Deformations of groups of gauge transformations

JOURNAL

OF FUNCTIONAL

ANALYSIS

Deformations

101,

(1991)

&)8Jt‘tl

of Groups

of Gauge Transformations

A. J. NICAS* Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada AND

C. W. STARK of Mathematics,

Department

Gainesville, Communicated Received

Florida

University 3261 I

of Florida,

by A. Cannes July 6, 1990

This article is concerned with the deformation theory of representations of a given group H into the group of gauge transformations of a principal G-bundle over a manifold X. We develop for gauge groups an analog of Andre Weil’s theory of deformations of representations of finitely generated groups into Lie groups, Explicit computations of the infinitesimal deformations and the associated higher cohomology are made. We extend the Weil theory to treat infinitesimal deformations arising from vector fields on the base manifold and examine the role played by the continuous cohomology of the gauge group. tK? 1991 Academic PXSS, 1nc.

In this article we investigate the deformation theory of representations of a given group H into the group of gauge transformations of a principal G bundle over a space .Y. The group of all gauge transformations can be viewed as an infinite dimensional Lie group when endowed with the appropriate topology. It is interesting to compare the theory of such groups with that of finite dimensional Lie groups. The representation theory of gauge groups has been examined by a number of authors [PS, Gel. Our point of view is somewhat different as we will focus on representations of groups into gauge groups. A deformation of a representation is a parametrized family of representations with prescribed properties * Partially Canada.

supported

0022-1236191

$3.00

by

the

Natural

Sciences

408 Copyright C 1991 by Academx Press, Inc. All rights of reproduction in any form reserved.

and

Engineering

Research

Council

of

GAUGE

409

TRANSFORMATIONS

containing the given representation. There is a well-developed theory of deformations of representations of finitely generated groups into finite dimensional Lie groups due to Andre Weil and others. We develop an analog of this theory in the context of gauge groups. Given a smooth principal bundle P with structure group G and base space X, let Aut(P) be the associated bundle of groups arising from the conjugation action of G on itself. Suppose &’ is a functor which assigns to a bundle an appropriate space of sections. The gauge group associated to k’ is JZ Aut(P). When P is the trivial G bundle and J%’ is the functor which assigns to a bundle the space of C’ sections where 0
1.4. Assume

X is a connected manifold and let ~2 be section functor. If f is a finitely generated group and if M Aut(P)) is such that

an appropriate

QE Hom(T,

H’(I-;

AdGopl,)=O,

for each x E X, then Q is locally rigid in Hom(T,

4 Aut(P)).

In the above theorem AdG is the adjoint representation of the structure group G on its Lie algebra and ~1,~is the representation determined by p of r into the copy of G in Aut(P) lying over x. The infinite dimensional Lie group J& Aut(P) also has an adjoint representation denoted AdM A”t(P). The computation of H*(J’, Ad-& A”t(P) 0Q) for an arbitrary representation Q: f -+ JZ Aut(P) is a difficult and interesting problem. For convenience we will often write H*(T, Q) for I-I*(l’, Ad.A(A”t(P)q). In the explicit computations of this cohomology which appear in section 2 and are summarized below, we will be primarily concerned with the case that P is the trivial bundle and J%! is the C” section functor. In particular we consider representations Q: r+ Map”(X, G) where 0 < m < co. Given x E X the homomorphism Q,~: r+ G is defined by the composition of Q with the evaluation homomorphism ev,: Map”(X, G) -+ G. PZ*(r, e,) will be used to denote the cohomology H*(T, Ad’oe,).

410

NICAS AND STARK

We observe that the methods of sheaf cohomology are applicable to the computation of H*(T, Q): THEOREM

2.2.

For 0
< 00 H*(T,

Q) z rxb!EF&(Z,

&Y9).

TX is the global sections functor, d m3gis the Zr module sheaf of C” 3 valued functions on X, and dXF * are derived functors of #0& in the category of Zr module sheaves. Associated to a homomorphism Q: r+ Map”(X, G) there is a natural map 4: X-t Hom(r, G) defined by Q(x) = ev,e = Q,. Recall that a group r is said to be FP, if Z has a Zr projective resolutions P, such that Pi is finitely generated for 0 d id n. We prove the following vanishing theorem. THEOREM 2.8. Suppose Q: r+ Map”(X, G), 0 < m < co, is such that the image of Q lies in a single orbit of the conjugation action of G on Hom(T, G). Let n > 0. Zf r is FP, and H”(T, Q .) = 0 for some x E X then H”(I’, Q) = 0.

Related to Theorem 2.8 is the following vanishing criterion for H’ : THEOREM 2.7. Let 0 6 m < cc and Q: T-r Map”(X, G) be a representation. Suppose r is finitely generated and for each x E X, H’(T, Q,) = 0. Then

H'(T,p)=O. It is possible to calculate H*(r, Q) if Q: r+ Map”(X, G), 0
e) Z C” sections p) is the vector

and the stabilizer H has the representations is completely

of Q*(c) xH H”(T, bundle

associated

u) to the principal

The hypothesis of this Theorem 2.8 and Theorem 2.14 will be satisfied in interesting casesas in the next result. PROPOSITION 2.16. Suppose connected semisimple Lie group

r is a discrete cocompact subgroup of a L. In addition assume that L has no compact

GAUGE

411

TRANSFORMATIONS

factors and that the real rank I of its Lie algebra is greater than 1. If G is compact, X is connected, and 0 6 m < co then for any representation

Q: r+

Map”(X,

G)

H’(I-,

@)= 0

H”(T,

e)= C” sections of h*(t)

xHH’(J:

e,)

for

n2 L

where x0 is any base point and 5 is the principal H-bundle lying over the orbit of e,, and H is the stabilizer of e,,. If, in addition, Ad’ 0eXOis irredicible and non-trivial then H”(T, Q) = 0 for 1 < n < r.

Several features of the deformation theory of representations into groups of gauge transformations are not encountered, or are of modest importantce, in the finite dimensional theory (i.e., when X is a finite, discrete space). These issues are considered in Section 3 of this paper. The notion of “rigidity” is subject to reappraisal, since one may study rigidity problems for Hom(T, G) with respect to any subgroup of Aut(G). For finite dimensional, connected semisimple Lie groups G, Inn(G), the group of inner automorphisms, is of finite index in Aut(G), but if G is a group of bundle sections then its automorphism group may also include elements defined by deforming the base and bundle. This phenomenon is not novel-the action of the Virasoro algebra of vector fields on the circle appears in the study of loop groups and related physical theories. If P is a flat bundle over X then we can produce infinitesimal automorphisms of P associated to vector fields on X by horizontal lifting. Given a flat connection of P we construct a homomorphism of the space of vector fields on X into H’( r; Ad-M A”t(P) 0 Q). Using this homomorphism we can enrich Weil’s cohomological criterion for infinitesimal rigidity with respect to inner automorphisms sufficiently to detect infinitesimal rigidity with respect to the full Lie algebra of infinitesimal automorphisms of a flat bundle P. Lemma 3.3 and Theorem 3.10 give appropriate generalizations of Weil’s cohomological results. Transversality conditions for infinitesimal rigidity are also discussed in Section 3, and these can be quite sharp if G is a compact group. The homomorphism from vector fields on X to H’(T, Q) mentioned above factors through the natural map H&,,(A Aut(P); Ad,“*“‘(‘)) + H*(T; Ad-K*W=) 0.~). In Section 3 we construct a sequence of homomorphisms VecY(X) -+ H&,,(& Aut(P); AdMAut(‘)) + H’(T, Q) where VecY(X) is the set of C” vector fields on X. Examples are computed to show that this composite can be non-zero. A generalization of the map VecY(X) + H&,,,(.Af Aut(P); Ad”*“‘(“) to higher cohomology using a product construction is also made. If QK is the based loop group of a compact Lie group K then the

412

NICASAND

STARK

existence of nontrivial deformations which detect the image of Vect(S’) --f HiO,,(OK; AdnK) shows that H,!,,,(QK; AdRK) is non-zero and even rather large (see Sect. 3, especially Example 3.6). By contrast if G is a finite dimensional Lie group which is compact or is semisimple and contains no S&(R) factors then H&,,(G; Ad’) = 0.

