Characteristic classes of super vector bundlesUgo Bruzzo and Daniel Hernández Ruipérez Citation: Journal of Mathematical Physics 30, 1233 (1989); doi: 10.1063/1.528606 View online: http://dx.doi.org/10.1063/1.528606 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/30/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Holomorphic vector bundles on Sasakian threefolds and parabolic vector bundles J. Math. Phys. 51, 063508 (2010); 10.1063/1.3442721 Stable vector bundles and string theory AIP Conf. Proc. 1130, 71 (2009); 10.1063/1.3146240 Lie algebroid structures on a class of affine bundles J. Math. Phys. 43, 5654 (2002); 10.1063/1.1510958 VectorBundle Classes form Powerful Tool for Scientific Visualization Comput. Phys. 6, 576 (1992); 10.1063/1.4823118 Graded manifolds and vector bundles: A functorial correspondence J. Math. Phys. 26, 1578 (1985); 10.1063/1.526921

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Characteristic classes of super vector bundles Ugo Bruzzo Dipartimento di Matematica, Universita di Genova, Via L. B. Alberti 4, 16132 Genova, Italy

Daniel Hernandez Ruiperez Departamento de Matematicas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain

(Received 7 September 1988; accepted for publication 4 January 1989)

A theory of characteristic classes for complex super vector bundles over supermanifolds is developed. Such characteristic classes are proved to fulfill the usual properties, and it is shown that, under suitable conditions, they can be represented in terms of supersmooth curvature forms.

I. INTRODUCTION

This paper is devoted to the development of a theory of characteristic classes for (complex) super vector bundles (SVB's) by generalizing the construction of the Chern classes of smooth complex vector bundles in terms of the cohomology of the projectivization of the bundle. Such a theory should be relevant to physics, since many techniques exploited in supersymmetric gauge theories or in superstring theory, mainly in connection with the anomaly problem, ac­tually involve nontrivial super vector bundles over super­manifolds and the study of their cohomology ring. I

We consider supermanifolds in the framework of the theory initiated by De Witt and Rogers,2--4 which was given a satisfactory setting by Rothstein.5 Even though we shall use a particular category of supermanifolds (which were called f§ supermanifolds), it should be noticed that the results in the present paper could be easily extended to the more gen­eral setting described by Rothstein, provided that the grad­ed-commutative Banach algebra B that enters the theory is finite dimensional and there is a surjective algebra morphism B-+F, where Fis the ground field.

The resulting category of supermanifolds contains ob­jects that are topologically richer than Berezin and Kos­tant's graded manifolds6

,7; this of course reflects On the co­homology of supermanifolds and on the properties of SVB's on them.

The basic algebraic object in supermanifold theory is a real Grassmann algebraBL = (BL)ofB (BL ). withLgener­ators. (In dealing with complex SVB's we shall also use the complexification of BL,CL = BL Ell Re.) The Cartesian product B 'E + n is graded (by "graded" we always mean "l2 graded") according to the rule

=:= [(BL );;'X (Bdn at [(BL );"X (BL)~] ,

Since there is a direct sum splitting BL = Rat NL, where NL is the nilpotent ideal of B L' maps a: B L -+ R (body map) and s: BL -+NL are defined.

The first step in introducing supermanifolds is to con­sider a distinguished class of functions f U C B 'E,n - B L; the most suitable choice seems to be given by the so-called GHoo functions introduced by Rogers. 4 One gets a sheaf f§ JY' of graded commutative B L' algebras on B 'E.n, where L' is a positive integer such that L - L '>n. The most natural defin-

ition of supermanifold now would seem to state that a (GHoo ) supermanifold is a pair (M, d) where M is a topo­logical space and d is a sheaf of graded commutative B L'

algebras, such that (M,d) and (B ';'"n,f§ JY') are locally iso­morphic as ringed spaces. However, it was shown elsewhere8

that the resulting category of supermanifolds is not suitable to develop a theory ofSVB's that parallels the ordinary theo­ry of vector bundles on real or complex manifolds, e.g., the graded tangent bundle to a GHoo supermanifold has no stan­dard fiber. A solution to this problem is obtained by replac­ing f§ JY' with the sheaf of graded commutative B L algebras f§ = f§ JY' at B L' B L> and defining f§ supermanifolds as pairs (M,d) locally isomorphic with (B 'E,n, f§ ).8,9

A particular class of supermanifolds is given by the so­called De Witt supermanifolds; they are characterized by the fact that they can be covered by means of f§ atlases A = {( Uj,ifj)} such that the images ifj (U,) are of the type if; (U;) = Vi X N 'E,n, where the Vi are open sets in R m

• It can be shown that an (m,n) De Witt supermanifold M is a fiber bundle over a smooth m manifold Mo (called the body of M) with standard fiber N'E,n. Therefore, M and Mo have the same integer cohomology. Moreover, it was proved in Ref. 10 that the structure sheaf d of a De Witt supermanifold is acyclic, which is not the case for a general supermanifold.

