On graded bundles and their moduli spaces

Vol. 23 (1986)

.

REPORTS

ON GRADED

ON

BUNDLES

MATHEMATICAL

No. 2

PHYSICS

AND THEIR MODULI

SPACES

J. CZY~ Institute

of Mathematics,

Polish Academy (Recrid

of Sciences, Warsaw,

Poland

May 24. lYH5)

Graded bundles are defined and analysed. It is proved that each smooth graded bundle is trivial, i.e. it can be split into w&lE, but there exist non-trivial analytic graded bundles. Dimensions of spaces of certain special graded bundles are estimated. The problem of graded bundles is partially solved in the case of concrete manifolds: P’ C, P”C, Riemann surfaces, tori, Stein manifolds, etc. Graded bundles are also considered as a method of non-linear superposition of configurations represented by holomorphic vector bundles. We apply this method in the Ogevietsky-Sokatchev model of supergravity and in instantons.

0. Introduction The object of our studies is the cohomology H’(M, GL(n, /1E)), where 1” E denotes a vector bundle over the base M having as fibres E,, XE M, 2” GL(n, /iE) is a bundle of groups over M having as fibres GL(n, M,), i.e. groups of all invertible matrices whose elements take values in the Grassmann algebras AE,, 3” the cohomology is meant in the sense of Tech, i.e. it is formed by classes of equivalent cocycles. Recall that for a covering (52,) of M a cocycle (gas) is a set of transition functions which are sections of GL(n, /iE) over Qafl := $2, n 52, satisfying the equalities Y&x= &I) - ’ 9 sap 9s: = gay

and two cocycles

(y,,,), (gJ

for sections

restricted

are said to be equivalent

to Q+ iff

&J = %&a SB l for a certain set of sections s,E~@, GL(n, MZ)). Any such class can be interpreted as a global structure over the base M which, when restricted to sufficiently small covering sets 51,, looks like a direct sum &I MZ. It is similar to a vector i=

bundle, because

1

Cl991

vector bundles

locally look like

200

J. CZU2

Q= x(&K), where K = R or C. But here the fibres of this structure are & M, so that they are provided with structures of Grassmann /iE,-moduli while fibres of vector bundles are “ordinary” vector spaces. Note that the Grassmann algebras /iE, carry their gradation into each element of H’(M, GL(n, /fE)). That is why we call these elements graded bundles (more exactly, classes of equivalent graded bundles). In the paper we study both single examples of graded bundles and sets of graded bundles satisfying some “initial conditions” and then we try to answer the question how “big” these sets are. The idea of graded bundles seems to be closely connected with physical studies concerning the boson-fermion duality. The graded bundles can be qualified as objects of so called “supergeometry”, developed since the early seventies. The aim of supergeometry is to adapt the usual geometry to the needs of particle physics and to construct geometric models including possible symmetries in the world of elementary particles, especially so called supersymmetry. In order to determine uniquely a graded bundle’ we should fix a class of manifolds and mappings occurring in the setup which defines this bundle. Recall that we have at our disposal classes of smooth, holomorphic, algebraic and G” (Grassmann-differentiable) functions and manifolds. In the first, smooth case, we prove that each graded bundle is trivial which means it can be put in a form FV@/iE or, equivalently, it admits a cocycle (gas) such that gUB(x)EGL(n, K) cGL(n, /1E,) for each XE!Z&. Thus the theory of smooth graded bundles seems to be only a slight extension of the theory of smooth vector bundles. In the remaining cases non-trivial graded bundles are proved to exist and they are our main objects of interest. In order to see the place of holomorphic graded bundles in global analysis let us look at the diagram. The vertical arrows in the diagram correspond to either specialization to particular groups or sheaves (1 = 2, 3 =z-5, 4 = 6) or to reduction in the sense of confining to gas and s, taking values in appropriate subgroups (2 S- 3, 2 -4). It follows from the diagram that the idea of graded bundles can be understood as a generalization of the Cousin problems and the idea of vector bundles. The most interesting seem to be those graded bundles which cannot be put either in block (3) or in block (4) of the diagram. We prove their existence in such a prosaic case as M = P’ C. We also briefly describe G-graded bundles whose bases are G”-manifolds and transition functions are G”-maps in the sense of the Alice Rogers definition. I regard the G “-graded bundles as the “most super” - spaces which I can imagine. Throughout the paper I will be asking the reader to help me to solve several problems. Especially difficult for me is the problem of a shape and an upper bound of dimensions of the aforementioned spaces of graded bundles for sufficient-

T 3

%

g ._ 5 c! 2

l-

linear bundle

I&!@, GL(n, 0)

Holomorphic

Holomorphic

ii

X(M,

c’)

P-bundle

* In order

not to frighten

physicists

we very rarely

Obstructions in construction of entire functions having given zero-sets and multiplicities of zeros which arose from Weierstrass theorem and were formulated by Cousin as his I problem

5.