NOTATION

AND TERMINOLOGY

Suppose G is an arbitrary group and I7 is a finitely generated group with a presentation n=

(yj(jEJ):

W,=f?(iEZ)),

where { 1/1:j E J} is a finite set of generators and { Wi = Wi(yj): i EZ} is a (possibly infinite) set of relations. A homomorphism h: ZZ-+ G is uniquely determined by its values on the generators and hence there is a bijection of sets: Hom(Z7, G) + { (8,) E Gj: Wi( g,) = 1 } given by h H (h(y,)). If G is a topological group then { ( gj) E GJ: Wi(gj) = 1 } inherits a natural topology as a subspace of the product topological space GJ. This topology is independent of the presentation as it coincides with the compact-open topology on Hom(U, G) where Z7 is viewed as a discrete space. We will use the notation R(U, G) for the set Hom(ZZ, G) endowed with this topology. When G is a Lie group R(ZZ, G) is a real analytic variety; furthermore, if G is the set of real points of a linear algebraic group then R(ZZ, G) has the structure of a real affine variety. R(Z7, G) is customarily called the representation variety of n in G (see [LM] as a general reference). The group of all continuous automorphisms of G, denoted Aut(G), acts on the left on R(Z7, G) via tl (8,) = (c((8,)). In particular G acts on R(Z7, G) by conjugation. This action is continuous (real analytic if G is a Lie group, algebraic if G is algebraic). If X and Y are smooth manifolds and if m is an extended integer, 0 d m < cc, then we let Map”(X, Y) denote the space of C” maps from X to Y, with the C” topology, i.e., the topology of uniform convergence on compact sets of functions and their derivatives up to order m (all orders if m = co). If G is a finite dimensional Lie group and X is a smooth manifold, then Map”(X, G) is an infinite dimensional Lie group under pointwise multiplication, and its Lie algebra is Map”(X, 59), the space of maps from X into the Lie algebra of G, under pointwise bracket [PS, MI]. Two variations of this construction are important for us. In the first instance, maps from X to G may be replaced by sections of suitable bundles. The bundles of groups of interest to us will always be the underlying bundles for groups of gauge transformations of principal bundles, so

GAUGETRANSFORMATIONS

413

we review these bundles here. Let G be a real finite dimensional Lie group which, unless otherwise stated, will be assumed to have finitely many connected components. Let rc: P -+ X be a principal G-bundle. The bundle of Lie groups Aut(P) := P x G G associated to P by the conjugation action of G on itself has as its space of C” sections Map”(Aut(P)) the group of C” gauge automorphisms of P. This group of principal bundle automorphisms has as its Lie algebra the space of sections Map”(Ad(P)) of the bundle of Lie algebras Ad(P) := P xG Y associated to P by the adjoint representation of G on its Lie algebra 9. We may also generalize Map”‘(X, G) by replacing Map” by other section functors with values in the category of Banach manifolds, notably by taking space of Sobolev or Holder sections. If rc: E + X is a fiber bundle, then we usually let AE denote any of these section functors, with the proviso that Holder sections will be considered only for compact manifolds X and that any spaces of Sobolev sections will be required to satisfy the hypotheses of the Sobolev multiplication theorem, as on page 114 of [FU], so that group operations are continuous. We will refer to [Pa] for the common properties of these functors and to [FU] for properties of gauge groups of Sobolev sections. 1. WEIL'S THEOREM AND LOCAL RIGIDITY FOR GAUGE GROUPS Weil’s Theorem on Cohomology and Local Rigidity

Andre Weil’s seminal argument relating local rigidity of subgroups of Lie groups to first cohomology generalizes to subgroups of Banach Lie groups of maps to finite dimensional Lie groups. The discussion below follows in outline the treatment of deformations by [Wel, We2, We31 in the account of [Ra]; our references for the differential topology of infinite dimensional manifolds are [La, Pa]. LEMMA

1.1. Supposethat Ef-F&G

is a sequenceof Banach manifolds and C’ maps, where h 0f(E) = c E G; supposea E E, b E F, and c E G satisfy

f(a)=b, h(b) = c; assume and assumethat this subspaceof T,,F is complemented, as are im(dhl,) and

414

NICAS

AND

STARK

ker(df I,). Then the image of an open neighborhood neighborhood of b in h ~ ‘(c).

of a in E contains a

ProoJ: This is the first lemma of [We31 or [Ra, Lemma 6.8, p. 913, translated into a Banach manifold setting. The argument begins with a few reductions. Since the statement is local, we may reduce to maps between Banach spaces by taking neighborhoods. The splitting hypothesis at G implies that by further cutting down our neighborhood of c we may assume that h is a submersion by composing with projections onto a subspace identified with the image of im(dhl,). By taking a subspace of T,E identified with a complement of ker(df I,), we may take a submanifold W of E transverse to the kernel of the differential at a; replace E by W in the diagram and the left-hand arrow has as its differential a continuous linear isomorphism onto a complemented subspace of T,F. After these reductions, we have a split, short exact sequence at the differential level. The implicit function theorem for Banach manifolds of [La, pp. 17-191, implies that h-‘(c) is a submanifold of F, at least near b, and that f(W) is also a submanifold of F containing b; the two tangent spaces coincide at b and f( W) E h -l(c). At this point in the finite dimensional argument Weil notes that the two submanifolds of E have the same dimension and so agree locally because of the inclusion of one in the other. The differential splitting hypothesis at E gives the Banach manifold version of the argument. The normal bundle of f( W) in h-‘(c) is well defined, since both have complemented tangent spaces. If f( W) and h-‘(c) were unequal near b then this normal bundle would be non-zero; our hypotheses force this normal bundle to vanish, so the two manifolds agree near b. 1

The next result is a cousin of Theorem 14.8 of [Pa]. We will let J%! denote any of the section functors satisfying the axioms [Pa, Chap. 43, e.g., &Z = Co, c’, or ~2’ is a Sobolev or Holder section functor. See Chapter 8 of [Pa] for his discussion of these examples, all of which give Banach manifolds of sections when applied to a smooth finite dimensional fiber bundle. If f: q + [ is a differentiable morphism of fiber bundles (so the bundles have a common base X, the map f: E, -+ E, on total spaces carries fibers to fibers, and f covers the identity on the base X) then the “vertical differential” Sf of f is defined to be the restriction of the differential off to the vertical tangent subbundles, Sf: Vert TE, + Vert TR,, Vert TE, := ker(dp,: TE, + TX), Vert TE, := ker(dp[ : TE, + TX).

415

GAUGETRANSFORMATIONS

LEMMA 1.2. Suppose n, c, and 8 are smooth fiber bundles over a connected mantfold X, that I], c, and X are finite dimensional, and that bundle maps ++@-+e

are given which are sufficiently C’ maps by composition

and are such that (h 0f )(E,) and y E A8 satisfy

smooth (e.g., C” works in all cases) to induce

is a section of 0. Suppose that a E An, BE A[,

f,(~,=B? h,(B) = Y and that for all XE X, im(df I,(,,) = ker(6hl&. Then the image of an open neighborhood of cI in An contains a neighborhood of B in (h,)-‘(y). Proof If the bundle morphisms are C” then Palais’ Theorem 14.8 and the lemma above give the conclusion immediately. The argument applies in slightly more generality, though, and it worth our trouble to assume the weaker hypothesis that our bundle maps induce morphisms of section spaces under composition. The derivative hypotheses of Lemma 1.1 are the crucial issue. The tangent space to Aq or to A’[ at a section is identified with a space of “vertical sections”: T,A?v] g &‘(a*

Vert TE,),

TB.A?[ z A@*

Vert TE,).

If the composition maps above are C’ then their differentials are compositions with the vertical differentials: d(f~)l.(o)l.x

= (W.&&a(x)),

d(h,)l~(~)I.x= (Wpcx,)g(+)l Under our exactness hypothesis, the rank of the kernel of the vertical differential of h along p is constant and these kernels form a vector sub-

416

NICAS

AND

STARK

bundle K of the pullback via /I of the vertical bundle. Pick a Riemannian metric and take orthogonal complements to find a complementary subbundle IC’ to K. Then

so the kernel for Lemma 1.1 is complemented, .Af!K.

since this kernel is exactly

1

The bundles of greatest interest to us in the discussion below will be bundles of groups whose spaces of sections are groups of gauge transformations. We recall that if P is a principal bundle over base space X with fiber G then its principal bundle automorphisms or gauge transformations are sections of the associated bundle of groups Aut(P) = P xG G, where G acts on itself by the adjoint action. We will also have occasion to consider the “adjoint bundle,” Ad(P) := P x L-‘9, the bundle associated to P by the adjoint representation of G on its Lie algebra 9. For our purposes the bundles Aut(P) resemble groups, and these bundles will have spaces of sections which are indeed groups. A few observations are in order. First, two sections of Aut(P) may be multiplied, resulting in another section of this bundle of groups. The crucial thing to check here is that if sections u and fl are given over a local coordinate neighborhood Ui z X by !xi, /Ii: Ui -+ G and if we define the restrictions to each Ui of the product section by y,(x)=a,(x) .Bi(x), then the effect of a transition function hivi: lJin U,+G is Y,i(X)=hi,,(X).Yi(X).h;,,(X)~’ =h,,i(x).cr,(X).D;(x)h;,.j(~)~’ =hi,l(x)ai(x).h,,(x)-’

.h,Jx)$;(x)

h,,(x)-’

so this form of pointwise multiplication is well defined. Second, essentially the same observation on transition functions shows that taking products over points in X defines a map Aut(P) @ Aut(P) + Aut(P), where the domain is the Whitney sum, and similarly taking inverses fiberwise defines a map Aut(P) + Aut(P). Both of these maps are as smooth as the bundle Aut(P). The third observation begins with any word W in letters g1> g;’ , ***’ gk, g; ‘. Just as W defines a map n 1Gi Gk G + G, so this word defines a map on the Whitney sum, hw: @r <,
GAUGE

417

TRANSFORMATIONS

just an elaboration on the second observation, since h, may be thought of as a combination of multiplication and inversion maps, and as above h, is as smooth as the underlying bundle. Our final observation of this sort is that Aut(P) has a well-defined neutral element in every fiber and a smooth section whose value over every point in the base is this neutral element. Combining this with the map hw: OlGi
1

UE @

Aut(P): h,(u)(,=el,foreachxEX,

I
ill

I

.

