, 374 (1963). 36. Formalisme Canonique et Quantization en Relativite Generale, Cahiers de Phys. 157, 357 (1963). 37. Note on Uniqueness of Canonical Commutation Relations (with R. Arnowitt), J. Math. Phys. 4, 615 (1963). 38. Conditions for Flatness in General Relativity (with R. Arnowitt), Ann. Phys. 23, 318 (1963). 39. Interaction Among Gauge Vector Fields (with R. Arnowitt), Nucl. Phys. 49, 133 (1963). 40. Ambiguity of Harmonic Oscillator Commutation Relations (with D. Boulware), Nuovo Cim. 30, 23 (1963). 41. External Sources in Gauge Theories (with D. Boulware), Nuovo Cim. 30, 1009 (1963). 42. Renormalization Group and Regge Behavior (with G. Purlan and G. Ma-
355
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311. New Results in D = 11 Supergravity, in Particles, Strings and Cosmology, p. 821, ed. P. Nath (World Scientific, 1999). 312. 'Good Propagation' and Duality Invariance Constraints on Scalar, Gauge Vector and Gravity Actions (with O. Sarioglu and J. McCarthy), Class. Quant. Grav. 16, 841 (1999) [hep-th/9809153]. 313. Counterterms/M-Theory Corrections to D = 11 Supergravity (with D. Seminara), Phys. Rev. Lett. 82, 2435 (1999) [hep-th/9812136]. 314. The Immortal Bel-Robinson Tensor, in Relativity and Gravitation in General, p. 35, eds. F. Atrio-Barandela and J. Martin (World Scientific, 1999) [gr-qc/9901007]. 315. Higher Derivative Chern-Simons Extensions (with R. Jackiw), Phys. Lett. B 4 5 1 , 73 (1999) [hep-th/9901125]. 316. Graviton-Graviton Scattering, Bel-Robinson and Energy (Pseudo)-Tensors (with J. Franklin and D. Seminara), Class. Quant. Grav. 16, 2815 (1999) [gr-qc/9905021]. 317. Nonrenormalizability of (Last Hope) D = 11 Supergravity, with a Terse Survey of Divergences in Quantum Gravities, in Fundamental Interactions: From Symmetries to Black Holes, eds. J.M. Frere, M. Henneaux, A. Sevrin and Ph. Spindel (Presses Univ. de Belgique, Brussels, 1999) [hepth/9905017]. 318. Two Outcomes for Two (Super-)Problems, in The Many Faces of Superworld, ed. M. Shifman (World Scientific, 1999) [hep-th/9906178]. 319. Infinities in Quantum Gravities, Annalen Phys. 11, 5 (2000). 320. Conformal Anomalies Revisited: Closed Form Effective Actions in D > 4, Nucl. Phys. B 8 8 (Proc. Suppl.) (2000). 321. Closed Form Effective Conformal Anomaly Actions in D > A, Phys. Lett. B479, 315 (2000) [hep-th/9911129]. 322. Tree Amplitudes and Two-Loop Counterterms in D = 11 Supergravity (with D. Seminara), Phys. Rev. D 6 2 , 084020 (2000) [hepth/0002241]. 323. Massive Spin 3/2 Electrodynamics (with A. Waldron and V. Pascalutsa), Phys. Rev. D 6 2 , 105031 (2000) [hep-th/0002241]. 324. Clebsch (String) Parametrization of 3-Vectors and Their Actions (with R. Jackiw and A. Polychronakos), Phys. Lett. A279, 151 (2001). 325. (Dis)continuities of Massless Limits in Spin 3/2-mediated Interactions and Cosmological Supergravity (with A. Waldron), Phys. Lett. B 5 0 1 , 134 (2001) [hep-th/0012014]. 326. Gauge Invariances and Phases of Massive Higher Spins in (A)dS (with A. Waldron), Phys. Rev. Lett. 87, 031601 (2001) [hep-th/0102166]. 327. Stability of Massive Cosmological Gravitons (with A. Waldron), Phys. Lett. B508, 347 (2001) [hep-th/0103255]. 328. Partial Masslessness of Higher Spins in (A)dS (with A. Waldron), Nucl. Phys. B607, 577 (2001) [hep-th/0103198]. 329. Null Propagation of Partially Massless Higher Spins in (A)dS and Cosmological Constant Speculations (with A. Waldron), Phys. Lett. B 5 1 3 , 137 (2001) [hep-th/0105181]. 330. A Century of Gravity: 1901-2000 (plus some 2001), in 2001: A Spacetime