Various problems in analytic functions: generalizations of RiemannRoch theorem, generalizations of &functions, algebraic maps in PC, etc. put on equal footing by Steenrod and Grauert in early 50’s

3.

1

_

use the general notion

-------

Problem

Object ------1 st cohomology __----and

GUn,

of sheaves.

Implication

from

paper

Equivalence class in generalized I Cousin

Holomorphic

u H;(M, c)

C-bundles

Obstructions in construction of meromorphit functions having given singular parts which arose from Mittag-Leffler problem and were formulated by Cousin as his II problem

6.

Generalization of I Cousin problem (below) for meromorphic sections of vector bundles

bundle

AE))

graded

Grauert

As in the present

K,(M,

2. Holomorphic

Frenkel

1st cohomology in analytic

202

J. CZYP

ly general bases and the question regarding the conditions for any holomorphic graded bundle to be algebraic. The paper is divided into three chapters. The first and the shortest one is devoted to the physical background of supergeometry and graded bundles. In the second chapter we describe graded bundles and analyse them by methods of global analysis. We prove there the triviality of smooth graded bundles and non-triviality of certain analytic, algebraic and G”-graded bundles. We recognize geometrical properties of graded bundles and certain topological features of spaces of graded bundles satisfying some “initial” conditions. This chapter contains also a review of manifolds admitting “many” and “few” graded bundles. There projective spaces, Riemannian surfaces, tori, Stein manifolds and other examples are considered. The reviewed examples suggest that sets of graded bundles are completely “unstable” with respect to changes of base spaces. In the third chapter graded bundles are applied in two domains of in particular in the Ogievetskymathematical physics, namely supergravity, Sokatchev-Schwarz model and instantons. In both cases the construction of graded bundles enables us to enlarge the class of systems represented by holomorphic vector bundles and to obtain new solutions in gauge field theory. In this paper some results given in my previous papers [21 J, [22], [23] and [24] are described in detail. It seems natural to pass from graded bundles to more general objects by replacing the Grassmann algebras AE, by e.g. Clifford algebras, Lie superalgebras and so on. We do not pursue this direction in this paper and try to concentrate on the case of Grassmann algebras. These algebras seem to be the best realizations of the duality between the relations of commutation and anti-commutation.

1. Why graded bundles? One of the most fundamental principles both in Nature and in mathematics as follows: can be called the “general duality principle”. It may be understood “For each thing in a very broad sense like for instance a particle, an animal, a physical phenomenon, an abstract object, a theory, an idea or a thought there The word “dual” here concerns many meanings. They range exists a dual thing”. from a denial to a complement and to various symmetries. And furthermore, “both things, the entity and the dual entity, are indispensable for a good insight and an understanding without illusions”. For a profound study of the general duality principle I recommend Manin’s article [59]. Physics satisfies the general duality principle very well, perhaps even better than other sciences [IS]. As a fundamental example one can point out the bosonfermion duality. This duality rightly can be considered the most principal law about the structure of matter. No exception from it has ever been observed and no

ON GRADED

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203

serious physicist nowadays questions the existence in Nature of these two sorts of particles only, in contrast to almost all other physical knowledge. Some information about properties of fermions and bosons can be found already in early papers of Pauli concerning spectra of atoms, which were written before quantum mechanics was discovered. Nowadays, in the famous “Feynman lectures on physics” [34] the boson-fermion duality is used as a first principle of quantum mechanics (this is not the case of many other lectures) and its exceptionality is strongly stressed by the author. The reflection of the boson-fermion duality in mathematical models describing the behaviour of both these types of particles is to include a symmetric and an antisymmetric relation and to place them on an equal footing. In particular this is the case of the spin-statistics theorem. Furthermore, if bosons and fermions are represented by vector fields then for bosonic fields there is an antisymmetric relation of commutation and for fermionic fields a symmetric relation of anticommutation. Such relations have been introduced by Dirac in the late twenties and since then they are an indispensable ingredient of quantum field theory. On the other hand, in differential geometry for two vector fields defined by means of the Leibnitz rule X(cp$) = X(cp)II/+cpX(tj),