As in the usual construction of the representation variety, even if Z7 is finitely generated but not finitely presented, a finite number of the relators Wi are sufficient to define the representation variety, by Hilbert’s basis theorem. (Here we note that the intersection with each fiber of R(Z7, Aut(P)) is exactly the usual representation variety, and that a finite generating set of words chosen for one fiber will suflice for all fibers, again thanks to the fact that transition functions act by conjugation in G.) The gauge group of the principal bundle P is Gauge(P) := J&’ Am(P), where we may take any of the Banach section functors appropriate for the preparatory discussion and the lemma above. We now observe that R(Z7, & Aut(P)) = AR(ZZ,

Aut(P))

since (a) every collection of k sections of Aut(P) which satisfy the defining relations must actually satisfy those relations “pointwise,” that is, in each fiber of the bundle of groups Aut(P), and (b) the topologies of this mapping space are inherited from @ , GiS k &! Aut( P) in both instances. We now recall the setup for Weil’s argument and the appearance of Iirst cohomology in a sufficient condition for local rigidity. If L7 is a finitely generated group, say Z7= (“pi (jEJ):

Wi=t? (ieZ)),

where J is a finite indexing set and I is not assumed to be finite, then the local rigidity problem for a representation u: L7 -+ G is analyzed via the diagram

418

NICAS

AND

STARK

with maps defined by

h:

(Uj)

l-b

( Wi(Uj)).

Note that f(e) = (~(7~))~ that h(u(y,)) = (e), and that h o.f(G) = (e)~ nit, G. The usual identification of the representation variety with a subset of the product njEJ G identifies Hom(Z7, G) with K’((e))c njG5 G, and an individual homomorphism such as u is identified with the product in n,,, G of its images of the Y,~.The representation u is locally rigid if and only if the image f(G) contains an open neighborhood of (I), so our problem lies in the context of Lemma 1.1. The differential off is

TCG -+TM,,,n G jtJ

but this differential identifies T,,,,,, IYIjEJ

is more easily understood G

with

T,,,

H,E

after right

translation

J G,

(A)

where

Z((I’,))=Adou and where Z is understood as the Fox Jacobian whose kernel is identified with Z’(Z7; AdGo u). Note that the image of B is B’(Z7; AdG 0 u). (See [LM, Ra], or [We31 for versions of this analysis.) Thus the vanishing of the cohomology group Hr(Z7; AdG 0 U) is exactly the hypothesis that

im(df I,) = ker(dhl cucy,d Weil’s theorem is proved by applying Lemma 1.1 to the setup described here to conclude that the orbit under conjugation of our model representation u contains a neighborhood of u in the representation variety. The extension of Weil’s argument to groups of gauge transformations regarded as Banach manifolds requires only rewriting Diagram A and

419

GAUGE TRANSFORMATIONS

imposing the splitting hypotheses required for the Banach manifold Implicit Function Theorem. Diagram A becomes the following commutative diagram in our setup. Here e denotes the “identity section” and u is our base section of R(17, Aut(P)): T, & Aut( P) A

T,,&

@

Aut(P)

dbl.

T,,,J%’ @ Aut(P)

l
itl

d(Ru-l)l,

I( T,k’ Aut( P) h

(B) I/

I

T,,,Jif

63 AW’)~

T,,, A’ g, Aut(P)

I
Thus we have the following generalization manifold evironment.

of Weil’s theorem to the Banach

THEOREM 1.3. Suppose that G is a finite dimensional, connected Lie group, and supposethat P is a principal bundle with structure group G, with Aut(P) the bundle of groups associatedto P by the adjoint representation of G in G. Let JH be one of the section functors cited above. If IT is a finitely generated group, if u E R(IT, J.&!Aut(P)) is such that

ff’(17; AdMA”‘(P)o u) = 0, and if ker(B), im(B) = ker(Z), and im(Z) in Diagram B are complemented subspacesof the tangent spacesappearing in the second row of Diagram B, then u is locally rigid in R(IT, JY Aut(P)).

Under some circumstances we can readily verify the splitting hypotheses needed for applying Lemma 1.l, for example if Lemma 1.2 is relevant. The next result is a down to earth instance of that and asserts that pointwise rigidity hypotheses can have global rigidity consequences for subgroups of gauge groups. THEOREM 1.4. AssumeX is a connected manifold and let Jt? be one of the space-of-sectionfunctors cited above. If IT is a finitely generated group and if u E R(IZ, .& Aut(P)) is such that

H’(I7;

AdCo ul,) = 0,

for each x E X, then u is locally rigid in R(I7, &! Aut(P)). Proof We begin with the observation that the finite dimensional Weil theorem shows that our vanishing hypothesis in first cohomology implies that each u(x) is a locally rigid representation of 17 in Aut(P)I, z G. It

420

NICAS AND STARK

follows that u(X) lies in a single G-orbit in the variety R(ZZ, G). This G-orbit is a smooth submanifold of the appropriate product of copies of G. The pointwise vanishing hypothesis asserts that in the pointwise version of Diagram A, ker(Z.,) = im(B,) for each x E X, since X is connected, this implies that the linear maps Z,r and B, have constant rank, and Lemma 1.2 gives the splitting we need. Thus the pointwise vanishing hypothesis implies the existence of the splittings needed for the application of Lemma 1.I ; the argument for Theorem 1.3 takes over at this point. 1 Open Mapping Results Most of the work in this paper drives at “local-to-global” conclusions, but there are also observations which should be made of a “global-to-local” character. This discussion will be simplified if we begin with some statements on evaluation maps in spaces of sections. PROPOSITION 1.5. Suppose F + E A X is a fiber bundle over a manifold X with a Euclidean neighborhood retract F as fiber. Given x E X, fix an ident@ation via a chart of the fiber over x with F, z- ‘(x) + - F. Define an evaluation map Ev,~: Co(E) --* F as the composite

Co(E) a

7~ ‘(x) IL,

F

s ++ W(x)). Then Ev., : C”(E) + F is an open mapping. Proof: X is not necessarily compact, so that space Co(E) of continuous sections of our bundle is given the compact-open topology. Pick an imbedding F c1\ RN for some N and consider the induced imbedding

of the preimage of a chart neighborhood of x. Given u E Co(E), we want to show that for each open neighborhood Z of u in Co(E), Ev,(Z) contains an open neighborhood of $(u(x)) in F. Let N(K, Y) be a subbasic open neighborhood of u in the compact-open topology, so K is a compact subset of X, Y is an open subset of E, N( K, Y) = {s E Co(E): s(K) E Y}, and u(K) E Y. We may assume that K contains a neighborhood of x and that u(x) E Y. Choose 6 > 0 so that the intersection of a tubular neighborhood of u(K) of radius 6 in U x RN with U x F lies in Y. Take a neighborhood W of F in RN which retracts to F via r: W + F and take a neighborhood V in X with compact closure such that x E Vc P c U n K. Given f. = $(u(x)) and

421

GAUGETRANSFORMATIONS

any point f, in F, within an RN-ball of radius 6, centered at fO and contained within W, construct a function h such that h:X+RN if

h(y)=0

Y# K

h(x) =fi -fo h(y)EW,

6)

for all

y EX

if if

y$V,and ye V.

by the Tietze extension theorem. Define s1 E Co(E) by Sl(Y) =

S(Y)

(s+roh)(y)

Then $ osI(x) = fi and s1 lies in N(K, Y). Since f, is ‘an arbitrary point of the open &ball centered at Jo, we conclude that Ev, is an open mapping. 1 R(I7, Aut(P)) is an example of a “varietal subbundle” of a smooth bundle, since this subset of the bundle @ Aut(P) is a bundle with smooth transition functions, but with fibers which are varieties. This imbedding of R(Z7, Aut(P)) in a bundle with manifold fibers smooths discussion of function spaces for R(Z7, Aut(P)). To discuss c’, Sobolev, or Holder versions of the proposition above, we want to make an assumption of “local smoothness,” which appears below as a manifold neighborhood hypothesis. R(Z7, Aut(P)) may also be studied as a stratified fiber bundle, since the Whitney statification of R(Z7, G) stratifies the fibers of R(ZZ, Aut(P)) naturally enough to produce a stratified bundle structure; from this point of view we are asking below that the image of our section lie in a single bundle stratum. More technical assertions on evaluation maps for the partially smooth section spaces of stratified bundles do not seem appropriate here. PROPOSITION 1.6. Suppose F+ E--S X is a sufficiently smooth fiber bundle over a sufficiently smooth manifold X. If A? is any section functor satisfying the Palais axioms and if u E A!E is continuous then

is an open map at each u E A’E such that u(X) has a marCfold neighborhood W in E. Proof. Use the A-homogeneity of (sufficiently smooth) manifolds to prove this claim. Given so = ($0 u)(a) E F, if& E F is sufficiently close to f.