The nonacyclicity of the structure sheaf of a generic su­permanifold M has the consequence that the de Rham-type cohomology of the complex of supersmooth forms on M is quite distin8 from the ordinary de Rham cohomology of M, and that a Cech-de Rham type isomorphism fails to exist. Indeed, introducing the sheaves Ok of k superforms on Mby letting

OD=d, OI=:=Der* d, n k = Akn l, for k> 1,

where A is the graded-antisymmetrized graded tensor prod­uct over d, and defining a sheaf morphism (Cartan's exteri­or differential) d: nk

-+ nk + I in the usual way, one gets the following resolution of the locally constant sheaf B Lover M:

(1.1 )

The exactness of ( 1.1 ) is a consequence of a Poincare lemma for f§ superforms, which was proved elsewhere. II The coho­mology of the complex of graded BL modules (rO*,d) ("supersmooth de Rham cohomology") will be denoted by HSDR (M) (r denotes the global section functor). Standard

1233 J. Math. Phys. 30 (6), June 1989 0022-2488/89/061233-05$02.50 © 1989 American Institute of Physics 1233

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cohomological arguments l2 entail the existence of a mor­phism

HSDR (M) --H(M,BL ) =HDR (M) !J7BL> (1.2)

which in general is neither injective nor subjective. However, if M is De Witt, so that the sheaves Ok are acyclic, the mor­phism (1.2) is bijective. 13

Within the category of [§ supermanifolds it is possible to develop a satisfactory theory of SVB's; the category of rank (r,s) SVB's on a fixed supermanifold (M,sf) turns out to be equivalent to the category of rank (r,s) locally free graded sf modules. It is the purpose of the present paper to develop a theory of characteristic classes for such bundles in the complex case. We shall consider characteristic classes with integer coefficients, and we shall study their representa­tion in terms of the curvature form of a connection on the bundle. More precisely, this paper is organized as follows. In Sec. II we introduce projective superspaces and analyze their integer cohomology; moreover, we review the definition of SVB and study the cohomology of the projectivizations of an SVB. It turns out that it is convenient to introduce two pro­jectivizations, an even and an odd one, and correspondingly in Sec. III we introduce even and odd Chern classes; a Whit­ney product formula is then proved. In Sec. IV we investi­gate whether the characteristic classes of an SVB E over M can be represented by means of cohomology classes in HSDR (M) defined in terms of the curvature of a connection on E (Chern-Weil theorem); we are able to answer in the affirmative in the cases where E has rank (1,0) or (0,1), or when E has arbitrary rank, but M is De Witt.

II. PROJECTIVIZATIONS OF A SUPER VECTOR BUNDLE

In this section we deal with some preparatory material that we shall need in the following section to introduce Chern classes of super vector bundles. We start by recalling the definition of SVB. 8

Definition 2.1: A rank (r,s) complex super vector bun­dle is a triple (E, M, p) where E and Mare supermanifolds and p: E __ M is a [g map, such that (i) M has a cover {Ui }

with [g diffeomorphisms

tPi: p-I(U)--UiXF (2.1)

satisfying pr20tPi = p, where Fis a rank (r,s) free graded CL

module; (ii) the transition functions defined by letting

tPi 0 tP;I(X,U) = (x,gij(x)u)

are morphisms of graded C L modules, i.e., they are [g maps gij: Ui n ~ --GL(r,s), where GL(r,s) is the super Lie group formed by the even automorphisms of the graded CL module C~+S.14.15 0

We recall that an XEGL(r,s) is an invertible matrix showing the block structure

X= (A B) CD'

where A and D have entries in (CL)o and are rX rand sXs, respectively, while Band C have entries in (C L ) 1 and are rXs and sX r, respectively.

