MODERN DIFFERENTIAL GEOMETRY For Physicists
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World Scientific Lecture Notes in Physics Vol. 32
MODERN DIFFERENTIAL GEOMETRY For Physicists Chris J Isham Theoretical Physics Department Imperial College of Science and Technology United Kingdom
VQ World Scientific Hi
Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd., P O Box 128, Fairer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
Library of Congress Cataloging-in-Publication Data Isham, C. J. Modern differential geometry for physicists/Chris J. Isham. p. cm. — (World Scientific lecture notes in physics; vol. 32) ISBN 9971-50-956-3 ISBN 9971-50-957-1 (pbk) 1. Geometry, Differential. I. Title. II. Series QA641.184 1989 516.3'6--dc20 89-16487 CIP
Copyright © 1989 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any forms or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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PREFACE
These notes are the content of an introductory course on modern, coordinate-free differential geometry which is taken by our first-year theoretical physics PhD students, or by students attending the one-year MSc course "Fundamental Fields and Forces" at Imperial College. The geometry course is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen with an eye to the way in which differential geometry is applied these days to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, nonlinear sigma-models and other types of nonlinear field systems that feature in modern quantum field theory. The course is in three parts dealing with respectively (i) introductory coordinate-free differential geometry, (ii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds, (iii) introduction to the theory of fibre bundles. In preparing the notes I was particularly conscious of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry. In particular, in the first part of the course I have laid considerable stress on the basic ideas of "tangent space structure"
which I develop from several different points of view: some geometrical, and others more algebraic. My experience in teaching this subject for a number of years is that a firm understanding of the various ways of understanding tangent spaces is the key to attaining a grasp of differential geometry that goes beyond what is, all too often, merely an acquiescence in the jargon of the field. Part two of the course is concerned with the theory of Lie groups and the action of Lie groups on differentiable manifolds. I have tried to emphasise here the geometrical foundations of the connection between Lie groups and Lie algebras, but the latter subject is not treated in any detail and readers not familiar with this topic should supplement the notes at this point. The final part of the course contains a short introduction to the theory of fibre bundles and their use in Yang-Mills theory. This is fairly brief since many excellent introductions to this subject aimed at physicists have been published in recent years and there seemed no great point in replicating the material in detail here. Finally, let me emphasise that what is presented here are precisely the notes as handed out by me to the students to avoid having to write too much on the blackboard. In the context of the course, this written material is naturally reinforced by my verbal comments. I must apologise if the reader finds that the absence of these expository remarks renders some of the material rather pithy in places.
VI
CONTENTS Preface
v
1. Differentiable Manifolds 1.1 Main Definitions 1.2 Tangent Spaces 1.3 Tangent Vectors as Derivations 1.4 Vector Fields 1.5 Integral Curves and Flows 1.6 Cotangent Vectors 1.7 General Tensors and n-Forms 1.8 DeRham Cohomology
1 1 6 12 24 31 40 46 53
2. Lie Groups 2.1 Basic Definitions 2.2 The Lie Algebra of a Lie Group 2.3 Left-Invariant Forms 2.4 Transformation Groups 2.5 Infinitesimal Transformations
61 61 67 82 87 101
3. Fibre Bundles 3.1 Bundles 3.2 Principal Fibre Bundles 3.3 Associated Bundles 3.4 Vector Bundles 3.5 Connections in a Principal Bundle 3.6 Parallel Transport
Ill Ill 123 135 151 154 164
Index
179 vii
1 DIFFERENTIABLE MANIFOLDS
1.1. MAIN DEFINITIONS Definition A (finite dimensional) differentiable manifold is a (connected) topological space with the properties: (i) M is locally homeomorphic to IRm for some m < oo. Thus, around any point/? e Mthere exists an open neighbourhood £/and a homeomorphism <> / from U into an open ball in Um.
(ii) If (Uu 4>\) and (U2,
2) are two such coordinate charts then the overlap (or transition) functions 2°\~1 '• 4>\{UlC\U2) -* 02 (Ui n U2) are smooth (i.e., infinitely differentiable, or C00)
l
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Modern Differential Geometry For Physicists
Comments (a) A point peUcM, has the coordinates (4>\p), m m 4>\p),..., 4> {p)) e U with respect to the chart (U, ) where the functions ': £/-»• IR, / = 1 , . . . , m, are the coordinate functions defined as ' (p): = u'{4> (p)) in terms of the slot function u': Um-+ U, u'{x): = x'. The coordinate functions are often written as x', i = 1, ..., m, and the coordinates of a particular point p as (x1 (p), x 2 (/?), (b) There are analogous definitions of Ck (k
Differentiable Manifolds
3
Definition A subset N of a differentiable manifold Mis a C°° submanifold of M if every point of N lies in some chart (U,fa)with faN(lU)
= 4>(U)nUk
where 0
Examples (a) The vector space R" is a differentiable manifold with a globally defined coordinate chart in which the coordinates of a vector x are defined to be the components (x\ ..., x"). Any open subset U of R" inherits a differentiable structure from U" and is a n-dimensional manifold. (b) The circle Sl := {(x, y) e U2\x2+y2 = 1} is a C^-manifold which is a one-dimensional submanifold of the vector space U2. An explicit set of coordinate charts is the following:
u^ x
Ui:={(x,y)\x>0} fa (x, y) := y U2 := {(x, y) | x<0 } fa (x, y) := y U3 := {(x, y) | y>0 } fa (x, y) := x U4:= {(x,y)\y<0} fa(x,y):=x
Note that although I have written the coordinate functions as functions of both x and y it is understood that x2+y2= 1 and the notation (x, y) means the point on the circle with this value of x and y, i.e., the circle is a one-dimensional (not two-dimensional) space.
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4
To see that the overlap functions are differentiate consider as an example the overlap of £/, and U3. In £/, n U3 we have i
y = (1—x )2 withO
^W _ 1 = U,(i-^) 2 ) and so 2.
01o«/,3-1(x) = ( l - x 22\2)
which is indeed infinitely differentiable on this range of values for x and y. (c) The ^-sphere is defined as S" := {x e IR"+I | x-x = 1} It can be given an explicit differentiable structure by means of stereographic projection from the north and south poles. Ux := S" - {North pole} U2 := S" - {South pole} 0, and 2 are respectively stereographic projections from the north and south pole. X,
\ \XU • • •» xn+l)
\ 1 X„+i =
X„+\
+^1—Xi
x„
x2
: _
1
Xn+\
. . . — X„j
1
x
n+\ ,
Differentiable Manifolds
5
Detailed consideration of the overlap functions shows that this gives S" the structure of a differentiable manifold and that this is a submanifoldofDr +1 . A very important general idea in mathematics is that of a morphism. This is a map between two spaces both equipped with the same "mathematical structure" and which preserves this structure. In group theory, for example, a morphism between two groups is a homomorphism, whereas in topology a structure preserving map is a continuous map between two topological spaces. In differential geometry, the role of a morphism is played by a "Cr-function" between two manifolds which is defined as follows:
Definition (a) The local representative of a function/(from a manifold M to a manifold N) with respect to the coordinate charts {U, 4>) and (V, \[/) on M and N respectively, is the map ^o/o^-' : (U) c um
•W .
(b) A function f:M -* N is a C-function if, for all coverings of M and TV by coordinate neighbourhoods, all its local representatives are C r functions. [Note. Fortunately, it suffices to show this for one system of coordinate charts for each manifold. It then follows automatically for all other coordinate systems. Exercise!]
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Modern Differential Geometry For Physicists
(c) A function/:M—>- ./Vis a C-diffeomorphism iff is a one-to-one, surjective map such that both / a n d its inverse are C r functions. Note, (i) Two manifolds that are diffeomorphic (i.e., there exists a diffeomorphism between them) can be regarded as being "equivalent" in an appropriate sense, i.e., they are concrete copies of a single abstract Platonic manifold situated somewhere in the realms above. This is in the same sense that two groups that are isomorphic are regarded as being the "same" group. (ii) The set of all diffeomorphism of a manifold M onto itself forms a group Diff(M). Groups of this type play an important role in several branches of modern theoretical physics. 1.2. TANGENT SPACES One of the most important ideas in differential geometry is that of a "tangent space" to a manifold. This is based in part on the intuitive geometric idea of a tangent plane to a surface: TxSn := {velir + 1 |x-v = o} is the tangent space to the sphere S" at the point xeR"+\ but tangent space structure is also deeply connected with the local differentiable properties of functions on the manifold and this gives a more algebraic slant to the idea. There are in fact several, superficially quite different, definitions of a tangent space, some of which emphasise the geometric picture and some the algebraic. For finite-dimensional manifolds it can be shown that these definitions are actually equivalent and it is largely a matter of taste (and the details of the problem in hand) which determines the approach to be adopted in any particular case. The situation with
Differentiable Manifolds
1
regard to infinite-dimensional manifolds is however somewhat different and many of the finite-dimensional definitions are no longer appropriate. One of the main aims of this part of the Course is to present these different approaches to the definition of a tangent space and to show that they are equivalent. Thus we are firmly in the world of finite-dimensional differential geometry but I will start the presentation with the definition that does generalise easily to the infinite-dimensional case. This particular approach is slanted heavily towards the geometric view of a tangent space as being "tangent" to the manifold, rather as in the picture above for the case of a sphere. The problem with this picture however is that it is just that — a picture. In particular, it involves embedding the sphere in a particular euclidean space 1R"+1 and then regarding the tangent space as a specific linear subspace of this vector space. However, one of the aims of modern differential geometry is to present ideas in a way that is intrinsic to the manifold itself — not dependent on an embedding in some (largely arbitrary) larger dimensional vector space. In particular, the basic definition of a differentiable manifold made no reference to such a space, and neither should the definition of a tangent space. The critical question is therefore "What should replace the intuitive idea of a tangent vector as being tangent to a surface in the usual sense and "sticking out" into the containing space?" The answer lies in the idea of a tangent vector being tangent to a curve in the manifold. In the figure above, it is clear that a number of curves can be drawn on the sphere with the property that, at the point xeS", their tangent vectors (in the usual sense of curves in euclidean space) are equal to the vector v. The crucial point here is that the curves themselves lie in the manifold S", not in the surrounding IR"+' and that therefore this idea of a tangent vector as being defined in some way as 'tangent to a curve' can be generalised to an arbitrary manifold without embedding it in a
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higher dimensional vector space. The general idea therefore is to actually define a tangent vector in terms of a curve to which it might pictorially be deemed to be tangent. However, things cannot be quite as simple as this because, for example, it is clear that in the case above of S" which is embedded in a convenient space, there are many curves on the manifold that are 'tangent' to the given vector v:
This leads to the idea of a tangent vector as being an appropriate equivalence class of curves, and this is the definition that we shall now present. Definition (a) A curve on a manifold M is a smooth (i.e., C°°) map a from some open interval (—£,£) of the real line into M. [Note. The "curve" is defined to be the map itself, not the set of image points in M. It is frequently important to remember this distinction.]
(b) Two curves ox and o2 are tangent at a point p in M if (i) (7,(0) = ff2(0) = p
(ii) In some local coordinate system (xl, x2,..., xm) around the point the two curves are 'tangent' in the usual sense as curves in IRm
Differentiable Manifolds
dx'
~dl
(ffi(0) r=0
dx1 (o (t)) ~dt 2
9
fori = l,...,m.
(1.2.1)
(=0
Note. If ax andCT2are tangent in one coordinate system then they are tangent in any other coordinate system covering the point peM. Thus the definition is an intrinsic (i.e., coordinate system independent) one. (c) A tangent vector at p e M is an equivalence class of curves in M where the equivalence relation is that two curves shall be tangent at the point p. We write the equivalence class of a particular curve a as [a]. (d) The tangent space Tp M to M at the point p is the set of all tangent vectors at the point p. The tangent bundle TM is defined as TM := [J TPM. There is a pe M
natural projection map n : TM -* M that associates with each tangent vector the point p in M at which it is tangent. The inverse image (i.e., the "fibre") of any point p under the map n is thus just the set of all vectors that are tangent to the manifold there, Tp M. This is a very special case of the general idea of a fibre bundle to which we shall return later. Comments (a) This definition of a tangent vector is consistent with the intuitive geometrical one when, as in the case of a sphere embedded in a euclidean space, there is a natural way of representing tangent vectors as elements of the containing vector space. This also extends to the tangent bundle TM which, in the case of the sphere, can be represented as:
Modern Differential Geometry For Physicists
10
TSn = {(x,v)elR''+1 X IR" +1 |x-x = 1 andx-v = o}. (b) A tangent vector v in TPM can be used as a "directional derivative" on functions/on A/by defining:
v(f) := -f{o
(1-2.2)
it)) (=0
where a is any curve in the equivalence class represented by v, i.e., v = [a].
[Exercise: Show that this definition does not depend on the particular choice of a in the equivalence class v.] The idea that a tangent vector can be regarded as a type of differential operator is a crucial one which lies at the heart of the other, more algebraic, definitions of tangent space structure. We shall return to this point later. In the heuristic, pictorial, representation of tangent vectors to S" embedded in IR"+I, it is clear that any two tangent vectors at a point p can be added to give a third, and that a real multiple of a tangent vector is itself a tangent vector. This suggests strongly that it should be possible to make TPM into a vector space in the general definition given above. Theorem
The tangent space TPM carries a structure of a real vector space.
Proof
In the diagram above, c, and a2 are two representative curves
Differentiable Manifolds
11
for the two tangent vectors i>, and v2 in TpM. We use a coordinate chart (U, 4>) aroundp e Mwith the property that <j>(p) = 0 e IRm and then consider the image curves 4>°al and °er2 which map an open interval into Um. We cannot of course "add" the curves cr, and c2 directly since the set M in which they take their values is not a vector space. However, the local space Um is a vector space and hence we can legitimately consider the sum t-~~~°o{(t) + 4)°o2{t) which is a curve in Um and which, like both er, and o2, passes through the null vector 0 when t = 0. It follows that the map t~~-4>~l°(4>°ox (t) + 4>°o2(t)) is a curve in M which passes through the point p when t=0. We then define: (i)
u, + u2 := [(A"'o (0off, + 0(72)]
(1.2.3)
and similarly, if v = [a], (ii)
rv :=["' o (rtfoo)]
for
all r in R.
(1-2.4)
It can be shown that [Exercise!] (i) These definitions are independent of the choice of chart (U, ) and representatives a{ and er2 for the tangent vectors Vi and v2. (ii) They make T^M into a real vector space. Note. This means that TM is a vector bundle. More details of this later. A tangent space can be regarded to some extent as a local "linearization" of the manifold and it is a fact of considerable importance that a map h between two manifolds M and TV can also be "linearized" with the aid of the tangent spaces and the vector space structure that they carry. This is the concept of the "pushforward" map h* that maps the tangent spaces of M linearly into the tangent spaces of N:
12
Modern Differential Geometry For Physicists
Definition If h : M -* N and v e TPM then we define h*(v) in Th{p) TV by h*(v) := [h°a]
where v = [a].
(1.2.5)
Exercise (a) Show that this definition is independent of the particular choice of the curve a in the equivalence class v e TPM. (b) Show that h* is a linear map, i.e., for all vu v2 e Tp Mand r e U, h* (t^ + t^) = h* (u,) + h* (v2)
(1.2.6)
h*(rv) = rh*(v).
(1.2.7)
(c) If M, Nand Pare three manifolds and if h:M-+ Nand k:N—*P then show that (k o h)* = k* h*.
(1.2.8)
1.3. TANGENT VECTORS AS DERIVATIONS We recall Eq. (1.2.2):
v{f) := JtfW))
where [o] = v
(1.3.1)
1=0
which enabled the equivalence class of curves v e TPM to act as a type of differential operator on the space C°°(Af) of real-valued differentiable functions on M. The first "algebraic" way of
Differentiable Manifolds
13
defining tangent vectors is to turn Eq. (1.3.1) around and define TPM as a space of "directional derivatives". This is developed by thinking of Cco(M) as a ring over IR, and of a tangent vector as a derivation map from this ring into IR: Definition (a) A derivation at a point p e M i s a map v:Cx{M) -*• IR such that (i) v(f+g) = «(/) + v(g) for a l l / g e C°°(M) »(r/) = rv(f) for a l l / e C°°(M) and r e U
(1.3.2) (1.3.3)
(ii) v(fg) = ftp) v(g) + g(p) v(f) for a l l / # in C»(M).(1.3.4) (b) The set of all derivations at /> e M is denoted Z)pAf. Comments (a) Eqs. (1.3.2-3) merely assert that v is a linear map from the vector space CX(M) to IR. The term "derivation" is used because of the crucial "Leibniz" rule in Eq. (1.3.4) which is clearly analogous to the rule in elementary calculus for taking the derivative at a fixed point of the product of two functions. (b) The space of derivations DPM can be readily given the structure of a real vector space by defining: (wi + « 2 ) ( / ) : = « i ( / ) + f2(/)
(1-3-5)
v(rf):=rv(f)
(1.3.6)
for a l l / „ / 2 , / in Cm{M) and r e U. (c) Eq. (1.3.1) defines a map fi: TPM-»• DPMwhich is clearly linear and injective (i.e., one-to-one). We will show below that it is
14
Modern Differential Geometry For Physicists
also surjective and is therefore an isomorphism. It is this fact that enables us to view tangent vectors geometrically as equivalence classes of curves, or algebraically as derivations of CX(M). A very important example of a derivation is provided with the aid of any coordinate chart (U, ) containing the point p of interest. Define
du"
f°
(1.3.7) Equation (1.3.7) defines a derivation at p e M. What we want to ( d\ show now is that fi — 1, .. ., dim (M) form a basis set for the vector space D„M. Lemma Let (U, 0) be a chart around p e M with associated coordinate functions (xl, x2,..., xm) and such that x* (p) = 0 for all fi= 1, .. ., m = dim(M). Then, for a n y / in C°(M) there exists/; e C°°(M) such that
(i)A(p)
-U?/
(1.3.8)
(ii) f(q) = f(p) + 2_j x"(Q)fn (
hood of p.
Differentiable Manifolds
15
Note. Eq. (1.3.8) means that the function/ can be written locally as
Proof Let F := fo (j)'1 defined on some open ball B around 0 e IRm. Then, for &e B, F(au a2,..., aj = F(au ..., aj - F(au a2,..., am-u 0) + F(a{,.. .,am-i,0)-F(a1,a2,..
.,a m _ 2 ,0, 0)
+ + + F(au 0,. . .,0)-F(0, 0,. .., 0)+F(0,0,. . .,0) (1.3.10) This identity can be rewritten as m
F{a{, a2,..., am) = F(0, 0,..., 0) + £ F(a„ ..., taM, 0,..., o) 11= \
= F(o, o,..., o ) + 2 ,
—\^-
• - ^ - i . ^ . - • ->°) a»dt
m
= F(0, 0,..., 0) + ^ a^F^a,, a2,..., a J 11=1
(1.3.11)
r SF . where
F^ (a)
:=
I (a b . . ., a^x, tam 0,. .., 0)dt is Jo du" C co -functionon£C[R m .
a
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Now define ftl:= F^o <> / which is defined initially on some open neighbourhood of p e M but which can then be extended arbitrarily to the rest of M. Then f{q) = if" (
(
m
= F (0) + J x" (g) F„ {xl (q), ...,xm (q)) m
and hence the desired representation certainly exists. But then,
(1.3.12) where, it should be recalled, f(p) refers to the constant function which assigns the same value f(p) to all points q e M. However, if v e DPM and if c:M -»• IR is a constant map (with the value c) then v(c) = cv(l) = cu(l.l) = c[l.t>(l) + l.w(l)] = 2cv(l) and hence v(c) = 0. Applying this result to Eq. (1.3.12) (and remembering that xv (p) = 0) we get
which proves the Lemma.
QED
Differentiable Manifolds
17
Corollary lfveDp(M)thenv
= ^v(xli){ fi=\
— \ \OX
.
(1.3.13)
Jp
Proof If x^ip) T£ 0 for jU=l,..., w then define new coordinate functions y" := xA—x"(/?) and apply the Lemma using this coordinate system. Then in
/()=/(/>)+ 5 > ( )fA ) m
= /(/>) + £ [ > ( )-x"(p)]fM( ). Applying the derivation v to both sides of this equation and remembering that v vanishes on the constant functions f{p) and x"(p), we get the result Eq. (1.3.13). QED ^ ( d \ Note that if D := > u" = 0 for some set of real numbers
£i \dx»Jp
[v',...,
vm) then 0 = v{x'1) = vM which means that: (d
\
= dim(M) The set of vectors u= 1,.. ., mare linearly independent and span the vector (1.3.14) space DPM. In particular, dim (DPM) == dim (M). The set of real numbers {vl, v2,..., vm} are called the components of the derivation v e DPM with respect to the given coordinate system. They uniquely determine the derivation, although it should
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be noted that the components of a given derivation will be different in different coordinate systems. We can use the result above to prove the critical fact that DPM and TPM are isomorphic vector spaces: Theorem The linear map JU: TPM —- DPM defined by [Eq. (1.3.1)] with [a] = v
[n{v)]{f):=-f{o{i)) r=0
is an isomorphism. Proof We already know that it is an injective and linear map, and therefore it remains to prove that it is surjective. So, let v e DPM and construct a curve ov: (—e, e) -*• M such that: (i)
av(0)=P at
t~o
Then, i f / e C°°(M),
(/)
x
J < ,
" ^l>%)/^?* "•' • ' But,
i-o
at
1= 0
«-o \dx")p
Differentiable Manifolds
-IT; (/••-)
19
( ° <*„) (= 0
0(P)
« ( / ) •
/= 0
Thus [o-„] (/) = u(/), and hence the map /u is surjective. QED Comments (a) It follows that since TPM ^ DPM we have in particular dim (TPM) = dim (M). (b) The fact that these spaces are isomorphic means that we can use either according to our particular interests and application under consideration. Any formula proved for tangent vectors defined as equivalence classes of curves in TpM must have an analogue within the definition as a derivation in DPM, and vice versa. In particular, given a tangent vector v e TpMwe can expand it in components by thinking of it as a derivation and then using Eq. (1.3.13) to give: m
v =
(1.3.15)
^ i ? " ,i=\
dx",
with V":=
u(x") =
where [o] = v.
—jc"off(f)
at
(1.3.16)
1=0
(c) Similarly, within the "algebraic" derivation language, the pushforward map h*, defined for equivalence classes of curves in Eq. (1.2.5), becomes:
20
Modern Differential Geometry For Physicists
lfh:M-> N, then h* : DPM -* Dh(p) N is defined on any v e DpM as the derivation in Dh (p) N satisfying (h*v)(f):=v(foh)
(1.3.17)
for all feC™(N). In particular, note that if we have a curve c : ( — e, e) -* M then
(1.3.18)
where -
is the derivation on C°°(IR) defined simply as
,dtl0
' dt
forall/eC°
(1.3.19)
1=0
(d) We can see now that the tangent bundle TM has a natural structure of a 2m-dimensional differentiable manifold. Essentially, the 2m coordinates of a vector v e TPM are assigned to be the m real numbers {xl (p),x2(p),. .., xm (/?)} fixing the point p in a local coordinate system, and the m real numbers {v(xl),v(x2),..., m v(x )} which are the components of the (derivation) v with respect to this same system. More precisely, if (U, ) is a local coordinate chart around p e M, then we define the bijection TM, \v [c]
4>(U)X
'c\
X Uf
— (4>(p), , [c]) where [c] e TPM
Differentiable Manifolds
21
with any of its tangent spaces TJim by associating with any vector w e Um the derivation w(/):=-/(x+rw) at
(1.3.20) (=0
i.e., w e IRm generates the curve t~~~~x+tw in IRm. We come now to some technical results concerning the tangent space structure of the product M X JVof two manifolds. These will be of importance to our later discussion of Lie groups and fibre bundles. Theorem On the product manifold M X N there is a natural isomorphism T(M)(MXN)^TpM@TqN v — {pru(v),pr2,(v))
(1.3.21)
where, prx : MX N' — M; (p,g)—p
pr2: M X N ^ N {p,q)~*q
(1.3.22)
are the usual projection maps. Proof Let a : ( — e, e) -* M X N be a curve such that [a] = v e T(M) M X N. Then there exist unique curves er, and o2 in M and N respectively such that o(t) = (er, (?), a2 (t)). Namely, a{{t) := prx o o(t) and a2(t) := pr2 ° a(t). Define a linear map x : T ^ MXN^
TPM 0 7^JV by
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Modern Differential Geometry For Physicists
X([<x]):=([<x,U<72])
(1.3.23)
which is clearly one-to-one and surjective. Thus % is the desired isomorphism. Furthermore,
CTl
* ( f ) 0 = (/?ri °CT)* ( ^ ) O = ^'* "* (^)o
and so [er,] = prx* [a] as claimed (where we have used Eq. (1.3.18)). QED Comments (a) It is convenient to define the injections (for each p e M and qeN), iq:M~*MX
N; j p : N^
x~~{x,q)
MX
N
y~~(p,y)
(1.3.24)
which satisfy the relations pri o iq = i d M ;
pr2o iq=
q()
Pr{ °jp = P( ) ;
pr2 °jp = idN
(1.3.25)
where id M [resp. idN] is the identity map of M [resp. N] into itself and q{ ) [resp. p{ )] is the constant map on N [resp. M] taking every point into q [resp. p]. Thus the map inverse to the isomorphism x in Eq. (1.3.23) is T„M(BTt,N-+TlM)MXN (1.3.26) (a, /3)~~~ iq*a +jp*/3
Dijferentiable Manifolds
23
(b) Let P be another manifold and let / : M X N-*• P. Then /* T(pq) M X N-* Tfipq) P factorises as T(M)MXN-
^TpM@TqN
f T
f* =f °X
(1.3.27)
P
f{p,Q)
where. (1.3.28)
and, foiq:M^P
•
fijp:N-+P (1.3.29)
(c) The diagonal map associated with a manifold M is the map A : M —»- M X M defined by A (x) := (x, x). In the chain of maps TXM
A
* »
T(xx) MXM
X
„ TXM 0 TXM
we have X ° A* (a) = (a, a) for all a e TXM.
(1.3.30)
In particular, iff: M X M — P we have/o A : M — P with/° A(x) = f(x, X) and (/°A)* (a) = /* (A*(a)) = f°x°A*{a) = f (a, a) for all a e TXM. Thus,
(fiA)* = {foQ* + (fijx)* .