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Odyssey, Proceedings of the Inaugural Conference of the Michigan Center for Theoretical Physics, Ann Arbor, Michigan, 21-25 May 2001, eds. Michael J. Duff and James T. Liu (World Scientific, 2002) [hep-th/0110027]. Newtonian Counterparts of Spin 2 Massless Discontinuities (with B. Tekin), Class. Quant. Grav. 18, L171 (2001) [gr-qc/0109091]. Inconsistencies of Massive Charged Gravitating Higher Spins (with A. Waldron), Nucl. Phys. B631, 369 (2002) [hep-th/0112182]. New Properties of Matter in (A)dS and their Consequences, Invited Lecture, Journees Relativistes, Dublin, Int. J. Mod. Phys. A17, 2645 (2002) [grqc/0201085]. Gauge Invariance with Mass: Higher Spins in Cosmological Spaces, in Strings and Gravity: Tying the Forces Together, Proceedings of the Fifth Franqui Colloquium, Brussels, 10-21 October 2001, eds. Marc Henneaux and Alexander Sevrin (De Boeck & Larcier, 2003) [gr-qc/0202053]. Massive, Topologically Massive, Models (with B. Tekin), Class. Quant. Grav. 19, L97 (2002) [hep-th/0203273]. Gravitational Energy in Quadratic-Curvature Gravities (with B. Tekin), Phys. Rev. Lett. 89, 101101 (2002) [hep-th/0205318]. Large Gauge Invariance and Non-Abelian Finite Temperature Effective Actions (with L. Griguolo and D. Seminara), Phys. Rev. D67, 065016 (2003) [hep-th/0212140], Energy in Generic Higher Curvature Gravity Theories (with Bayram Tekin), Phys. Rev. D67, 084009 (2003) [hep-th/0212292]. Arbitrary Spin Representations in deSitter from dS/CFT with Applications to dS Supergravity (with A. Waldron), Nucl. Phys. B662, 379 (2003) [hepth/0301068]. Some Remarks on Dirac's Contributions to General Relativity, Proceedings, Dirac Centennial, Florida State University, 2002, gr-qc/0301097. Energy in Topologically Massive Gravity (with B. Tekin), Class. Quant. Grav. 20, L259 (2003) [gr-qc/0307073]. Shortcuts to High Symmetry Solutions in Gravitational Theories (with B. Tekin), Class. Quant. Grav. 20, 4877 (2003) [gr-qc/0306114]. Novel Properties of Massive Higher Spin Fields, in Renormalization Group and Anomalies in Gravity and Cosmology, eds. M. Asorey and I.L. Shapiro, Ouro Preto, Brazil, March 17-23, 2003; Nucl. Phys. B 1 2 7 (Proc. Suppl.) (2004) 36 [gr-qc/0306048]. A Note on Matter Superenergy Tensors, Class. Quant. Grav. 20, L213 (2003) [gr-qc/0304095]. Stress and Strain: TM„ of Higher Spin Gauge Fields, Johns Hopkins Workshop (2003), to appear in Proceedings JHEP, hep-th/0403059, The Many Dimensions of Dimension, invited lecture, Nordstrom Commemorative, Helsinki, Finland, 2003, Comment. Phys. Math. Soc. Sci. Fenn. 166, 65 (2004) [physics/0402105]. A Note on Stress-Tensors, Conservation and Equations of Motion, A. Peres Festschrift, Found. Phys. 35, 11 (2005) [gr-qc/0402107]. Schwarzschild and Birkhoff a la Weyl (with J. Franklin), Am. J. Phys. 73,