422

NICASANDSTARK

then there exists an A-automorphism h: W-+ W which is close to the identity, which is fiber-preserving, and which has h(f,) =f,. Then ho UE AE and Ev,(hou)=f,. fi THEOREM 1.7. Let P, X, Aut(P), ~2’ be as above. Supposex EX and pick an ident&ation via a coordinate chart of the fiber z~&~,(x) with G. Let u E R(I7,4 Aut(P)) and let

u., : II -+ G,

be the element of R(I7, G) obtained by evaluation at p. If u is locally rigid in R(l7, & Aut(P)) and u(X) has a manifold neighborhood in R(il, Aut(P)) then u, is locally rigid in R(IZ, G). Proof: If u is locally rigid in R(l7, .k Aut(P)) = A’R(IZ, Aut(P)), then the orbit of U, JZ Aut(P) . u is open in R(Z7, & Aut(P)), and our evaluation map results show that Ev,(& Aut(P) U) = G. u(x) is open in R(Z7, G). [

If Theorem 1.7 applies, then for y sufficiently near to x in X, uy is equivalent to u,, further rigidifying the pointwise representations.

2. COMPUTATIONS

OF COHOMOLOCY

In this section we consider the problem of computing H*(T, Q) for a representation e : r -+ J# Aut( P). We will be primarily concerned with the case where P is a trivial bundle. Let X be a paracompact space and m an extended integer 0 0 assume that X is a smooth manifold. There are sheaves I”, 8’ m,‘, given over an open set U c X by am(U) = C” real valued functions on U &Yg( U) = C” 9 valued functions on U. 8”’ is a sheaf of algebras and d m,Y is a 6” module sheaf. If A is the same symbol will also be used to denote the sheaf of locally A valued functions on X. Given a representation e: r-+ Map”(X, G), the sheaf 6”“,’ module sheaf over the sheaf of rings Zr (actually, a Zr-8” sheaf). The action is defined as follows: Forf E&Fs(U) and yeZT

(rf J(u)= AdG(e(y)(u))f(u).

a set then constant is a left bimodule

423

GAUGETRANSFORMATIONS

Let Ext’,,( -, - ) denote the jth derived functor of Horn& abelian group of homomorphisms of Zr module sheaves. LEMMA

2.1. Forj>O

Hj(T, e)=Ext’,,(Z,

-, - ), the

&,,‘).

Proof. For each XGX the stalk a”‘,“(x) can be embedded in an injective Zr module Z(x). For an open set U E X define a sheaf 9’ by FO( U) = I-I,, u Z(x) with the obvious restriction maps. According to [Go, Theorem 7.1.11 8’ is an injective Zr module sheaf. Applying this construction to 9’/~9’~.~ and successive quotients we obtain a resolution b”~ ’ -+ P* in the category of Zr module sheaves and thus

Ext’,,(Z,

~9~2~) = @(Horn&Z,

9*)).

Since Z is a sheaf of locally constant functions on X it is immediate that the natural map Hom,,(Z, 9*) + Horn&Z, P*(X)) is an isomorphism. Noting that 9*(X) is an injective resolution of the Zr module E”,“(X) = Map”(X, “) we obtain Exti,,(Z,

&,,‘)

The right side is by definition lemma. 1

= Exti,,(Z, W(r,

Map”(X,

Q) completing

8)). the proof of the

Next we consider B%Y-‘,,( -, - ), the jth derived functor of &‘COJ%‘~,-( -, - ) where for Zr module sheaves d and %I’, %O&‘z,-(&, ~8’) is the sheaf whose sections over an open set U are given by X~J&-,(&‘, a) = Horn&&l “, gl U). If 59 is a Zr- &‘” bimodule sheaf then X0&‘&d, 9) and &!XYj,,(&‘, @) are 8” module sheaves. THEOREM

2.2. For j> 0 Hj(T, Q)= TxdXT&(Z,

J?~).

Here we are using r, to denote the global sections functor. ProoJ: There is a spectral sequence [Go, term is given by EFq = HP(X, &%F~,(Z,

Theorem 7.3.31 whose E, 6’“~“))

and whose E, term is the bigraded object associated to a filtration of Ext&(Z, cP~). Since CP is a soft sheaf and 8X94 &Z, c?‘,~) is a 8” module sheaf it is also soft. Hence EFq = 0 for p > 0. Also

424

NICAS

AND

STARK

It follows that E2Y = Ez

= Ext;,(Z,

8,,‘).

The result now follows from Lemma 2.1. 1 Remark 2.3. 1. The real analytic category is somewhat different as the sheaf of germs of real analytic functions is not soft. This case will be treated elsewhere. 2. There is a change of rings isomorphism: 6XF&(Z,

&m39) z &Lit-9-&(R,

c?‘,~‘).

3. Theorem 2.2 is also valid if we replace Map(X, G) with the group of gauge transformations of a possibly non-trivial principal G bundle 5 over X and 8’“. ’ by the sheaf of Cm sections of 5 xG AdG. As a ZT module sheaf, the sheaf Z has a locally free resolution. the stalks of SXS&(Z, &‘m,y) are given by ~22°Fk,,(Z,

F*“)(x)

= Extk,,(Z,

Hence

bmxY(x)) = !?=(r, 8”,“(x)).

These groups can be computed when a suitable local rigidity is condition is imposed. PROPOSITION 2.4. Assume that f is FP,. Suppose x E X and there is an open neighborhood V of x such that y E V implies Q? is conjugate to Q,. Then there is an isomorphism

H”(I’,

&“‘,“(x))

g H”(T,

Q,) @ 67(x).

Proof: Let ,u = Q, and let D = D(,u) = { gpg - ’ 1g E G} be the orbit of p under the conjugation action of G on R(f, G). By hypothesis there is an open set V c X containing x such that G(V) c D. The map n: G + D given by n(g) = gpg- ’ is the projection of a smooth fiber bundle and in particular is locally trivial. Hence there is an open neighborhood U of x contained in V such that ilU: U-+ D lifts to a C” map 6: U-r G. Let L: U-+ Aut(Y) be the C” map defined by L(u) =Ad’(G(u)-‘). Then for every UE U and every y ET, AdG(e(y)(u)) = L(u)-’ Ad’@(r)(x)) L(u). L induces an isomorphism of real vector spaces, also denoted by L, L: 8’“,“(x) + C?-“(X). Given a germ f: (U, x) -+ 9, L(f) is the germ represented by the function L(f)(u) = L(u)f(u) (U E U). The inverse of L is induced by u I-+ L(u)-‘. The vector space $9 has r module structure determined by Q,. Consider the tensor product of Q over R with &Y’(x) viewed as a trivial r module. There is an isomorphism of real vector spaces: $: 9 0 &m(x) + F”z~(x)

425

GAUGE TRANSFORMATIONS

t+k(u@f) = fu. The map II/- ‘L: &m3Y(x) -+ $3@B”(x) is an isomorphism of Zr(~-‘L) &Y(x)H ,(r bimodules. b” “( )) ie(y9 z;‘; )) irfrduces an isomorphism *. . 1 9 ’ x + i 3 X mX is then a consequence of the next lemma. [

or all j > 0. Proposition

2.4

LEMMA 2.5. Let k be a field of characteristic 0 and Van arbitrary vector space over k. If r is FP, and M is any left kT module then the natural map H”(T, M)@, V+ H”(T, Mak V) is an isomorphism.

If dim, V< co then no hypothesis on r is required.

Remark.

Proof. All kT modules will be left kT modules and all tensor products will be over k. The natural transformation @: Hom,,( -, M) @ V -+ Hom,( -, MO V) is defined as follows. For any kT module N @(f@u)(p)=f(p)@u where fEHom,,(N,M), UEV, and PEN. Clearly @: Hom,,(N, M) @ V + Hom,,(N, M@ V) is a monomorphism. Suppose for the moment that V is finite dimensional. Then for any kT module Q a choice of basis for V determines an isomorphism of kT modules Q @ V 2 @y= 1 Q where m = dim, V. Since the functor Horn&N, - ) commutes with finite direct sums, we see that in this case @ is an isomorphism. If N is finitely generated over kT and V is arbitrary then any f E Horn&N, MO V) lies in some k subspace of the form Hom,,(N, M@ IV) where W is a finite dimensional subspace of V. It follows that @ is surjective an hence an isomorphism. If r is FP, then there is a kT projective resolution P, of k such that for 0 d i 6 n Pi is finitely generated as a kT module. There is a commutative diagram

Hom(P,-,,

M)O V-

Hom( P,, M) 0 V @,.

@n-l I

Hom(P,-,,

Hom(P,+,,

I MO V)-

Hom(P,,

MO V) -

HWP,

WC3 V

@??+I I + 1, MO V)

with @, _ I and @, isomorphisms and a,, + , a monomorphism. A diagram chase reveals that @, induces an isomorphism of the corresponding n-coboundaries and n-cochains and hence an isomorphism in cohomology. m Proposition

2.4 has a number of consequences.

COROLLARY 2.6. Assume that r is FP,. Supposex E X and .o, is locally rigid. Then there is an isomorphism

H"(T, a","(X))~HH"(r,@,)oam(x).

426

NICAS AND STARK

Proof: In the notation of Proposition 2.4, the orbit D of Q, is open in R(f, G) because pr is locally rigid. Since 4 : x + R(f, G) is continuous there is an open set V such that d(V) c D and so the conclusion follows

from Proposition

2.4.