(1.3.31)
24
Modern Differential Geometry For Physicists
1.4. VECTOR FIELDS So far we have dealt only with tangent vectors defined at a single point in the manifold. We come now to an important generalization of this idea in which we consider a "field" of tangent vectors in which a tangent vector is assigned to each point of M (or, more generally, to each point in some open subset of M). A critical requirement here is that the tangent vectors should vary "smoothly" as we move around the manifold — a concept that is formalised in the following definition: Definition A vector field X on a C°°-manifold M is a "smooth" assignment of a tangent vector Xp e TPM for each point p e M. "Smooth" is defined to mean that, for a l l / e C " ^ ) , the function M »R p — (Xf)(p):=Xp(f)
(1.4.1)
is smooth (i.e.,C°°). Thus a vector field can be regarded as a map X: C^iM) —* K C (M) in w h i c h / — X(f) is defined in Eq. (1.4.1). The function X(f) is often known as the Lie derivative of the function/along the vector field X, and is then usually denoted Lxf or sometimes £xfComments (a) A vector field can also be defined as a smooth cross-section of the tangent bundle TM, i.e., TM \n M
\X
n°X=\&u
(1.4.2)
Differentiable Manifolds
25
In this case, "smooth" means that X is a C^-map from the manifold M to the 2m-dimensional manifold TM (with the structure defined in Sec. 1.3). It is a straightforward matter to show that this definition of a smooth assignment agrees with the one above. (b) The map/--* X(f) has the following properties with respect to the ring structure on Cm{M): (i)
X(f+g) = X(f) + X(g) for a l l / g e C»(M)
(1.4.3)
(ii) X{rf) = rX{f)
for all/e C°°(Af) and r e R(1.4.4)
(iii) X(fg) = /*(
for a l l /
(1.4.5)
which should be contrasted with the Eqs. (1.3.2-4) for a derivation in DPM. Equation (1.4.3) shows that X is a linear map from the vector space C°°(M) into itself, and Eq. (1.4.5) shows that X is a derivation of the ring C°°(M) in the usual sense of the word. (In this respect, the use of the term "derivation" in relation to Eq. (1.3.4) is potentially confusing, but it is conventional). (c) This procedure can be reversed in the sense that it is possible to define a vector field X as a derivation of the ring of functions Cm(M) satisfying the three equations (1.4.3-5). Such a derivation assigns a derivation (in the sense of Sec. 1.3) to each point/? in M denoted Xp and defined by Xp{f):=[X{f)](p)
(1.4.6)
for each/in CX(M). It is easy to see that Xp satisfies the three conditions Eqs. (1.3.2-4). But then the map p ~~~~ Xp assigns a field of tangent vectors in the sense of the first definition above, and it is clear that this assignment is smooth in the sense of Eq. (1.4.1). Thus we see that a
26
Modern Differential Geometry For Physicists
completely equivalent way of defining a vector field in the first place is to regard it as a derivation on the ring C°(M). This alternative approach (in which we go from vector field to derivation at a point p e M, rather than the converse) is useful in many situations and we shall employ it wherever it is appropriate. (d) Let (U,
(Xf)(p) = Xp{f) =
2*P(**)
I
A
—
-2^'M-Af
\
f
(..4.7,
which we write as, (1.4.8) This expression, interpreted as Eq. (1.4.7), shows the precise sense in which a vector field can be thought of as a first-order differential operator on the functions on a manifold. (e) The functions {Xx", n= 1,..., m) are defined on the open set U that defines the coordinate chart with the coordinate functions x" and are known as the components of the vector field Xwith respect to this coordinate system. These component functions will of course depend on the coordinate chart in use. For example, let (U, 4>') be another coordinate chart with the property that U'D U =£ 0. Then we can meaningfully compare the components Xx*1 and Xx,tL on the open
Differentiable Manifolds
27
set f/D If. Thus, using Eq. (1.4.8) with respect to both systems of coordinates, rn
X-
rn di ni 1 = > Xx" . _Xx axr =i M dx" t=i dx'" ~
> Xx" n= i
(1.4.9)
Now we can apply Eq. (1.4.8) again, this time to the vector d expanded with respect to the coordinates x'". Thus dx" => dx" Hdx"
(x'v)
.
(1.4.10)
,v
dx
Let us define the components of Xin the two coordinate systems as X" := Xx" and X"' := Xx'". Then, substituting Eq. (1.4.10) into Eq. (1.4.9) and equating terms, we find v
,
f,
dx'v
X = > X"-— p[ dx"
(1.4.11)
which is a familiar expression in elementary, coordinate dependent, approaches to differential geometry. In this sense, Eq. (1.4.11) plays a key role in understanding the precise connections between these older approaches to differential geometry and the more modern, coordinate-free, formalism that is being developed in this Course. In this context it is important to appreciate that the precise meaning of the rather formal expression dx'v/dxfl is the function on the overlap U P\ U' defined by dx'v dx"
d \ \dx" L
d du"
.
(1.4.12)
28
Modern Differential Geometry For Physicists
Now we come to the rather important question of whether or not we can "multiply" together two vector fields Xand 7 to get a third field. Since X and Y can be viewed as linear maps of C°°(M) into itself we can define the composite map X° Y: CX(M)^> Cm(M) as X° Y(f) := X{ Yf) and this is certainly linear, i.e., it satisfies Eqs. (1.4.3-4). However, it is not a vector field since it fails to satisfy the crucial "derivation" property in Eq. (1.4.5). Indeed, XoY(fg) = X(Y(fg))
= X(gY(f)
+ fY(g))
= X(g) Y(f) + gX(Y(f))
+ X(f) Y(g) + fX(Y{g))
which does not equal gX{ Y{f) + fX( Y(g)) which it would have to if it were to be a derivation. However, we also have YoX(fg) = Y(g)X(f)
+ gY(X(f))
+ Y(f)X(g)
+
fY(X(g)).
Subtracting these two expressions we find
(X°Y-Y°X)(fg)
=
g(X°Y-YoX)(f)+f(X°Y-YoX)(g)
which illustrates the vital point that X ° Y— Y ° X is a vector field even though the individual composites X ° Y and Y ° X are not. This new vector field X ° Y— Y ° X is called the commutator of the vector fields Xand Y and is usually written as [X, Y]. Comments (a) In terms of the local representation Eq. (1.4.8) of vector fields as differential operators it is clear that the simple composition X ° Y cannot be a vector field since it includes second-order differential operators as well as first-order ones. When we subtract Y <> X from X ° Y these second-order terms cancel and only the first-order differential operators are left, and hence it is a vector
Differentiable Manifolds
29
field. The components of the commutator [X, Y] can easily be computed [Exercise!] to be m
[X, Y]« = ]T {Xv Y,$ - 7V X,i)
(1.4.13)
where, as is conventional, the expression Y,% means the derivative of the functions y with respect to the coordinates xv in the sense of Eq. (1.4.12). (b) If X, Y and Z are any three vector fields on M then their commutators satisfy the so-called Jacobi identity: [X, [Y,Z]\ + [Z, [X,Y]] + [Y, [Z,X]] = 0.
(1.4.14)
This means that the set of all vector fields on M forms a real Lie algebra. This algebra is infinite-dimensional and can, with care, be regarded as the Lie algebra associated with the infinite dimensional group Diff(A/) of difFeomorphisms of M. This is related to the, somewhat heuristic, assertion that a vector field is the "generator of infinitesimal transformations" on the manifold. We will return to this idea in a little while and also to the question of when the infinite-dimensional Lie algebra of all vector fields possesses a non-trivial finite-dimensional subalgebra. In Sec. 1.3 we defined the push-forward h*:TpM'—- Th(p)N induced by a map h:M^-N. This was a linear map between the vector spaces TPM and Thip) Af and the question arises of whether or not it is possible to define in a similar way an induced map from the vector fields on M to those on N. Given a vector field on M and a smooth map h from M to N, a natural choice for an induced vector field h*X or\ N might appear to be: [h*X]Hp) := h*(Xp)
(1.4.15)
30
Modern Differential Geometry For Physicists
but, as we can see with the aid of the following diagram, this may fail to be well-defined for two reasons:
(i) If there are points xy and x2 such that h(xl) = h(x2) (i.e., if h is not one-to-one) then the definition in Eq. (1.4.15) will be ambiguous if h*{Xx^j # h*(XX2). (ii) If h is not surjective then Eq. (1.4.15) does not specifiy the induced field outside the range of h. Note however, if h happens to be a diffeomorphism from M to TV then neither of these objections apply and we can define an induced vector field h*X via Eq. (1.4.15). It is of course always possible that in certain cases the definition will work, even if h is not a diffeomorphism, and this motivates the definition: Definition Let k.M-^-N and let X and Y be vector fields on M and TV respectively. Then, X and Y are said to be h-related if, at all points x in M, h*(Xx)=
Yh(x)
and we then write Y = h*X. Comments (a) If Xand Fare A-related and i f / e CX{N) then (/z*A p )(/)= Yh{p)f
(1.4.16)
Differentiable Manifolds
i.e., and so
31
[X(fo h)] (p) = Xp(/o h) = (Yf)(h (p)) X(foh) = (Yf)oh
(1.4.17)
which is a useful way in which to express the quality of /j-relatedness.
(b) lfh*Xx = y, and h*X2 = Y2 then, for all / i n CX(M\ we can use Eq. (1.4.17) to write ([y„ Y2]f) oh = (Yl (Y2f)) o A - 1 ** 2 = * i ( ( ^ 2 / ) ) o /i -
= X, (X2 (foh))
1 ** 2
-1++2
= [XuX2](f°h)
(1.4.18)
which is to say that h* ([Xu X2]p) = [h* Xu h* X2]Hp) for all p in M. Thus [Xi, X2] is,j/z-related to [7,, Y2] with h*[X1,X2] = [h*Xl,h*X2].
(1.4.19)
1.5. INTEGRAL CURVES AND FLOWS In this section we will discuss the precise sense in which a vector field can be said to be the generator of an "infinitesimal"
32
Modern Differential Geometry For Physicists
coordinate/active transformation of a manifold. Expressions such as Sx"^X"(x)
(1.5.1)
appear frequently in the more physics-oriented literature but there is a deceptive simplicity about this equation and it covers up a fair amount of subtlety. We start by recalling that we have shown the equivalence of regarding tangent vectors as equivalence classes of curves and of derivations defined at a point in the manifold. It seems intuitively clear that if we have a smooth, non-intersecting family of curves on a manifold then the tangent vectors at each point can be taken together to form a vector field on the manifold M:
More interesting however is the converse question. Namely, given a vector field X on M is it possible to "fill in" M with a family of curves in such a way that the tangent vector to the curve passing through any particular point p in M is just the vector field evaluated at that point, i.e., the derivation Xpl We start by considering the simpler case of finding a single curve passing through a specified point pe M and such that the tangent vector at every point along this curve agrees with the vector field evaluated at that point. This idea is contained in the formal definition: Definition Let Xbe a vector field on a manifold M and let p be a point in M. Then an integral curve of X passing through the point p is a curve / — a(t) in M such that
Differentiable Manifolds
33
(i)
(1.5.2)
for all s in some open interval (— e, e) of U. Comments (a) In local coordinates we can write X as (Eq. (1.4.8))
x=y
(Ax")— pi
dx"
and, in our case, (**") (ff(0) = XffW (x") = a* R J x" = -^ x"off(0 and hence Eq. (1.5.2) implies that the components XM of X are to determine the integral curve t ~*-CT(0according to the ordinary differential equation X" {a(t)) = — x""a(t) at subject to the boundary condition Jt"off(0) = x" (p)
n=\,...,m
n=\,...,m
(1.5.3)
(1.5.4)
(b) The discussion of the existence and uniqueness of solutions to the system of first-order differential equations in Eq. (1.5.3) with initial data (1.5.4) can be tackled by any of the familiar analytical methods. For example, we can write the equivalent integral equation
34
Modern Differential Geometry For Physicists
x"
(CT(0)
= x" (p) + I X" (a(s)) ds
(1.5.5)
Jo
which may be solved uniquely for some range of values for t using the Picard iterative method. By this, or similar means, we can demonstrate the existence of an integral curve passing through the specified point p and defined for t lying in some open interval ( — e, e) of IR. It is however, as always, a different matter when it comes to discussing the existence of solutions that are defined for all values of the parameter /. Solutions in overlapping open sets can be patched together to some extent but this still does not guarantee the existence for all values of t e IR. An important example of a "pathological" behaviour of this type can occur when M is noncompact in which case it is possible for a locally-defined integral curve to "run-off" the "edge" of M after only a finite range of t values. This motivates the important defintion: Definition A vector field l o n a manifold M is complete if, at every point p in M, the integral curve passing through p can be extended to an integral curve for X that is defined for all t eU. Comments (a) It can be shown that if X is a vector field on a manifold M that is compact then X is always complete. Since it is complete vector fields that can most readily be identified with the Lie algebra of the diffeomorphism group it follows that the study of Diff(M) is considerably easier when M is compact than when it is not. (b) When using differential geometry in theoretical physics, a situation that arises not infrequently is to have a vector field X defined on a manifold M that is also equipped with a volume element/measure dp, in such a way that iX, regarded as a
Differentiable Manifolds
35
differential operator on the Hilbert space L2 (M, dfi), is symmetric. It transpires that iX is a genuine self-adjoint operator however if and only if X is a complete vector field. This result is highly relevant in discussions of the quantization of a classical mechanical system whose configuration space is a non-trivial differential manifold (as, for example, in a system with constraints, or in a gauge theory). Examples/Exercise (a) The vector field d/dx is complete on U but not on the open submanifold IR+ of positive real numbers. In wave mechanics, this corresponds to the well known difficulty in defining a self-adjoint momentum operator p in the case of an "infinite potential barrier" at x = 0 that confines the particle to the positive real axis. This is closely related to the fact that, in any tolerably well-behaved pair of self-adjoint operators x, p satisfying the canonical commutation relations [x,0] = i
(1.5.6.)
it can be shown that the eigenvalue spectra of both operators must be the whole real line. (b) The vector field xd/dx is complete on IR+. In the problem of quantizing on U+ this means that it is much better to consider the observables x and xp than the usual pair x, p that are appropriate for quantum mechanics on the whole real line. If n := xp then the analogue of the CCR Eq. (1.5.6) is the so-called "affine commutation relations" [x, n] = ix
(1.5.7)
for which, unlike Eq. (1.5.7), genuine self-adjoint operators can be found with the spectrum of x being just U+ as required.
Modern Differential Geometry For Physicists
36
This particular example gives just the vaguest hint of how one might need to amend the usual quantum mechanical rules when dealing with a situation in which the configuration space is nontrivial. In the general case there is no natural analogue at all of the conventional CCR of Eq. (1.5.6) which are closely locked to systems in which the configuration and phase spaces are simple vector spaces. However, whenever there is some generalization of Eq. (1.5.6) it is almost invariably a more complicated version of Eq. (1.5.7) in the sense that a canonical operator (i.e., not the unit operator) appears on the right hand side. We wish now to consider the sense in which a vector field, complete or otherwise, can be regarded as the generator of "infinitesimal" transformations on the manifold. The appropriate starting point is the following definition: Definition A local, one-parameter group of local diffeomorphisms at a point p in M consists of: (i) An open neighbourhood U of p. (ii) A real constant £ > 0. (iii) A family {J 11 \ < s} of diffeomorphisms of U onto the open set 4>t(U) CM with the properties (i)
The map ( - e , e) X U-*M (t,q)-~t(q)
(1.5.8)
is a smooth function of both t and q. (ii) Whenever t, s e U are such that \t\, \s\ and |*+.s| are all bounded above by e, and for all points q e U such that t{q), 4>s{q) and <j>t+s{q) also belong to U, then 4>s (,(
4>s+t(q)-
(1.5.9)
Differentiable Manifolds
37
Comments (a) The first occurrence of the word "local" in the definition refers to the fact that the difFeomorphisms , are only defined a priori for t lying in some small open interval ( — e, e) of the real line. The second occurrence of the word refers to the fact that the difFeomorphisms are only defined locally on M, viz., they are only defined on the (perhaps 'very small') open subset U C M. (b) The term "one-parameter group" in the definition refers to Eq. (1.5.9) which shows that the map f-*~0, can be thought of as a representation of part of the additive group of the real line by local difFeomorphisms of M. In this context note that Eq. (1.5.9) implies that <j>0 (q) = q for all points q e U. (c) Through each point q e M we can consider the local curve t"^4>t (q). Since there is such a curve passing through any point in U we can obtain a vector field on U by taking the tangents to this family of curves at each point.
The resulting vector field is said to be induced by the family of local difFeomorphisms and is defined precisely as
Xt(f):=jtf(.MQ))
(1.5.10) f=0
Modern Differential Geometry For Physicists
38
for all points q in Uc M. The following proposition seems intuitively obvious but is nevertheless worth proving properly: Theorem For all points qsU, the curve /-*-(/>,(
for 111 < e. Proof Define the curve 4>q • ( —e, e) -»• M by 4>q (t) :== <j>, {q). According to the definition of an integral curve in Eq. (1.5.2) we want to show that
rx^
^[jt
(1.5.11)
for \s \< e.
But,
/ d(
*U/"* *' ( 0 )
Jtf(M,))
. (1.5.12) s
Let t = s+u. Then the right hand side of Eq. (1.5.12) is
du
= -j- f(4>u fate)))
f(s+u (
"»
u=0
QED. We have seen how to go from a local group of local diffeomorphisms to a vector field. But the problem of interpreting an equation like (1.5.1) involves the converse procedure, viz., going from a vector field to a group of diffeomorphisms. The central concept here is of a "local flow".
Differentiable Manifolds
39
Definition Let Xbe a vector field denned on an open subset [/of a manifold M and let p be a point in UcM. Then a localflowof X at p is a local one-parameter group of local difFeomorphisms denned on some open subset Vof U such that p e V C U and such that the vector field induced by this family is equal to the given field X. Comments (a) Local flows always exist, and are unique, by virtue of the usual existence and uniqueness theorems of ordinary differential equation theory. (b) If X is a complete vector field then we can always choose V to be the entire manifold M, and the interval ( — £,£) can be extended to the real line. In this case, if <\>f denotes the family of difFeomorphisms that is the flow for X, then the map t—^4>? is an isomorphism from the entire additive group of the real line into the group Diff(M) of difFeomorphisms of M. We can say that this defines a oneparameter subgroup of Diff(M) and hence an element of its Lie algebra (see later for a full discussion of the relation between Lie groups and Lie algebras). This is the sense in which a vector field is said to belong to the Lie algebra of Diff(M). (c) In a local coordinate system, the local flow <$>? satisfies the differential equation (cf. Eq. (1.5.3)) X»{tf{q))=~x»{?{q)) at
(1.5.13)
around any point q in the open set U C M. If we perform a Taylor expansion of the coordinates x* {? {a)) of the transformed point 4>f (q) around the value t=0 we get:
40
Modern Differential Geometry For Physicists
x" (? (q)) = x" (q) + tX"(q) + 0(t2)
(1.5.14)
which can be regarded as a reasonably rigorous version of the heuristic Eq. (1.5.1) and the claim that a vector field generates "infinitesimal transformations". (d) If X and Y are two vector fields on Mit can be shown that [Exercise!]
at
= [X, Y].
(1.5.15)
/=0
1.6. COTANGENT VECTORS We come now to the topic of cotangent vectors and differential forms — ideas that are of the greatest importance, both in the mathematical development of differential geometry and in its applications to theoretical physics. Definition (a) A cotangent vector at a point p in a manifold M is a real linear map k from TPM into U. We will write the value of k on the tangent vector v e TPM as (k, v) or (k, v)p if it is necessary to emphasise the point in the manifold with which we are concerned. (b) The cotangent space at p e M is the set TP*M of all such linear maps, i.e., it is the dual of the vector space TPM. It follows from general linear algebra, that this dual space is itself a vector space and that dim(r*M) = dim{TpM) [ = dim(Af)]. (c) The cotangent bundle T*M is the set of all cotangent vectors at all points in M:
Differentiable Manifolds
41
T*M:= \J T*M. peM
It is a vector bundle and, as in the case of the tangent bundle TM, it can be equipped with a natural structure of a 2m-dimensional differentiable manifold (cf. Sec. 1.3). (d) A one-form co on M is a "smooth" assignment of a cotangent vector cop to each point p e M. "Smooth" means in this context that, for any vector field X on M, the real-valued function on M (co,X){p):=(wp,Xp)p
(1.6.1)
is smooth. Comments (a) A very important application of these ideas is to classical Hamiltonian dynamics. If such a classical mechanical system has a configuration space Q then the phase space (i.e., the space of classical states) is defined to be the cotangent bundle T*Q. (b) Care needs to be taken when interpreting Eq. (1.6.1). It might look at first sight as if this were asserting that a one-form is a dual of the space of vector fields in the same sense that a cotangent vector is an element of the dual space of a space TPM of tangent vectors. However, this is certainly not the case if the set of vector fields is regarded as a vector space over U. since (co, X) as defined in Eq. (1.6.1) is a function on M, not a real number. The analogy does become closer if the vector fields are regarded instead as a module over the ring C^iM), but even then care is needed as these are infinite-dimensional spaces and topological subtleties can arise concerning the precise structure of a dual of such a space. It is safest for our purposes just to accept (1.6.1) as it
42
Modern Differential Geometry For Physicists
stands without trying to reinterpret it in the full language of duals of topological vector spaces. (c) As we saw in Sec. 1.3, any local coordinate system around a point p e M generates a basis set
(£), (£),
6 2)
°' '
of derivations for the tangent space TPM = DPM. As usual in linear algebra, there is an associated basis for the dual space T*M that is denoted (dx%,. .., (dxm)p
(1.6.3)
and defined by
{{dX%
)p =K
\w)
'-
(L6 4)
-
Thus, any k e T*M can be expanded as
k = 2 K (dxflX t=i
with
K - <*» \ — ))P • d-6-5) \dx"Jp
It also follows that the pairing (k, v) can be written entirely in terms of these local coordinate components as m
<M) = 2 V
(L6-6)
where (Eq. (1.3.13)) v" = v(x"). (d) These remarks extend to a local coordinate representation for one-forms and we write
Differentiable Manifolds
M=
m
W
2 H ( ) <**"'
Le
W =
"' P
43
m
2 ^ ^ ^dX^P ( L6 - 7 )
where the components o^ ( ) of the one-form &> are the functions defined on the open set U of the coordinate chart by coM(q):=(co,(-£A
),
(1.6.8)
for all q in U C M. Now we come to a most important property of one-forms. Namely, that even though it is not true in general that a map h:M'—- Nbetween two manifolds M and ./V can be used to "push forward" a vector field on M, it can be used to "pull back" a oneform on N. This feature of forms is of the utmost significance and, for example, is deeply connected with the way in which global topological structure of a manifold can be coded into the DeRham cohomology groups of M defined using differential forms, and the fact that fibre bundles also "pull back", not "push forward" (see later). The first step is to define the pull-back map h*:T*hip)N-> T*Mon the individual cotangent spaces: Definition (a) If h:M-* Nwe have the linear map h*:TPM->• Th(p) N, and then the map h*:T*ip)N-^- T*M is defined in the usual way as the algebraic dual of h*. Thus, for all k e T*h(p) N and v e TPM, (h*(k),v)p:={k,h*{v))h(p)
(1.6.9)
44
Modern Differential Geometry For Physicists
(b) It is a matter of the greatest importance that this definition can be extended to one-forms as follows. If co is a one-form on TV then the pull-back of co is the one-form h*co on M defined by (h*co,v)p:=(co,h*(v))h(p)
(1.6.10)
for all points p e M and all tangent vectors v e TPM. Comments (a) If M, N and P are three differentiate manifolds with maps h:M^Nand k:N-+P, then (cf. Eq. (1.2.9)) (hh)* = h*k*.
(1.6.11)
(b) The pull-back map h* is only defined on those cotangent spaces T*N for which the point qeNis such that there exists peM with q = h(p). It is therefore not strictly correct to think of h* as inducing a map from the complete cotangent bundle T*N into T*M. An important special case of the pull-back operation occurs when we have a vector field X on the manifold M with an associated local, one-parameter group of local diffeomorphisms t -*• (j>f. For each / e IR, we can define the pull-back one-form (also on M) 4>f*(co) of a given one-form co on M, and this one-parameter family of forms can be used to describe the way in which co changes along the flow lines of X as follows. Definition The Lie derivative Lx of a one-form co with respect to a vector field X on the manifold is defined to be the rate of change of co along the flow lines of the one-parameter group <$>? of local diffeomorphisms associated with X:
Differentiable Manifolds
Lxoi
: = — {?*co)
45
(1.6.12)
(=0
Comments It can be shown that [Exercise!]
(a) -{tf*co)
= 0** (Ljfi))
(b) Lx(co, Y) = (Lxco, Y) + (co, LXY) for all vector field X and Y. Recall that, on a function f, X{f), and that LXY := [X, Y].
(1.6.13) (1.6.14) Lxf:=
(c) In a local coordinate system, the components of the one-form Lxco are related to those of co and X by m
(Lxco)/J=X^,v^v+wvXv,,.
(1.6.15)
v=l
The Lie derivative Lxf:= X(f) of a function/along X arises also in the following definition of the "exterior derivative" of/— a concept that generalises to a general n-form (see later; it is convenient for this reason to think of a function as a "0-form") and which is of the greatest significance: Definition I f / e CX(M) then the exterior derivative off is the one-form df defined by
(df, X) := X(f) = Lxf for all vector fields X on M. In local coordinates:
(1.6.16)
Modern Differential Geometry For Physicists
46
W = X
77 ^ A -
(1-6.17)
Exercise Show that exterior differentiation commutes with pull-back in the sense h*{df) = d(fih)
(1.6.68)
for all/in C00 (A/). 1.7. GENERAL TENSORS AND n-FORMS In order to proceed from tangent and cotangent vectors to general tensors and n-forms it is necessary to introduce in some way the idea of the tensor product V <£> W of two vector spaces V and W. The most general and powerful approach is via the "universal factorization property" in which a tensor product of V and W is defined to be a vector space, written V® W, and a bilinear map fi: VX W -* V® W with the property that, given any other vector space Z and a bilinear map b : VX W-* Z there exists a unique linear map bs: V® W-+Z such that KX W JU / ^ . V® W
b /^°*
>Z ^ b = b*on v (8) w := ju(v, w).
(1.7.1)
It is easy to show that the pair (V ® W, /u) is unique up to isomorphisms. This definition is very general, but it is also rather abstract, and for our purposes it is useful to exploit the fact that,
47
Differentiable Manifolds
for finite-dimensional vector spaces, there is a natural isomorphism; : V® W-+ B(V* X W*, U) where B{V* X W*, R) is the vector space of bilinear maps from the Cartesian product V* X W* of the duals of V and W into the real numbers IR: j : V® W-+ B(V* X W*, U) :
j(v®w)(k,l):=(k,v)(l,w).
(1.7.2)
This can be regarded as an extension of the well-known fact that, if V is finite-dimensional, there is an isomorphism j : V -* (V*)* given by j : V -* V** j(v)(k):=(k,v)
(1.7.3)
for all vectors v e V and k e V*. Applying these ideas in the context of differential geometry we arrive at the definition: Definition A tensor of type (r, s) at a point p e M is an element of the tensor product space
77(M):=
TPM
®T*M
(1.7.4)
Comments (a) The notation ® V means the vector space formed by taking the tensor product of Fwith itself Mimes. It is convenient to allow this to include the value r = 0, with the product in this case being defined to be IR. In using this convention it should be remembered
48
Modern Differential Geometry For Physicists
that there is a canonical isomorphism W <8> IR s W for any real vector space W. (b) Using Eq. (1.7.2) and Eq. (1.7.3) for our finite-dimensional spaces TPM and T*M it follows that an alternative definition of a tensor of type (r, s) at the point p e M is as a multilinear map X T*M x X 7;M
• U .
(1.7.5)
from the Cartersian product of T*M r-times, and TpM s-times, into the real numbers IR. (c) Special cases of type (r, s) tensors include (i) T°/ (M) = T*M (ii) lj-° (M) = (T*M)* * TPM (iii) Tp° (M) is called the space of r-contravariant tensors. (iv) Tfs (M) is called the space of s-covariant tensors. (d) The definition extends at once to include the idea of a (r, s) tensor field on M. This is simply a "smooth" assignment of a tensor of type (r, s) to each point p e M. [Exercise: What would "smooth" mean in the case when the tensor product is defined by Eq. (1.7.4), and in the case when Eq. (1.7.5) is used instead?] (e)In the abstract definition (1.7.1) of the tensor product V® Wot two vector spaces F a n d W, the tensor product of v e F a n d we W is defined to be the element fx(v, w) e F ® W. In the more concrete definition (1.7.2), the tensor product o f f and w is defined to be the unique bilinear map from V* X W* into IR which takes on the values (k,v) (l,w) on (k,l) e F* X W*. This definition extends to multiple tensor products, and in particular If a e Tf (M) and a' e T/
(M) then a <8> a' e Trp+r''s+s' (M)
Differentiable Manifolds
49
is defined to be the multilinear map a®a'{ku.