cp, rj are local functions,

the commutator is well defined but the anticommutator does not make any sense (as a vector field). This asymmetry can be proved to be a consequence of using a commutative field K (usually C or R) as a set of values of local coordinates instead of an algebra admitting both commutation and anticommutation. That is why axiomatic geometry of manifolds fails with respect to the needs both of elementary particle physics and the general duality principle. This anomaly of course did not discourage physicists from developing the theory of elementary particles. It was the tensor, spinor and operator calculus which provided them with the necessary symmetries and antisymmetries. Nevertheless, the boson-fermion duality still found no reflection in definitions of parameter spaces. A more direct link between particle physics and mathematical models has been proposed by F. A. Berezin in the sixties in his formulation for Fock spaces. His idea is as follows: “replace everywhere the field C (or R) with a complex (or real) Grassmann algebra”, cf. [ 133. The discovery of a large number of elementary particles and their investigation (in particular collecting them into multiplets) gave in the early seventies a broad scope for Berezin’s idea. At first similarities between certain bosonic and fermionic multiplets suggested some kind of symmetry, a so called supersymmetry, which

204

J. CZYi

exchanges bosons and fermions. Furthermore, gauge theories with non-Abelian gauge groups, developed since the middle fifties, have been used more and more as a method for analysing multiplets and reactions of particles. As a consequence, anticommuting vector fields on space-time appear more and more frequently in the physical literature. In order to explain supersymmetry by the method of gauge fields AkulovVolkov [l] and Wess-Zumino [S2] have introduced a combined algebra of generators of the Poincare group (and the conformal group as well) and fermionic vector fields, see also [33]. In this algebra, apart from the well-known relations of Poincare algebra and Dirac fields the following new symmetric relation has appeared :

;Qay Qci; = Pai up to a suitable constant

factor,

where Q,, Qd are the fermionic fields, Pai is a space-time translation and ( -, . ) denotes the “anticommutator”. A class of algebras with such a bracket had not been distinguished by mathematicians before so that this and related physical concepts brought to the attention of mathematicians algebras having partially symmetric and partially antisymmetric “Lie” brackets. In a few years a consistent theory of so-called Lie superalgebras and their representations was developed, in which the main contributions are due to V. Kac [52] and I. Kaplansky [53]. One of the achievements is a classification of simple finite dimensional Lie superalgebras, which is much more wealthy than Dynkin’s classification in the non-super case. A novelty here is a double parametrization. Another interesting phenomenon are finite dimensional analogues of some of Cartan’s infinite dimensional simple Lie algebras. It turns out that Whitehead algebras in homotopy theory and a number of algebras known before in general topology are Lie superalgebras, see [61]. Nevertheless, Lie superalgebras in some sense also fail the general duality principle. This is because the duality between linear algebraic relations of Lie algebras and globally geometric and analytic properties of Lie groups, in which duality is induced by the functional “exp”, as yet has not been extended to the domain of Lie superalgebras. (Lie supergroups, which are used often in mathematical physics, are either purely formal objects or ordinary Lie groups “super’‘-structure on the level of local functions provided with an additional algebras.) Note that the uncertain status of fermionic vector fields in geometry of supersymmetry is not too serious a problem because the base space-time in supersymmetric gauge field theory is usually flat so that linear algebra seems to be sufficient to express the desired formulae (though one can meet opposite opinions, see [3]). But soon, in the middle seventies, supergravity that is a unified gauge theory of local space-time symmetries and supersymmetry was invented, see [27],

ON GRADED

BUNDLES

so that space-time CW, C351,L-631

AND THEIR

MODULI

in the “supertheories”