With this definition, the sheaf of germs of sections of a

1234 J. Math. Phys., Vol. 30, No.6, June 1989

complex SVB E over a [g supermanifold (M,sf) is a locally free graded Y module, where Y = sf !J7 R C may be identi­fied with the sheaf of germs of [g maps M __ C L'

9 and one indeed can show that the category of rank (r,s) complex SVB's on M and the category oflocally free graded Y mod­ules are equivalent. 8

Let us recall some further details in the case of complex super line bundles (CSLB's).16 These are defined as com­plex SVB's of rank (1,0) [equivalently, one could consider the rank (0,1) case, since the two types of bundles have the same standard fiber and structure group]. Here the transi­tion functions are [g maps gij: Ui n ~ -- (Cd~, where (CL);l' =GL(1,O) =GL(O, 1) is the group ofinvertible ele­ments of (CL )0; thus the set of isomorphism classes of CSLB's over a given supermanifold M is in a one-to-one cor­respondence with HI (M,Y;l'), where Y~ is the invertible subsheaf of Yo' There is an exact sequence

O--l--Y o--Y;l' --0,

and the associated cohomology sequence contains the seg-ment

{)

H I(M,Yo) -H I (M,Y;l') _H 2(M,l).

If EEH I(M,Y~), we define 8(E)EH 2 (M,l) to be the ob­struction class of E. Unless M is De Witt, 8 is not necessarily injective, so that CSLB's are not classified by their obstruc­tion class.

Projective supers paces. After fixing non-negative inte­gers r,s,h,k with h<r and k<s, we define GL(h,k;r,s) as the subgroup of GL(r,s) whose elements are matrices of the form

(~ b

e

q

c o I o

where the blocks have the following dimensions, both hori­zontal and vertical: h,r - h,k,s - k. Here, GL(h,k;r,s) is a De Witt supermanifold with body GI(h;r) XGl(k;s), where Gl(h;r) is the subgroup of matrices in Gl(r;C) (ordinary Lie group) of the form

(~ ~). Hence it follows that the quotient

Gh•k (r,s) = GL(r,s)/GL(h,k;r,s)

is a De Witt supermanifold, of even dimension h(r - h) + k(s - k), odd dimension k(r - h) + h(s - k), and body Gh (r) X Gk (s), where Gh (r) is the Grassmann manifold of h planes in C'. It is otherwise obvious that Gh•k (r,s) parametrizes the rank (h,k) free graded sub-CL -

modules of C ~ + s.

Now, let Wbe a rank (r,s) free graded CL module, and define

P 1.0( W) = space of rank (1,0)

free graded sub-CL -modules of W,

po, I (W) = space of rank (0, 1 )

free graded sub-C L -modules of W.

U. Bruzzo and D. H. Ruiperez 1234

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From the previous discussion it follows that P I,O( W) and pO,1 ( W) are both De Witt supermanifolds, with dimensions (r - 1,s) and (s - l,r), respectively, and have bodies iso-• morphic with the complex projective spaces pr - I and ps - I • It follows that P 1,0 (W) [resp. P 0,1 ( W) ] has the same integer cohomology as pr - I (resp. ps - I ).

Tautological exact sequences. On P I.O( W) we may de­fine a tautological bundle So, which is the rank (1,0) subbun­dIe of P I.O( W) X W formed by the pairs (u,v) such that VEU;

analogously, one defines a rank (0,1) tautological bundle SI onPO,1 (W), which isa subbundleof pO,1 (W) X W. Now, let V be the body of W, i.e., the vector space V = W EEl c C,

I.

where C is given a C L -module structure by means of the body map u: C L -+ C; V is graded, V = ~) EEl VI' Denoting by Sf, i = 0,1, the tautological bundles of the projective spaces P( Vi)' the body of Si (in the sense of De Witt supermani­folds) in just Sf, whence one has commutative diagrams

0-+ Si -+ pl-i,i( W) X W-+ Qi -+ 0,

, , l i = 0,1, (2.2)

° -+ Sf -+ P( Vi ) X Vi -. Q f -+ 0,

where Qi and Q f are by definition the quotient (super) bundles. The following theorem is a straightforward conse­quence of (2.2) and of classical results concerning the coho­mology of projective bundles. 17

Theorem 2.1: The integer cohomology of P I.O( W) is freely generated over Z by {1,x,x2

, ••• ,xr- I}, where x is the

obstruction class of So. Analogously, the integer cohomo­logy of pO.1 ( W) is freely generated over Z by {1,t,t 2