. .,kr+r'\ vu. . . , t w ) := a(ku. . .,kr; vu. . ., v) X a v/cr+],. . .,kr+r'',vs+l, f
. . .vs+s,) (1.7.6)
(f) If dim(K) < oo and dim(W) < oo then d i m ( F ® W) = dim(F) dim( W). This is clear from the fact that if {eu e2,. . ., en} is a basis set for V and {fi,f2, • • -,fm} is a basis set for W (so « = dim(F) and m = d\m{W)) then a basis set for F <8> W is { e , ® ^ | / = 1 , . . . , n a n d _ / ' = l , . . . , m). In our case, this means that, with respect to a local coordinate system, a basis set for Tr/{M) is all vectors of the form —
| ®...<8>| — |
and the expansion coefficients respect to this basis are
<8» (dxv\
<8>... ® (dxVs)p
(1.7.7)
c/' "^... V j of a e Trp's{M) with
in which a e Tr/{M) is being viewed as a multilinear map from the product of T*M r-times and TPM s-times. A very important example of a tensor field is an rc-form where 0
X2,. . ., Xn) = ( - l) d e g ( P ) co (XPW, Xn2), . . ., Xp{n))
(1.7.8)
50
Modern Differential Geometry For Physicists
where Xu X2,..., X„ are arbitrary vector fields on M and deg(P) is the degree of the permutation P, i.e., it is 0 if P is even and 1 if P is odd. We will write the set of all «-forms on M asA"(M). If we have an «-form &>, e A"' (M) and to2 e A"7(M) then cox <8> co2 is a («[ + n2)-covariant tensor field but it will not be a («, + «2)-form since it does not satisfy the alternating property with respect to all its "indices", i.e., (eo, <8> co2) (Xu . .., Xn{; X„x + , , . . . , X„t + „2) will not in general be antisymmetric with respect to permutations of the indices that take an index k in the range l^k^rii into the range nl + l^k^n] + n2- This is remedied with the aid of the following definition: Definition If o)xeAn\(M) and co2eA"2(M) then the wedge product, or exterior product of w, and eo2 is the {nx + n2)-form cox- co2 defined by
cor co2 := -^— w
( - l)deg(/>) [
J
(1.7.9)
l - "2* PermsP
where, if co is any «-form, the "permuted" n-form cop is defined to be of (Xh X2,. .., X„) :=
OJ(XPW,
XP(2),...,
XP(n))
(1.7.10)
for all vector fields Xu X2,. .., Xn on the manifold M. Comments (a) This wedge product has many important properties with respect to the other structures that can be imparted to differential forms. For example, the pull-back of a 1-form defined in Eq.
Differentiable Manifolds
51
(1.6.10) can be generalised in an obvious way to an arbitrary nform. (In the case of a function/0-form, the pull-back of/e C°°(./V) by h:M-*N is defined to be the function h°f in CX{M), cf., Eq. (1.6.18) in this context.) This generalized pull-back has the very significant property of "commuting" with the wedge product. More precisely, if h :M^-N and if a and J3 are differential forms on N, then h*(a - 0) = (h*a) - {h*P).
(1.7.11)
Another major example of structure preserved by pull-back will appear below when we discuss the exterior derivative of a general rc-form. (b) The wedge product turns the vector space A{M) := dim(AO
® A" (M) into a graded algebra:
n= 0
w
i ' w2 = ( — I)"1"2 co2 ~ coi
(1.7.12)
if co{e A"' (M) and co2 e A"2 (M). (c) It can be shown [Exercise!] that a basis set for the n-forms evaluated at a point p e M is all vectors of the form (dxf,%'(dx%^...(dx'1% (1.7.13) and that if a differential form co is expanded locally in, terms of such elements then the coefficients in the expansion cofci2 ^are skew-symmetric under the exchange of any pair of indices. In Eq. (1.6.16) we defined the exterior derivative dfof a 0-form/ (i.e., in CX{M)) as a 1-form and showed that it commuted with pull-backs in the sense of Eq. (1.6.19). We wish now to extend this definition to an arbitrary «-form and to construct an "exterior
52
Modern Differential Geometry For Physicists
derivative" that will convert it into a (n + 1 )-form and in a way that still commutes with pull-backs. Definition If co is a «-form on M with l
dco{Xu .. ., Xn+l) := X (~ l) i + 1 X, (co(Xu . . ., %, . . ., Xn+X) +
^(-l)i+j(o([Xi,XJ], X\, • • ., Xh . . ., Xj,. . ., Xn+l)
for all vector fields X{, X2,. . ., Xn+l .
(1.7.14)
The symbol X means that that particular field in the series is ommited. Comments (a) There are no «-forms for n > dim(M) and therefore it is conventional to define dco := 0 if co is a dim(M)-form. (b) A case of particular importance in the theory of fibre bundles and Lie groups is when co is a 1-form. The somewhat complicated formula in Eq. (1.7.14) then simplifies considerably and the 2form dco acting on any pair of vector fields X, Y is dco(X, Y) = X((co, Y)) - Y((co, X)) - (co, [X, Y]}.
(1.7.15)
Note that the notation X((co, Y)) means the effect of acting with the vector field X on the function (co, Y) in CX(M).
Differentiable Manifolds
53
Comments/Exercise (a) If we expand a n-form a> locally as co = 1
2
dx" - dx" ....
coW2. .^
dx"" (with all indices summed over) then dco = co^j^.
Un, v
Jx v - dx"1 - dx"2...-
dx"".
(1.7.16)
(b) The exterior derivative behaves rather nicely with respect to the wedge product and the associated graded algebra structure on A(M): d{cox - co2) = da>i - a)2 + (—l)degtUl co, ~ dco2.
(1.7.17)
(c) With the definition (1.7.14) of exterior differentiation, we have commutativity with the act of pulling back. Thus, if h :M-^-N and to is a n-form on N then d{h*oS) = h*{dco).
(1.7.18)
1.8. DeRHAM COHOMOLOGY One of the most important mathematical applications of differential forms stems from the observation that if the exterior operator d is applied twice to any n-form co then the resulting (« + 2)-form d(d(co)) is identically zero. This result is usually written succinctly as d2 = 0
(1.8.1)
and is of the greatest significance. It can be proved directly from the general formula (1.7.14) for the exterior derivative; in local coordinates it is a simple consequence of the fact that mixed partial derivatives commute. We can then write down the sequence of real vector spaces:
54
Modern Differential Geometry For Physicists
Q^*Cco{M)^Ax{M)^A\M)
d
—
d
— Am(M)-» 0
(1.8.2)
in which the product of any two consecutive maps is zero by virtue of Eq. (1.8.1). Note that, for convenience, I have effectively introduced a space A~l(M) := {0} with d: A"'(M) — A°(M) = C^iM) being the trivial map. This sequence of maps and spaces is called the DeRham complex of the differentiable manifold M and is a special example of the more general concept of a differential complex which is defined to be any sequence of maps and vector spaces (unconnected with differential geometry in general): b
c
d
(1.8.3) with the property that the composition of any consecutive pair of the linear maps is zero, i.e., b°a = c°b = d°c = e°d = fie =
= 0
(1.8.4)
or, equivalently, lm(a) C Ker(Z>), Im(b) C Ker(c), Im(c) c Ker(rf).. . (1.8.5) i.e.,
The sequence of inclusions in Eq. (1.8.5) has a direct parallel in the case of differential forms with the observation that Eq. (1.8.1) implies in the sequence (1.8.2), Im(d--A"-i(M)^A"(M))cKer(d-A"(M)^An+\M)). Equation (1.8.6) motivates the definition:
(1.8.6)
Differentiable Manifolds
55
Definition (a) A n-form w is said to be closed if da) = 0. The set of all closed nforms is denoted: Z"(M):= Ker (d A"
(M)^An+\MJ)
= {coeA"(M)\dco = o}. A closed «-form is also said to be a n-cocycle for the DeRham cohomology. (b) A n-form a) is said to be exact if there exists some (n — 1 )-form /? such that dp = co. The set of exact «-forms is denoted: Bn(M):=
\m{dAn~x{M)^An{M))
= {weAn{M)\co
= dp for some fi e
A"-\M)}.
An exact «-form is also said to be a n-coboundary for the DeRham cohomology. It is clear from Eq. (1.8.6) that every exact n-form is closed and hence that the vector space B"(M) is a subspace of Z"(M). But when are they equal? This question of when a closed form is necessarily exact has far reaching implications since the problem of proving that a given closed n-form is exact involves global topological properties of the manifold. Indeed, it can be shown (Poincare's lemma) that on a euclidean space IRm every closed form is exact and hence the failure of Z"(M) to equal B"(M) does reflect in some way the degree to which the manifold M is not topologically trivial like the space U.m. It is desirable to find some precise mathematical construction that faithfully reflects the departure from exactness, and this is contained in the following definition:
56
Modern Differential Geometry For Physicists
Definition The DeRham cohomology groups HOR(M), 0
of the
(1.8.7)
Comments (a) Since A~l(M) := {0} it follows that B°(M) = {0}. On the other hand, a function fe A ° (M) is closed if and only if its partial derivatives vanish in any coordinate system. Since M is connected (by assumption) it follows that a closed 0-form is simply some constant function, i.e., a real number. Thus: fi°(M)^{0}andZ°(M)^[R
(1.8.8)
and hence, H°DR(M)^U
(1.8.9)
for all connected manifolds M. (b) On M = U, the most general 1-form can be written as cox = g(x) (dx)x with respect to the global coordinate system on U. But then, d da> = — g{x) dx-dx + g{x) d(dx) = 0 dx
(1.8.10)
and so every 1-form is closed. However, if we define f(x):=
\Xg{y)dy Jo
(1.8.11)
Differentiable Manifolds
57
then df= g(x)dx and hence the 1-form co is necessarily exact. This implies that HXDR{M) ^ {0} and indeed, since Poincare's lemma implies that for any manifold Um we have Z"(Um) =s B"(Um), it follows that for any IRm, HnDR {Um) =s {0} for n > 0 ^Uforn = 0.
(1.8.12)
(c) Now consider the one-dimensional manifold S1. Since this is connected, we have H%R (Sl) ^ IR as before. And, since Sl is onedimensional, it is also true that every 1 -form is necessarily closed so that A\Sl) s Z[(Sl). However, unlike the case M = IR it is not true that every 1-form is exact. For example, in angular coordinates, the one-form dB is not exact (not withstanding its appearance) since the angular variable 9 is not continuous or differentiable when considered as a function on the entire space Sl, and there is no differentiable function on the circle such that d9 = df. It is not difficult to deduce from this that HlDR(Sl)^U
(1.8.13)
which reflects the most important topological difference between the circle and the real line (the former is not simply connected but has 7T, (S*1) s Z).
(d) We saw in Sec. 1.7 that exterior differentiation commutes with the act of pulling-back in the sense that (Eq. (1.7.18)) d{h*w) = h*(daf)
(1.8.14)
where h :M-^N and co is a «-form on N. This equation has a very significant implication for the cohomological use of differential forms since it implies that
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Modern Differential Geometry For Physicists
(i) If a) is closed, then so is h*co. (ii) If co is an exact form with co = dp then h*(dp) = d{h*P) and hence h*co is also exact.
(1.8.15) (1.8.16)
Now if &>! and co2 belong to the same equivalence class in the quotient space Z"(N)/B"(N) then there must be some (n— l)-form P such that ft>i — co2 = dji. But then Eqs. (1.8.15-16) imply that the pull-backs h*(cox) and h*(co2) lie in the same equivalence class in Z"(M)/B"(M). Summarising: A differentiable function h:M -*• N induces a linear map h* :
HDR(N)
-* HoR(M)
This feature is an intrinsic property of any useful cohomology theory: the cohomology groups/vector spaces probe the topological properties of the manifold or topological space with which they are associated, and the properties of the maps between such spaces are reflected in the induced homomorphisms between the groups attached to the spaces. (e) We recall that two continuous functions / and g between topological space X and Y are said to be homotopic if they can be "deformed" into each other continuously. More precisely, there must be some continuous function F: [0,1] X X—- Ysuch that F(0, x) = f(x) and F{\, x) = g(x). A characteristic property of traditional cohomology theories (using simplicial decompositions for example) is that under these circumstances the associated homomorphisms J*:H"(7)
— H"(X) and g*-Hn(Y) ~* Hn(X)
(1.8.18)
are in fact equal. Precisely this property can also be shown to hold for the
Differentiable Manifolds
59
DeRham cohomology of a differentiable manifold. Thus, i f / a n d g are two differentiable functions from a manifold M to a manifold jVthen they induce identical homomorphisms (1.8.17) if they are homotopic to each other. (f) If follows from the above that the DeRham cohomology theory for differentiable manifolds has many general properties that are shared by the usual simplicial, or singular, cohomology structures. This is no coincidence. Indeed, one of the most remarkable and powerful results in the theory of the global structure of differential manifolds is DeRham's theorem which asserts that the DeRham cohomology groups are actually isomorphic to the real cohomology groups computed using singular or simplicial theory. This really is very striking since the singular/simplicial groups depend only on the underlying topological structure of a manifold (i.e., they know nothing of its differential structure of coordinate charts etc.) while, on the other hand, differential forms are basically defined in a local way that rests heavily on the differential structure: the ability to probe global properties stems entirely from the structure and special properties of the operation of exterior differentiation. This identification of the DeRham and "topological" cohomology groups can be used either way. The question of the existence of closed but non-exact «-forms can be answered at once if the(say) real, simplicial group H" (M, M) is known. Alternatively, there may be cases where the simplicial groups can be computed using the differentiable forms on the manifold. From all perspectives, DeRham's theorem is of major importance. (g) Another satisfying link between the various cohomology theories concerns the "cup product". In the conventional singular/ simplicial theory this is a pairing homomorphism: U : HP{X) <8> H"{X) — Hp+q{X)
(1.8.19)
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Modern Differential Geometry For Physicists
and it is the existence of this entity that makes cohomology more powerful than homology for many purposes. Is there an analogue of this cup product for the DeRham theory? A clue is given by the skew commutativity of the product in the simplicial/singular case: a\Jb = {~\)P9b\Ja
ifaeHp(X)
and beH9(X).
(1.8.20)
This is strikingly reminiscent of Eq. (1.7.12) which states that differential forms have a similar graded structure under the wedge product. What is suggested therefore is that the wedge product plays the role of the cup product in DeRham cohomology. This possibility is strongly supported by Eq. (1.7.17) which means that the conditions of closedness and exactness are preserved under the wedge product in such a way as to induce a linear map A : HpDR(M) ® HUM) - HpD+R"(M)
(1.8.21)
which satisfies Eq. (1.8.20). The full version of DeRham's theorem confirms this conjecture and shows that the wedge product is indeed an exact analogue for differential form cohomology of the cup product in the other theories.
2 LIE GROUPS
2.1. BASIC DEFINITIONS Lie groups are of the greatest importance in modern theoretical physics and the subject can be approached from a variety of perspectives. The discussion to be presented in what follows is elementary and is mainly concerned with the geometrical relation between a Lie group and its Lie algebra, and those aspects of Lie group theory that are especially appropriate for the third section of the Course dealing with fibre bundle theory. Definition (a) A real Lie group G is a set that is (i) A group in the usual algebraic sense. (ii) A differentiable manifold with the properties that taking the product of two group elements, and taking the inverse of a group element, are smooth operations. Thus, the maps fi-.GXG^G
and
v.G^G 9~*9~~X
{9I,9T)-~9I92
are both C°°. (Note that it suffices to consider (gu g2) -*• 9\ g^-) 61
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Modern Differential Geometry For Physicists
(b) A topological subgroup H of G that is also a submanifold is called a Lie subgroup of G. (c) The right and left translations of G are the diffeomorphisms of G labelled by the elements g e G and defined by: rg: G -*• G,
lg: G —>- G
g'^g'g
g>W gg'
(2.1.1)
Comments (a) One can also talk about real analytic Lie groups and complex analytic Lie groups in which the underlying manifold is required to be a real or complex analytic manifold respectively. It is a very deep and non-trivial theorem that every real Cx Lie group admits a real analytic structure, (i.e., a set of coordinate charts covering G can always be found with the property that the associated overlap functions are C"). This famous theorem is due to Montgomery, Gleason and Zippin; it constitutes the solution to Hilbert's fifth problem. (b) Note that the left and right translations satisfy the relations
^
2
= VV,
(2-1-3)
Furthermore, both maps g~*~lg and g-~-rg are injective and hence define respectively, a isomorphism and an anti-isomorphism from G into the group Diff(G) of difFeomorphisms of G. In general, a homomorphism between two Lie groups Gx and G2 (i.e., a morphism in the category of Lie groups) is required to be a smooth map between the two underlying differentiable manifolds as well as being a group homomorphism in the usual algebraic
Lie Groups
63
sense. This also applies to the maps G -*• Diff(G) afforded by the left and right translations in so far as the infinite-dimensional group Diff(G) can be equipped with the differential structure of a genuine Lie group. Examples (a) When equipped with its natural differential structure, the vector space U" becomes an ^-dimensional (abelian) Lie group. (b) The group of Mobius transformations of the complex plane is defined to be the subset of diffeomorphisms of the complex plane C of the form z—•
az + b cz + a
where a, b, c, a e C
with ad—bc= I. (2.1.4)
This is a 6-dimensional real Lie group and plays an important role in the geometrical approach to complex analysis, and in the theory of relativistic, interacting strings. (c) The circle Sl ^ {ew | 0 ^ 6 < 2n} can be given the structure of a real one-dimensional Lie group by using the usual differentiable structure and defining the group combination law as multiplication of complex numbers. This group is usually denoted U(l) (or just U\) and belongs to a series U(n),n=l, 2,. .. of Lie groups (see below). (d) The real, general linear group in n-dimensions is defined as GL(n, R) := {AeM(n, U) \ det(A) * 0}
(2.1.5)
and is thus the subset of all real, nXn matrices M(n, U) made up of those matrices that have an inverse. 2
The space M(n, U) is in bijective correspondence with IR" and
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Modern Differential Geometry For Physicists
as such inherits a natural topological and differentiable structure. Note that the map det: M{n, IR)~*~IR, A—^det(A), is continuous, and hence d e t - 1 {0} is a closed subset of M(n, IR). The subset GL(n, U) is the complement of this set, and is hence open. As such it acquires both a topological and a differentiable structure from M(n, IR), and it is with respect to this structure that it is a Lie group. It follows that the dimension of GL(n, R) is the same as that of M(n, IR), viz., n2, and that it decomposes into two disjoint components according to whether det(^4) is greater than, or less than, 0. (e) The complex, general linear group in ^-dimensions is defined as GL(n, C) := {AeM(n, C) | det04) # 0}
(2.1.6)
and, as an open subset of the 2n2 (real) dimensional manifold M(n, C), it acquires the structure of a 2n 2 -dimensional Lie group. Notice that, unlike GL(n, U), GL(n, C) is a connected space since a matrix A such that det(A) > 0 can be connected continuously to a matrix B for which det(5) < 0 by a path in GL(n, C) on which the determinant is a complex number. (Put in another way, removing the single point 0 from the real line IR disconnects it into two pieces whereas the same operation performed on C leaves a connected space.) (f) The connected component of GL(n, IR), (i.e., the component containing the unit element 1) is denoted GL+(n,U):=
{A e GL(n, U) \ det (4) > 0}
(2.1.7)
and is of course itself a differentiable manifold of dimension n2. To show directly that it is a subgroup of GL(n, U) note that: (i) d e t ( l ) = 1 so that the unit element 1 belongs to GL+(n, U),
65
Lie Groups
(ii) dQl(AB) = det(A)det(B) and hence A, B e GL+(n, U) implies that ABeGL+(n,U), (iii)det(/l~1) = [det(^)] _1 and hence A e GL+{n, U), implies that A~'eGL+(n,U), which are precisely the three conditions for a subset of a group G to be a subgroup of G. (g) The special linear groups are defined as the following subgroups of the general linear groups: SL{n, U) := {A e GL(n, U) | det(A) = 1}
(2.1.8)
SL(n, C) := {A e GL(n, C) | det(^) = 1}.
(2.1.9)
(h) The real orthogonal group is defined as a specific subgroup of GL(n, U) by 0(n, R):= {AeGL(n, U)\AA' =l\.
(2.1.10)
To see that this is a subgroup note that: (i) 11' = I s o that the unit element belongs to 0(n, U). (ii) If AA'=t and BB'=1 then (AB) (AB)' = ABB'A' = 1 and hence 0(n, U) is closed under multiplication. (iii)If AA'=± then A'1 = A'. Hence A"l(A~')' =A'A = A~lA = 1 and hence AeO(n,U) implies that A~l e 0(n, U). In terms of the matrix elements of the members of 0(n, U), the defining constraint AA'=l.is equivalent to the equations n
^AikAjk
= dy
ij=l,
•• ., n
(2.1.11)
k=\
which impose (n2 + n)/2 polynomial constraints on the matrices regarded as elements of M(n, U). The definition of 0(n, U) in terms of these \n2 + n)/2 polynomial
Modern Differential Geometry For Physicists
66
constraints plus the fact that &\m(GL(n, U)) = n2, suggests that 0(n, U) is a submanifold of GL(n, U.) of dimension n2 — {n2 + n)/2 — n(n— l)/2, and a more careful investigation shows that this is indeed so. However, even without embarking on such a detailed analysis, there are several important pieces of information that can be extracted from the general form of Eq. (2.1.11). For example, consider the chain of maps idx T
A
GL{n,U) A
-+GL(n,M)XGL(n,U) ~--
(A, A)
-* GL (n, U)XGL(n, U) ~~-
(A, A')
-^ GL(n, U) — - AA'
(2.J.12)
where T: GL(n, U) — GL(n, U) is defined by T(A) := A 'and// is the "multiplication" function /x: GL(n, U) X GL(n, R) — GL(n, U), H{A, B) := AB. Now, the composite function/? := ju ° (idX T) ° A is a continuous map from GL{n, U) into itself with the property that 0(n,U)=p-l@}.
(2.1.13)
But the inverse image of a closed set (such as {!}) under a continuous map is itself a closed set, and hence we deduce that 0(n, U) is in fact a closed submanifold of GL(n, U). Furthermore, Eq. (2.1.11) implies that n
n
so that 0(n, U) is actually a closed subset of the sphere
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67
so
-i) c Mfo U)^W . This in turn means that 0(/i, R) is a closed and bounded subset of the vector space M(n, U) and hence, by the Heine-Borel theorem, we deduce the very important result that the Lie group 0(n, IR) is a compact manifold. This has profound implications in many different directions; for example, the representation theory of a compact Lie group is far more tractable than that of a non-compact group. (i) In a very similar way the unitary group U(ri) is defined by U(n):= {AeGL(n, C)\AAf = l }
(2.1.15)
and is a compact Lie subgroup of GL(n, C) with real dimension n2. The special unitary group SU(n) is defined as SU(n) := {A e U(n) \ det(A) = 1}
(2.1.16)
and is a compact Lie group of real dimension (n2— 1). 2.2. THE LIE ALGEBRA OF A LIE GROUP One of the most important features of a Lie Group G is the fact that there is always an associated Lie algebra which accurately encodes many of the properties of the group. Rather remarkably this even includes certain global topological properties of G such as, for example, whether or not it is a compact manifold. The crucial property of a Lie group that enables this to occur is the existence of the left and right translations of G onto itself which can be employed to map local, tangent space related structure around the entire group manifold. The key definition is the following: Definition (a) A vector field X on a Lie group G is left-invariant if it is /^-related to itself for all g e G (cf.Eq. (1.4.16)) i.e.,
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Modern Differential Geometry For Physicists
lg*X = X
for all geG,
(2.2.1)
or, equivalently, lg*(X,) = X„.
for all g, g' e G.
(2.2.2)
(b) Similarly, Xis said to be right-invariant if it is rs-related to itself for all geG, i.e., r,*X
X
for all geG,
(2.2.3)
or, equivalently,
rAXg) = X,g
for all g, g' e G.
(2.2.4)
Comments (a) The set of all left-invariant vector fields on a Lie group G is denoted L{G). It is clearly a real vector space. (b) The critical observation now is based on the general fact (Eq. (1.4.19)) that if two vector fields Xx and X2 on a manifold M are /z-related to two vector fields Y{ and Y2 on a manifold N, where h:M -»• iV, then the commutator [X{, X2] is /z-related to the commutator [Yu Y2]. This means that if X, and X2 are two left-invariant vector fields on Gthen lg* I X X2] = [lg*Xu lg*X2], from Eq. (1.4.19) = [X[,X2].
(2.2.5)
Thus I , , I 2 e i ( G ) implies that [Xu X2] e L{G), and so the set L{G) of all left-invariant vector fields on G is a sw£> L/e algebra of
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69
the infinite-dimensional Lie algebra of all vector fields on the manifold G. It is called the Lie algebra of G. This results seems, and indeed it is, very elegant. But are there any left-invariant vector fields on any given Lie group G. And if there are, how many of them are there? Or to be more precise, what is the dimension of the real Lie algebra L(G)1 The answer to these vital questions is given comprehensively by the following theorem: Theorem There is an isomorphism of L(G), regarded as a real vector space, with the tangent space TeG to G at the unit element e e G. Proof We will construct an explicit isomorphism i: TeG —*• L{G) by exploiting the left-translation operation. Specifically, if A e TeG define i(A) := LA where LA is the vector field on G defined by LAg := lg*A
for all geG.
(2.2.6)
Note that, for all g, g' e G, /,.* {14) = lg,*(t9*A) = lg,g*A = LAg,g
(2.2.7)
so that LA is indeed a left-invariant vector field. It is clear that A -»• LA is a linear map and so it remains to prove that it is one-to-one and onto. (i) If LA = LB then in particular LA = Lf. But LA = le*A=A, and similarly Lf = B. Hence the map /: TeG -»• L(G) is injective. (ii) Now let L be any vector field in L(G) and define ALe TeG by AL := Le( = I -\*Lg for any g6 G).
(2.2.8)
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Clearly, lg*AL = lg*(Le) = Lg since L is left-invariant. i(AL) = L and hence i:TeG —- L{G) is surjective.
Thus QED
Note. It is clear that L -> AL is the inverse of i:TeG -»• L(G) so that lg*(AL) = Lg and lg-^(LA)
= A.
(2.2.9)
Corollary The dimension of the vector space L(G) is equal to dim(TeG) = dim(G). Thus since G is being assumed to be a finite-dimensional manifold it follows that L(G) is a finite-dimensional, non-trivial subalgebra of the Lie algebra of all vector fields on G. A significant implication of this theorem is that it should be possible to regard the tangent space TeG as the Lie algebra of the Lie group G. The idea of a commutator is defined only for vector fields, (i.e., not for vectors at a fixed point in G) and hence the Lie bracket on TeG has to be constructed from the commutator on the left-invariant vector fields L(G) with the aid of the isomorphism / : TeG —- L(G) discussed in the theorem. Specifically, it A, A' e TeG then the Lie bracket [AA'\ e TeG is defined to be the unique element in TeG such that L1AA']
= [LA,LA'}
(2.2.10)
[AA']:=[LA,LA']e.
(2.2.11)
or, equivalently,
The power and elegance of this way of viewing the Lie algebra of a Lie group is reflected nicely in the following result.