1

2.7. Let 0 6 m < co and Q: I-+ Map”(X, G) be a representation. Suppose r is finitely generated and for each x E X H’(L’, 4,) = 0. Then H’(T,p)=O. THEOREM

Proof: By Weil’s theorem [We31 the condition H’(I’, Q,) = 0 implies that Q, is locally rigid. According to Proposition 2.4 the stalks of the sheaf &YYk,(R, &““T~) vanish. Applying Theorem 2.2 we conclude H’(T, Q) = 0. [

The proof of the next vanishing theorem, Theorem 2.8, is similar. SupposeQ: I--+ Map”(X, G), 0 0. Zf r is FP, and H”(T, Q,~)= 0 for some x E X then H”(T, Q)= 0. THEOREM

2.8.

Proof Note that if d(X) lies in a single orbit H*(r, Q,) E H*(f, Q,) = 0 for all y E X. The conclusion

Proposition

of R(T’, G) then now follows from

2.4 and Theorem 2.2. 1

The following simple homological Theorem 2.14.

lemma will be need for the proof of

LEMMA 2.9. Let U be an additive category and V a full abelian subcategory with the property that any morphism ,f: A + B in U with A E Obj(V) has a kernel ker( f) -+ A, a cokernel B + coker( f ), and an image Im(f) + B (i.e., a kernel for B -+ coker(f )); furthermore, ker(f ), Im(f) E Obj(V). Suppose F: U -+ W is a functor to an abelian category which preserves monomorphismsand the restriction F(, is exact. Then for any cochain complex (C*, d*) in U such that cj~Obj(V) for j
F(H’(C*))

r H’(F(C*))

for j d n. Proof: For j< n the hypothesis that F preserves monomorphisms is redundant and the proof is standard. For j=n note that ker(d”), Im(d”) E Obj(V) and there is a commutative diagram:

0 .-----+ F(ker(d”)) a I Oker(F(d”))

-----+

F(C”)-

F(Im(d”)) F(i) I

II + F(Cn) F(d”)) F(C”+‘)

0

GAUGE

421

TRANSFORMATIONS

The top row is exact because Flv is an exact functor and the bottom row is exact by definition. c1 is the unique morphism which exists by virtue of the definition of ker(F(d”)). F(i) is a monomorphism because i: Im(d”) + C”+’ is a monomorphism and F preserves such. A diagram chase reveals c1is an isomorphism. There is another commutative diagram: 0-

F(Im(d”-

F(ker(d”))

‘)) -

---+

F(H”(

C*)) -

0

----+

I H”(F(

C*)) -

0

CI I

I

0 ----+

Im(F(d”-‘))

--+

ker(F(C”))

The exactness of I;[, implies that the first vertical arrow is an isomorphism and that the top row is exact. The bottom row is exact by definition and another diagram chase shows that F(H”(C*)) -+ H”(F(C*)) is an isomorphism. 1 Let Z be a smooth manifold and Y a real topological vector space. The following definitions are suitable for the purpose of this section. DEFINITION 2.10. A left action of a Lie group H on Y is a continuous map H x Y + Y written (h, Y)H hy such that for each h E H the map L,: Y -+ Y defined by Lh( y) = hy is linear; furthermore, ly = y and (hk) y = h(ky) for all y E Y and h, k E H. DEFINITION 2.11. A mapf: Z + Y is C” if it is continuous and for every continuous linear functional L: Y + R the composite Lf is Cm. DEFINITION 2.12. A left action of a Lie group H on a topological vector space Y is Cm if for any smooth manifold Z, any C” map f: Z + Y, and any h E H the composite L, f is C”.

We will use the following notation: Map”(Z, If Z has a C” right H-action Map”(Z,

Y)={f:Z-+YlfisC”}. and Y has a C” left H-action

Y)H = {f E Map”(Z,

Y) 1f(xh) = h-If(x)

define

for all h E H}.

Let Uz be the additive category whose objects are topological vector spaces with C” left H-actions and whose morphisms are H-equivariant continuous linear maps A: Y, + Y,. Let V; be the full subcategory whose objects are finite dimensional vector spaces with a left H-action. Note that a continuous linear map A: Y, + Y2 has a kernel, namely A -l(O) -+ Y,; furthermore, if A( Y,) is closed then A has a cokernel Y, + Y,/A( Yi). The

428

NICAS AND STARK

hypothesis on A( Y,) is needed because we require topological vector spaces to be Hausdorf. If Y, is finite dimensional so are ker(A) and Im(A) and in particular Im(A) is closed. Thus V; is an abelian category and every morphism Y, + Y2 in UG with Y, l Obj(Vg) has a kernel and image. Given a smooth manifold 2 define a functor F UM, + Vector Spaces by F(Y) = Map”(Z, Y)H and if A: Y, + Y, is morphism F(A) is given by composition F(A)(f) = Af: Clearly F preserves monomorphisms. 2.13. If every finite dimensional representation qf H is completely reducible then Flv; is exact. PROPOSITION

Proof. Let 0 + V, 5 V, 8, V, + 0 be an exact sequence in VE, Im(a) is H invariant and V, is finite dimensional. The hypothesis on H implies that Im(cc) = ker(/?) has an H invariant complement W. Let rt: V, + Im(cr) be the corresponding projection and s: V, + V, the composite s = ~1~’ ft. Note that ~(slr~(~) is the identity. Since /5x = 0 and F is a functor F(b) F(or)=O and thus Im(F(cr))c ker(F(fl)). To show the reverse inclusion let f E ker(F(/?)). Then fif = 0 implies Im(f) c ker(/?) = Im(cr). Let g= sf: Then F(g) = f proving that ker(F(/?)) c Im(F(a)). i

Now suppose that e: r-+ Map”(X, G) is a representation such that the imageofg:X~R(f,G)liesinasingleorbitD(~)={g~gLg’Ig~G}ofthe conjugation action of G on R(T, G). Let H be the stabilizer of p: r+ G, i.e., H= (hEGIhp=ph}. If V is a left RI- module define Cj(r, V) to be the vector space of all j-cochains, i.e., maps c: rj+ V with a left r structure given by (Y ’ c)(Y 1T...3 Yj) = Y . tc(Y 13 ...5y,)). The coboundary operator d: Cj(r, p) + C* + ‘(r, p) is defined in the usual fashion: (dc)(Yo, ...* Y,) = YoC(YI t ...gYj) +

c O
+(-1)“’

(-l)if’C(

YO,

...t YiYi+ I > ...2 Yj)

C(Yo, .... Yj- I),

Cj(r, ,u) will be used to denote C’(r, Y) where 3 has the r-module structure given by AdGp. Let B, be the inhomogeneous bar resolution of R over RT (see [HS, p. 2161). Then there is an isomorphism of cochain complexes Hom,,(B,, Y) r (C*(T, p), d). This is essentially a tautological consequence of the definition of B,; the I- action on Y is given by Ad” ,u. Given a group r and left RT modules P, Q such that Q is finite dimensional over R we make Hom,,(P, Q) into a topological vector space by giving it the topology of pointwise convergence, i.e., the smallest topology

GAUGE

429

TRANSFORMATIONS

such that the evaluation maps ev,: Horn&P, Q)- Q defined by ev,(f) =f(x) are continuous. In particular this applies to Horn&B,, 9). C*(T, p) will also be given the topology of pointwise convergence. There is a left action of H on 3 given by restricting the G action. Since H is the stabilizer of p this action commutes with the action of r on 99 given by AdG p. For any RT module P the vector space Horn&P, 9) has a left H action given by (hf)(x) = hf(x). Similarly, CJ(T, p) has a left action of H; furthermore, the coboundary homomorphisms of C*(T, cl) and Horn&B,, 9) are H-module homomorphisms. The isomorphism Horn&B,, 9) 2 (C*(T, p), d) is a topological isomorphism of H-module cochain complexes. Note that with the given topology the left action of H on Horn .,-(B,, 9) (or equivalently C*(T, p)) is C” in the sense previously defined. The map n: G + D(p) defined by n(g) = gpg- ’ is a smooth principal H-bundle which will be denoted by 5. Let 6*( be the pullback via the map 6: X+ D(p). 8*< is also a principal H-bundle whose total space is Th e right action of H is given by E= {(x, g) E XxGIi(x) = n(g)}. (4 &TV = (x9 gh). Recall that Map”(X, 9) is a r-module via the action (y .f)(x) = AdG(&y)(x)) f(x) where x E X, y E r, and f~ Map”(X, 3). There is a natural isomorphism Map”(E,

Cj(r, p))“s

defined as follows: Let fe Map”(E, given by W-)(Y)(X)

Cj(r, Map”(X,

3))

Cj(r, p))“. For XE X and y E r’

!P is

= AdG(g).f-(x, g)(y),

where g E G is such that B(x) = gpgg’ = n(g). It is easy to verify that this is well defined. The inverse of !P is given explicitly as follows: Let CE CJ(r, Map”(X, $9)). Then for (x, g) E E and y E rj

Y”-‘(c)(e, g)(y) = Ad”W’) C(Y)(X). It can be routinely verified that the following diagram is commutative: Map”(E,

Map”(E,

Cj(r, p))” A Cd’)* I Cj+ ‘(r, PL))~ &

Cj(r, Map”(X,

9))

dJ

I Cj+ ‘(r, Map”(X,

8))

430

NICAS AND STARK

Hence !P induces an isomorphism H*(Map”(E, Now, by definition, analyze H*(Map”‘(E,

C*(T, 11))“) 2 H*(C*(f, H*(T, e) = H*(C*(T,

Map”(X, Map”(X,

C*(T, ~1))~) g H*(Map”(E,

3))).