, ••• ,t S-

I}, where t is the obstruction class of SI' Projectivizations 0/ super vector bundles. Let us define

the super Lie group

PGL(r,s) = GL(r,s)/(CL )tI

together with the canonical projection ;L: GL(r,s) -+PGL(r,s); PGL(r,s) acts in a natural way on pl.O( W) andpo,l( W). Given an SVB p:E-+M, whosetran­sition functions relative to a fixed cover are gij' we define its even and odd projectivizations as follows: P I,O(E) [resp. pO,1 (E)] is the bundle on Mwhose standard fiber over xEM isP l.o(Ex ) [resp. pO,1 (Ex)] and whose transition functions are the maps ;L0gij' We shall denote by 1Ti: P 1- i,i(E) -+M, i = 0,1, the bundle projections. The operation of taking the projectivizations is functorial, in the sense that iff M -+ N is a morphism of f!J supermanifolds, and E is an SVB over N, there are f!J maps F

i: P I - i,i ( / - I E) -+ P I - i.i (E) such that

the following diagram commutes:

F,

pI -I,i( /-IE),-+P 1- i,i(E),

i = 0,1. f

M N,

P l.o(E) and po.! (E) carry tautological bundles defined in the obvious way; So(E) -+p I,O(E) has rank (1,0), while SI (E) -+pO.1 (E) has rank (0,1). There are two tautological exact sequences,

O-+Si (E) -+ 1Ti- IE -+ Qi (E) -+0, i = 0,1.

1235 J. Math. Phys" Vol. 30, No.6, June 1989

The assignment of the tautological bundles is functorial as well, i.e., there are commutative diagrams

In order to get information about the integer cohomo­logy of the projectivizations of E, we must use the Leray­Hirsch theorem. 18 We need it in the following weaker form than the one given in Ref. 18: if p: Q-+M is a locally trivial topological bundle, with standard fiber F, K is a principal ring, and there are cohomology classes {a l " ·aq } that when restricted to the fibers generate freely over K the cohomo­logy of the fibers with coefficients in K, then H( Q,K) is a free H(M,K) module generated by {a l " ·aq }. Ifwe consider the bundles pI - i,i(E) over M, the hypotheses of the Leray­Hirsch theorem are fulfilled as a consequence of Theorem 2.1, so that we have the following theorem.

Theorem 2.2: The following isomorphisms ofZ modules hold:

H(P \- i,i(E),Z)

=H(M,l.) ® zH (P 1- i.i( C;: +'i),Z), i = 0,1.

III. CHARACTERISTIC CLASSES OF SUPER VECTOR BUNDLES

Given a rank (r,s) SVB p: E-+M, we can straightfor­wardly introduce its even and odd Chern classes as follows: if x and t are, respectively, the obstruction classes of the even and odd tautological bundles of the projectivizations of E, we let (with reference to Theorem 2.2)

, s

x'= - L CJ(E)x'-j, t S = - L Cl(E)t S -k, j=1 k=1

(3.1 )

so that CJ(E) and C 1 (E) are well determined elements in H 2j(M,Z) and H 2k (M,Z), respectively. Correspondingly there are two total Chern classes,

r S

CO(E) = L CJ(E), CI(E) = L Cl(E). (3.2) j=O k=O

According to this definition, a rank (r,s) SVB has r even and s odd Chern classes. The normalization and functoriality properties of these classes are readily proved.

Theorem 3.1: If E has rank (1,0), then

CO(E) = 1 - 8(E); (3.3)

if E has rank (0,1), then

CI(E) = 1-8(E). (3.4 )

Proof If rank E = (1,0), then E has only an even projec­tivization; moreover, So(E)=E, so that (3.3) follows. A similar argument applies to the rank (0,1) case. D

Theorem 3.2: Iff M -+ N is a morphism of f!J supermani­folds, and E is an SVB over N, then

Ci(/-IE) =/*Ci(E), i= 0,1.

Proof This property follows from the functoriality of the projectivized and tautological bundles. D

U. Bruzzo and D, H, Ruiperez 1235

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In order to prove a Whitney product formula, we need some further constructions. Recalling that Y is the sheaf of germs of ~ maps M - C L' and denoting by CC the sheaf of germs of smooth complex-valued functions on M, we can define a morphism y: Y - CC by letting y( j) = (IDj for all jEY( U). By means of y, the sheaf cr; can be given an Y­module structure. Given a rank (r,s) SVB E over M, we let

E# = E® yc. (3.5)

E # is a rank r + s graded smooth complex vector bundle on M, which splits canonically into E # = E 1! ffi E f, where the two summands have rank rand s, respectively. The bundles E # can be described in terms of their transition functions as follows. If {g,) is a set of transition functions for E, then the matrices {aogi) have the block structure

(h.. 0)

aogij = ; kij

EGl(r;C) XGI(s;C),

and the sets {hij} and {k ij } provide transition functions for E 1! and E f, respectively.