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71
Theorem Let / : G -* 77 be a smooth homomorphism between the Lie groups G and 77. Then the induced map f*:TeG-* TeH is a homomorphism between the Lie algebras of the groups. Proof Since/is a homomorphism we have, for all g, g' e G,f(gg') = Ag)f(Q'), and hence, for all g e G, M = W>/
(2.2.12)
Let A be in TeG with the associated left-invariant vector field LA e L{G) (so that A = LA) as defined in Eq. (2.2.6). Then, /* {L?) = f*lg* (A) = (filg)*(A) =
(lfl9ff)*(A)
= lf(g)*MA) = Lff?X)
(2-2.13)
which says that the vector fields LA and jJ*(A) are /related. But then, from Eq. (1.4.19), MAA'] = /* ([LA, LA'\) = = [l/*«\ Lf*(A\
[MLA),MLA'))f(e) = [f*(A)MA')]
(2.2.14)
and hence/*:TeG -^ TeH is a Lie algebra homomorphism. QED Comments (a)lf{Eu E2,. .., En}, n = dim(G), is a basis set for L(G) s 7^G, then the commutator of any pair of these vector fields must be a linear combination of them. Thus we can write
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[EmEfi] = ^Ca/Ey
(2.2.15)
for some set of real numbers C„/, known as the structure constants of the Lie group/algebra. (b) A similar construction to the above can be carried out with a set of right-invariant vector fields associated with the elements of TeG. Thus, the analogue of Eq. (2.2.6) is the definition RA:=rg*A
(2.2.16)
with the "inverse" AR := Re( = r -i*(Rg) for any g e G). It can be shown that [Exercise!] (i) [RA, RA'] = R[A'A] = R -{AA']
(2.2.17)
(ii) [RA,LA'] = 0
(2.2.18)
(in)RA = (Adg-J*LA
(2.2.19)
where the adjoint map Ad s : G—-G is defined for each g e G by Adg(g'):=gg'g~l.
(2.2.20)
We come now to a result that is of central importance in the theory of Lie groups and their associated Lie algebras. First we recall that an integral curve t -»• ax(t) of a vector field X is said to be complete (Sec. 1.5) if it is defined (or if it can be extended to a curve that is defined) for all values of the parameter t. We know that every vector field on a compact manifold is complete but that there are plenty of examples of incomplete vector fields on a noncompact space. The key result in Lie group theory is that every leftinvariant vector field on a Lie group G is complete, even if G is a
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73
non-compact manifold. Before proving this however we need a preliminary result concerning the "infinitesimal" form of the product group law on G. Thus, let JX:GXG —*• G be the group product/composition map, so that ^(gi,g2)'-= QxQi- We wish to consider the induced tangent space map H*'-T(gug2) GXG -> TgigiG and we will analyse this with the aid of the discussion in Sec. 1.3. In particular, there is a map x from T^ j) GXG onto the space TgG® Tg2G through which the induced map ji* factorises: T(9h92)GXG
>Tg{G®Tgfi x
/
jl
where JU* =
fi°x
(2.2.21)
T
9X92G
From Eq. (1.3.27) we have fl (a, fi) = in°ig)* a + (//%)* P
(2.2.22)
where a e TglG,fie Tg2G, iff2(g) = (g,g2) and jg>(g) = (gx,g) e GXG. Thus n°igi (g) = n(g,g2) = gg2 and n°jgi (g) = /i(g\,g) = g\g; which can be rewritten as / K 2 = r92 a n d A*% = hv
(2.2.23)
Hence, Eq. (2.2.22) becomes fi(a,fi) = re2*(a) + lgi*(fi)
(2.2.24)
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for all (a, /?) e T9x G 0 T?2 G. This is the result that we shall need in the proof of the main theorem. Theorem If X is a left-invariant vector field on a Lie group G, then X is complete. Proof (a) Let t -* (7*(/) be an integral curve of Xthrough ee G which is defined for all | /1 < e for some e > 0. Thus axJ —)= \dt)s
X x, foral\\s\<s ° <')
(2.2.25)
and hence,
which shows that t —- lg°ox(t) is an integral curve of X through the point g e G. (Thus we have used the left translation operation of the group G on itself to "pull" the original local integral curve to all over G.) It follows that if we define 4>?(g):= g er*(/),then / -»• ^is a local flow (Sec. 1.5) for the vector field X. This means that < ( < ( # ) ) = <+, 2 (#) and so ox{tx + t2) = ox{t,)ox{t2)
(2.2.26)
for all 11{ | < e, 1121 < £ and 11{ +12 \ < s. This equation can be read as saying that the map t -* ox(i) is a "local" homomorphism from the additive group R of the real line into the Lie group G.
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75
(b) In general, if t -> ax(t) is an integral curve for a vector field X on any manifold then it can be shown that [Exercise!] arX (t) = ax (rt)
for all r e U.
(2.2.27)
This suggests a means whereby the integral curves on G can be extended consistently using the group structure. Namely, if 11 \ < 2e then define
ax(t):= K U j j
( 2 - 2 - 28 )
which is consistent with the original definition if 11 \ < e since then, using Eq. (2.2.26), = {oxn (t)f = axl1 (20 = ax{t).
(2.2.29)
Clearly this procedure can be iterated to 11 \ < ne for any positive integer n, and hence to the whole real line. Thus the crucial task now is to show that the curve defined in (2.2.28) is in fact an integral curve of X. (c) If, temporarily, we denote the curve defined on the left-hand side of Eq. (2.2.28) by ax, then we need to show that for |s| < 2e, a% (d/dt)s = Xax(s). To do this we factorize the expression (2.2.28) as ax = /LI°A°OX°H where H: IR — U is defined by H(i) := t/2, and A:G — GXG is the usual diagonal map A(#) = (g,g). Thus sX
*\Jt),= ^*A*aX* \jjs
=
^* A *fc<J = >* oA *(* v J
where we have used the factorization from Eq. (2.2.21). Then, from Eq. (1.3.30),
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Modern Differential Geometry For Physicists
^{jtj^^i^ny^ism) (2.2.30)
where we have used Eq. (2.2.24). However, the left and right translations of a group commute, (i.e., rg°lg = lg°rg for all g e G) and hence the first term on the right-hand side of Eq. (2.2.30) can be written as (using also the fact that the vector field X is leftinvariant):
But, from Eq. (2.2.26), rax°crx = lax°ox and hence x x r x,„*X (r x°(T — = (ra x°o — w e = v w )*\ w )*\ " " ' \dtJo ' \dtj o
= U * ^ = ^>-
( 2 - 2 - 32 )
Thus the right-hand side of Eq. (2.2.31) is x x = x > anc t m s s t n e n r s t t e r m o n lo um*Xo (s/2) X[o (s/2)\ -> * * the righthand side of Eq. (2.2.30). But the second term on the right-hand side of this equation is clearly the same, and hence Eq. (2.2.30) reads
^*(i)=^w = x ^>
<2-2-33)
which is precisely the statement that dx defined by Eq. (2.2.28) is
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Lie Groups
an extension of the original integral curve ax from the interval 111 <e to 111 < 2e. Integrating the procedure then shows that the integral curve of X can be extended to all values of t e IR and hence the left-invariant vector field X is complete. QED Definition A
A
(a) The unique integral curve t—*oL (t),A = o* (d/dt)0 of the left-invariant vector field LA that is defined for all t eM. by virture of the above theorem is written as t — exptA
AeTeG.
(2.2.34)
(b) The exponential map exp : TeG -* G is defined by exp (A) := exp tA |, =1 .
(2.2.35)
Comments (a) It can be shown that the exponential map is a local diffeomorphism from the tangent space TeG at the unit element ee G into G. Thus there is some open neighbourhood Vof 0 e TeG such that the exponential map restricted to Fis bijective, and both it and its inverse (defined on the image of V in G under exp) are smooth. (b) If G is a compact Lie group then the exponential map is a surjective function from TeG onto G. It is then of necessity not an injective map as else it would establish a diffeomorphism between the non-compact vector space TeG and the compact group G. The statement in (a) still applies however and hence the map is one-toone in some neighbourhood of ee G. (Note. I have assumed here that G is connected).
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If G is not compact then the exponential map may not be surjective (there are some famous simple examples of this phenomenon). On the other hand it is now possible for it to be globally injective. (c) A one-parameter subgroup of a Lie group G is a smooth homomorphism ^ from the additive group of the real line U into G, i.e., for all tu t2e U, we have
nUx + Q = nUi) M-
(2.2.36)
It follows from Eq. (2.2.26) (and its equivalent for the fully extended integral curve) that t—^exp tA is a one-parameter subgroup of G. The following proposition shows that the converse is also true, viz., every one-parameter subgroup of G is of the form ?—*-exp tA for some A e TeG = L(G). Thus there is a one-to-one association between one-parameter subgroups of the Lie group G and its Lie algebra. The neighbourhood of ee G onto which exp:TeG -»• G maps diffeomorphically is "filled" with the images of these subgroup maps. Theorem If /j.: U -> G is a one-parameter subgroup of G then, for all t e U, p,(t) = exp tA where A := /u*(d/dt)0. Proof Eq. (2.2.36) implies that /u°ls = l^ffi for all s e IR. Hence
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79
and hence t~^-fi(t) is an integral curve for LA e L(G). The result follows from the uniqueness of such curves. QED Corollary Let f:G —*- H be a smooth homomorphism between two Lie groups G and H whose exponential maps are denoted expG ;TeG-+ G and expH : TeH —» H respectively. Then we have the commutative diagram:
exp„
ForalMe^G,
expH(f*A) = f(expGA).
(2.2.37)
Proof Define n:R — Hby fi(t) := /(exp G tA). Then /i(ti + t2) =f(expG(t{
+ t2)A) = /(exp G /,^ expGt2A)
= /(exp G txA)f(expGt2A) = //(/,)//(f2) so that fi is a one-parameter subgroup of //. It follows from the theorem that ju(t) = expHtB
where B:= }i*\—) eTeH. \dtJo
(2.2.38)
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Let keCx(H). Then, from Eq. (1.3.17), B(k) = lj)
(/co//) = j t (k(M(t))) \,=o = j t (W(exp c tA)) t=0
A
= L e{kof) where the last step follows because t — expGtA is an integral curve for the left-invariant vector field Z/. But L? = A hence B(k) = A(kof) = (/* A)(k), i-c, B = MA). Thus Eq. (2.2.38) reads, /(exp c L4) = n{t) = exp„ (tf*{A))
(2.2.39)
which, with t=\, proves the result in Eq. (2.2.37). QED Corollary The adjoint mapping of G onto itself is Adg(g') := gg'g~l for each g eG. Then, exp(Ad,*(5)) =gexpBg-1
for all B e TeG. (2.2.40)
Proof For each j e G , the mapping Adff: G -»• G is a homomorphism of the Lie group onto itself. Therefore, from the corollary above, exp (Adg*(B)) = Ad? (exp B) = gr exp B g ~' .
QED
Note. The map # —*- Ad,,* gives a representation of G on 7^G = L(G) known as the adjoint representation. Note that its kernel is the centre of G.
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Lie Groups
Example (a) Let us consider an explicit example of a set of left-invariant vector fields for the connected component GL+ (n, U) of the general linear group GL(n, M). This group appears as an open subset of the linear space M{n, U) of all real nXn matrices and the tangent space at any point in G (and especially at ee G) can be identified in a natural way with the vector space M(n, U), which can therefore in turn be associated with the Lie algebra of GL+(n,R). A natural system of coordinates on GL+(n, R), which are certainly valid in some neighbourhood of the unit matrix 1, are the "matrix-element coordinates" defined by x'J (9) '•= 9>J
where g e GL+ (n, U) and /, j= 1, . . ., n.
Let AeTeG ^ M{n, U). Then the coordinate representation of the left-invariant vector field associated with A is
L^tpL^'i(~)t
(«-4.)
and {LAxu)g = — {x'J (g exp tA))t.Q.
(2.2.42)
However, A e M(n, R) is a matrix and hence we can consider the curve t-~-elA in GL + (n, U) where etA refers to the usual exponential of a matrix. But the tangent vector to this curve at t=0 is obviously the matrix A and, furthermore, it is clear that it defines a one-parameter subgroup of GL + (n, U). Hence e'A = exp tA
for all * e U and AeTeG^
M(n, U)
and, in particular, Eq. (2.2.41) can be rewritten using
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LAx'J =
at
-x'J{ge'A)
I^TM") dt
1=0
k=\
1=0
n
k=l
Hence we can write the left-invariant vector field as ik
jkj
gmAKJ\—-
(2.2.43)
in which the indices /, j , k are all summed from 1, .. ., n. (b) In a similar way, one can show [Exercise!] that the rightinvariant vector field on GL+ (n, U) can be written in coordinate form as Rf = Aikgki\
dxIJ,
(2.2.44)
It should be noted that the representation (2.2.43) gives \LA', LA\ = L[A,yA] where [A', A] is the usual matrix commutator, i.e., the Lie algebra structure induced on TeGL+ (n, U) ^ M(n, M) is just the commutator of the matrices. A natural basis for M(n, U) is the set of matrices Ei} defined as (Ej)ki := Sik Sjh and the associated left-invariant vector fields are Li = 9k'
k
(2.2.45)
tdx
2.3. LEFT-INVARIANT FORMS The concept of left-invariance for vector fields on a Lie group G has a natural analogue with respect to the dual structure of
Lie Groups
83
differential forms. Two of the most important applications of leftinvariant forms are in the development of a group invariant DeRham cohomology for Lie groups, and in the theory of connections in fibre bundles. The latter topic has been much discussed by theoretical physicists in recent years in relation to the general mathematical framework for Yang-Mills theories and, in particular, in the investigations of anomalies in the quantization of such systems. The discussion of invariant differential forms given in this Course is not very comprehensive and is mainly concerned with developing the minimal set of tools necessary to set up the theory of connections in bundles. Definition An n-form co e A"(G) on a Lie group is said to be left-invariant if, for all g E G, l*to = co,i.e.,l*(cog) = co-t , for all #'e G.
(2.3.1)
A right-invariant differential form is defined in an analogous way. Comments (a) From Eq. (1.7.18) we have I* (dco) = d(l*co) and hence if co is left-invariant so is its exterior derivative dco. This is one of the starting points for the development of an invariant version of the DeRham differential form cohomology. (b) We showed in Sec. 2.2 that there is an isomorphism i:TeG ~^L(G) in which i(A) = LA. Similarly, there is an isomorphism between T*G and the set L*(G) of left-invariant one-forms on G. This associates to each de T*G the left-invariant one-form: Xdg:=lt-x*(d)eT$j.
(2.3.2)
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In this context, T*G ^ L*(G) is called the dual Lie algebra of G. However, it should be noted that this nomenclature is potentially confusing in so far as L*(G) is not itself a Lie algebra, it is only the dual to the underlying vector space of the Lie algebra L(G). Note also that the left-invariant one-forms defined in Eq. (2.3.2) stand in a dual relationship with the left-invariant vector fields LA according to the equation {Xd, LA)g = id, A) for all g e G
(2.3.3)
where (d, A) refers to the pairing between A e TeG and d as an element of the algebraic dual vector space T*G. (c) In Eq. (2.2.15) we introduced the idea of structure constants with respect to a particular basis {Eu E2, • .., En}, n = dim(G) of left-invariant vector fields: [Ea, Efi] = Ca/Er
(2.3.4)
We can define a dual basis \col, a>2,..., co"\ for L*(G) by (of, Ep) := 8$.
(2.3.5)
In order to derive an analogue of Eq. (2.3.4) for this basis of oneforms it must first be noted that, whereas the commutator of two vector fields is itself a vector field, the natural combination of a pair of one-forms (with a wedge-product) leads, not to a one-form, but a two-form. However, a two-form can also be reached from a one-form by exterior differentiation, which suggests that both this operation and the operation of taking a wedge product may be involved in the analogue of Eq. (2.3.4) for a basis of left-invariant one-forms. We can substantiate this intuition by examining the two-forms dof via their effect on a pair of the basis elements for L(G) using Eq. (1.7.15):
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Lie Groups
dof (E,,, E) = E„ {{of, E)) - Ey {(coa, Efi)) - (of, [Efi, Ey}) (2.3.6) But (of, Ey) = d; which is a constant and is hence annihilated by the vector field Ep. Thus, using Eq. (2.3.4) in the last term of Eq. (2.3.6) we find dtoa(ElhEy)=
-C^.
(2.3.7)
However, from the basic definition of the wedge-product in Eq. (1.7.9) we have of - of (E„, E) = cos® coe (Ep,
Ey)-fi**y
= s$s;-s*sep which can be combined with Eq. (2.3.7) to give the famous CartanMaurer equation: dof + jC,}ya co11- ojy = 0.
(2.3.8)
Definition The Maurer-Cartan form S is the L(G)-valued one-form on G which associates with any v e TgG the left-invariant vector field on G whose value at g e G is precisely the given tangent vector v. Specifically, if (S, v) denotes this left-invariant vector field then, <S,i>> (?'):= W , - i * w )
veTgG.
(2.3.9)
Comments (a) The Cartan-Maurer form is left-invariant. (b) On the left-invariant vector fields LA, Eq. (2.3.9) reads {E,Lf){g')
= Lt.
(2.3.10)
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86
(c) Since L(G) = TeG it is also possible (and potentially less confusing perhaps) to regard the Cartan-Maurer form as taking its values in TeG, rather than in L{G). With this interpretation, Eq. (2.3.10) becomes {z,Lf)
=A
(2.3.11)
which, in fact, serves to define S precisely. In particular, when G = GL(n, U) with TeG ^ M(n, U), we can think of E as a M(n, IR)-valued, (i.e., a matrix-valued) differential form on the group. Using Eq. (2.3.11), and the explicit expression Eq. (2.2.43) for LA in the case of GL(n, U), we see that Zv = (g-iyk(dx%
(2.3.12)
(d) Let Q:M—>- G where Mis some differentiate manifold (if Mis a space-time or physical space, Q could be thought of as a "gaugefunction" associated with a Yang-Mills theory using the Lie group G). Then Q.*E is a L(G)-valued one-form on M. When G is a matrix group, (i.e., a subgroup of some GL(n, IR)) then we can use (2.3.12) to write
s,
(<"* "(^l)
= (E,j, a*
dxr
a(x)
(n->(x)yk(dxk%ix),n*\ = ( Q - ' ( x ) ) ' M ^ - | (xkj)
=
(si-1(x))ik^-x»(n(x))
dx\
aw
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87
and so,
(n*3)j< = (
(2.3.13)
which is often written rather symbolically as Q*Z = £l~'dQ.
(2.3.14)
Exercise Show that under the action g-~~- rg of right translation of G onto itself, the Cartan-Maurer form transforms as r„*(S) = Ad,_1*(3)
(2.3.14)
in the sense that, for any v e Tg,G, (rg*(E), v) is the element of TeG given by Ad,_, *((S, t>)). 2.4. TRANSFORMATION GROUPS In almost all applications of group theory in theoretical physics the groups arise as groups of transformations of some other space. This motivates the following sequence of definitions: Definition A group G is said to act on a set M on the left, or to have a left action on M if there is a homomorphism g-~~-yg from G into the group Perm(M) of bijections of M. Thus (i) ye(P)=P (ii) ygi°ygi = ygm
forallpeM for all gu g2 e G.
(2.4.1) (2.4.2)
Comments (a) The image point yg (p) is usually written as simply gp, in which case Eq. (2.4.1-2) become
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(i) ep = p (ii) ffiidiP) = (gigi)p
forallpeM. for all gu g2 e G.
(2.4.1') (2.4.2')
(b) There is an analogous definition of a right action of a group G on a set M. This is an anti-homomorphism g —— 5g from (7 into Perm(Af). Thus (i) de{p)=p
forallpeM.
(2.4.3)
(ii) V ^ 2 = < U
f o r a11
2 4 4
#" ^
G G
-
(-->
Writing the image point Sg(p) in this case as pg, Eqs. (2.4.3-4) become (i) pe = p (ii) {pgx) g2 = p{gig2)
forallpeM for all gu g2 e G.
(2.4.3') (2.4.4')
(c) If Af is a topological space (respectively differentiable manifold) then it is natural to require that the bijections involved are restricted to the subgroup of Perm(M) of homeomorphisms (respectively diffeomorphisms) of M. However, if G is a topological, or Lie, group then in accordance with the usual concept of a morphism it would be natural to put a topological (respectively differential) structure on the group of homeomorphisms (respectively diffeomorphisms) of M and then to require that the maps g~~»- yg or g~~~-Sg are continuous (respectively differentiable). This is not impossible, but it is a difficult and complicated business in general since these groups are infinite-dimensional and hence one becomes involved with the non-trivial subtleties of infinite-dimensional topology/differential geometry. This seems an unnecessary complication, especially since most of the time the Lie groups G with which we will be concerned are themselves finite-dimensional. The way out of this problem is contained in the following definition.
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89
Definition A left action of a Lie group G on a differentiable manifold M consists of a homomorphism g—~yg from G into the group of diffeomorphisms Diff(Af) of M, so that Eqs. (2.4.1-2) hold, and such that the map r : G X M ->• M defined by V:GX
M^
M
(9,P)-~Yg(p)
= 9P
(2-4.5)
is a smooth map from the differentiable manifold G X M into the manifold M.
Comments (a) The definition of a left action is often given directly in terms of this map r : G X M—• M, in which case the basic homomorphism conditions in Eqs. (2.4.1-2) become (i) T(e, p) = p for all p e M (ii) r(gu r(g2, p)) =T(gl g2, p) for all £„ 02 e G.
(2.4.6) (2.4.7)
(b) There is of course an equivalent definition for a right action of a Lie group on a differentiable manifold. Note that, in practice, there is no need to consider left- and right- actions separately since they are in one-to-one correspondence. Thus, given a left action g •~*-y„, a right action can be defined by Sg := y _, and vice versa. (c) There is a well-defined concept of a morphism between a pair of group actions of a Lie group G on manifolds M and M'. Specifically, a map f:M—M' is said to be equivariant with respect to the group actions g ~*- yg and g ~~*~ y'g on M and M' respectively if the following diagram commutes;
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y'g°f = H
(2.4.8)
for all geG.
Note that Eq. (2.4.8) is equivalent to the statement: Sf(p) = f(gp)
for all geG and p e M.
(2.4.9)
A more sophisticated version of this idea allows for two Lie groups G and G', one acting on M and the other on M'. A morphism in this case consists of a homomorphism p.G^-G' and a map f:M -*• M' such that GX M-
-*-G' X AT
pxf r M-
>M'
r°Pxf=f°r
(2.4.10)
i.e., for all g e G and pe M, P(9)f(p)=f(gp).
(2.4.11)
We come now to a series of definitions concerned with various aspects of the action of a Lie group G on a differentiable manifold M. Comments and examples are interspersed with the definitions proper. Definition (a) The kernel of the G-action is the subgroup of G defined by K:= {geG\gp
= p for allpeM}.
(2.4.12)
The group action is said to be effective if K = {e}. The kernel measures the "part" of the group that is not represented at all in the G-action on M. It is a normal subgroup of
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91
G and the action of G on M passes naturally to an action of the quotient group G/K. The action of G/K on M is effective. An example is afforded by the adjoint action of G on itself in which Adg(g'):=gg'g-1.
(2.4.13)
The kernel of this action is the centre C(G) of G. Thus, for example, in the adjoint representation of SU{3) on itself the Z3 centre is represented trivially and it is the quotient group SU(3)/T3 that acts effectively. The same remarks apply to the induced linear representation Ad,,* on the vector space TeG. Thus the eight dimensional representation of SU(3), (i.e., the "octet" representation) is a faithful representation of S£/(3)/Z3. (b) The G-action is free if, for all pe M,{g\gp = p} = {e}. Thus in a free action every point in M is moved away from itself by every element of G, with the exception of the unit element e only. In particular this means that, given any pair of points p, qe M, either (i) there is no g e G such that p = gq, or (ii) there is a unique g e G such that p = gq. Note that a free action is effective but the converse is not true. For example, any faithful linear representation of G is effective but it is never free since, if 0 is the null vector in the representation vector space then #0 = 0 for all g e G, by linearity. A well-known example of a free action of G is left (or right) translation on itself. On the other hand, the adjoint action in Eq. (2.4.13) is not free since e e G is left fixed by every g e G, i.e., Adg (e) = e for all g e G. (c) The G-action is transitive if any pair of points p, q e M can be "connected" by an element of the group. More precisely, for all p, qeMthere exists ge G such that p = gq.
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In a transitive action one could say that, in some appropriate sense, the whole of M can be "probed" by the G-action. This is complementary to the idea that in an effective action, the whole of G can be "probed" by the action on M. A very important example of a transitive G-action is when M = G/H where H is a closed subgroup of G. The left action of G is defined by yg(g'H):=gg'H.
(2.4.14)
This action is transitive for any pair of groups G and H, irrespective of whether they are Lie, or even just topological, groups. However, in the Lie group case, with H a closed subgroup, it can be shown that there is a unique analytic manifold structure on the coset space G/H such that G is a Lie transformation group on this space (e.g., Helgason's book). Note that this particular action is effective if and only if H contains no normal subgroup of G. It is never free since h{eH) = eH for all heH. Note also that a linear representation is never transitive since the null vector 0 cannot be "joined" to any other element in the vector space by the action of any g e G. (d) Given that a generic G-action will not be transitive, it is useful to define the orbit 0pof the G-action through pe M as the set of all points in M that can be reached from p: 0P := {q e M | there exists geG with q = gp}. (2.4.15) It is clear that the orbits through any pair of points in M are either equal to each other or are disjoint. This leads to an equivalence relation on M in which two points are defined to be equivalent if and only if they lie in the same orbit. As usual, M is partitioned by this equivalence relation:
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93
The set of equivalence classes is called the orbit space of the G-action on M and is denoted M/G. This space carries a natural quotient topology, but it is frequently rather unpleasant (eg., non-Hausdorff). Note that the orbit space of the right action of a subgroup H of a group G on G is just the space of left cosets G/H referred to above. In the Lie case, with H a closed subspace, the orbit space does of course have a decent topological/differential structure, namely the unique analytic structure mentioned in (c). (e) The stability group!isotwpy group!little group Gp of the action is denned at each peMby: Gp := {g e G \ gp = p)
(2.4.16)
which, in the Lie case, is a closed subgroup of G for each pe M. Note that the kernel of the action is related to the stability groups by K=
n GP.
(2.4.17)
peM
If p and q lie on the same orbit of the enaction on M then there is some g e G such that p = gq. It follows that: Gp = gGqg~' so that the stability groups along an orbit are always conjugate.
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This concludes the batch of definitions relating to the various features of an action of the Lie group G on the differentiate manifold M. Now we come to a theorem of considerable significance, especially for the use of global features of Lie groups in theoretical physics. We have seen above that the action of G on the coset space G/H is transitive. The theorem we are referring to asserts the converse: if M i s any space on which G acts transitively then it is effectively a space G/H for some H. The word "effectively" covers up rather a lot of sins and it is best to state first the result from a purely group/set theoretic perspective, with no reference to topology or differential structure: Theorem Let G be a group that acts transitively on a set M. Then, for each pe M, there is a bijectiony^ : G/Gp —»• M, where Gp is the stability group at the point p, defined by jp:G/Gp-*M gGp-~gp.
(2.4.18)
Proof (a) First we must ensure that the map in Eq. (2.4.18) is welldefined. This is always necessary when, as here, a function is defined on some space of equivalence classes in terms of a particular representative in each class; i.e., one needs to show that the definition is independent of the particular representative that is selected. In the present case we note that if #, and g2 are in the same coset then gx = g2h for some h e Gp and hence g{p = g2hp = g2p. Thus the definition in Eq. (2.4.10) is indeed independent of the representative chosen from the coset.