9)))

Hom,,(B,,

so we must 3))“).

Now suppose that r is FP,. Then there is a projective resolution P, + R of the trivial Rf module such that for j< n the module P, is finitely generated over RT. Since B, + R is also a projective (in fact free) resolution there is a chain homotopy equivalence 4: B, + P,. I$ induces an H-equivariant chain homotopy equivalence $*: Hom,,( P,, $9)-+ Hom,,(B,, 3). Since for x EB, (@)* ev, = ev4(,) the map d* is continuous with respect to the topology defined earlier and iff: X+ Hom,,(P,, 9) is C” so is 4*f: Hence d* induces an isomorphism H*(Map”(E,

Hom,,(P,,

F?))H)~ H*(Map”(E,

Hom,,(B,,

9))“).

If every finite dimensional representation of H is completely reducible we can combine Lemma 2.9 and Proposition 2.13 to conclude that ifj< n then Hj(Map”(E,

Hom,,(P,,

Now H*(Hom,,(P,,

9))~ Map”(E,

9))“) z Map”(E,

H’(Hom,,(P,,

3)))“.

H*(T, p) so we have proved that for j
Suppose q is any principal H-bundle with bundle projection rc: T-, B, V is a left H-module, and f: X-r B is a C” map. According to [Hu] there is an isomorphism between the C” sections of the vector bundle Vrf*(qxH V) and the vector space Map”(f*T, V)H given f*(r) ‘H explicitly by (u: f*T -+ V) H s where s(x) = [(x, e), u(x, e)] and rr(e) = f(x). Applying this to the principal bundle 5, f = 4, and V= H’(T, p) we obtain Theorem 2.14: THEOREM 2.14. Suppose n 2 0, r is FP,, and the stabilizer H has the property that each of its finite dimensional representations is completely reducible. Then there is an isomorphism

H”(f,

Q) z C” sectionsofd*(r)

xH H”(T, p),

where i*(5) xH H”(T, ,a) is the vector bundle associated to the principal H-bundle Q*(t).

431

GAUGE TRANSFORMATIONS

The above theorem will be applicable in interesting cases. We first make the observation: LEMMA 2.15. Zf Q: Z+ Map”(X, G) is such that X is connected and for every x E X Q, is locally rigid then the image of 6: X-t R(Z, G) lies in a single orbit of the conjugation action of G. Proof: Choose x0 E X and let D be the orbit of Q,, under the conjugation action of G on R(Z’, G). Let I’= {XE X~Q.~E D}. Clearly I/ and X- V are both open. Since V is not empty and X is connected the result follows. i

We can now prove: PROPOSITION2.16. SupposeZ is a discrete cocompact subgroup of a connected semisimpleLie group L. In addition assumethat L has no compact factors and that the real rank r of its Lie algebra is greater than 1. Zf G is compact, X is connected, and 0
Q: r+

Map”(X,

G)

W(r, Q)= 0 w(r, Q)= C”

sectionsof Q*(t)

xH

w(r, Q,,)

for

n>l,

where x0 is any basepoint and 5 is the principal H-bundle lying over the orbit of Q,.~and H is the stabilizer of Q,. Zf, in addition, AdGo@, is irreducible and non-trivial then H”(T, .o) = 0 for 1 d n < r. Proof: According to Corollary VII 4.6 of [SW] under the above hypothesis on L and G we have for any x E X H’(Z, Q,) = 0. In particular by Weil’s theorem [We3], each Q, is locally rigid and Lemma 2.15 implies that the image of 6 lies in a single orbit of the conjugation action. Also Theorem 2.7 implies that H’(Z’, Q) = 0. Note that a discrete cocompact subgroup of a connected semisimple Lie group is FP,, i.e., FP, for all n. Since G is assumed to be compact, the stabilizer H of any Q, is also compact and so any finite dimensional representation of H is completely reducible. Thus Theorem 2.14 applies. If for some x,, the representation AdGo@+ is irreducible and non-trivial then Corollary VII 4.4 of [SW] implies H”(Z’, Q,) = 0 for 1
3. NOTIONSOF RIGIDITY IN GAUGE GROUPS We consider an extension of the definition of rigidity which makes explicit the role of a group of automorphisms, the inevitable appearance of

432

NICAS

AND

STARK

this extension in our study of representations into gauge groups, and conclude with remarks on product obstructions to deformations. A. Generalizing Rigidity and Remarks on Cohomology The notion of rigidity for a homomorphism i: H + G of groups can be served in different flavors, corresponding to subgroups A of Aut(G). We will say that i: H + G is “A-rigid” if each allowable j: H -+ G agrees with i after composing with an element a of A H-‘-G

IdH I H’-G

LI I

so the map of A to the appropriate space of homomorphisms R(H, G) defined by a E A H a 0 i has all of R(H, G) as its image. The most familiar treatments of rigidity problems use either the subgroup of inner automorphisms (leading to “Inn(G)-rigidity,” in our terminology), as in Chapter VI of [Ra], or the full automorphism group of G (giving what we call “Aut(G)-rigidity”) as in [MO]. We will say that i: H -+ G is “locally A-rigid” if the orbit of i under the action by composition of A on the appropriate space of homomorphisms R(H, G) is open, and i: H + G will be said to be “infinitesimally A-rigid’ if the tangent space at i to the A-orbit through i equals T,R(H, G). Abusing d-rigid” for a Lie language, we will also say that i is “infinitesimally subalgebra d of the Lie algebra of Aut(G) if the image of d under the differential of the Aut(G)-action at i equals T,R(H, G). The local and the infinitesimal notions coincide if a suitable implicit functions theorem is available, but we should be careful with these issues in the context of manifolds of maps. In the language suggested here, we have already seen that Weil’s argument gives a sufficient condition for infinitesimal Inn(G)-rigidity and shows that infinitesimal Inn(G)-rigidity together with splitting hypotheses setting up the implicit function theorem implies local Inn(G)-rigidity in the setting of finitely generated subgroups of (possible infinite dimensional) Lie groups. Our next observation is quite general. If G is a Lie group (perhaps infinite dimensional) and f, : G -+ G is a differentiable one-parameter family then the crossed homomorphism of automorphisms with f0 = Id,, cc gH (Rg),((d/dt) f,(g)) is an element of ZL,,,(G; AdG), the group of continuous one-cycles on G; if h,: H h, G A G is a deformation of a homomorphism h: H + G obtained by composing h with fi, then the crossed homomorphism on H corresponding to h, is h*(a)E Z&,,(H; AdG 0h). We have established the following result.

GAUGE TRANSFORMATIONS

433

PROPOSITION 3.1. Let G be a Lie group (possibly infinite dimensional) and h: H + G a C’ homomorphism and let 5 E TT Aut(G). The crossed homomorphismon H corresponding to composition of h with the infinitesimal deformation 5 lies in Zk,,,(H; AdGo h) and is the restriction to H of an element of Zf,,,(G; Ad’).

See [SW, p. 2601, for more on continuous cohomology. For the finite dimensional Lie groups G most often considered in rigidity problems, Inn(G) is of finite index in Aut(G) and thus deformations of h obtained by composition with deformations of Id, are necessarily inner. For mapping groups J(X, G) and groups of sections, however, automorphism groups are richer and we find that the class of natural deformations obtained above by deforming Id, must be accounted for, and considered “neocoboundaries.” B. Deformation in A(X, G) Rigidity questions for subgroups of a group of maps &‘(X, G) may be posed over a number of subgroups of Aut(&‘(X, G)), and while our first treatment of infinitesimal or local deformations considered gaugeequivalence (“Inn(.M(X, G))-rigidity”), we find that we must also consider some of the other natural subgroups of Aut(&(X, G)), or, more generally, of Aut(& Aut(P)). We begin with 4(X, G), since several features are clearer here than for spaces of sections. Composition with a diffeomorphism 4 of X defines an automorphism of &Z(X, G). The best view of this imbedding of Diff(X) in Aut Jlt(X, G) observes that the semidirect product

acts from the left on the mapping (A(X,

group 4(X,

G) xl Diff(X)) x &(X,

G), via

G) + JY(X, G)

so &(X, G) xl Diff(X) maps to Aut(&(X, G)) with image Inn Jz’(X, G)) x1 Diff( X). Suppose that 17 is a finitely generated group and consider R(Z7, .M(X, G)) % M(X, R(Z7, G)). The Diff(X) action on &(X, G) induces an action on R(fl, &Y(X, G)); differentiating this action at a base representation i gives a map from Vect(X) = ATX to the tangent space

434

NICAS ANDSTARK

TJ?(K Ji’(X, G)) z &(i*TR(H, G)), or, following Weil and the discussion above on continuous cohomology, maps z’(n; A&“(X.“)” i) Vect(.Y) Z&,,,(H; Ad,“(X”)o i) II Vect(X) -

H&,(n,

I Ad-//(X,G)o i) -

I H’(LT; Ad4U(X,G)o i)

PROPOSITION 3.2. The map Vect( X) + H’( n; Ad.“(X,G) Oi) factors through the continuous cohomology ofA(X, G) with Ad-//(xG) coefficients, via