The definition (3.5) en tails the existence of vector bun­dle maps E - E f; these can be lifted to maps between the projectivized bundles pI - i.i(E) _ peE f) and between the tautological bundles, so that one obtains commutative dia­grams

O-Si (E) -1Ti- IE -> Qi (E) -0, t t t i = 0,1,

O-S(ET) -1TT -IE_Q(ET) -0,

where 1TT is the bundle projection E T - M. The commutati­vity of these diagrams implies that, for fixed i, Si (E) and SeE T) have the same obstruction class. This in tum implies the following lemma.

Lemma 3.1: CO(E) = c(E!!), CI(E) = c(Ef). 0 It is now possible to prove Whitney's formula. Theorem 3.3: IfO-E -F - G-O is an exact sequence of

SVB's, then

Ci(F) = Ci(E)Ci(G), i = 0,1, (3.6)

where the product in the right-hand side is the cup product inH(M,Z).

Proof Since O-E-F-G-O is an exact sequence of locally free modules, by tensoring it with CC one gets an exact sequence of smooth vector bundles over M,

O-E#-F#-G#-O,

which splits, thus giving isomorphisms F f =E f ffi G f. The ordinary Whitney formula then yields c(Ff) = c(Ef)c(G f), which, together with Lemma 3.1, implies the thesis. 0

It should be noticed that we have stated the Whitney product formula in terms of exact sequences ofSVB's rather than in terms of direct sums ofSVB's, since, due to the nona­cyclicity of the structure sheaf of the base supermanifolds, not all exact sequences ofSVB's split (see Sec. IV).

We conclude this section by introducing the Chern character of an SVB; as we shall see in Sec. IV, the represen­tation of the characteristic classes of an SVB in terms of curvatures is most simply exhibited by means of the Chern character. For a given rank (r,s) SVB E over M, by means of the formal factorizations l9

1236 J. Math. Phys., Vol. 30, No.6, June 1989

r r

I CJ(E)xj = IT (1 + yjx),

j=O j=1

s s

I C1 (E)t k = IT (1 + 8k t), k=O k= I

we define the even and odd Chern characters of E

ChO(E) = ± el'j, ChI (E) = ± e\ j= I k= I

and the total Chern character

Ch(E) = Cho(E) - ChI (E).

Of course Ch(E)EH(M,Z), and there is a decomposition

'" Ch(E) = I Chi(E), Chi (E)EH 2i(M,Z);

i=O

in particular, one has Cho (E) = r - s (we assume that Mis connected) .

The analog of the Whitney product formula for Chern characters reads as follows: if O-E-F-G-O is an exact sequence of SVB's, then

Chi(F) =Chi(E) +Chi(G), i=O,1. (3.7)

IV. REPRESENTATION OF CHARACTERISTIC CLASSES IN TERMS OF CURVATURE FORMS

Let E be a complex SVB of rank (r,s) on a supermani­fold (M,d); a connection II on E is an even morphism of sheaves of graded C L -modules

ll: E-Hom(TM,E) =:;0 1 ®;/'E,

satisfying (recall that Y = d ® R C)

ll(f5) =jll(t) +dj®t, VjEY(U), tEE(U),

and V open UCM.

In contrast with smooth bundles, and in analogy with holo­morphic bundles, an SVB does not always carry a connec­tion (a more detailed discussion of this point is to be found in Ref. 20). This is due to the nonacyclicity of the structure sheaf of a generic supermanifold; indeed, in the case of a De Witt base supermanifold, connections always exist.

Let

(4.1 )

be an exact sequence of complex SVB's over a supermanifold (M,d); general arguments21 show that the sequence (4.1) is split if and only if a certain cohomology class in H I (M,Hom (G,E») vanishes. Since Hom (G,E) is a locally free graded Y module, it is acyclic if M is De Witt,20 so that we have that all exact sequences ojSVB 's on a De Witt super­manifold split. This implies that an SVB over a De Witt su­permanifold always admits connections; indeed it can be shown that the connections on E are in a one-to-one corre­spondence with the splittings of the exact sequence ofSVB's overM

0-0 1 ®E-D(E) -E_O,

where D(E) is E ffi (0 I ® E) endowed with the structure of graded Y module given by

j<tffia) =f5ffi (ja + dj®t),VjEY(U),

tEE( U), aE(!l1 ®E)( U),

and V open UCM.