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95
(b) Suppose j(g{Gp)=j{g2Gp). Then gxp = g2p and hence g\XgiP = P. which means that g^xg2 belongs to the stability group Gp, which is precisely the condition that gxGp = g2Gp, (i.e., gx and g2 belong to same right G^-orbit). Thusjp is one-to-one. (c) Since the G-action is transitive, every point in Mean be written as gp for some g e G. Hence jp is surjective. QED Comments (a) When G is a Lie group acting smoothly and transitively on a differentiable manifold M, the crucial question is whether or not the bijection jp in Eq. (2.4.18) is a diffeomorphism. The two major results in this direction are: (i) The stability group Gp is a closed subgroup of G and hence G/Gp can be equipped with the unique analytic manifold structure referred to before, (ii) If M is locally compact and connected, and if G is a locally compact Lie group, then the bijection j p is a diffeomorphism from G/Gp onto M. (b) If the G-action on Mis not transitive then M decomposes into a disjoint union of orbits, on each of which G does act transitively and with a specific stability group. It follows that each orbit Opis in bijective correspondence with the coset G/Gp where Gp is the stability group for peOpC M. The critical question in the Lie group case is then to decide when the spaces G/Gp can be regarded as embedded submanifolds of M; which first requires finding out when the isotropy groups of the different orbits are closed. This is a complicated question and we refer to the literature for further information on these topological matters. We will now give several important examples of familiar manifolds that admit a transitive group of transformations by a
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Lie group G and are hence diffeomorphic to a coset space G/H for some subgroup H of G.
Theorem
The n-sphere S" is diffeomorphic to 0(n+1, U)/0(n, U).
Proof
(a) According to the main theorem we must find a transitive action of the group 0(n + 1, IR) on the n-sphere and then show that the stability group at any conveniently chosen "fiducial" point is a 0(n, U) subgroup. «+i
Let (v, w) = X vt w, denote the usual inner product on the real vector space [R"+1 where [vt | /= 1 , . . . , n+ 1} are the components of the vector v with respect to the usual orthonormal basis set {(1,0, ..., 0), (0,1, .. ., 0), . .., (0,0, .. ., 1)}. Then the key observation is that the linear action of <9(n+l, IR) on this vector space leaves invariant the inner product, and hence in particular the set of unit length vectors: Sn:={veUn+l\(v,v)
= l}.
This is the 0(n+ 1, U) action on the n-sphere that we shall use, and therefore the first task is to show that the group acts transitively. (b) Let v° := (1, 0, . . . , 0) e S" c Un+i and let t; be any other unit length vector. Choose a set of orthonormal vectors {e\ e2,.. ., e"} such that the complete set {v, e\ e2,. .., e"} is an orthonormal basis set for Un+1. Then we have the matrix equation v = Av° :
Lie Groups
97
»i
V2
e{
e{
el+x
J
n+\
(2.4.19) The choice of the vectors {e\ e2,. .., e"} is such that the matrix A satisfies the equation AA' = A'A = l a n d hence A e 0(n+ 1, IR). Since v was any unit vector in the n-sphere, this shows that the action of 0(n+ 1, IR) is transitive on this space. (c) The isotropy group at the fiducial vector v° is clearly all matrices of the form
t\
0
0 0
0 . . ..
o\
X
0
The nXn matrix X is any such satisfying XX' = X'X = l „ x „ . Hence the isotropy group is this particular 0(n, U) subgroup. Comments
(a) If the vectors {e\ e2,. .., e"} are chosen such that the orientation on IR"+I induced by the orthonormal basis set {v, e\ e2,. . ., e"} is the same as that arising from the usual basis, then the matrix A in Eq. (2.4.19) satisfies det(^4) = 1 and hence belongs to the SO(n+l, U) subgroup of 0(n+\, M). It then follows that
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98
Sn^SO(n+l,U)/
(2.4.20)
SO(n,
(b) In a similar way, we can consider the invariance of (v, w) = £ vfWi under the linear action of U(n+ 1) on C
. This
i=i
action will map into itself the (2n+ l)-sphere 2n+l
,+
S
= {veC
l
n+l 2
\(v,v)=^j\vi\
=
1}
/=i
and, as in the case above, it can be shown that this action is transitive. The stability group is a U(n) subgroup, which proves the result S 2n+l
U(n+l)/U(n)
A particular example is when n=\. that S 3 s U(2)/U(l)
^ SU(n+
\)/SU{n)
Since SU{\)
(2.4.21)
= [e\ it follows
(2.4.22)
^ SU(2).
This result, that the group space of SU(2) is diffeomorphic to a 3sphere, is important in a number of different ways. It can also be obtained directly from a study of the implications of the defining relations 'a
b\ (a*
c*}
,c
d)\b*
d*j
• for a matrix
a \c
b\ d
(::)
and Det
'a
^
x
di
to belong to SU{2). [Exercise!]
= 1 (2.4.23)
Lie Groups
99
(c) Another important example of a transitive group action can be derived by considering the linear action of SU(n+1) on C" +1 . This clearly maps complex lines into complex lines, and hence passes to an action on the complex projective space CP" which is defined to be the space of all such lines in C" +1 . Unlike the original linear action, this SU(n+ 1) action is transitive, and the little group is a U(n) subgroup. Hence: CP"^SU(n+\)/U(n)
(2.4.24)
A particular example is CP 1 ^ SU(2)/U(l) which, since CP1 =s S2 (the complex Riemann sphere) proves the famous result S2^SU(2)/U(\).
(2.4.25)
(d) Similarly, the SO(n+ 1, U) linear action on IR"+1 generates the result UP"^SO(n+l,U)/0(n,U).
(2.4.26)
Theorem Let S„ := {nXn, real, positive-definite, symmetric matrices} Then
Sn^GL
+
(«, U)/SO(n, U).
(2.4.27)
Proof (a) We define the left action of GL + (n, U) on M(n, U) by yg(A) := gAg', geGL + (n,U) and AeM(n, U).
(2.4.28)
The first step is to show that the subspace S„ C M(n, U) is mapped into itself by this action.
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Modern Differential Geometry For Physicists
(i) If A = A' then [gAg']! = (g')'A'g' = gAg', so that the image of a symmetric matrix is symmetric. (ii) To say thatv4 is positive-definite means that (v, Av) > 0 for all v e M." such that D ^ O , where we regard A as a linear operator on the real vector space IR" with the inner product (v, v) defined as usual. Then (v, (gAg')v) = (g'v,A(g'v)) since the transpose of a matrix is defined such that (v, Bw) = (B'v, w). But ge GL +(n, IR) is invertible, and hence (g'v,A(g'v)) > 0 for all v =£ 0 if and only if (v, Av) > 0 for all v =£ 0. Hence if A is positive-definite, so is gAg' for all ge GL+(n,U). These two statements show that S„ is indeed mapped into itself under the GL+(n, U) action defined in Eq. (2.4.28). (b) To see that the action is transitive we note that, since a symmetric matrix S is diagonalizable, we can write S = ODO' for some Oe SO(n, U) and diagonal matrix D = diag(W,, d2,. . ., dn). Furthermore, since A is positive definite we must have dj > Ofor all / = 1,. . ., n, and hence we can take the square root of D and get the expression S -\ODVt\ODV'.
(2.4.29)
But Eq. (2.4.29) is of the form Eq. (2.4.28) and, since ODl/2 e GL + (n, IR), we do in fact have precisely a transformation of that type. This shows that every S e S„ can be obtained from the unit matrixle S„ (the "fiducial" matrix for this action), and hence the GL+(n, U) action on the subspace S„ C M(n, U) is transitive. (c) The isotropy group at the fiducial matrix 1 is the set of all geGL+(n, R) such that yg(±) = gig' =31.But this is precisely the subgroup SO(n, IR), and hence we deduce the result of the theorem. QED
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101
Comments (a) This result plays a central role in the fibre bundle approach to The theory of Riemannian metrics on a manifold. (Roughly speaking, a Riemannian metric on a manifold M assigns an element of S„ to each point in M.) (b) There is a similar result SnA s GL + (n, U)/SO(n— 1,1) where the matrices in SnA have signature ( — , + , + , + , . . . , + ) . 2.5. INFINITESIMAL TRANSFORMATIONS We now come to a topic of considerable importance for any programme that involves the use of Lie transformation groups, namely the properties of "infinitesimal transformations" and the way they relate to the full, globally-defined group action. The basic idea is as follows. If a Lie group G acts as a group of transformations on a differentiable manifold M then each one-parameter subgroup of G will produce a manifold-filling family of curves, and hence a vector field on M that is tangent to this family everywhere.
An orbit of a one-parameter subgroup of G as it acts on M.
In this way we arrive at a map from the Lie algebra element that generates the one-parameter subgroup into the infinite-dimensional Lie algebra of all vector fields on M. The interesting question is the extent to which this linear map reflects the details of the action of the Lie group G on M; as we shall see below this is a generalisation of the way in which the Lie algebra of G reflects the properties of G itself. In the formal development of this idea that follows we shall use a right action, rather than the more usual left action, of G on M. The
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reason for this choice will emerge later, but it is not a real restriction and all results that we will derive have exact analogues for a left action. Definition Let G be a Lie group that has a right action g -»• 8g on a differentiate manifold M. Then the vector field on M induced by the action of the one-parameter subgroup t —- exp tA, AeTeG is defined as
Xt(f):=-f(pexp(tA))
(2.5.1) (=0
where fe C^iM) and, as usual, we have written Sg(p) as pg.
Comments (a) If the orbit Op through a point peM smoothly embedded submanifold, then Tp(Op)^{x^\AeTeG}.
of the G-action is a
(2.5.2)
(b) The induced vector field XA has a flow f (p) = p exp(L4) = 4xP(M) (/»), i-e., the flow is 4>t = ^exp(M)-
(2.5.3)
(c) The analogue of Eqs. (2.4.6-7) for a right action is a smooth map r : MX G -» M with (i) V(p, e) = p for all peM, (ii) T(r(p, gx), g2) = T(p, g{ g2) for all peM and g ug2 e G.
(2.5.4) (2.5.5)
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103
Then each ge G defines a map M-*• Mby <5?(/?) := T(p, g) = /?#. However, there is another way to use T : MX G—-M, namely to each p e M w e associate the map Mp: G —- M defined by Mp (g) := T(p, g) = pg
(2.5.6)
in terms of which, Eq. (2.5.5) is the equivariance (Eq. (2.4.8)) ofM p : Mp°rg = Sg°Mp
for all g e G and pe M,
(2.5.7)
where rg:G —*• G is the usual right translation rg(g') := g'g. Now, if LA is the left-invariant vector field on G associated with A e TeG then, for a l l / e C,co(M), (MP*(LA)) if) = LAg {foM) = (lg*A) (foM) = A(f°Mp°lg). (2.5.8)
But Mpo/? = Mpg
for all p e M and # e G
(2.5.9)
and hence, Eq. (2.5.8) implies (Mp* (LA)) if) = A{foM„) = — (/oM w ) (exp tA) (=0
=
"T/CW
at
t=0
Thus, Mp*LA = XAg
(2.5.10)
exp (tA))
for all ge G
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which shows that, for each pe M, the vector fields LA on G and XA on M are A/p-related. Note that, since LA = A e TeG, it follows from Eq. (2.5.10) that, for all pe M, XA = MP*(A)
(2.5.11)
which is a useful alternative definition of the induced field XA. This is used effectively in the next result which deals with the infinitesimal version of the equivariance of a group action of the Lie group G on a pair of manifolds M and M' (cf. Eq. (2.4.8)).
Theorem Let G be a Lie group that acts on the right on manifolds M and M' with induced vector fields XA and X'A respectively, and let f:M -> M' be equivariant under this action, i.e., f{pg)=f{p)g
forallpeMand^eG.
(2.5.12)
Then the vector fields XA and X'A are/-related for all A e TeG. Proof From Eq. (2.5.11) we have XA = Mp* (A). Also, the equivariance condition Eq. (2.5.12) implies that, for all g e G and peM, {foM) (g) = f(pg) = f(p)g
= M'f(p) (g)
(2.5.13)
and hence MXA)
=f*Mp*{A)
= (foMp)*(A)
= M'flp)*(A)
= X'?(p)
(2.5.14)
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105
which is precisely the condition that the vector fields XA and X'A be /-related. QED. Now let us consider a very special case of a right action of G on a manifold M, namely when M is G itself and 8g = rg for all g e G. Then, XA = Gg* (A) = lg* (A) = LAg
(2.5.15)
since Gg{g') = gg' = lg{g'). In other words, the left-invariant vector field LA is induced by the right translation of G on itself. Thus, for all /eC°°(G),
LA(f) =
-f(gexv(tA))
(2.5.16) /=0
Similarly, the right-invariant vector field RA, A e TeG, is induced from the left translation operation: RA(f)
=
dt
-f((exptA)g)
(2.5.17) r=0
This way of looking at the vector fields LA and RA is useful in a variety of situations. A good example is the following theorem which deals with the relation between the commutator of a pair of Lie algebra elements and the "push-forward" Ad9* of the adjoint action of G on itself. Theorem Let A, B e TeG with Lie bracket [AB]. Then,
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[AB] =
d_ dt
—AdexptA*(B)
(2.5.18) (=0
Proof In general, if <$>? is a flow for a vector field X on a manifold M and if Y is any other vector field then (Eq. (1.5.15)) [X,Y\
d_ dt
lim[Y-d>f*(Y)]
f*(Y)
t-o
/=o
t
(2.5.19)
Now, according to Eq. (2.5.16), a flow for the left-invariant vector field LA, A e TeG, on the Lie group G is ? = rexplA. Thus, using Eq. (2.5.19) [AB] = [LA, LB\
lim = ll
\.l"e
r
e\ptA* l^exp-M/J exp tA* V-^exp-
t lim [B ' e x p IA* —
r^O
—
r—0
t _ lim [B ~ Ad exp _ M * (B)] t
lim LAdexpM* (B) - B] —
(—0
t
which proves the result. QED Note. We know from Eq. (2.2.40) that exp hdg*{B) = g exp B g ~' for all g e G. Using this result and Eq. (2.5.18), one can deduce that exp tA exp B exp-tA = exp (t[AB] + 0(t2)).
(2.5.20)
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107
Theorem If the Lie group G acts on a manifold M then, for all g e G and A e TeG, XM**w = 5g-{*(xA).
(2.5.21)
Proof We have, x^gM)
= Mpi,[Adgi,(A)]
= (M p -Ad,)*U).
(2.5.22)
But, MpoAdg(g') = p(gg'g~l) = M w °r,_,(0') = V . ^ „ ( ? ' ) (2.5.23) so that Mp°Ads = <5s-,oM0. Then Eq. (2.5.22) implies that XAVU>
= Sg_^Mp^{A)
=
Sg^{XApg). QED
We can now use some of this work to prove what is undoubtedly one of the central results in the infinitesimal theory of transformations and which explains why the subject is so important in many applications. This asserts that the map that associates the vector field XA with A e TeG is actually a homomorphism of Lie algebras, i.e., the Lie algebra of the Lie group G is "represented" by the vector fields on the manifold M on which G acts: Theorem Let M be a manifold on which a Lie group G has a right action. Then the map A ~~•* XA, which associates to each Ae TeG the
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induced vector field XA on M, is linear and a homomorphism of L(G) = TeG into the infinite-dimensional Lie algebra of all vector fields on M: [XA, XB] = X[AB]
for all A, B e TeG * L(G). (2.5.24)
Proof It is clear from Eq. (2.5.11) that the map A—~-XA is linear so the key task is to prove Eq.(2.5.24). Since (Eq.(2.5.3)) a flow for XA is dexptA w e c a n use Eq. (2.5.19) to write
[XA, XB] = S lXB-S«*«*M\
(2.5.25)
Hence, using Eq. (2.5.21) we have A
lim[xB
B
[X , X ]
- XAd^~'A*{B)] t
_ lim X
l^txp
— t^o
IA*(B)~B]
t
(2.5.26)
where the last step follows from linearity: XA — XA' = X(A A'\ and -XA = X~A. Since TeG is a finite-dimensional vector space the linear map A -*- XA is continuous and we can move the lim ?—>-0 inside the square bracket to get
[xA xB] = =
(l lMa A lB) B]/,) x
Z "' -
J^UB]
by virtue of Eq. (2.5.18). QED
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Comments (a) If we had followed the same procedure throughout but with a right, rather than a left, action of G on M, then the result in Eq. (2.5.24) would have become [xA, XB] = X[BA]
(2.5.27)
so that the linear map A —*• XA would be an anti-homomorphism from TeG = L(G) into the Lie algebra of vector fields on M, rather than a homomorphism. Basically, this is the reason why we have used a right action throughout this section. (b) Several properties of the G-action on M are faithfully reflected in the homomorphism //: TeG -> {Vector fields on M}, ju(A) := XA. (i) Suppose the action is effective and let A e Ker(//) so that fx{A) = 0. Then the one-parameter subgroup £-~*-exp tA acts trivially on M, which is consistent with the effective nature only if A = 0. Hence we conclude: "If G acts effectively on M then the map A -> XA is an isomorphism from TeG ~ L(G) into the Lie algebra of vector fields on M". (ii) Suppose the action is free and that A e TeG is such that XA vanishes at some point pe M. This means that p e M is left fixed by the one-parameter subgroup £-*-exp tA, which is a contradiction of the free nature of the action unless .4 = 0. Hence. "If G acts freely on M then the induced vector field XA is nowhere zero for all A e TeG, Aj^O." As a final remark on this whole issue of infinitesimal transformations one might wonder when the process discussed in this section can be reversed. That is, we have started with a group action of G on M and derived a homomorphism from the Lie algebra of G into the vector fields on M. But, if we find that a
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manifold M and Lie group G are such as to admit a homomorphism from L(G) into the vector fields on M, when is it true that there is a global group action of G on M such that the given homomorphism is precisely the one that derives from this action? One obvious problem is that more than one Lie group usually share a given Lie algebra, being different global coverings of each other. It is thus sensible to start with the assumption that the group we are talking about is the unique simply-connected Lie group with the given Lie algebra. We also need to restrict our attention to the cases where the vector fields in the range of the homomorhism /n are complete since the fields induced by a group action always have this property. A famous result encompassing some of these features is Palais' Theorem. "If jx is a homomorphism from L(G) into the vector fields on M, and if G is simply connected and M is compact, then there exists a unique (j-action on Msuch that fx(A) = XA — the induced vector field for that action."
3 FIBRE BUNDLES
3.1. BUNDLES In Part I, we have already seen an illustration of the use of fibre bundles when we "glued" together the tangent spaces at all points of a manifold M to form a new manifold TM that could be regarded as a bundle of vector spaces over M. The basic bundle with which TM is "associated" is the bundle of frames (i.e., sets of basis vectors for a tangent space) over M and, as we shall see, this can be used to give a very geometrical way of thinking about general tensor structures on a manifold. More directly connected with physics is the fact that fibre bundles are the correct mathematical framework in which to discuss the Yang-Mills field and the associated idea of gauge transformations. In this case, the spaces that are glued together are copies of the internal symmetry group, and the ways in which these fibres twist around the spacetime (typically Euclideanised and compactified) reflect topological properties of the gauge theory such as instanton number. A connected application is the use of bundles in the context of Kaluza-Klein theories and dimensional reduction. For all these reasons, there has been a steady increase in recent years in the use of fibre-bundle language by theoretical physicists, and the aim of this third section of the course is to provide an introduction to some of the basic ideas. In the original discussions of in
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fibre-bundles, fairly heavy use was often made of the bundleanalogue of coordinate charts; a classic reference in this context is N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, 1951. However, most treatments these days employ a more intrinsic approach, and this is the path we will follow here. A text I find especially useful in this respect is D. Husemoller, Fibre Bundles and much of what follows is based on this reference. First, the basic definition of a bundle: Definition A bundle is a triple (E, n, M) where E and M are topological spaces and n : E —- M is a continuous map. E is called the total space or bundle space and M is the base space. The map n is known as the projection map. The inverse image n ~' (x), xe M, is the fibre over x. If, for all xe M, n~l(x) is homeomorphic to a common space F, then F is known as the fibre of the bundle and the bundle is said to be a fibre bundle. This condition will always apply in the cases of interest to us and from now on we will assume that it is satisfied. A Cm-bundle is as above but with E and M being C^-manifolds and with n a C^-map. It will be understood in what follows that at any point where a map is required to be continuous it will also be required to be differentiable if the spaces concerned are differentiable manifolds. Comments (a) By common consent, the total space E is often called the "bundle" even though, strictly speaking, this refers to the triple (E, 71, M).
(b) We often denote a bundle (i.e., the triple) with a Greek letter like £, n etc.; in this case the total and base spaces of a bundle £ are denoted £ ( f ) and M(£) respectively.
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(c) If (E, n, M) is a bundle with fibre F then this is commonly indicated by writing F^-E—-M or F^-E
M Examples (a) One of the simplest examples of a fibre bundle is the product bundle over M with fibre F. This is just the triple (M X F, pruM).
(b) A famous example of a fibre bundle is the Mobius band which is a twisted "strip" whose base space is the circle S\ Thus, for example, the fibre could be taken to be the closed interval [0,1], but note that the total space E is not the product space 5"' X [0,1] (and neither is it homeomorphic to it). It can be represented as follows, where the two points " 6 " (respectively "c") on the two sides of the strip are to be identified:
-b
c \.
^ -^s
1
s'
-
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(c) Another famous example is the Klein bottle. This is rather difficult to draw as it cannot be embedded in Euclidean 3-space, but it can be represented with the aid of the following diagram which shows how it is obtained from a cylinder by reflecting in the diameter d-e and then identifying: b
c
e(\d
[e
c
b
) '
a.
.a S'
From the viewpoint of application to theoretical physics, one of the most important ideas in fibre bundle theory is that of a "crosssection". This is a sort of "twisted" function which is defined on the base space and which takes its values in the fibre of the bundle. We saw in Part I how vector fields and difFerentiable one-forms could be regarded as cross-sections of the tangent bundle and cotangent bundle respectively, and we shall see later how this idea can be extended to incorporate general tensor fields. In Yang-Mills theory, all fields (with the exception of the Yang-Mills field itself) are cross-sections of various vector bundles that are associated with the fundamental Yang-Mills "principal" fibre bundle. Crosssections of vector bundles always exist but this is not necessarily the case for a general fibre bundle; an important example of this phenomenon will be given below in the context of principal bundles. Definition A cross-section of a bundle (E, n, M) is map s : M — E such that the image of each point xe M lies in the fibre n~l(x) over x, i.e., nos = id A/ .
(3.1.1)
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In the case of a product bundle, a cross-section defines a unique function s.M^-F given by Eq.(3.1.2): s(x) = ((x, s(x))
VxeM.
(3.1.2)
In the case of a Mobius band, a cross-section can be represented with the aid of an identification diagram:
i
/
i
"5
In discussing the general ideas of differentiable manifolds in Sec. 1, it was point out that many important non-trivial differentiable manifolds arise naturally as sub-manifolds of other, more tractable, manifolds. A similar situation arises for fibre bundles and leads to the idea of a "sub-bundle" of a fibre bundle (E, n, M) with fibre F. This is a subspace (or submanifold) of E with the extra property that it always contains complete fibres of E , and is
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therefore itself a fibre bundle with fibre F defined over some subspace of M. The formal definition is as follows. Defintion (a) A bundle (£", n', M') is a sub-bundle of a bundle (E, n, M) if (i) E' C E (ii) M' C M (iii) K' = n{E (b) Let N be a subspace of M. Then the restriction of (£, n, M) to A" is defined to be the bundle (n~l(N), n.-i.N), N). It should be noted that if (E, n, M) is a sub-bundle of the product bundle (MXF, pru M) then cross-sections of the former have the form s(x) = (x, $(x))
(3.1.3)
where s : M -* F is a function such that, V x e M, (x, s(x)) e E. Examples (a) Let H be a closed Lie subgroup of the Lie group G. Define the map n:G -* G/H by n(g) := gH. Then (G, n, G/H) is a bundle with fibre H. In general there will be no (smooth) cross-sections for bundles of this type since a necessary condition for the existence of such sections is that G be diffeomorphic to G/H X H, which is usually not true. (b) The tangent bundle TS" of the n-sphere S" can be represented rather nicely as the specific sub-bundle of S" X IR"+1 E(TS")*{(x,y)eS"
X U"+'\x-y = o}.
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Similarly, the normal-bundle v(S") of S" is defined to be the set of all vectors in U"+l that are normal to points on the sphere: E{v{S")) := {(x, y) e S" X U"+l | 3 ke U s.t.y = kx}. A cross-section of TS" is of course simply a vector field on S"; a cross-section of v(S") is called a normal-field of S" as a submanifoldoflR" +1 . Now we come to the important idea of a "morphism" between a pair of bundles (E, n, M) and (E', n', M') which, as usual, is defined to be a structure-preserving map between the relevant spaces. In the present context, this will involve continuous maps from E to E' and M to M' which map fibres over M into fibres over M' (as usual, we will require these maps to be smooth if E, E', M, and M' are differentiate manifolds). More precisely: Definition A morphism between a pair of bundles (E, n, M) and (E', n',M') is a pair of maps (u,f) where u : E-* E' a n d / : M ~* M' such that the following diagram is commutative
i.e., n'°u = fin.
(3.14)
Comments (a) Eq.(3.1.4) implies precisely that u{n~\x)) C n' V x e M , i.e., the pair of maps (u, f) is fibre preserving.
'(/(*))
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(b) Since n is surjective it follows that the map f is uniquely determined by the map u. (c) The definition generalises in the usual way to give the notion of an isomorphism between a pair of bundles (E, n, M) and (E', %', M'), i.e., this is a morphism (u, f) from (E, n, M) to (E', n', M') and another morphism (u',f ) from (E', n', M') to (E, n, M) such that u'°u = id £ ; u°u' = id£fof=idM;fif
=id M ,.
(3 L5)
-
(d) The composition of the morphism (u, f):(E, n, M) -+ (E', n', M') and the morphism (u',f): (E\ n', M') — (E", n", M") is the morphism between the bundles E and E" (u'°u, f'°f) : (E, n, M) -* (E", TI",M"). The following diagram is commutative: E—^E'—*~E U'
u
71
\ f
71'
\) f '
(e) If M = M', a morphism (u, id^) is called a M-morphism. Fibre bundles are usually thought of as being some sort of "twisted product" of the base space with the fibre — a rather pictorial concept which implies that, in some appropriate sense, a fibre bundle looks like a product "in the small". This restriction is not contained in the definition of a bundle given above (although all the examples cited are in fact of this type) and we need to define what it means for two bundles to look like each other locally (so that, in particular, we can require a general fibre bundle to look locally like a product bundle).
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119
Definition (a) Two bundles £ and rj with the same base space M are said to be locally isomorphic if, for each xe M, there exists an open neighbourhood U of x such that £\v and t]^ are ^/-isomorphic. (b) A bundle (E, 71, M) is trivial if it is M-isomorphic to the product bundle (M X F, prx, M) for some space F. It is locally trivial if it is locally isomorphic to the product bundle. Note (i) The relation of being locally isomorphic is an equivalence relation on the set of all bundles over the topological space (or differentiable manifold) M. (ii) When a bundle is trivial there is not in general any distinguished, or uniquely determined, trivialization. Examples All the previous examples are locally trivial bundles. Note that a locally trivial bundle is necessarily one with a fibre. In fact an alternative definition of a fibre bundle is a triple {E, n, M) such that, for each xe M, there exists an open neighbourhood U C M and a homeomorphism h: U X F -* n~\U) such that n{h(x,y))=x
Vxe[/,VyeF.