Vect(X) + H&,,(A(X, + H’(A’(X,

G); Ad”(X.G)) G); Ad.“(X,G)) + H’(Z7; Ad”(X,G)o i),

where the second and third homomorphismsare induced by change of ficients and inclusion, respectively.

coef-

This observation is useful mainly as a constraint on the image of Vect(X), but it is striking that continuous cohomology makes a more important appearance in this infinite dimensional context than in finite dimensional deformation problems. Earlier in this paper we saw that Weil’s theorem gives a sufficient condition for infinitesimal and (by the implicit function theorem, assuming splitting hypotheses) local Inn(&‘(X, G))-rigidity of representations of finitely generated groups to &(X, G). Unfortunately, if X is of positive dimension and h: I7--+ &(X, G) is non-constant in the sense that the corresponding function X + R(ZZ, G) is not constant, then the Diff(X)action on R(ZZ, &(X, G)) produces deformations of h which are not Inn(&(X, G))-equivalent to h, so Weil’s theorem in its original form produces no examples of rigid representations. We want a criterion analogous to Weil’s for infinitesimal (and perhaps local) Diff(X)-rigidity and for rigidity under the combined actions of Inn(&(X, G)) and Diff(X). The cohomological test for infinitesimal versions of rigidity under the action of d(X, G) x1Diff(X) is easily stated. LEMMA 3.3. Suppose IT is a finitely generated group, X is a smooth manifold, and G is a Lie group. h: IT + A(X, G) is infnitesimally rigid under the action of A(X, G) x Diff(X) zf the map of Vect(X) into H’(n, AdMM(X,G)) induced by h is epic. In practice, J&(X, G) x Diff(X)-rigidity is more easily detected by a transversality condition than by the cohomological test. FROP~SITION 3.4. SupposeIT is a finitely generated group, X is a smooth manifold, and G is a finite dimensional Lie group. Let h: II+ A(X, G) be a

GAUGE TRANSFORMATIONS

435

homomorphismand let s: X + R(17, G) be the corresponding map of X to the representation space of Ii’ irt G. Assume that s(X) lies in the smooth points of R(l7, G). Let 9 denote the singular foliation of R(I7, G) by orbits of the adjoint action. h: II-, .44(X, G) is &(X, G) ~1Diff(X)-infinitesimally rigid if and only ifs is transverse to 9. Proof. The tangent space T,R(I7, ;I;e(X, G)) E As*TR(IZ, G). Since G is finite dimensional Lie group, its Lie algebra 9 has a finite basis a,, .... ak; abusing notation we also write a,, .... ak for the corresponding constant maps X -+ 9. These constant maps aj induce elements GI of &Zs*TR(Z7, G) (s(x))) and at each x these sections restrict by cj:X++ (XT(dldt)lt=o Aexp(za,f to span s*(F)l,. A vector field V on X produces an element s”(V) of As*TR(17, G) defined by S(V): XH (x, s*( V,)), and this defines a pointwise map SIx: T, X + s*TR(I7, G)I,. s is transverse to 9 if and only if for each x E X, s*TR(Z7, G)l,=Sl,(T,X)+s*(9)l,. Since each vector field on the representation variety which is tangent to .F is a finite linear combination with function coeffkients of the images of the aj, our transversality condition is equivalent to &?s*TR(ZI, G) = s(Vect(X)) + C (A( X, R) ii,.). The last equality asserts that the tangent space at e of Jz’(X, G) MDiff(X) surjects onto the tangent space &s*TR(l7, G) via the differential of the action map, i.e., h is A(X, G) x1Diff(X)-infinitesimally rigid. 1 COROLLARY 3.5. Let X, Il, and G be as in the Proposition. Let h : II + &?(X, G) and assumethat the corresponding map s: X -+ R(lI, G) has its image in the smooth points of R(IZ, G). h is infinitesimally Diff(X)-rigid lj” and only ifs is a submersiononto its image.

These observations leads to the following examples. Here C, denotes the infinite cyclic group, G is a finite dimensional Lie group, X is a finite dimensional manifold, and 17 is a finitely generated group. EXAMPLE 3.6. (1) Suppose X is a smooth component of R(Z7, G) and that h: l7+ J&(X, G) is the representation corresponding to Id,E A( X, R(Z7, G)). Then h is infinitesimally Diff(X)-rigid and infinitesimally JZ( X, G) >aDiff(X) rigid. (2) Suppose that G is a compact Lie group with a maximal torus T< G and that s: X -+ T is a smooth submersion. If we identify R(C,, G) z G, then the representation h: C, + &(X, G) corresponding to s is infinitesimally ..4!(X, G) N Diff(X) rigid.

The preceding example generalizes to a rigidity criterion in the following way.

436

NICAS

AND

STARK

PROPOSITION 3.7. Assume that G is a compact Lie group with a maximal torus T and Weyl group W. The quotient map q: G--t G/A, of G to the quotient for the adjoint action restricts to T so that q(T) = T/W, a ji’at orbifold, and so that q(T) = G/A.. A representation h: C, -+ A(X, G) is infinitesimally &(X, G) x Diff(X) rigid if and only if the corresponding map s: X + G has the property that q 0s: X + T/W is a submersion.

Here the appropriate orbifold sense of “q 0 s is a submersion” is that over manifold points of T/W we have a submersion and over exceptional points with neighborhoods of the form Bk/Q for a finite group Q, our map factors through a submersion to Bk. Both of the essential observations here are unlike finite dimensional deformation theory. The existence of infinitesimally rigid representations of the infinite cyclic group is unexpected. A smooth component of R(IZ, G) will not usually be a closed manifold, and the problem of characterizing closed manifolds X for which R(Z7, J&(X, G)) contains a rigid representation seems to be a challenging problem in foliations. The obvious problem to consider at this point is the extension of these infinitesimal results to local A&(X, G) XIDiff(X)-rigidity. This question is more subtle than local &‘(X, G)-rigidity due to the complexities of the diffeomorphism group and of composition operators, and we expect to return to the matter in another paper. C. Deformations in ~2 Aut(P)

When properly formulated, results of the sort we have obtained for mapping groups J%‘(X, G) ought to extend to groups ~2 Aut(P) of gauge transformations, but some care is required. The main observation is that a connection on P is wanted to produce an action of Diff(X) (rather, an infinitesimal action of Vect(X)) on spaces of sections of Aut(P). The first difficulty is that this infinitesimal action does not always give a Lie algebra analog of the semidirect product observation we exploited in mapping groups. A second problem appears when we attempt to restrict such an infinitesimal action to sections representing homomorphisms l7+ ~2’ Aut(P), since it is not obvious that the vertically projected vector fields we produce will be tangent to the representation subvariety of @ Aut(P). Both of these difficulties can be resolved if we assume our connection on P is flat, since curvature is the obstruction to solving both problems. Suppose that P + X is a principal G-bundle, where G is a finite dimensional Lie group and X is a finite dimensional manifold, and let Aut(P) = P x G G be the bundle of groups with fiber G associated to P by the adjoint action, so that ~5’ Aut(P) is the group of gauge transformations of P.

GAUGE

437

TRANSFORMATIONS

A connection on P, i.e., a G-invariant horizontal distribution in the tangent bundle TP, induces horizontal distributions in T Aut(P) and in T GjEJ Aut(P). Our main requirement is the family of vertical projections TAut(P) + Vert(TAut(P)) = ker(w,: TAut(P) -+ TX), T eiEJ Aut(P) + Vert( T @ js J Aut( P)) associated to the connection. The infinitesimal principal bundle automorphisms of P are the G-invariant vector fields, while the infinitesimal principal bundle automorphisms of P fixing the base of P are the vertical G-invariant vector fields. A connection on P decomposes the space of G-invariant vector fields on P as a direct sum of vertical and horizontal G-invariant vector fields, and (Hor( TP))G

x* ) TTX

is an isomorphism. Although a connection on P induces splittings of tangent spaces to Aut(P) and @ Aut(P), this correspondence between vector fields on X and a family of vector fields on the total spaces of Aut(P) and @ Aut(P) has an analog. Infinitesimal automorphisms of the bundle of groups Aut(P) restrict on each fiber to infinitesimal isomorphisms of fibers and project to infinitesimal diffeomorphisms of X. The kernel of this projection, the set of vertical vector fields preserving the bundle of groups structure, will agree with the image of & Ad(P) = T,Jl Aut(P) under the differential of the adjoint action of & Aut(P) on Aut(P). One might expect the Lie algebra of infinitesimal automorphisms on Aut(P) to split as a semidirect product of adjoint, vertical vector lields on the total space of the bundle plus vector fields on X, but recall that the curvature of a connection obstructs such a splitting. If our connection is flat, then in analogy to the untwisted case considered above, we have a splitting of this Lie algebra of automorphisms. If s is a section of one of the bundles E = P, Aut(P), or @ Aut(P), and if Y is a vector field on X, then we want to produce a vector field W along s associated to V which makes sense as an infinitesimal variation of s among sections. Thus WE Ts*( TE) should be a vertical field, and we take the natural candidate, W= Vert(s,( I’)) E r Vert(s*( TE)) = rs*(Vert(

TE)).