205

SPACES

could

not always

be

assumed to be flat. Definitions of supermanifolds in the sense of manifolds admitting also fields satisfying Berezin’s postulate which are anticommuting vector mathematically rigorous have been given by K. Gawedzki [37], B. Kostant [56], D. Leytes [SS], M. Bather01 [ll] and quite recently by A. S. Schwarz [74]. Omitting slight differences one can say that a supermanifold M in the sense of their definitions looks as follows: as a piece of a ,,mathematical substance” (i.e. as a “smooth” topological space) M is identical with an ordinary manifold but the algebra of local functions C,z(M,K) is replaced by an algebra r,Z(M, AE) of local sections of a Grassmann bundle AE. Note that AE can be non-trivial in contrast to MxK. Most general models in supergravity are considered in so-called extended supergravity, where iV fermionic fields are considered simultaneously, so that N copies of the auxiliary spinor bundle E are used to formulate it. (The case N = 8 turns out to be extremely interesting; many physicists see here the best candidate for a- unified theory of all four fundamental interactions: gravitational, electromagnetic, strong and weak, see [19], [63].) The geometric nature of gravity and the desire to have a coherent and a complete theory of supergravity have stimulated searches for a geometric formalism suitable for all interactions occurring in the extended supergravity, in particular those between different fermionic fields. The present paper may be considered as one of these attempts. Another interesting result in supergravity is due to Ogievetsky-Sokatchev [64] whose work is continued by A. S. Schwarz er al. [38], [74], [75]. They have found a transformation which maps the supergravity defined as a dynamical system on the smooth space-time manifold into a global holomorphic structure of a complex superspace. It looks like an example of the Penrose transformation, see [66]. But after a closer insight one can see the following novelty: we have obtained a complex structure starting from a non-elliptic system (gravity is hyperbolic) while the “pure” Penrose transformation maps systems of elliptic equations into holomorphic structures (Cauchy-Riemann equations). It seems that the extension produced by a supergeometry is that which compensates the non-ellipticity. If this is true then a large number of dynamical systems with spin should be coded into holomorphic superspaces. The second domain, where analytic structures in superspaces seem to be possible and useful are super Yang-Mills fields, especially anomalies which have been discovered recently, see [79]. Summarizing, we can say that our intentions are determined by: the belief in the genera1 duality principle, the wish to develop the Berezin idea, the need to geometrize supergeometry and supergravity (in particular extended supergravity) and the hope of making progress in the Penrose theory.

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As a conclusion let us propose: (1) to try to adapt for supergeometry (in Berezin’s spirit) the category bundles, (2) to use more analytic structures in supergeometry.

of linear

2. Graded bundles: definition, properties and examples 2.1. Graded bundles: to

the notion

We remember well that each vector bundle is locally trivial, i.e. when restricted sufficiently small covering sets Q= of its base space it is isomorphic to

s2, X(C$ K). We shall apply

Berezin’s

idea to vector

bundles

so that

we should

replace K by a Grassmann algebra nV over K, dim, I/ = r. But there is a difference between structures of K and /1V since the field K possesses distinguished linear coordinates determined by the points 0 and 1 while nV admits an infinite set of linear systems of coordinates with no distinguished system; this set is parametrized by points of the linear group GL(r, K). A special case is when the same system of coordinates is distinguished at each point of the base space because nothing prevents this system from changing, when local coordinates in the base space are changing. Thus the ingredient which we need is not a single copy of a Grassmann algebra like nV but a congruence of Grassmann algebras, say AE,, indexed by points x of the base space M. In other words, a vector bundle K: E -, M, which will be called from now on the auxiliary bundle, is given and any aimed graded bundle

locally looks like 6 n Elfi,, i.e. it is isomorphic

there to the

direct sum of n copies of ,4Eln,. Let us compare algebraic structures of vector bundles and graded bundles. We know that fibres of vector bundles are vector spaces. In the case of graded bundles the fibres are free nE,-moduli over the Grassmann algebras AE,. Thus graded bundles are provided with more subtle algebraic structures than vector bundles. In order to determine a graded bundle globally a system of transition functions (y,,,) defined on the intersections fiaa : = 52, n Q, must be given. These functions describe changes of coordinates in fibres through points XEC&, if we replace the local coordinate system of the base space attached to s2, by that attached to Sz,. The following equalities are satisfied: galygBa = id0a,V gas gar = gnu so that

on

QaP, : = s2, n Qb n Q2,-

(g=$ is a cocycle. Let us observe that transition functions of bundles of these two types are different sorts of objects. As for vector bundles, they are functions taking values in

ON GRADED

BUNDLES

GL(n, K) or, in other words, local M xGL(n, K). In the case of graded sections of the bundle of groups nl: GL(n,

AND THEIR

sections bundles, x;‘(x)

/lE) -+ M,

MODULI

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of the trivial bundle of groups the transition functions are local

= GL(n,

‘XE,),

XEM.