(3.1.6)
From now on we will assume that all bundles under discussion are locally trivial. To finish this first general section we come to an idea of the greatest importance. We recall that one of the significant features of differential forms (when compared with vector fields) was that, given a pair of manifolds M and N and a m a p / : M^-N, any differential form co on TV can be "pulled back" to give a differential form f*(co) on M. It is highly significant that fibre bundles can also be
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pulled back by a map between two spaces. For example, this shows the deep consistency with the use of differential forms (characteristic classes) on the base space M of a bundle to describe the topological structure of the twists of the fibres around the base space. As might be expected, a central role is played here by the cohomology groups of M and their relation to differential forms via DeRham's theorem. Definition Let £ = (E, n, M) be a bundle with base space M and let/: M' -* M be any map from M' to M. Then the induced bundle (or pullback) of £ is the bundle f*(£) over M' with the total space E' = {(x', e)ehf' X E\f(x')
= n(e)}
and the projection map n':E' —>• M' defined by n'(x', e) := x'. Comments (a) If7 : N -»- M is the inclusion map for a subspace TV of M then the restriction £\N is yV-isomorphic to the induced bundle j*(£). (b) There is a natural morphism from the induced bundle (E', n', M') to the original bundle (E, n, M) defined as:
E(f*(0)
h
E(i) fzix', e) •= e
M'
M
f Note that n~l(x')
= {(x\ e) e {x'} X E \ f(x')
n(e)}
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Fibre Bundles
= {x'} X n~\f{x')) and hence / maps each fibre n~\x') C E(f*(£)) homeomorphically onto the fibre n~\f(x')) C Eg). In particular, this shows that each fibre of the pull-back /*(£) is homeomorphic to the fibre F of the original bundle £. (c) Ifg : M" ~* M' a n d / : M' — M, then if £ = (£, TT, A/) is a bundle over M, the pull-backs obey the same type of functorial property as do the pull-backs of differential forms. More precisely, the bundles g *(/*(%)) and (f°g)*(0 are M"-isomorphic. (d) It should be noted that cross-sections pull back too. Thus if s : M —• E(£) is a cross-section of £, an associated cross-section s: M' —• £ ( / * ( £ ) ) can be defined by
£(0
£(/*(£)) /< 71'
s(x') := (x', so/(x'))
71
M
M'/
This diagram commutes in the sense t h a t / °s = s°f. (e) There is a deep algebraic-topological connection between the classification of bundles over some fixed base space and the pullback operation. This starts with the observation that if £ = F —>• E —- M is a fibre bundle, a n d / a map from N into M, then it seems intuitively clear that the pull-back bundle /*(£) will not change dramatically if/ is varied smoothly. More precisely, if/ g are a pair of maps from TV into M t h a t are homotopic (i.e., there exists some continuous map K: N X [0,1 ] -* Msuch that, VxejV, A/x, 0) = f(x) and i^(x, 1) = g{x)) then it is fairly easy to show that /*(£) and #*(£) are isomorphic bundles over N. Thus if 55f(./V) denotes the set of isomorphism classes of F-bundles over N, any Fbundle £ over M generates a map
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at: [N, M] -* ^(iV),/—/*(£)
(3.1.7)
where [N, M] denotes the set of homotopy classes of maps from N into M (strictly speaking, these maps have to be base-point preserving, i.e., we choose points x0 e N,y0e Mand require that all the maps /from N to M being considered satisfy/Cx0) = yo)In general, the map in (3.1.7) is neither one-to-one nor onto. However, for a wide class of spaces N (e.g., CW-complexes, which includes all differentiable manifolds) it can be shown that there exist universal bundles UF = F -* EF -> BF with the property that the map aUF:[N,BF]^<8F(N)
(3.1.8)
is both one-to-one and onto, i.e., (i) Iff, g : N -* BF then/*(UF) * g*(VLF) if, and only if, / and g are homotopic maps. (ii) If r\ is any F-bundle over N, there exists some/: JV-»- BF such that rj is isomorphic to f*(VLF). Thus the set of isomorphism classes of ^-bundles over N is in bijective correspondence with the set of homotopy classes of maps from N into BF. This procedure is rendered meaningful by the observation that different universal F-bundles have base spaces that are necessarily homotopically equivalent. Hence the problem of classifying bundles over TV "reduces" to the problem of computing the homotopy set [N, BF] — a task that is tailor-made for study with the aid of algebraic topology. The general idea of characteristic classes can also be seen clearly from this construction. More precisely, let a> be an element of any cohomology group of BF and let n be a bundle over some space N induced by a map cp: N -*• BF, i.e., n « q>*(UF). Then the characteristic class of n corresponding to the "universal" class co is defined as co^:= (p*{co). This object is unambiguous since any pair
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123
of maps (px and q>2 from N into BF inducing isomorphic bundles over N are homotopic and hence, in particular, the induced cohomology classes (p*{oj) and (p2*{oS) are equal. Note also that if xP:M^N, then ( ^ ) * ( U F ) « t*()
= ^ * ( ^ * ( f O » = ($»o^)*(fo) =
Q)(MnuF)
Thus the association of a characteristic class con with a bundle rj is "covariant" with respect to the pull-back operation of bundles and cohomology classes. 3.2. PRINCIPAL FIBRE BUNDLES We must start now to consider the wide-ranging question of how fibre bundles are to be classified and how the different types can be constructed. A crucial idea is that of a "principal fibre bundle" which is a bundle whose fibre is a Lie group in a certain special way. These bundles have the important property that all nonprincipal bundles can be constructed from (or are "associated" with) a specific principal bundle. It transpires that the twists in a bundle associated with a principal bundle are uniquely determined by the twists in the latter, and hence the topological implications of fibre-bundle theory are essentially coded into the theory of principal bundles. The crucial definitions are as follows: Definition A bundle (E, n, M) is a G-bundle if E is a right G-space and if (E, n, M) is isomorphic to the bundle (E, p, E/G) where E/G is the orbit space of the G-action on E and p is the usual projection map.
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Note that the fibres of the bundle are the orbits of the G-action on E and hence, in general, will not be homeomorphic to each other. If G acts freely on E then (E, it, M) is said to be a principal Gbundle and G is the structure group of the bundle. Note that, in this case, the freedom of the G-action implies that each orbit is homeomorphic to G, and hence we have a fibre bundle with fibre G. Examples (a) The product bundle (M X G, pru M) is a principal (/-bundle under the trivial G-action (x, g0)g := (x, g0g). One might expect a "trivial" G-bundle to be defined as one which is isomorphic to the product bundle. However, this requires a definition of a morphism between two principal bundles, and this is not the same thing as a bundle morphism since the G-action is also required to be preserved in some appropriate sense; we will return to this later. (b) Let G be the cyclic group Z 2 = {e, a} with a2 = e, and let this group act on the ^-sphere S" by exchanging the antipodal points: xe := e xa := —x This action is free and gives rise to a principal Z2-bundle whose base space is diffeomorphic to the real projective space UP" « Sn/Z2(c) One of the most important examples of a principal bundle occurs when / / i s a closed Lie subgroup of a Lie group G. Then H acts on the right on G by group multiplication. This action is
Fibre Bundles
125
clearly free and the orbit space is simply the space of cosets G/H. Thus we get a principal //-bundle (G, n, G/H) whose fibre is the group H. When this is a locally-trivial bundle (see below) we can say that G is the "twisted product" of H with G/H. For example, the U{\) action on SU(2) gives rise to a bundle U(l) -> SU(2) — SU(2)/U(l) which is the bundle Sl — S3 — S2. This famous fibering of the 3-sphere by the circle is known as* the Hopf bundle. Other examples of principal bundles obtained in this way are
U(n)-+SU(n+l)-+CP" SO(n)^SO(n+l)^S". (d) In Yang-Mills theory with the internal symmetry group SU(2), the bundle corresponding to instanton number one is a principal >St/(2)-bundle with a base space which is a 4-sphere regarded as the one-point compactification of "Euclideanised" Minkowski spacetime. We recall that the group space of SU(2) is a 3-sphere, and in fact this instanton bundle is of the form
(e) Another example of a principal bundle can be obtained by considering the action of the multiplicative group C* of non-zero complex numbers on the space C"+1- {0} given by (z l5 z 2 , . . . , z„+i) X : = (z{ X, z2 X,. . . , z„+i X) V X e C*.
This is a free action and the orbit space is the space CP" of complex lines in the vector space C"+1. Thus we get the principal bundle C* — C + , - { 0 } - > C P "
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in which the projection map associates with each n+1-tuple (z[, z 2 ,. .., z„+1) the point in CP" with the homogeneous coordinates (z,;z 2 ;. . .;z„ +1 ). A closely connected bundle is obtained by restricting to the U{\) subgroup of C* which acts freely on the real (2n+ l)-sphere embedded in C"+1-{0}. This gives rise to the principal £/(l)-bundle C/(l) — S2n+] — CP". (f) An example of a principal fibre bundle of major importance in differential geometry is afforded by the bundle of frames attached to an ^-dimensional differentiable manifold M. A linear frame (or base) b at a point x e Mis an ordered set (bx, b2, • .., b„) of basis vectors for the tangent space TXM. The total space B(M) of the bundle of frames is defined to be the set of all frames at all points in M and the projection map n: B(M) -> M is defined to be the function that takes a base into the point in M to which it is attached. A natural free right GL(n, [Reaction on B(M) is defined by (6„ b2,..., bH) g := (bjgn, bjgj2,. .., b^jj where g e GL{n, U). The bundle space B(M) can be given the structure of a differentiable manifold as follows. Let ( / c M b e a coordinate neighbourhood with coordinate functions Then any base b for the vector space TXM, x e U, can be expanded uniquely as
"•-Pi-t),-'-1--" for some non-singular matrix b{ in GL(n, U). Thus we can define the map
Fibre Bundles
h:UX
127
GL(n,U)-^n~l(U) (x, g)—(glJ(dJ)x,...,
gj(dj)x)
and then use (x\ . .., x"; g{) as the coordinates for a differential structure on B(M) which, thereby, becomes a manifold with dimension n + n2. (g) Another important bundle in Yang-Mills theory is obtained by considering the set U of all Yang-Mills potentials in the theory. If the underlying (/-bundle over spacetime M is trivial, then the corresponding gauge group © is the set of all smooth maps from M into G; more generally it is the group of automorphisms of the principal bundle (see below). In either case, the physical configurations of the theory are identified with potentials that are related by gauge transformations, i.e., they lie in the same orbit of the action of the gauge group © on U. With some care in the precise selection of© and/or U it can be shown that this action is free and hence that U is a principal ©-bundle with base space U/©. A related example arising in general relativity is the space RiemM of Lorentzian (or Riemannian, if that is appropriate) metrics on M on which the diffeomorphism group of M acts as the gauge group ©. Both these examples differ from the previous ones in that the spaces concerned are all /n/zm'te-dimensional and hence some subtlety arises in the detailed constructions. After this fairly extensive set of examples we must return to the general theory and in particular to the generalization of the idea of a bundle morphism to include the preservation of the group actions in principal bundles. Definition A bundle morphism (u,f) between a pair of principal G-bundles (P, n, M) and (P', n', M') is said to be a principal morphism ifu:P —- P' is a (7-space morphism, i.e.,
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u{pg) = u{p)g
V pe P,
V g e G.
This can be generalised to the case in which (P, it, M) is a Gbundle, (P', n', M') is a G'-bundle, and A : G -* G' is a homomorphism. We then require, u(pg) = u(p) A(g)
V /? e P a n d V g e G.
This situation arises, for example, when discussing spinors on a differentiable manifold. In this case G' is the orthogonal group SO(n, U) acting on the bundle of orthonormal frames and G is its double covering group Spin(n, IR) acting on the bundle of spinor frames. We come now to a theorem that is of some significance, although it is easy enough to prove. Theorem Let (u, idM) be a principal M-morphism between a pair of principal G-bundles (P, n, M) and (P', n', M). Then u is necessarily an isomorphism.
Proof (a) First we will show that u is one-to-one. Thus let u{px) = u(p2). Then ii{pi) = n'(u(ps)) = n' (u(p2)) = 7c(p2) and therefore p{ and p2 belong to the same fibre. Hence 3 g e G such that Pi = Pig, and thus u{px) = u(p2g) = u(p2)g = u(p{)g. But G acts freely on P and hence g = e. Thus /?, = p2, which proves that u is one-to-one.
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(b) Next let us show that u is surjective. Thus let p' eP'. Choose p e P such that Tc{p) = n'(p'). Then n'(u(p)) = n{p) = n'(p') and hence p' and u(p) are in the same fibre. Therefore 3 g e G such that p' = u(p)g = u(pg), and so u is surjective. (c) Finally, it must be shown that the inverse map u~x is continuous (or smooth in the case of difFerentiable manifolds). This is left as an exercise! QED Comments (a) Armed with the idea of a principal morphism we can now give a proper definition of what it means to say that a principal G-bundle (P, n, M) is trivial. Namely that there is a principal morphism from (P, n, M) to the product bundle (M X G, pru M). The idea of local triviality can then also be defined in the obvious way. All of the specific principal bundles we have considered above are in fact locally trivial, although this is not always a minor matter to prove: for example, the statement that H -* G -» G/H is locally trivial when G is a finite-dimensional Lie group is a famous theorem. (b) It follows from the above that the set of all principal morphisms from a principal G-bundle £ = (P, n, M) to itself forms a group. This is called the automorphism group of the bundle and is denoted Aut(£). In Yang-Mills theory, in a non-trivial instanton sector, the automorphism group of the bundle plays the same role as the familiar group of "gauge transformations" in the instantonzero sector (when the bundle is trivial). Indeed, when £, is trivial, Aut(£) is isomorphic to the group Cm(M, G) of gauge transformations: Theorem Let £be the product bundle (M X G, pru M). Then the set of all automorphisms u: MXG^-MXG of this principal bundle are
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in one-to-one correspondence with maps x : M u(x, g) = (x, x(x)g).
G such that
Proof Let u be such an automorphism of £. Then u(x, g) = (x,f(x, g)) for some/: M X G—» G. Define a map x:M—>- Gby x(x) :=f(x, e). Then u(x, g) = (u(x, e))g = (x, x(x))g = (x, x(x)g). Conversely, given x define u(x, g) := (x, %{x)g). Then (u(x, g))g' = (x, %{x)gg') = u(x, gg'). QED We have remarked already that one of the significant features of fibre bundles is the way in which, like differential forms, they can be pulled back via a map between a pair of manifolds. The next result is important since it shows that the property of being a principal bundle is preserved in this process. Theorem Let £' = (P', n',M')bea principal (7-bundle and let/: M-> M'. Then the induced bundle /*(£) is also a principal G-bundle. Proof Lei f*(0
=
(P,n,M)
PIG Define a G-action on M X P' by (x, p')g := (JC, p'g). This induces an action on P since n'(p'g) == n'(p') = f{x). Clearly this action is free. Now define h : P/G — Mby h([p]G) := n{p). Then,
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(i) h is injective, since if /z([Pi]G) = h([p2]G) then n{p{) = n{p2) and hence 3 g e G such that /?, = p2g, which means precisely that [Pih = [Pih(ii) h is surjective since ifxeMthen, since 7r is surjective, 3 peP such that n(p) = x. But then h([p]G) = x. (iii) Finally, it can be shown that [Exercise!] h and h~l are continuous. Thus the pull-back bundle /*(£) = (P, n, M) is a principal G-bundle. QED Comments (a) This shows in particular that the restriction §v of £ to a subspace U of M is itself a principal bundle. (b) Since the twists of any fibre bundle are those inherited from the parent principal bundle, it follows that particular interest is attached to the universal (/-bundles G —» EG -*• BG. As for all universal bundles, these are characterised by the property that EG is a contractible space. For example, the infinite covering of 5 ' by the real line produces a bundle Z -*• IR -> S1 that is a universal Z-bundle. Thus the principal Z-bundles over any manifold M are classified by the homotopy set [M, S1] a H\M, Z). A more sophisticated example is afforded by the sequence of principal U( 1 )-bundles C/(l) — S2n+l^CPn. Taking the limit n -*• oo in an appropriate sense we get the bundle t/(l) —S 0 0 —CP 00 .
(3.2.1)
The fact that n{(S") a 0 for i < n, generalises to the feature that all the homotopy groups of S™ vanish which, since it is a CW complex, implies in this instance that Sx is a contractible space.
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Thus (3.2.1) is a universal l/(l)-bundle, and hence t/(l)-bundles over an arbitrary manifold M are classified by the homotopy set [M, CP™]. A similar argument to that employed for the infinitesphere, shows that n2(CPco) « Z and that all other homotopy groups vanish. This in turn implies the famous classification result 55y(,)(Af) « [M, CPm] « H2(M, Z). Note also that the set of Yang-Mills potentials, and the set of Riemannian metrics, are both contractible spaces. Thus the fundamental "gauge-bundles" © — 91 — %/& and DiffM -* RiemM —- RiemM/DiffM are universal bundles for the gauge group © and the diffeomorphism group DiffM respectively. Finally we come to a theorem that is of considerable significance in a variety of applications of fibre bundle theory in differential geometry and theoretical physics, especially in those areas where global topological properties are important. This theorem states that, contrary to what is sometimes felt to be intuitively "obvious", a principal fibre bundle will not in general admit any smooth cross-sections unless it is untwisted! More precisely: Theorem A principal (7-bundle (P, n, M) is trivial (i.e., M-isomorphic to (M X G, pru M)) if and only if it possesses a continuous crosssection. Proof Let a: M —> P be a cross-section of the bundle. Then, for any point p e P, there must exist x(p) e G such that p = o(n(p))x(p). The resulting function x : M -* G is unique because the G-action on the bundle space P is free. Clearly x(pg) = x(p)g. Now define u.P^MXGby u{p):- (n(p), x(p)). Then,
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(i) u is a bundle morphism; and (ii) u(pg) = (n(p), x(pg)) = (n(p), x(p)g) = (n(p), x(p))g = u(p)g. Hence u is a principal morphism and therefore, by the theorem above, an isomorphism. The converse statement is trivial. Namely, ifh:MX G —>- Pisa principal morphism then we can define a cross-section o: M ~* P by o{x) := h(x, e). The proof of the theorem can be finished by showing that all the various maps are continuous (or smooth, as appropriate); this is left as an exercise. QED Comments (a) The inverse of the map u : P -*• M X G is h : M X G -* P defined by h(x, g) := o(x)g. (b) It is clearly of some importance to know when a principal bundle is trivial. This is primarily a global problem and involves a delicate interplay between the topological structures of the fibre and base space. One well-known result is that every principal bundle defined over a contractible (paracompact) base space is necessarily trivial. (c) An example of a non-trivial principal bundle arises in YangMills theory and is responsible for the so-called "Gribov" effect. This is the bundle © -* % -> %/® referred to earlier in which 51 is the set of all Yang-Mills potentials and the group © is the gauge group of the theory. Choosing a cross-section of this bundle corresponds to choosing a gauge. However, as Singer showed, this particular principal bundle is definitely not trivial and hence no smooth cross-sections exist. A similar situation arises in the corresponding situation in general relativity with the principal bundle DijfM -»• RiemM -*• RiemM/DiffM. The Gribov-Singer
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effect means that there is an intrinsic obstruction to choosing a gauge that works for all physical configurations; whether or not this has any real significance in the theory is still a matter for debate. (d) The precise extent to which a global cross-section of a principal bundle does not exist can be used to give a topological classification of the twists in the bundle and hence to relate to the theory of characteristic classes. One approach (which is applicable for any fibre F) involves triangulating the base space sufficiently finely that the bundle is trivial over any simplex, and then trying to construct a cross-section by first defining it on the O-simplices (this is always possible since the O-simplices are just points) then extending it to the 1-simplices, then to the 2-simplices, and so on. At some point, if the bundle is non-trivial, a dimension n will be reached for which the cross-section cannot be extended to all the simplices of the next higher dimension. Thus there will be some nsimplex A and a map a from A into the fibre F which cannot be extended to the interior of A. But A is homeomorphic to a ^-sphere, and the inability to extend a is equivalent to the statement that the element in nn(F) corresponding to the homotopy class of a is nontrivial. It can be shown that this association of an element of n„(F) to A depends only on the cohomology class of the (n+ l)-cell interior of A, and hence generates an element of H"+i(M, nn(F)). This gives the primary obstruction to constructing a continuous section of the bundle. In general, this analysis shows that potential obstructions to constructing a cross-section lie in the cohomology groups Hk(M, nk-x{F)), k = 1, 2, . . . . In particular, this analysis can be applied to a universal principal bundle G -»• EG -*• BG and yields the basic universal characteristic classes for all principal Gbundles.
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3.3. ASSOCIATED BUNDLES In this section we will discuss a method of constructing a great variety of fibre bundles that are associated in a precise way with some principal bundle. The basic idea is that, given a particular principal bundle (P, n, M) with structure group G, we can form a fibre bundle with fibre F for each space F on which G acts as a group of transformations. First we must introduce the concept of a "G-product" of a pair of spaces on which G acts. Definition Let Zand Y be any pair of right G-spaces. Then the G-product of Xand Y is the space of orbits of the G-action on the Cartesian product XX Y. Thus we define an equivalence relation on X X Y in which (x, y) = (x\ y') iff 3 g e G such that x' = xg and y' = yg. The G-product is denoted X XGY and the equivalence class of (x, y) e X X Y is written as (x, y)G or [x, y]. Note that, given any right G-space Y, the group G can itself be used as the right G-space Xto give the G-product G X G Y. It is relevant for our considerations later that this space is homeomorphic to Y: Lemma The map i:G XGY^-Y homeomorphism.
defined by i([g,y\) := yg~l is a
Proof (i) This map is well-defined since [guyi] = [g2, y2] implies that 3 g e G such that g2 = gxg and y2 = yxg. Therefore y2giX = y\gg~{g\x = j ^ r 1 . (ii) The map i is surjective because i([e, y]) = y.
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(iii) The map i is injective since if i([gu y,]) = i([g2, y2]) then and y\Q\X = yidV and so (g2, y2) = {gxg^ g2,yxg^g2), hence (g2,yi) = (9i,y\)QED Finally, we should show that the map i and its inverse are continuous. As usual, this is left as an exercise. Now we come to the crucial definition of an associated fibre bundle. Definition Let £ = (P, n, M) be a principal (/-bundle and let F b e a left G-space. Define PF:= P XFGwhere (p, v)g := (pg, #~'u)and define a map nF : PF^-M by nF([p, v]) := n(p). Then £[F) := (PF, TiF, M) is the fibre bundle over M with fibre F that is associated with the principal bundle £ via the action of the group G on F. Note that nF is well-defined since if [pu vr] = [p2, u 2 ]then 3 geG with (p2,v2) = (Pig,g~lVi) and then n{p2) = n(pxg) = n{p{). Note also that, in order for the definition to make sense, we should show that £[F] really is a fibre bundle. The crucial step is contained in the next theorem. Theorem For each x e M, the space nF ' (x) is homeomorphic to F. Proof For each point p e P{£) there is an associated map ip:F^» PF defined by ip(v) := [p, v]. Now, Tip[p, v] = n(p) and hence ip(F)C KFx{n{p)). Choose some particular point p0 e n~l(x) so that iPo(F) C nF ' (x). Now, if [p, v] e TiF ' (x) define j P o : nF ' (x) -* F as the map which takes [p, v] into i(p0,p)v where x(p0,p) is the so-called
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translation function of the principal bundle £,, whose value is defined as the unique element in G such that p = p0?(Po, p)- The mapj^ is well-defined since x{p0, pg)g~lv = x(p0,p)v. But, (i) irfjpo ([P, v]) = [p0, T(A>, p)v] = [p0 T{PO, p),v] = [p, v]
Hence fPo and 7^ are inverses of each other, and therefore jPo is a homeomorphism. QED It is clear that the "twists" in an associated bundle £[F] are determined by the twists in the underlying principal bundle t, = (P, n, M) and also by the way in which the group G acts on the fibre F. In particular, if G acts trivially on F(i.e., V g e G, V v e F, gv = v) then £[F] is M-isomorphic to the product bundle (M X G, pr{, M) via the map [p, v] —»- (7r(/?), V). However, within the class of associated bundle morphisms, £[F] is only deemed to be trivial if the underlying bundle £ is trivial as a principal G-bundle (see below for further remarks on this potentially confusing nomenclature). Examples (a) Let £ be the principal Z2-bundle formed from the action of the group Z2 = {e, a] on the circle Sl in which a e Z2 sends x e S1 into —x. The base space Sl/J.2 is also difFeomorphic to S\ and £ is simply a double covering of the circle by a circle:
CZ^
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A number of interesting bundles can be formed from this particular bundle by the process of association: (i) Let Z2 act on F = [—1, l]by a e ^ t a k i n g re [— 1, 1] into itself (i.e., the trivial action). Then the associated bundle £[F] is just a cylinder. (ii) Let Z2 act non-trivially on [ — 1, 1 ] by a e Z2 taking re[—l, 1 ] into —r. Then the associated bundle £[F] is the Mobius strip. (iii) Let Z2 act on F = S1 by a e Z2 reflecting the elements of the fibre circle in a diameter. In this case the ensuing associated bundle is the Klein bottle. Thus we see that some of the most "popular" examples of fibre bundles are obtained from what is arguably the simplest nontrivial principal bundle. (b) The next example is of major importance in differential geometry as it provides a "bundle-theoretic" way of viewing general tensorial structures. The principal bundle of interest in this case is the bundle of frames B(M) on the ^-dimensional manifold M. The structure group G is GL(n, U) and of course this can act on a variety of spaces, especially vector spaces. The simplest case is when F = W and the action of GL(n, U) is just the usual linear group of transformations. Then the associated bundle B(IR") = B(M)X can GL(H,R) K " t>e identified with the tangent bundle 7"Mvia the map that takes the equivalence class [b, r] into Z"=1 btr' e TXM where b is a base for the tangent space T^Mand r = {r\ r2,..., r"). (c) More generally, let p : GL(n, U) -»• Aut Vbe any representation of GL(n, U) as a group of linear transformations of the real vector space V. Then B[V] is called the bundle of tensors of type p, and cross-sections of this vector bundle are tensor fields of type p. For example: (i) Let V = U and let p be the identity transformation p{a) = 1 V a e GL(n, M). Then the corresponding associated bundle is the bundle of absolute scalars. The bundle space is diffeomorphic to
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the product M X U and the cross-sections are in one-to-one correspondence with the usual real-valued functions on M. (ii) Now let V = IR again but this time we let p(a) = (det df for some (JO e IR. The ensuing associated bundle is called the bundle of scalar densities of weight p. (iii) More generally let V = IR"m and let a e GL{n, IR) act on v e IR"m by i\- • • • in
(p(a)v) . J[-
/ : , . . . . k„
. := (det a)10 a,,kt... a. k ahti,... ah , v
• • • Jm
A2]. . . .