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261 (2005) [gr-qc/0408067]. 349. Shortcuts to Spherically Symmetric Solutions: A Cautionary Note (with J. Franklin and B. Tekin), Class. Quant. Grav. 2 1 , 5295 (2004) [grqc/0404120]. 350. Conformal In variance of Partially Massless Higher Spins (with A. Waldron), Phys. Lett. B603, 30 (2004) [hep-th/0408155]. 351. Cotton Blend Gravity pp Waves (with R. Jackiw and S.-Y. Pi), Acta Phys. Polon. B36, 27 (2005) [gr-qc/0409011]. 352. Duality Invariance of all Free Bosonic and Fermionic Gauge Fields (with D. Seminara), Phys. Lett. B607, 317 (2005) [hep-th/0411169]. 353. How Special Relativity Determines the Signs of the Nonrelativistic, Coulomb and Newtonian, Forces, Am. J. Phys. 73, 6 (2005) [gr-qc/0411026]. 354. Free Spin 2 Duality Invariance Cannot be Extended to GR (with D. Seminara), Phys. Rev. D 7 1 , 081502 (2005) [hep-th/0503030]. 355. Curvature Invariants of Static Spherically Symmetric Geometries (with A. Ryzhov), Class. Quant. Grav. 22, 3315 (2005) [gr-qc/0505039]. 356. Birkhoff for Lovelock Redux (with J. Franklin), Class. Quant. Grav. 22, L103 (2005) [gr-qc/0506014]. 357. Graviton-Form Invariants in D — 11 Supergravity (with D. Seminara), Phys. Rev. D 1 5 , 027701 (2005) [hep-th/0506073]. 358. Conserved Charges of Kerr-AdS Spacetimes for Various Gravity Theories (with I. Kanik and B. Tekin), Class. Quant. Grav. 22, 3383 (2005) [grqc/0506057].
Books Edited 1. Co-editor, Brandeis Summer Institute in Theoretical Physics Proceedings, 1964, 1965, 1966, 1970 (Plenum, Prentice Hall, MIT Press). 2. Recent Developments in Gravitation (with M. Levy) (Plenum, 1978). 3. Co-editor, Themes in Theoretical Physics / ( N o r t h Holland, 1979). 4. Co-editor, Themes in Theoretical Physics II (World Scientific, 1989). 5. Particles, Strings and Cosmology (World Scientific, 1990).
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PHOTOGRAPHS
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S. Deser and G. Kane
R. Arnowitt, S. Deser and M. Duff
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M. Porrati and C. Nunez
C. Hull and E. Sezgin
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G. Gibbons and R. Arnowitt
H. Schnitzer and A. Waldron
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A. Greenspoon and L. Smolin
A. Milliken and R. Marquez
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R. Woodard and T. Dereli
i i M. Duff, R. Arnowitt, J. Nelson and C. Misner
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Stanley and Elsbeth Deser with Mike and Lesley Duff
MM
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Stanley and Elsbeth Deser
Richard and Young-In Arnowitt
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S. Ferrara and S. Deser
C. Pope, G. Gibbons, K. St. M. .md C. Hull
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C. Misner
T. Damour, S. Deser and S. Ferrara
This volume comprises the contributions to the proceedings of Deserfest - a festschrift in honor of Stanley Deser. Many of Stanley Deser's colleagues and longtime collaborators, including Richard Arnowitt and Charles Misner of "ADM" fame, contribute insighted article. Ranging from lower dimensional gravity theories all the way to supergravity in eleven dimensions and M-theory, the papers highlight the wide impact that Deser has had in the field.
YEARS OF P U B L I S H I N G !
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