Note that no trivialization of this bundle is given. The dependence of parametrizations of E, on C&3 x induces a dependence of parametrizations of the structural group at the points XEM on the neighbourhoods U,. This is not so in the case of vector bundles, where the structural group GL(n, K) is identical at all points of the base space. The equivalence relation of graded bundles can be defined analogously to that of vector bundles. Furthermore, in the inductive limit we can obtain all graded bundles over a given base M for which E is the auxiliary bundle. In this way we are led to a I- 1 correspondence between the set of all such graded bundles and the first cohomology H’(M, GL(n, LIE)) in the sense of Tech. It is a particular case of the first cohomology with coefficients in a sheaf of non-Abelian groups, see

WI, L-401, WI. The step by step construction

t

the Grassmann bundle AE over E m+2

m+2’+’

+

of a graded

bundle

the bundle CL@, IIE) of invertible matrices with matrix elements in the bundle II./? mfn22’

looks as follows:

+

the graded bundle, i.e. a class of cocycles in a set of sections of the previous bundle GL(n, kE)

m+n22”+’

m+n2

Let us say a few words about the structural group GL(n, group of all invertible n x n matrices having matrix elements algebra /II/: The group elements are matrices of the type A = A(O)+ . :.. +A”’ 3

m+n2’+’

AV). This is the in a Grassmann

A’k’ = [a!“‘] 11 ’

where ~6%) E Ak V k = 1 1 . . . . r : = dim, V, i, j = 1, . . ., n, such that A”’ # 0. The ‘hquality’ A(” = I n determines a normal subgroup which will be denoted

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J. CZYt

HLh

AV). One can check inductively that HL(n, AV) is a nilpotent Lie group HL(n, AV) is solvable). a BY direct calculation one can prove that GL(n, AV) is a semi-direct product of GL(n, K) and HL(n, AV). Thus the Lie algebra of GL(n, AV) splits into a direct sum of gl(n, K) and the Lie algebra of HL (n, AV). (hence

2.2. Gr&ed

bundles: the definition

DEFINITION. We say that we have a graded bundle if the following system is given ‘G(E) = (CG, x,,, M, (E, X, M), (52,), (&), (go&), where 1. (E, K, M) is an auxiliary vector bundle x : E -, M with fibres E, of dimension r. 2. Y is a manifold, called a (total) space of C+(E), and x8: (4 2 M is a n projection such that each fibre xi1 (x) is isomorphic to @ AE,. The number n is

called the rank of the graded bundle 9(E). The AE,-module structure in each fibre ~6’ (x) is distinguished. 3. (a,) is a covering of the base space M and (d,) da: 71;’ (s2,) -+ 6 AE,, is a set of isomorphisms

E, : = Eln,

such that

4 d; ’ (e,) = sol@ (4 e,,

e,E&

AE,,

x&b

and the sections g,,, of the bundle GL(n, AE,,) form a cocycle as they did in the previous section. The set (d,) is called an atlas of the graded bundle C%‘(E). The graded bundle Y(E) is of a class e.g. Co, C” (that is smooth), c” (analytic), if all objects in the above setup of B(E) algebraic, G” (G rassmann-smooth) (manifolds, mappings, the auxiliary bundle) are of a given class. Note that each graded bundle is completely determined by spaces % and M, the projection ns and trivializations (da), similarly as in the case of vector bundles. The above definition in fact is due to “graded vector bundles”, but there is no problem to define also “graded principal bundles”. We say that graded bundles %‘I(E’) and FS2(E2) are equivalent if there exists an isomorphism i: E’ --*E2 and a family of sections (s,) of (GL(n, AEf)) such that 9.2p,= i(s,&spl), where i: GL(n, A??) + GL(n, AE2) is the induced map. This condition is equivalent to the existence of an isomorphism I of suitable differential structures of total spaces and A&L-moduli structures, I: Y’(E’) + 9’ (E2) given by e, -+ i(s,(x) e,).

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The set of equivalence classes of graded bundles %(E) having a given auxiliary bundle E up to equivalence and trivializations assigned to a covering (Q,) is in a natural 1- 1 correspondence with the Tech cohomology H’ ((QJ, GL(n, AE)) associated to the covering (&). By passing to the inductive limit we can identify the space of all graded bundles of rank n with a fixed M and E up to the equivalence with the first Tech cohomology space H’(M, GL(n, LIE)). If all transition, functions take values in subgroups HL(n, AE,) c GL(n, AE,) for all XE M, then we say that T(E) is a graded HL-bundle. The class of all graded HL-bundles having a given M and E can be identified with H’(M, HL(n, AE)). A graded bundle T(E) is said to be trivial if it is equivalent to a graded bundle having a cocycle U,, 3x-+ G,,(x)EGL(~,