Hm
The associated bundle is known as the bundle of tensor densities of weight co, contravariant of rank n and covariant of rank m. This example is quite useful in so far as the definition of densities in a "non-bundle" way tends to be a little awkward. In various places in the above, we have implied that the total space £[F] of an associated bundle is "like" some other space Z that is well known. Generally speaking we merely mean by this that £[F] is diffeomorphic to Z. But this does raise the question of what might be meant by a morphism between two associated bundles. This could of course be defined simply as a bundle morphism, but this would not take into account the "associated" status of the bundles and their corresponding relations to the underlying principal bundles. The most useful definition is when one starts with a principal morphism between these underlying bundles and then uses this to construct a bundle morphism between the two associated bundles. Not every bundle morphism is necessarily of this type and it follows therefore that this definition is more restrictive than the first one. The critical definitions are as follows: Definition (a) Let (u, f) be a principal morphism between a pair of principal G-bundles £ = (P, n, M) and £' = (P', n', M'). Then a bundle mor-
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phism between the associated bundles P X G.Fand P' X GF can be denned by uF([p, v]) := [u(p), v\. This is well-defined since uF([pg, g~lv]) = [u(pg), g~lv]=
[u{p)g, g~lv]=
[u(p), v],
and clearly it is a bundle morphism [Exercise!] (b) An associated bundle morphism between a pair of associated bundles £[F] and - M. Then there is a bundle M'-
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isomorphism between (i) the pull-back f*(£[F]) of £[F] and, (ii) the bundle obtained by first pulling back £ to give the principal bundle /*(£) and then forming the associated bundle f*(£,\F\ Proof Construct a map j :/*(&F]) CM' X (P X GF) -* f*(£)[F] C (AT X P) XGF by ;'(x, [/?, v]) :== [(x, p), i;].Then it is a straightforward exercise [!] to show thaty is an isomorphism. QED As a simple corollary to the above, we have the result that the restriction €[F]\N to a subspace Nc M o f the associated bundle £[F] is N-isomorphic (i.e., as bundles) to the bundle €\N[F] which is associated to the restriction ^ to N of the principal bundle £,. Comments (a) Let £ = (M X G, prx, M) and let Fbe any G-space. Then the associated bundle £[F] is M-isomorphic to the product bundle (M X F, pru M) with a map r.(M X G)G X F — M X Fdefined by i([(x, g), v]) := (x, gv). (b) A principal bundle £ is locally trivial, and hence, for each xe M, there exists an open neighbourhood U CM of x and a bundle isomorphism h: U X G —- P(i)\v- If F is a G-space, we can define h' : U X F — F f ^ y /z'(x, v) := [h(x, e), v] which is clearly a bundle £/-morphism. (Note. If h comes from a local section a: U —*• P(£) then h' (x, v) = [a(x), v]). Similarly we can define a map u' : Pf^—* U X f by u'([h(x,g),
v}) := (x, gv).
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It is easy to see that h'°u' = i d ^ ^ and u'°h' = id(/X F. Thus h' defines a bundle ^/-isomorphism from U X F to PF| v and hence, as a bundle, £,[F\ is locally trivial. If Ux and U2 are a pair of neighbourhoods of x e M over which £ is trivial, there will exist a pair of corresponding maps h'Ux : £/, X F -* /V| y, and /t' u2: £/2 X F — PF] Vl together with the pair of principal bundle local trivializations hVl: Ux X G —- P\Ux and hVl: U2 X G -* P| rj2. On the overlap £/, n U2 there is therefore a principal bundle morphism h^h^ : ((7, f l [ / 2 ) X 6 ^ (C7, n U2) X G. Thus there exists a map ^ t ^ '• {U\C\U2) -* G such that h'u}°h'U{ (x, v) = (x, gU2Ui (x)v). These functions gUlU{ are known as the structure functions, or transition functions of the bundle. They satisfy the well-known relations (i)
Qvx u2 (•*) = e
(ii) gUlU2(x) = {gU2uxix)YX (^) 9uiu2(x)gU2u3(x) = gulU}(x) for all UUU2,U3 with [/,n[/2nt/3^ 0. These relations are characteristic in the sense that it can be shown that any family of G-valued functions satisfying these relations are necessarily the transition functions of some fibre bundle. This is the foundation of the approach to fibre bundle theory based on local bundle coordinate charts. (c) According to the definition above, the natural definition of an automorphism of an association bundle £[F] is a morphism uF defined by uF([p, v]) := [u(p), v] where u e Aut £ is an automorphism of the principal bundle £. In particular, if £ = (M X G, pru M) is the product bundle, then every automorphism of t, is of the form u(x, g) = (x, x(x)g) for some map x : M -»- G. Then, on the associated bundle £[F], we have uF{[(x, g), v]) = [(x, %{x)g), v] or, using the bundle-isomorphism of PF with M X F defined
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above, i([(x, g), v]) = (x, gv), we can write uF as the map from M X Finto itself defined by (x, u)~—(x, x(x)v). This is relevant to understanding how a gauge transformation in Yang-Mills theory acts on the matter fields in the system. (d) Let £ be any principal G-bundle. Then G is itself a left G-space under left multiplication. The ensuing associated bundle £\G] is Misomorphic to £, by the map defined by /: P XG G —» Pwith i([p, 0\) '•= PQ-
More generally, if H is a subgroup of G and if r\ is a principal Hbundle, then we can form the associated bundle r][G] = P XHG which is a bundle with fibre G. As we shall see below it is in fact a principal G-bundle. (e) If £ = (P, it, M) is any principal (/-bundle then a very important class of associated bundles is formed by considering the action of G on the coset space G/H, where H is a closed subgroup of G, and hence constructing PXG G/H. This is closely related to the bundle £/H whose total space is defined to be the orbit space P/H of the //-action on P{£) and whose projection map onto M is defined as n'([p]H) := n(p). There is an Af-isomorphism j:PXG G/H — P/H defined by J(P, [9\H)G •=
[P9h
whose inverse k : P/H —- P XGH is given by k{[p]H):={p,[e]H)G. The example in (d) above, where a principal //-bundle r\ was converted into a bundle rj[G] with fibre G, is of considerable importance and merits further discussion:
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Definition Let H be a closed subgroup of G and let £ and r\ be principal Gand //-bundles respectively, defined over the same base space. Suppose there exists a principal bundle morphism with respect to the subgroup H, u: P{rf) — P(£), i.e., u(ph) = u(p)h for all heHcG. Then, (i) r\ is called a H-restriction of £. (ii) £ is called a G-extension of 77. Comments (a) Let 77 = (/>, n, M) be a principal //-bundle. Then there always exists a G-extension of r\ for any group G containing H as a closed subgroup. We simply form the bundle P XHG mentioned above and note that this can be given a G-bundle structure by defining [P, ffo\ff :== [P, doffl Then a principal morphism is defined by u:P-»PX„G,u(p):=[p,e]. (b) Let £ be any principal G-bundle with an //-restriction rj. Then there is a principal isomorphism of £ with t][G] defined by 1: P(£) — P(r]) XHG, i(p) := [u(q), g] where u : P{rj) — ,P(£) is the embedding morphism of rj in £ and (q, g) e P(?/) X G is any pair of elements such that p = u(q)g (the map 1 does not depend on the choice). Thus the G-extension of an //-restriction of a bundle is isomorphic to the bundle. More generally, if ^ is a principal Gbundle with an //-restriction rj, and if F is a left G-space (and hence, a fortiori, a left //-space), there is a bundle A/-morphism of t][F] with ^[F] in which (g, v)H is mapped to (g, v)G. Now we must return to the difficult question of deciding when a principal G-bundle admits a restriction to any specific subgroup H C G. In Yang-Mills theory this is related to the question of the spontaneous breakdown of the internal symmetry group from G to //; some of the more mathematical implications will be discussed shortly. A theorem of major importance is the following:
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Theorem A principal G-bundle £ has a restriction to a //-bundle if and only if the bundle £/H has a cross-section. Proof Let u : P(rj) -* />(£) be a restriction map. Then define a crosssection of £/H by a: M -*• P(£)/H, a{x) := [u(p)]H where p is any element in n~'(x). Conversely, let a : M — P(0/H « P(£) XH G/H be a crosssection of £///. />(£)
„ *P(i)/H
If p(p) := [p]H we have a principal //-bundle on />(£)///, a := (/>(<*), />, />(£)/#), and hence a* (a) is a principal //-bundle over M. Thus, as the bundle space of the //-bundle we are seeking, we define P{rj) := {{x, p) e M X />(£) I
Finally, define the bundle embedding morphism u : P(r]) —- P(£,) by u(x, p) := p. QED Comments (a) Assuming that M is paracompact it can be shown that any fibre bundle whose fibre is a cell (i.e., is homeomorphic to IR"for some n) always admits a cross-section. This follows from the theory discussed briefly in Sec. 3.2 where we argued that the obstructions to constructing a cross-section of a fibre bundle lie in the cohomology groups Hk(M, nk-i(F)), k = 1,2, If F is a cell then all its homotopy groups vanish, and hence so do the potential obstructions. In particular, if / / i s the maximal compact subgroup
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of a non-compact Lie group G then it is known that G/H is a cell. Hence a bundle whose structure group G is non-compact can always be reduced to its maximal compact subgroup. We shall see an important example of this shortly. (b) This example can be generalised in a very significant way. Specifically, using obstruction theory, it follows that if M is a ^-dimensional differentiate manifold, and if the homotopy groups of G/H satisfy HG/H)« 0 V / = 1,.. ., n-\, then the structure group can be be reduced from G to H. If these groups are nonzero then the obstructions to constructing a cross-section of £[G/H], and hence of reducing the group from G to H, are represented by elements of the cohomology groups H'(M; 7r,_, (G/H)), i = 1, . .., n. [We recall that if M has dimension n then H'{M) = 0 for all / >n] An example where the reduction can take place arises in considering an "instanton" Yang-Mills principal bundle (P, n, M) over an arbitrary 4-dimensional, compact (and therefore "Euclideanised") spacetime manifold M. Since the cohomology groups H'(M) vanish when / > 4, the only non-vanishing obstructions to reduction must come from nt{G/H), i = 1,2,3. For example, suppose that G = SU(3) and we are interested in the reduction to the usual SU(2) subgroup. Then we know that SU(3)/SU(2) « S5; but, in general, n^S")« 0 for / < n, and hence there is no obstruction to reducing the group from SU{3) to SU(2). This is basically the reason why, in 4-dimensional spacetimes, there are no topological properties of SU(3) Yang-Mills theories over and above the ones that already arise for the group SU(2). Notice in this context that, since SU(2) « S3 and n3(S3)« Z, there is a potential obstruction to reducing the structure group from SU(2) to the trivial group {e} (i.e., there is an obstruction to constructing a cross-section of the principal SU(2)-bund\e). This comes from the non-vanishing cohomology group H4(M; n3(S3)) = H4(M; T) ss Z; in fact the instanton number is precisely the value of this integer.
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Using the same line of argument one can see that, on a 3dimensional manifold L, there are no obstructions to reducing SU(2) to {e}. This is of relevance when considering the canonical quantization of Yang-Mills theory on the physical 3-space £. Now we come to an example of great significance in differential geometry. This shows how Riemannian metrics arise within the context of the bundle of frames. Theorem There is a one-to-one correspondence between the Riemannian metrics on a n-dimensional differentiable manifold M and the reductions of the structure group of the bundle of frames B(M) from GL(n, U) to 0(n, U). Proof A Riemannian metric g is a symmetric, positive definite, 2covariant tensor field. Thus if X and Y are any vector fields on M we have, (i) g(X,Y) = g(Y,X) (ii) g(X,X) > 0 and = 0 only if X = 0. Note. In component form we have g^ix)
= ^((d^)*, (dv)x) =
Suppose that the group of the principal bundle B(M) is reducible from GL{n, U) to 0{n, U) and let O(M) denote the reduced bundle. For each frame b e B(M) there is an injection ib: Un - * B(M) X GL{ „ R) W defined by ib(r) := [b, r]. We also have the isomorphism of this associated bundle with the tangent bundle TM via the map [b, r]—*-S"=1 rlbt. Combining these two maps we can evidently regard ib as a map from U" to TXM, where x e M i s the point at which b e B(M) is a base; i.e., ib(r) = Zf=1 r'bj.
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Now regard 0(M) as a subbundle of B(M). If u, v e TxM choose a base b e 7r0(W) (X) and define ffx(u,v):={ibi(u),ib1(v)) where (,) denotes the usual Cartesian inner product on the real vector space W. The object thus defined is clearly bilinear and positive definite, and the invariance of the U" inner product under 0(n, U) transformations means that it is independent of the particular choice of the base b for TXM [Exercise!]. Thus we have a well-defined Riemannian metric on M. Conversely, if g is a Riemannian metric on Mdefine O(M) to be the subset of B(M) made up of all sets of orthonormal basis vectors at all the points of M. This is clearly an 0(n, (R)-bundle (since this subgroup of GL(n, U) preserves the orthonormality) and the required bundle map /: 0(M) —»• B(M) is simply the subspace embedding. QED Comments (a) If b e B(M) then ( r,, r 2 ) = gx(ibru ibr2) for all r, and r2 in W. (b) If b is a base at x e Mthen b, = Z"=1 b?{x) (d^)x. If b is extended to a local section of B(M) then the set of local vector fields {bu b2,..., b„} is called an n-bein. (c) Since 0(n, R) is the maximal compact subgroup of GL(n, U) it follows that we can always construct cross-sections of the associated bundle above and hence find Riemannian metrics. (d) We could also consider the reduction of GL(n, U) to the noncompact group 0(n—l, 1) which would correspond to finding a Lorentzian signature metric on M. However, the coset space GL(n, U)/0(n—[, 1) is not a cell and hence there may be
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topological obstructions to the construction of such pseudoRiemannian metrics. In fact, this coset space is homotopic to the real projective space UP"+l which has many non-vanishing homotopy groups. This topological property was exploited in the theory of metric "kinks" — one of the earlier examples of the use of topological methods in theoretical physics. Finally we come to a result concerning the cross-sections of associated bundles that has been widely used in induced representation theory (in which vectors in the Hilbert space are crosssections of certain vector bundles) and in recent studies of anomalies in quantized gauge theories. Theorem The cross-sections of an associated fibre bundle (PF, nF, M) are in bijective correspondence with maps q>: P(^) —- F satisfying (p{pg) = g~y
(3.3.1)
The cross-section s9 corresponding to such a map cp is defined by sv(x) := [p, (p{p)} where p is any point in n~l(x). (3.3.2) Proof Given - PF be a cross-section and define
(3.3.3)
where p e n~l(x) and ip: F ->• nFl(x) is the injection of F into PF
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(and onto nF '(x)) defined previously by ip(v) := [p, v]. Now, ip(v) = [p, v] = [pg, g~l v] = ipg(g~lv), and hence q>s(pg) = I'^six)) = ipgx°h(
(3.3.4)
= bl([P>
Comments (a) If a: U
(3.3.5)
Now, (3.3.4) implies that s(x) = [a(x), (ps(a(x))], V x e U
(3.3.6)
and hence, using the local trivializations h: U X G -*• 7r~'(£/), h{x,g):= a(x)g, and h' : £/X F - * rcf'(I/),/z'(x, u) := [CT(JC), u], we see that s(x) and %(x) are related by h' (x, Su(x)) = s(x),
V x e U.
(3.3.7)
(b) If a 1: Ux -*• P and cr2: U2-* P are two local sections of P with £/, n U2 # 0 , then there exists some local "gauge function" Q : C/|flC/ 2 ^G such that o2{x) = al(x)Q.(x) V x e U1 n U2. Then su2(x) =
= %(ffi(x) Q (x)) = Q W V ^ f f . W )
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and so the local representatives sUt and sU2 are related by the gauge transformation Sul(x)
= Q(x)sU2(x),
Vjcet/,n[/,
(3.3.8)
Note that the associated trivializations satisfy hx(x,g) = ax{x)g and h2(x, g) = a2(x)g. Hence V°M*,<7) = h2-\ox{x)g)
= h2-l{o2(x)Cl(xyyg)
= (x,
Cl^g)
and so Q(x) is just the transition function gUxu2(x) discussed earlier. 3.4. VECTOR BUNDLES Vector bundles are of considerable importance in theoretical physics because the space of cross-sections of such a bundle carries a natural structure of a vector space and, as such, it can often replace the more familiar linear space of functions on a manifold. From one perspective, a vector bundle is simply a rather special case of an associated bundle in which the fibre is a vector space. However, because vector bundles arise so often, it is also quite common to present them in a framework that makes no direct reference to an underlying principal bundle. We will start this short section with this approach; the associated-principal definition will be given at the end. Definition (a) A ^-dimensional real (respectively complex) vector bundle (E, n, M) is a fibre bundle in which each fibre possesses the structure of a n-dimensional real (respectively complex) vector space. Furthermore, for each x e M there must exist some neighourhood
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U d M of x and a local trivialization h\ U X IR" -*• n \U) such that, for all y e U, h:{y) X•W -*• n~l(y) is a linear map. (b) A vector bundle morphism between a pair of vector bundles (E, n, M) and (E', n', M') is a bundle map (u, f) in which the restriction of u : E —*• E' to each fibre is a linear map. (c) The space T(E) of all cross-sections of a vector bundle (E, n, M) is equipped with a natural module structure over the ring C(M) of continuous, real-valued functions on M, defined by: (i) (s{ + s2) (x) := s{(x) + s2(x) V x e M ; su s2 e F(E) (ii) ( M defined by n{x, v) := x, (d) There is a "canonical" real line bundle y„ (i.e., a onedimensional vector bundle) over the projective space UP" with total space E{yn) := {([x],v)eUPn
X
R"+1|D
=- Xx for some X e U}
in which [x] denotes the line passing through x e U"+l. The
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projection map is n: E(y„) -> UP" defined by n([x], v):— [x]; thus n~l ([x]) is the line in IR"+1 that passes through x (and —x). Local triviality is demonstrated by noting that if U C S" is any open set that is sufficiently small that it contains no pair of antipodal points, and if U' is the corresponding set in UP", then a local homeomorphism h: U' X IR ->• n~l(U') can be defined by h([x], X) := ([x], Xx). Now let us show that vector bundles can also be discussed within the associated-principal bundle framework introduced in the last section. Theorem Let £ = (P, n, M) be a principal GL(n, (R)-bundle and let GL(n, U) act on IR" in the usual way. Then the associated bundle £[1R"] can be given the structure of an ^-dimensional real vector bundle. Proof The map ip:W -* n~„l(x), ip(v) := [p, v] where p e n~\x), is a homeomorphism from IR" onto n~n[(x). To give 7t~nl(x) a vector space structure, choose any p e n~l(x), and define (i) ip(Vl) + ip(v2) := ip{v{ + v2) Vvuv2eM" (ii) Xip{v) := ip{Xv) V v e IR", V X e U. Up' e 7r~'(x) is any other choice such that ip{v') = ip(v) for some v' e U", then ip.(v\) + ip,{v'2) = ip,{v\ + v'2) = ipgig^'v, + g~lv2) for some g e GL(n, IR) = * w ( ^ V i + v2)) = ip(vl + v2) = ip(vl) + ip(v2) and hence the vector space structure is independent of the choice
ofpen~l(x).
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To see that this bundle is locally trivial we define, as for any associated bundle, a map h' : U X IR" -— n^{U), h'{x,v) := [h(x, e), v] where h: UX GL(n, IR) -»• n~l(U) is a trivializing map of P over U C M. It is clear that the restriction h' :{x) X IR" -* 71R1(X) of the local trivializing map h' is linear for each xe UcM. QED Comments (a) The converse statement is also true, i.e., every vector bundle is bundle isomorphic to an associated bundle of this type. (b) If (u, / ) : £ - *
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fields on P are those which arise from the right action of G on P. According to the discussion in Sec. 2.5, to each element^ e L(G), there corresponds a vector field XA on P (cf. Eq. (2.5.1)) which represents the Lie algebra homomorphically in the sense that (Eq. 2.5.24) [xA, XB] = X[AB]
for all A, B e L(G).
(3.5.1)
Furthermore, since the action of G on P is effective, the map i: L(G) -» VFlds(P), A-~XA (3.5.2) is an isomorphism of L(G) into the Lie algebra of all vector fields on P. However, these fields are not suitable for our purposes since they do not point from one fibre to another. On the contrary, the vectors Xj are tangent to the fibre at p e P, i.e., they point along the fibre, rather than away from it. Technically, Xp is said to be a vertical vector, i.e., it belongs to the subspace VPP of TPP defined by VpP:={zeTpP
|7T*T
= 0}
(3.5.3)
where n : P —- M is the projection in the bundle. It is easy to see that the map/1~*-Xp is an isomorphism of L(G) onto VPP; in particular, dim(Fp) = dim L(G) = dim(G). What we need is some way of constructing vectors that point away from the fibre, i.e., elements of TPP that complement the vertical vectors in VpP. This motivates the definition of a connection as: Definition A connection in a principal bundle G —- P —• M is a smooth assignment to each point p e P of a subspace HPP of TPP such that
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(i) TPP « VPP © HPP for all p e P, (ii) St*(HpP) = HpgP for all g e G, p e P where Sg(p) = pg denotes the right action of G on P.
(3.5.4) (3.5.5)
Comments (a) Equation (3.5.4) implies that any tangent vector T e TPP can be decomposed uniquely into a sum of vertical and horizontal components lying in VpP and HpP. These components will be denoted by ver(z) and hor(r) respectively. (b) Equation (3.5.5) implies that the vertical/horizontal decomposition of the tangent spaces of P is compatible with the right action of G on P. (c) The projection map n: P-> Minduces a map n*\ TPP-- Tn(p)M whose kernel is VPP. Thus n*:HpP —»• Tn(p)M is an isomorphism. (d) The connection can be associated with a certain L(G)-valued one-form co on P in the following way. If T e TPP, we define wp(T) = r{(ver(T))
(3.5.6)
there i is the isomorphism of L(G) with VpP induced by (3.5.2). Thus: (l) cop(XA) = AVpeP,Ae L(G) (3.5.7) (ii) (d*co) = Ad9-> co, i.e., (Sg*co)p(r) = Ad ff -.(w p (T)), V T e TPP (iii) T e // P P if and only if wp(x) = 0.
(3.5.8) (3.5.9)
This introduction of a connection one-form is very useful, and in fact an alternative way of defining a connection is as a L{G)valued one-form on P satisfying Eqs. (3.5.7-8). A horizontal vector
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is then defined to be one satisfying (3.5.9), which leads once more to the direct-sum decomposition of TPP in (3.5.4). (e) If co, and co2 are a pair of connection one-forms on P, then so is the affine sum (f°n)a)i
+ (1
-f°n)co2
for any smooth function fe C°°(M). In particular this is true for constant functions, i.e., the set of all connection one-forms is a cone in the real vector space of all L(G)-valued one-forms on P. Now we come to the crucial question of what a connection oneform co "looks" like. A Yang-Mills field is usually written in the form AM", where p is a spacetime index and the index 'a' ranges from 1, . . . , dim G, corresponding to the fact the A transforms according to the adjoint representation of G under "rigid" (i.e., spacetime independent) gauge transformations. Thus, if we write A := AfEadx^ where E{,. .., EAm(G) is a basis set for L(G), we see that the Yang-Mills field can be regarded, at least locally, as a Lie algebra valued one-form on M. The precise relation between this field and the one-form connection on the bundle space P, is given by the following theorem.
Theorem Let a: U C M —• P be a local section of a bundle G —>• P •-* M which is equipped with a connection one-form co. Define the local o-representative of to to be the L(G)-valued one-form toL on the open set U C M given by coL := a* to. Let h: U X G -+ n"\ V) C P be the local trivialization of/"induced by a according to h(x, g) := a(x)g. Then, if (a, (i) e T(xg)(U XG)xTxU® TgG (cf. Eq.(1.3.26)), the
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local representative h*co of coon U X G can be written in terms of the local "Yang-Mills" field w^as (h*G>)^g){a,P) = A
(3.5.10)
where S is the Maurer-Cartan L(G)-valued one-form on G defined in Eqs.(2.3.9-10). Proof Factorthe map h: UX G^
PasU
X G"^
PXG^
P.
(x, g) — (<X(JC),
g)-~o(x)g
Then, (h*co) (a, P) = ((a X id)*S*co) (a, P) = (S*aj\aUU)
(a*a, fi)
= <*>a(x)g((<>oig)*a*a + (#0ja(x))*P) where, by Eq.(1.3.24), ig:P-+PX PX G,g-~(p,g). Thus
G, p—~(p,
g), and
jp:G-+
S°ig{p) = S(p, g) = pg, i.e., d°ig = d„:P-*
P
S°jp(g) = S(p, g) = pg, i.e., S°jp = PP:G^
P (cf. Eq. 2.5.6).
Therefore, ih*(o) (a, P) = (S*coa(x)g) (o*a) + OJa{x)g{Pa(mp).
(3.511)
Now, 6*coa(x)g = Adg-i(a>a(x)), according to the characteristic property (3.5.8) of a connection one-form. We also know that p = Lg
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for some A e L(G); in fact A = H,(/?) according to the definition of the Maurer-Cartan form given in (2.3.11). Furthermore, according to (2.5.10), we have Pa(x)*{Lg) = X„(x)g, and of course co(XA) = A e L(G). Putting together these results with (3.5.11) we finally obtain: (h*co) (a, P) = Ad„-.(coff(x)((T«a)) + 3,(0) = Ad,-.«(«))+3,(0).
QED
Thus a connection one-form co can be decomposed locally as the sum of a Yang-Mills field on M plus a universal L{G)-valued oneform on G. Hence, at least locally, specifying a connection is equivalent to giving a Yang-Mills field. This raises the interesting question of how the familiar gaugetransformations arise in the global fibre bundle formalism. There are two subtly different answers, depending on whether the gauge group is thought of in an active, or a passive, way. In general, a gauge transformation in the principal bundle G —- P -*• M is defined to be any principal automorphism of the bundle. If
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X \x)=Yx»(x)-— t=[
dxv'
(3.5.12)
^ ' dx"
defined on UD U' between the components X*1 and XM' of a vector field on the local coordinate charts U and U' respectively. This can be related to the 'active' view by noting that if (U, cp) is a coordinate chart on M, and iff: M -* M is a diffeomorphism near enough to the identity map that U n f{U) •£ 0 , then a new coordinate chart can be defined as the pair (f(U), f°f~x). I shall leave as an exercise (!) the task of investigating the exact relation between the active transformation X—-f*(X) and the corresponding local coordinate transformation (3.5.12). The precise statement for the Yang-Mills case is contained in the following theorem. Theorem Let co be a connection on the principal bundle G —»• P-* Mand let ox : U{ —» P and a2: U2 —* P be two local trivializations on open sets Ui,U2CM such that £/, n U2 # 0 . Let A^ and Af denote the local representatives of co with respect to o{ and o2 respectively. Then, if Q : t/, n U2 -»• G is the unique local gauge function defined by o2(x) = a,(x)Q(x)
(3.5.13)
the local representatives are related on Ux D C/2 by Af\x)
= Adfl
, ( 4 1 ' W ) + (Q*S),(x).
(3.5.14)
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Proof If d^ denotes the local vector
field
, then Af\x) := (of co)v (d „)
to'' and factorising the relation (3.5.13) as Uin c/2 — J — • X
PXG
"~ (fT,(x), Cl(x))
• ^
p (T,(x)Q(x)
we get A?\x) = ((ol X n)*<5*<w),W = (d*co)(aiMtaM)(aMd,d„
tt*(W
which, by the same type of argument as used in the proof of (3.5.10), can be rewritten as Af\x)
= Ad„ w -. (A«\x)) + H n w (Q*(
Corollary If G is a matrix group, this result can be rewritten using (2.3.13) as Af{x) = n(xylA™{x)
Q(JC) + Q(x) H a,a(x).
(3.5.15)
Comments (a) Equation (3.5.15) looks just like the familiar Yang-Mills gauge transformation which, in a sense, it is. But note that it relates different local Yang-Mills fields on different local regions of M, it does not refer to a single Yang-Mills field.
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(b) The relation with the transformation for a single field is as follows. If a: U->Pisa local section of G -»• P -* M with A := o* (co), then an active gauge transformation cp: P —>• P induces a transformation A—~CJ*(
(x) + Q(x)dftQ(xyl.