U. Bruzzo and D. H. Ruiperez 1236

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The representation theorem has been proved elsewhere in the case of complex super line bundles. Let E be a CSLB over a (generic) supermanifold M, and assume that E ad­mits a connection a, with curvature form !l. The Bianchi identity states that !l is closed, and one can prove22 that the cohomology class [!l] EH ~DR (M) ® R C is independent of the choice of the connection. Letj: H(M,'l) -+H(M,CL ) be the morphism induced by the inclusion of sheaves Z - C L'

and let a: HSDR (M) ® R C-H(M,CL ) be the morphism in­duced by (1.2); then we have l6

j(C? (E») = (i/21T)a([!l]). (4.2)

Equation (4.2) refers to the rank (1,0) case; obviously, in the rank (0,1) case we getj(C: (E») = (i/21T)a([!l]). In terms of the Chern character, in both cases one has

(4.3)

where Str denotes the supertrace of a matrix in GL(r,s).14 Equation (4.3) follows from the fact that Str !l = !l if rank E= (l,O),whileStr!l= -!lifrankE= (0,1).

Weare able to generalize this result to the case of a rank (r,s) SVB, provided that the base supermanifold is De Witt.

Theorem 4.1: Let E be a rank (r,s) SVB over an (m,n) dimensional De Witt supermanifold M. For any connection a on E with curvature !l, one has

j(Chk (E»)=(i/21T)k[Str!lk], 1 <;k<;mI2. (4.4)

For k> m12, both sides of (4.3) vanish identically. Before proving this theorem, we need the following re­

sult, which holds also when the base supermanifold is not De Witt.

Lemma 4.1: Letp: E-Mbe a rank (r,s) complex SVB. There is a [1 supermanifold morphism f N - M such that: (i)j*:H(M,'l) -H(N,Z) is a monomorphism; (ii) thereisa chain of morphisms of complex SVB's over N, gj: Fj _ I -+ Fj, withj = 1" 'r+ s, Fr+s =j-IE, and Fo = M x {O}, such that any quotient superbundle Fj I Fj _ I has either rank ( 1,0) or (0,1).

Proof This lemma is proved by double induction on the rank of E. If rank E = (1,0) or (0,1) the result is trivial. Suppose now that rank E = (r + l,s) and consider the even projectivization of E, 1To: P 1,0 (E) -M; the cohomology map ~: H(M,Z) -H(P I,O(E),Z) is injective by Leray-Hirsch. The pullback bundle 1TO-IE-P 1,0(E) has a tautological su­per line subbundle So (E) -+ P \,0 (E), and the quotient super­bundle Qo(E) has rank (r,s). By the induction hypothesis, there is a [1 map g: N -P 1,0 (E) satisfying the properties in the statement of this lemma. Then the compositionj = 1T oog: N - M yields the required map. The induction on the odd rank is proved in the same way. 0

Finally, we may prove Theorem 4.1. As a consequence of Lemma 4.1, and using the fact that on a De Witt super­manifold all exact sequences of SVB's split, there is a [1 morphismf N -M such that

j-IE=L I ffi'" ffiL r ffiKI ffi'" ffiKs' (4.5)

where the L 's have rank ( 1,0) and the K 's have rank (0,1). Using a "tubular partition of unity" over M (Ref. 20) it is

1237 J, Math, Phys" Vol. 30, No, 6, June 1989

possible to define a connection a on E which when pulled back toj-IE "splits" in accordance with (4.5),i.e., it defines connections I:::.j on Lj and 0 h on K h , j = 1 .. · r, h = 1" . s. Then we have [recalling that Str(!l~)k, where!l~ is the cur­vature of 1:::., can be regarded as a superform on M]

j* Str(!l~)k = ± (!l~j)k _ ± (!lElh)k. (4.6) j= I h= I

Now Eqs. (3.7), (4.4), and (4.6) yield

j*oj(Chk (E») = (i/21T)kj*oa( [Str!lk]).

Since j* is injective, this implies Eq. (4.4).

ACKNOWLEDGMENTS

We would like to thank C. Bartocci and J. Munoz Mas­que for useful discussions and valuable suggestions.

This work was done within the joint Italian-Spanish CNR-CSIC research project "Methods and applications of differential geometry in mathematical physics." It was also supported by "Gruppo Nazionale per la Fisica Matematica" of CNR, by the Italian Ministry for Public Education through the research project "Geometria e Fisica," and by the Spanish CICYT through the research project "Geome­tria de las teorias gauge."

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