The difficulty which appears now is that if the section s corresponds to an element of R(n, ~2’ Aut(P)) and if V is a vector field on X, then we want the vertical vector field W produced along s from V to be tangent to the subvariety R(17, Aut(P)) s oi Aut(P), and vertical projection will not ordinarily carry s*( I’), which is tangent to the representation variety, to a vertical vector which is also tangent to R(Z7, Aut(P)). However, if our connection on P is flat, then vertical projection will have this property.

438

NICAS AND STARK

PROPOSITION 3.8. Suppose that 7~:P + X is a principal bundle with fiber G and a flat connection w. Let Aut(P) be the bundle of groups associatedto P by the adjoint action, let ZZ= (a, (jeJ): Wi(ai)=e(iEZ)) be afinitely generated group, and let s E J&‘( mj Aut(P)) correspond to an element h E R(ZZ, Jl Aut(P)), so that under the map of bundles associated to the given presentation for ZZ,

@ Aut( P) -L

Gj Aut( P),

IEJ

itl

s maps to the neutral section of the second bundle of groups. Zf s(X) lies in the set of smooth points of R(ZZ, Aut(P)) = h-‘(e), zf VE Vect(X) is a vector field on X, and tf W= Vert(s,( V)) is constructed as above, then h,(W) = 0, i.e., W correspondsto an infinitesimal deformation of h in R(ZZ, J Aut(P)). Proof: Since P is given a flat connection, the associated bundles of groups @ Aut(P) carry induced flat structures. Take flat coordinate charts on the bundles E0 = ejeJ Aut(P) and E, = @ IE I Aut(P), so that in these coordinates over suitable open U G X, the horizontal section of E,, E, over U have the form UH (u, constant). The representation variety, restricted over suitable open UC X, is the preimage of the neutral section under

UxnG-

‘UxnG

i6I

jEJ (4

(gj)JH

t”5

( wi(Sj)))

Observe that if qE R(ZZ, Aut(P)) E E, and nE,,(q) E U, then the horizontal subspace of E, through q lies entirely in the variety R(ZZ, Aut(P)). Therefore, if s is a section of E, representing an element of R(Z7, J%’ Aut(P)), that is, a section of R(Z7, Aut(P)) viewed as a bundle over X, then Hor(T,,,,E,) is tangent to R(ZZ, Aut(P)). Assuming that s(X) lies in the smooth points of R(ZZ, Aut(P)), if we know that s*(V) and Hor(s,( V) are tangent to R(ZZ, Aut(P)), then we conclude that Vert(s,( V)) is also in this tangent space. 1 COROLLARY

3.9. Let

P,

X,

ZZ, and s be as above. The map

VH Vert(s,( V)) defines a continuous homomorphism Vect(X) -+ Z;,,,(d Am(P); Ad A A”‘(P)) -+ Z;,,,( ZZ; Ad, A/A”t(P)0h), associating to V the crossed homomorphism corresponding to the appropriate infinitesimal deformation.

439

GAUGETRANSFORMATIONS THEOREM

3.10.

fiber G, that Aut(P) action, that ZZ= (aj and that s E JY( ej h E R(Z7, &! Aut(P)). epimorphism

Suppose that z: P -+ X is a flat principal bundle with is th e bundle of groups associated to P by the adjoint (je J): Wi(aj) = e (iEZ)) is a finitely generated group, Aut(P)) is a section corresponding to an element Zf the map of the corollary above composesto give an

Vect(X) + H’(Z7; Ad~‘@AU’(P)oh) then h: II -+ & Aut( P) is infinitesimally rigid under the (infinitesimal) action of A Ad(P) xl Vect( X).

The transversality criterion for infinitesimal rigidity offered in the preceding subsection has an analog for bundles. The point in that discussion which requires care is transversality, as adapted to sections of bundles. As above, the new ingredient we require is a projection operator defining horizontal subspaces. DEFINITION 3.11. Supposethat 9 is a (possible singular) distribution on the total space of a fiber bundle [ = (F + E -% B), with a projection operator Vert: TE + ker(n,). A section s of 5 is “vertically transverse” to B if and only if for each b E B,

Vert(s,( ThB)) + Vert(F&J

= ker(x,)l,

3.12.

Let PA X be a flat principal G-bundle, let be a finitely generated group, and let W,(a,)=e(iEZ)) s E &‘( @ j Aut(P)) be the section corresponding to a homomorphism h: 17 + J%!Aut(P). Let 8 be the singular distribution on Qj Aut(P) defined by the adjoint action of G. Then h is infinitesimally J# Ad(P) )(IVect(X)-rigid if and only zf s is vertically transverse to Y. THEOREM

ZZ=(g,(jeJ):

D. Products of I-cocycles Let F be a (topological) group, V a (topological) left F module and 4: V” -+ V a (continuous) n-multilinear map which is F-invariant, i.e., for all geF and v1 ,..., v,~Vgd(vr ,..., u,)=&gv ,,..., gu,). Suppose cj:F+ V for i = 1, .... n are (continuous) 1-cocycles. Then it is routine to verify that the map $: F” -+ V defined by *(ET I,

.*.,

g,)=~(cl(gl)?

g,c*kJ,

g1g,&J*

...? gl’..gn-,cn(&))

is a (continuous) n-cocycle; furthermore, if any ci is a (continuous) 1-coboundary, i.e., of the form ci( g) = (1 - g) v0 for some v0 E V, then II/ is a (continuous) n-coboundary.

440

NICAS

In particular

this construction

STARK

defines a multilinear

(b* : H’(F, or in the topological

AND

map

V)” + H”(F, V)

case

Consider the case F= MG = Map”(X, G), where V= M’S and MG acts on M’S via the adjoint representation. Let vector space of G-invariant n-linear forms Y” + Y. In the example, Q*(Y) contains the Lie bracket [ , 1: Y2 -+ 9. Combining the product defined above with the Vect”(X) + Nb,,,(MG, Ad) we obtain a (n + 1 )-linear map Vect”(X)”

x Map”(X,

VecP(X)”

natural

map

with e* : H&,(MG,

Ad)

Q,,(9)) + Hz,,,(MG,

Given a representation e: r -+ MG composition + H”(T, e) yields a (n + 1)-linear map x Map”(X,

= Map”(X, 3) Q,,(Y) be the case n= 2, for

Ad).

Qn(‘3)) + N”(T, e).

REFERENCES CBWI WI

CFUI IGel [GoI WI

WI lJa1 CKNI PaI CLMI WI

A. BOREL AND N. WALLACH, “Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups,” Annals of Math. Studies, Vol. 94, Princeton Univ. Press, Princeton, NJ, 1980. P. FLEMING, Structural stability and group cohomology, Trans. Amer. Marh. Sot. 275 (1983), 791-809. D. S. FREED AND K. K. UHLENBECK, “Instantons and Four-Manifolds,” MSRI Publications, Vol. 1, Springer-Verlag, New York, 1984. I. M. GELFAND et al., “Representation Theory: Selected Papers,” London Math. Sot. Lecture Note Series, Vol. 69, Cambridge Univ. Press, Combridge, 1982. R. GODEMENT, “Topologie Algt-brique et Thkorie des Faisceaux,” Hermann, Paris, 1958. P. J. HILTON AND U. STAMMBACH, “A Course in Homological Algebra,” SpringerVerlag, New York, 1971. D. HUSEMOLLER, “Fibre Bundles,” Springer-Verlag. New York, 1966. N. JACOBSON, “Lie Algebras,” Wiley-Interscience, New York, 1962. S. KOBAYASHI AND K. NOMIZU, “Foundations of DiNerential Geometry, Volume I,” Wiley-Interscience, New York, 1963. S. LANG, Differential Manifolds,” Addison-Wesley, Reading, MA, 1972. A. LUBOTZKY AND A.R. MAGID, Varieties of representations of linitely presented groups, Mem. Amer. Math. Sot. 336 (1985). J. W. MILNOR, Remarks on infinite dimensional Lie groups, in “Relativity, Groups and Topology II” (B. S. de Witt and R. Stora, Eds.), pp. 1009-1057, Les Houches Session XL, 1983, North-Holland, Amsterdam, 1984.

GAUGE [MO]

[Pa] [PS] [Ra] [St]

[Well [We21 [We31

TRANSFORMATIONS

441

G. D. MOSTOW, “Strong Rigidity of Locally Symmetric Spaces,” Annals of Math. Studies, Vol. 78, Princeton Univ. Press, Princeton, NJ, 1973. R. S. PALAIS, “Foundations of Global Non-linear Analysis,” Benjamin, New York, 1968. Oxford Univ. Press (Clarendon), A. PRE~~LEY AND G. SEGAL, “Loop Groups,” London/New York, 1986. M. S. RAGHLJNATHAN, “Discrete Subgroups of Lie Groups,” Springer-Verlag, Berlin, 1972. C. W. STARK, Deformations and discrete subgroups of loop groups, in “Geometry of Group Representations” (W. M. Goldman and A. R. Magid, Eds.), pp. 301-309, Contemporary Mathematics, Vol. 74, Amer. Math. Sot., Providence, RI, 1988. A. WEIL, Discrete subgroups of Lie groups I, Ann. of Math. (2) 72 (1960), 369-384. A. WEIL, Discrete subgroups of Lie groups II, Ann. of Math. (2) 75 (1962), 578-602. A. WEIL, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149-157.