(3.5.16)
Note that, if the bundle is trivial, a cross-section a can be defined on all of M, and then (3.5.16) refers to a globally-defined L(G)valued one-form on M. (c) If the principal bundle is non-trivial, it is not possible to describe the connection co in terms of a single Yang-Mills field on M. Instead, one must cover M with local trivializing charts, and then the local Yang-Mills fields associated with any pair of overlapping charts £/,, U} will be related on Ut D Uj by (3.5.15) with the corresponding local gauge function ilu(x) satisfying the relation <7,(x) = Gj(x)Cljj(x). Note that these functions Qy: Ut D Uj—»G are precisely the bundle transition functions dicussed in Sec. 3.3. (d) Conversely, if {[/,•} is a set of trivializing open sets that cover M with corresponding local sections er, :£/,•—»• P and transition function Q.^: t/, n Uj ->• G, a "patching" argument shows that a unique connection one-form on P is defined by any set of local L((j)-valued one-forms A(,) on [/,• C M that are related pairwise by (3.5.15). Example An interesting example of the constructions above is a connection in the principal GL(n, [R)-bundle B(M) of frames on a n-
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dimensional manifold M. Any local coordinate chart (U, q>) on M provides a local section cr: £/ —>- B(M) by associating with xeUC M, the local frame (d,, d2,. .., dn)x. If co is a connection one-form on B(M), let r := o*a> denote the associated L(GL(n, [R))-valued one-form on U, and consider the relation between r and the local one-form V associated with another coordinate chart (C/',
attxeUnU'
dxv/dx,fl.
Then, T'pix) :=
(
=
J"ll(x)(o'*co)Ada)
= /«„(*) (/" : (x) r o (*)/(*) + J"• (x) ao7(x))
(3.5.17)
where we have used the general result (3.5.15) for a connection in a bundle whose structure group is a matrix group. If Gxx is some basis for the Lie algebra M(n, U) (the set of all n X n real matrices) of GL{n, W) then we can write the matrixvalued one-form r„ as ( I ^ = Tl^{G7^fi. In particular, if we pick the natural basis set (Gxx)es := <% Sxs then (3.5.17) becomes the wellknown transformation law for the components Y^ of an affine connection on M: dxa dx" dx'£ dx'" dx's dx*
P
dx'e d2xx dxx dx'"dx
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3.6. PARALLEL TRANSPORT We can now describe what is meant by "parallel transport" in a principal bundle £ = (P, n, M) equipped with a connection co. First, there is a technique for generating horizontal vector fields, i.e., fields whose flow lines move from one fibre into another. Definition Since n*: HPP^- Tnip)Mis an isomorphism, to each vector field X on M there exists a unique vector field X] on P such that, for all peP, (i) n*{x$ = XK{p) (ii) ver{xj,) = 0. The vector field X^ is known as the horizontal lift of X. Comments (a) The act of horizontal lifting is G-equivariant in the sense that dg*(Xl) = X\g. (b) A sufficient condition for a vector field Y on P to be the horizontal lift of a field on M is that (i) ver(Y) = 0, and (ii) Sg*(Yp)= Ypg VpeP,geG since we can then define unambiguously Xx := n*(Yp) for any pen~l(x). Then 7 = Xf. (c) Horizontal lifting has several pleasant algebraic properties: (i) (Xf + Y]) = (X + Y)] (ii) If fe C°° (M), then (/X) f = fin X] (iii) Using the general rule (1.4.19) for "/z-relatedness" we see that
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n*{hor[X\ r f ]) = n*([X\ 7 f ]) = [n*X\ n*Y]] = IX,Y] from which it follows that [X, YV = hor([X\ Y]]). Definition Let a be a smooth curve mapping the closed interval [a, b]cU into M(i.e., a is the restriction to [a, b] of a smooth curve defined on some open interval containing [a, b]). A horizontal lift of a is a curve a- : [a, b] ->• P which is horizontal (i.e., ver[a*] = 0) and such that 7i(a](t)) = a(t)\/ te [a, b]. Theorem For each point p e % ~' (a(a)), there exists a unique horizontal lift of a such that a\a) = p. Proof Extend a to the open interval / := (a — e, b+e) for some e > 0. Then a*(£) is a principal G-bundle over / and a4*(a>) is a connection one-form on P(a*(£)) [Exercise!]. P(a*(i))
°^—*P n
/
2
^M
a((t, p) := p
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Let 0:1—* P(a*(£)) be the integral curve passing through (a, p) e P(a*(£)) C / X P of the unique horizontal lift of the vector field d/dt on /. Define a} := a(°p. Clearly n(a\t)) = a{i) and tt^Qa1])
= ^{[affi)
= CDJ(t){a(*{/3}) = (ae*Q))m([fi]) = 0
where the last equality follows since a*oy is a connection one-form on P(a*(£)) and [/?] is horizontal (/? is the integral curve of a horizontal vector field). Thus af is horizontal. If y: [a, b] —*• P is any other horizontal lift of a, if can be pulled back by a { to give the curve in P(a*(£)), (af1oy) (t) := (t, y(t)) with a((t, y{t)) = y{t). Hence (afco) ([aj^y]) = 0 and so the lift t — (/, y(t)) to P(a*(£)) of the curve t — t in / is horizontal. But this is clearly unique and equal to ji. Therefore af is unique. QED Comments (a) In order to make full use of the ideas above (e.g., when defining covariant derivatives) it is necessary to have as explicit an expression as possible for the horizontal lift a] of a curve a in terms of the connection one-form co. In practice, this usually means making a comparison between the horizontal lift a^ and some other "natural", but non-horizontal, lift of a. Suppose that/?: [a, b] -»• Pis such a curve, i.e., n(fi{t)) = a{t) V t e [a, b]. Then there exists some unique function g : [a, b] —* G such that a\t) = P(t)g(t)
Vte[a,b].
We factor this relation as [a,b]-P*L8> t
—
PXG (P(t),g(t))
S
—
>
P P(t)g(t)
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and then, l<*\v>gw = [3°(P Xff)]p(t)gU)= S*([Pl lg])(fi(t),g(D) = (d°ig(t))*[P]l)«) + (^°JpU))*[g]g(ty Then using the type of argument following (3.5.11), and the fact that o> ([«*]) = 0, we find 0 = Adg(i)-icom([0]))
+ H„w ([?]).
(3.6.1)
But [g)g(t) = g*(d/dt), and 3, ( 0 (g*(d/dt),) = (g*E), (d/dt). In particular, if G is a matrix group we can rewrite (3.6.1) as 0 = g{t)-x
(3.6.2)
This is the differential equation which determines the horizontallift function g{t) in terms of the connection co. (b) If a: U -*• P is a local section of £, associated with a coordinate chart U C M, a natural choice for a lift of a curve a: [a, b] -* t/ is /?(;) := <7(a(/)). Then [/?] = a*[a], and hence <*>p(,)([P]) = (<7*w)a(o ([«])• But a*co is the local representative eou, which we have identified earlier with a Yang-Mills field A. In terms of this field, (3.6.2) becomes
0 = g{tyx AMt))d(t)^-{a{t))
+0(t)'l^(t)
(3.6.3)
where x" are the local coordinates in this chart. If the boundary conditions on g are g(a) = g0eG, (3.6.3) can be re-expressed as the matrix-valued integral equation Qit) = g0-
I A„ («(*))«*(*) g(s) ds
(3.6.4)
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where we have written the components of the tangent vector [a] as d» := dx"(.a(t))/dt. This integral equation for g(t) can be solved as the path-ordered integral g(t) = (Pexp - J = [1-
I A„ (a(s))aM(s) ds
+ J ds, I Ja
-
Afl(a(s))afl(s)ds\g0
ds2 A^ ((«(*,)) A,2 ((«&)) of1 (s{) a"2 (s2)
Ja
W
(3.6.6)
Thus the final local expression for the horizontal lift a} of the curve a:[a,b]-^UcMis a\t) = a{a{t)) fpexp - J A, (a(s))d«(s) ds)g0.
(3.6.7)
(c) If A is subjected to a gauge transformation of the type in (3.5.16) then it is fairly easy to see [Exercise!] that the path ordered integral transforms homogeneously as Pexp — I ^ ( a ( s ) ) ^ ( j ) d j | — £l(a(t))-1 Pexp - J ^ ( a ( s ) ) d " ( s ) ds) Q(a(a)).
X (3.6.8)
We can now give a precise meaning to the concept of parallel translation:
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Fibre Bundles
Definition Let a : [a, b] -* Mbe a curve in M. The parallel translation along a is the map T: A - 1 (a(a)) -*• n~l(a(b)) obtained by associating with each point pen~l(a (a)) the point af (b) where a] is the unique horizontal lift of a passing though p at t = a. Comments (a) Since a horizontal curve is mapped into a horizontal curve by the right action 8 of G on P we have r°dg = <59°T V ge G. In particular, if px = x(p) then pxg = T(/?)# = z(pg), which implies that T is a bijection of fibres. (b) An interesting case is when a is a closed curve (i.e., a loop) in M. There, is no reason why the horizontal lift should also be closed, and in general we get a non-trivial map from n~\a{a)) onto itself given by p— p
Pexp - (ju„ (a(s))af(s) ds
.
(3.6.9)
Thus we have a natural map from the loop space of M into G. The subgroup of all elements in G that can be obtained in this way is called the holonomy group of the bundle and plays an important role in understanding the relation between the connection and certain topological properties of M. This expression also appears a great deal in lattice-space type approximations to Yang-Mills gauge theory. The next important step is to extend the ideas of connections and parallel transport to associated fibre bundles. This is performed by means of the following definition of the vertical and horizontal subspaces of a tangent space to the associated bundle.
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Definition Let co be a connection in the principal G-bundle £ = (P, n, M) and let £[F] = (PF, nF, M) be the bundle associated to £ via a left action of G on F. The vertical subspace of Ty(PF), yePFis defined as (cf. (3.5.3)) Vy(PF) :=
{T
e Ty(PF) \TIF*T
=
0}.
Let kv : P(£) ->• PF, v e F, be defined by kv(p) := [p, v]. Then the horizontal subspace of T[p V](PF) is defined as H[PtV](PF) := kv*(HpP). Comments (a) Since k _i °dg = km the definition of HlP v] (PF) is independent of the choice of elements (p, v) in the equivalence class of y e PF. (b) Let a : [a, b] -* Mand let [p, v] be any point in nFl (a(a)). Let af be the unique horizontal lift of a to P{£) such that af (a) = p. Then the curve
4 ( 0 := K («f(0) = \ot\t\ v]
(3.6.10)
is the horizontal lift of a to PF passing through [p, v] at t = a. This leads to the concept of parallel translation (or transportation) in the associated bundle as the map xF : nFl(a(a)) -*• 71/T1 (a(6)) obtained by taking each point y e nFl (a (a)) into the point aF (b), where t~~~aF(t) is the horizontal lift of a to PF passing through y. (c) If a: U -»• P(^) is a local section of the principal bundle, the natural choice for a lift of a to P(£) is P{t) := cr(a(/)), and then a] = P(t)g(t) where #: [a, b] —» G obeys the differential equation
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171
(3.6.3). As usual, this local section generates the trivialization map h:UXG -» n~l(U), h(x, g) := a(x)g, with inverse u: l n~ (U) -» U X G defined as u(p) := (n(p), x(p)) where %:P^-G is the unique map satisfying/7 = o(n(p)) x(p) V p e n~l(U). Then the image of a* in U X G is u*a\t)
= (a(t), x(P(t)g(t)))
= (a(t), x ( ^ ) ) ^ ( 0 ) = ( « ( 4 0(0)
Similarly, using the local trivialization u' :rcf'(£/) —* U X .F, the image of ajt(/) is the curve t—~(a(t),g(t)v)eUXF.
(3.6.11)
Now, at last, we can consider the way in which parallel transportation is used to define the covariant derivative of a crosssection of a bundle whose fibre is a vector space V. The basic problem of defining a derivative of a cross-section \p : M —>• Pvis the lack of any canonical way of comparing the values of \p at any pair of neigbouring points in M. These values lie in two different fibres and although these can be compared using a local bundle trivialization, the resulting answer will in general depend on the trivialization chosen. However, when the bundle is equipped with a connection this can be used to "pull-back" the fibre over the second point to the fibre over the first, and then a subtraction can be performed unambiguously. The precise definition is as follows. Definition Let <^ = (P, n, M) be a principal G-bundle and let Vbe a vector space carrying a linear representation of G. Let a: [0, e] -*• M, £ > 0, be a curve in M such that a(0) = x0 e M and let \p '• M —- Pv be a cross-section of the associated vector bundle. Then the covariant derivative of \p in the direction a at x0, is
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V[«]^:= i-o I
jeny
(x0)
(3.6.12)
where xv is the (linear) parallel transport map from the vector space ny l(a(t)) to the vector space 7iyl(x0). Comments (a) In a local bundle chart, the horizontal lift of a(t) to U X Fpassing through (x0, v) at t = 0 is given by (3.6.11) as the curve /—— (a(f), g(t)v) where g(f) satisfies the differential equation (3.6.2/3) with the boundary condition #(0) = i . Then, if the local representative of \p is \pu: U-* Fsatisfying (3.3.7), i.e., h'(x, \pv(x)) = ^(x), the element of Vrepresenting xv\p(a(t)) is g(t)~l$u(a(t)). Thus the local representative of V w ^ is the element of V given by:
jt(g(trltu(v(t)))\,=o
/-0
(WiAxo) + Mxo) *u) —^ («(0)
(3.6.13) /=o
where, in deriving the last expression, we have used the result (3.6.3) relating g{t) to the local Yang-Mills field, and also the identity d(g(t) g(t)~x)/dt = 0. Definition (a) If v G TXM, the covariant derivative of the section \j/ of/V along v is defined to be Vv\p := V w ^ where a is a curve in M belonging to the equivalence class of v.
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173
(b) If X is a vector field on M the covariant derivative along X is the linear operator Vx: r(Pv) -»• T(PK) on the set r P K of cross-sections of the vector bundle Pv defined by V^(x):=V^.
(3.6.14)
Comments (a) A particular case of (3.6.14) is VM := Va for which, with the aid of (3.6.13), Eq. (3.6.14) becomes the familiar expression (V„*) (x) = d^(x) + A^x)Hx).
(3.6.15)
(b) Using the notation in (3.3.1), the covariant derivative of the cross-section \p can be expressed in terms of the equivalent function % : P -* Vsatisfying tp^pg) = g~y(p^{p), as V, x * = i„(A"j?v)
(3.6.16)
where ip is the usual map from V^» Pv, ip(v) := [p, v], and Xf is the horizontal lift of the vector field X to P [Exercise!]. Thus the function from P to V corresponding to the cross-section Vx\p is simply X f (^). (c) Not only is Vx a linear operator on T{PV), it also possesses a "derivation" property in the form Vx(M = / V ^ + X(f)f, V / e C°{M).
(3.6.17)
It is also linear in the subscript X(i.e., with respect to the set of vector fields on M regarded as a module over Cco(M))in the sense that (i) Vx+Yt=Vxt+VYt. (ii) V ^ ) = / V ^ .
(3.6.18) (3.6.19)
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174
Equations (3.6.17-19) are characteristic in the sense that one possible way of defining a covariant derivative on a vector bundle Pv is as any family of linear maps V^ : T{PV) -> T(PV), X e FFlds (M), satisfying (3.6.17-19). One can then track backwards and eventually arrive at the idea of a connection one-form on the underlying principal bundle £. All that remains to be done in relating the heuristic theory of Yang-Mills fields to the theory of connections, is to explain how the well-known Yang-Mills field strength F^ arises in this geometrical approach. We know that F^ will turn out to be a sort of "covariant curl" of A^ arranged to transform covariantly under the gauge transformation in (3.5.16). The discussion above of covariant differentiation is not applicable here since the connection/ Yang-Mills fields is not a cross-section of a vector bundle but rather a one-form defined on the principal bundle space. The new idea that must be invoked is a special extension of the exterior derivative to incorporate the vertical/horizontal splitting of the tangent spaces. Definition (a) If co is any A>form on P(£), the exterior covariant derivative of a; is the horizontal {k+ l)-form Dco defined by Deo := dcoohor Dco(Xu X2,..., Xk+l) = dco {horXu horX2,..., for any set Xu X2,...,
(3.6.20) horXk+1)
Xk+l of vector fields on P(£).
(b) If co is a connection one-form on P(£), G := Dco is the curvature 2-form of co. The relation of Dco to the familiar Yang-Mills field strength is contained in the following famous Cartan structural equation:
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175
Theorem If G = Deo is the curvature 2-form of the connection co, then on an arbitrary pair of vector fields X and Y on P{£,), GP(X, Y) = dcop(X, Y) + [cop(X) cop(Y)l V p e />(£)
(3.6.21)
where the bracket [cop(X) cop(Y)] refers to the Lie bracket in L(G) between the Lie algebra elements cop(X) and cop(Y). Proof Since both sides of (3.6.21) are linear functions of X a n d Yit suffices to prove the relation for the three choices: (i) X and Y both horizontal; (ii) X and Y both vertical; (iii) one of the pair X, Y is horizontal and the other is vertical. (i) If Z a n d Y are both horizontal then co(X) = co(Y) = 0 and Dco(X, Y) = dco(X, Y). Hence (3.6.21) is satisfied. (ii) If X p and Yp are both vertical then there exists some A, B e L(G) such that X„ = XA and Yp = XB. Now, a tensor expression like dcop(X, Y) is independent of which local vector fields are chosen to extend Xp and Yp away from p e M. In particular, we could evaluate this expression using XA and XB. Then the right hand side of (3.6.21) becomes dcop(XA, XB) + [cop(XA) oop(XB)}.
(3.6.22)
Now, from (1.7.15), dco(XA, XB) = XA(co(XB)) - XB(co(XA)) A B B co([X , X )). But co(X ) is the constant Lie algebra element B, and hence XA(co(XB)) = 0. Similarly, XB(co(XA)) = 0. By (2.5.24), [XA, XB] = XAB] and hence co([XA, XB]) = co(XAB]) = [AB]. Thus, since cop(XA) = A and top{XB) = B, we see that (3.6.22) vanishes. However, the left hand side of (3.6.21) vanishes identically since X and F a r e vertical. Thus (3.6.21) is satisfied.
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(iii) If X is horizontal and Y is vertical, then G(X, Y) = 0 (because Y is vertical) and [cop(X)cop(Y)] = 0 since co(X) = 0. Thus it remains only to show that dco(X, Y) = 0. Evaluating this tensorial object at p e P by invoking the same argument as above, we can replace Y with XA for some A e L{G). Now, X(co{XA)) = 0 since co{XA) is the constant Lie algebra element A, and co(X) = 0 since Xis horizontal. Thus dcop(X, Y) = — cop([X, XAX). However, if X generates the flow ^ o f diffeomorphisms of M, we know from (2.5.19) that [X, Y] = ^
(Y -
[x,XA] = lim(6aptA*(X)-
X)/t.
r—0
Thus, if Xis horizontal, so is [X, XA] and hence co[X, XA] = 0. QED Comments (a) If £•,,..., EAim(G) is a basis for the Lie algebra L(G), we can write co = coaEa, and then (3.6.21) becomes 1 Ga = dcoa + - CaPyco^ coy
(3.6.23)
where C01^ are the structure constants of L(G). This expression for Ga should be contrasted with the Maurer-Cartan equation (2.3.8). (b) If er: U -*• P is a local section, the local representative A :=a* co of co is supplemented with the local representative F := er*G of the curvature 2-form. It then follows from (3.6.23) that Fa = dAa + \CapyAB~Ay. Or, inserting coordinate indices, f; = - ( ^ - ^ + C V ^ ' ) .
(3.6.24)
ill
Fibre Bundles
(c) It is easy to prove the Bianchi identity DG = 0. (d) If CTJ : C/i -*• P and rr2: U2~* P are a pair of local sections with Ux n C/2 ¥" 0 , there exists some local gauge function Q : £/, D U2 -* G such that er2M = er^x)^*). Correspondingly, there are two local representatives for the curvature 2-form G namely Fw := ox*G and F{2) := o*G. Using an analysis very similar to that employed in the derivation of the gauge relation (3.5.14/15), it can be shown [Exercise!] that these curvature representatives are related by F%\x) = Q(xylFjil)(x)Q(x),
V x e U{ n U2.
(3.6.25)
This completes the derivation of the basic relation between the mathematical theory of connections in principal and associated bundles, and the physicists' familiar theory of the Yang-Mills field and its gauge transformations. Note that, although we have talked above about the Yang-Mills field, the same analysis also applies to the Remannian connection in the GL(n, [R)-bundle of frames B(M). In this case, the parallel transport and covariant derivatives coincide with the familiar operations from elementary Riemannian geometry, and the curvature 2-form taking its values in the Lie algebra of GL(n, U) is nothing but the usual curvature tensor in a non-holonomic basis.
INDEX adjoint action 91,105 adjoint map 72,80 adjoint representation 80,91,157 affine commutation relations 35 affine connection 163 algebraic topology 122 anomalies 83 anti-homomorphism 88 associated bundle 135,138,139, 141,151,153 associated bundle morphism 137, 140 associated fibre bundle 136,149, 169 automorphism 142 automorphism group 129
complex manifold 2 complex projective space 99 connected component 64 connection 154,155 connections in fibre bundles 83 coordinate chart 1,3,14,20,26 coordinate functions 2 coordinate system 20 coordinates 2 coset 94,125 coset space 92,143,148 cotangent bundle 40,114,152 cotangent space 40,43 cotangent vector 40 covariant derivative 166,171, 172,174 covariant differentiation 154 cross-section 24,114,121,132,134, 138,145,148,149,151,152, 162,171 cup product 59 curvature 2-form 174,17 5 curve 8,10,32 CW-complex 122,131
base 126,138,148 base space 112 Bianchi identity 177 bundle 112 bundle map 152 bundle morphism 124,127,140 bundle of frames 126,147,162 bundle space 112
DeRham cohomology 53,83 DeRham complex 54 DeRham's theorem 120 derivation 13,14,17,19,25,28, 32,42 diagonal map 23 diffeomorphism 6,29,30,39,62, 77,88,95,160 diffeomorphism group 34,127, 132 differentiable manifold 1,20,41, 61,64,86,88,101,112,122,126, 146,152 differentiable structure 64 differential complex 54 differential form 43,82,83, 119,121 differential operator 10,28
canonical commutation relations 35 Cartan structural equation 174 Cartan-Maurer equation 85 Cartan-Maurer form 85 Cartesian product 2 centre 80,91 C r -function 5 characteristc classes 122,134 ChristofFel symbol 154 closed form 55 cohomology 55,59 cohomology class 134 cohomology group 43,56,120, 122,134,145 commutator 28,70,71,82,84,105 complete 34,35 complete vector field 39,72 179
180 differential structure 93,94 dimensional reduction 111 directional derivative 10,13 dual Lie algebra 84 dual vector space 84 effective action 90,92,109 equivariant 89 exact form 55 exponential map 77 exterior covariant derivative 174 exterior derivative 45,52,53,83 exterior product 50 fibre 112 fibre bundle 43,52,111,119, 136,151 flow 106,176 frame bundle 138 free action 91,109 gauge gauge gauge gauge
function 86,150,160 group 127,132,133,159 theory 111,149 transformation 129,143, 157,159,168 G-bundle 123 general linear group 63,81 general relativity 133 G-extension 144 G-product 135 graded algebra 51,53 Gribov effect 133 group action 89,110,127 G-space 127.135.136,141 Hamiltonian dynamics 41 Hilbert space 35,149 hoionomy group 169 homeomorphism 88 hornomorphism 71,78,79,80,87, 89,107,109 homotopic maps 58 homotopy group 131,146,149 Hopf bundle 125 horizontal lift 164,165,166, 169,172
Index horizontal subspace 170 horizontal vector 156 h-related vector fields 30,68,164 H-restriction 144 induced bundle 120 induced representation theory 149 infinitesimal transformation 29,36,40,101,109 instanton 111,125,129,146 integral curve 32,34,38,72, 74,75,77,78 internal symmetry group 125,144 isotropy group 93,95,97 Jocobi identity
29
Kaluza-Klein theories kernel 80,90,93 Klein bottle 114,138
111
left action 87,89,99,101 left translation 62,67,74,91 left-invariant vector field 67,81,83,84,85,103,105 Leibniz rule 13 Lie algebra 29,39,61,67,69,70, 78,81,82,84,105,107,109,155, 175,176 Lie bracket 70,105,175 Lie derivative 24,44,45 Lie group 52,61,70,72,74,77,78. 83,86,88,90,92,96,10!, 110, 123,124,129 Lie subgroup 62 line bundle 152 linear frame 126 little group 93 local flow 39 local hornomorphism 74 local representative 5,150,158 local trivialization 152 local, one-parameter group 36,44 locally isomorphic groups 119,140 locally trivial bundle 119,129
Index loop space 169 Lorentzian metric
149
Maurer-Cartan one-form 158 M-morphism 118 Mobius band 113,115 Mobius strip 138 Mobius transformations 63 morphism 5,62,88,89,117,120, 124,142 multilinear map 49 n-bein 148 rc-coboundary 55 H-cocyle 55 rt-form 49,52 normal bundle 117,152 normal subgroup 90,92 ^-sphere 4,96,124 obstruction 134,145 obstruction theory 146 one-form 41,44,45,57 one-parameter subgroup 39,78, 81,101,109 orbit 92,93,95,101,124,127,135 orbit space 93,125,143 orthogonal group 65,128 orthonormal frame 128 overlap function 2,4 Palais'theorem 110 parallel translation 169,170 parallel transport 1 54,164, 169,172 path-ordered integral 168 Picard iterative method 34 Poincare's lemma 55,57 primary obstruction 134 principal automorphism 159 principal bundle 125,127,129, 130,135,137,140,151,155, 159,164 principal bundle morphism 144 principal fibre bundle 123 principal morphism 127,133,139
181 product bundle 113,115,119,124, 129,137,141,142 product manifold 21 projection map 9 projective space 124,149,152 pseudo-Riemannian metric 149 quotient group
91
Riemannian metric 101,127,132, 147 right action 88,89,101,102,107 right-invariant vector field 68,72,82,83 right translation 62,67,87,105 scalar density 139 self-adjoint operator 35 simplex 134 slot function 2 special linear group 65 special unitary group 67 spinor frame 128 stability group 93,94,95 stereographic projection 4 structure constants 72,84 structure functions 142 structure group 124,135,138, 146,147 subbundle 115,148 subgroup 65,67,92,95,143,144 submanifold 2,3,62,95,102,115, 152 tangent bundle 9,20,24,114,116, 138,147,152 tangent space 6,9,10,67,70,73,77, 81,111,126,138,156 tangent vector 9,10,19,24,32,81 tensor 47 tensor field 48,49,138,147 tensor product 46,48 topological group 92 topological space 1,88 topology 93 transformation group 87
182 transition function 1,142,151, 162,163 transitive 95 transitive action 91,94,96 translation function 137 trivial bundle 119,140 twisted function 114 twisted product 118,125 unitary 67 universal bundle 122,131 universal characteristic class 122,134 universal factorization property 46
Index vector bundle 11,41,114,149,151 vector bundle morphism 152,154 vector field 24,25,26,28,29,32,37, 38,67,72,74,81,101,102, 104,106,147,159 vertical subspace 170 vertical vector 155 wave mechanics 35 wedge product 50,51,53,60,84 Yang-Mills field 157,159,161, 167,174 Yang-Mills theory 83,86,114,125, 127,129,133,143,144,154