K) c GL(n,

LIE,),

i.e. in a suitable basis, transition functions are scalar-valued. If the cocycle (Gap) determines the vector n-bundle M! then the trivial graded bundle may be decomposed into S’(E) = W@AE. Each trivial graded HL-bundle is clearly of the type %(E) = 6 /lE. We see that the problem of the existence of a non-trivial graded question “to be or not to be” for the idea of graded bundles.

bundle is the

2.3. Smooth graded bundles THEOREM ct. Each smooth graded

bundle over any paracompact

base manifold is

trivial. Proof: In the first step we prove the theorem Since for hl, h, E HL(n, /iE), one has (h, h,)“’

= hi”+ h’:‘,

for smooth

h(l) is a AlEcomponent

graded

HL-bundles.

in LIE

and therefore (H’(M,

H(n, A,))(”

2 H’(M,

A(n, K)@n’E)

g A(n, K)@H’(M,

E) = 0

(A(n,

K) denotes the additive group of n x n K-matrices) since every sheaf of smooth sections of a vector bundle is fine, see [SO], [60]. Thus for a cocycle (h,,,) of our graded HL-bundle one can choose a collection of sections s,~r(H(n, AE,)) such that s, h,&’

= s(l) (I + h$-s;’

Let us put hi,, : = s, h,, s; l. The sheaf of smooth sections of ,4’E of HL(n, LIE,) such that (s;)(l) = 0 for each 2,

= 0.

is also fine, hence we can find sections s,’ hence (si h&s;; l)(l) = 0

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210

and (7; hip s;- l)@’ = (s;)‘2’ + (h&)(2 - ($‘)(2 = 0 Repeating this procedure r times we get (h,,,) rr (I&) = (I,) so that every smooth graded HL-bundle is trivial. In the second step consider an arbitrary smooth graded bundle S(E) with a cocycle (gap). Then gas decomposes in the following manner: gap = G, jiDls 7 Gx,,~ f (Q,, xC?L(n, cocycle but ($,) is usually (i;,,) are as follows:

whcrc

K))and h,,,~r(HL(n, not any cocycle.

AI&)). The family (GE,‘) forms a The conditions of compatibility for

j;lla = G,, (ii,,)- ’ G,,, i;pl;.= G,, i& G,., i;lly. Hence g$’ = G,, g$ + g;;’ G,, .

Let us introduce

a family s, of sections

of HL(n, A&) such that

s:” : = + 1 j, (G,, g;;’ -g;;’

Gya),

where ci,) is a smooth partition of unity subordinate to the locally finite covering (52,). By virtue of the compatibility conditions, we get (s, gap SP ‘)( ” = g$’ + $ ’ G,, - G,, sb’ ’ = sib’ + i c j, (G,, ski’ G,, - gx, G,, - G,, g:;’ + G,, gg; G,& =

s$’+ f

1 j,( -

g$’ + G,,

g!,:’ G,, G,, + G,, G,, gg)

Gyp)

Y

=

sib’+ i

c j,( - s$’- db’G,, -

= gik’+_t C j,(-g$'-g$')

G,, &‘I

= g$‘-g$’

= 0.

Y

Repeating

this procedure

for (s, gap sj 1)(2’, . . . , (s, gap s; 1)(r’, one

proves

that

(g,,) - V-Q. QED A similar proof can be carried out in the case of graded bundles of classes cp, p = 0, 1, . . . Note that a different proof is possible by checking certain conditions for groups which have been given in Frenkel’s paper [36]. Observe that if the bundle E is trivial then Theorem r becomes a simple consequence of the Steenrod reduction theorem, see [SO], [78].

ON GRADED

2.4. Holomorphic

graded

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bundles

In this section we will consider holomorphic graded bundles. Therefore K = C, all manifolds will be complex and all vector bundles will be holomorphic. Let us denote by SYE,, the set of graded bundles %‘(E)EH’(M, GL(n, NZ)) having a cocycle (gap) such that the cocycle (g$‘) determines the vector bundle M/: When 9(E) E S%E,,, we will also write C!&,(E)instead of W. THEOREM f3. 1” There

are non-trivial holomorphic

graded

bundles. In particular

if W = & W’, rank W’ = 1, then Sg,,, d =

contains a C-like

[3 ;

h’(M,

space, where

E).

Here h’( .) := dimcH’(.) and the cohomology sheaf of sections of the bundle E. 2” If moreover rank E = r = 1, then

is considered

in the holomorphic

S$ E,W -- cd’, where d’ = n2 h’(M, 3” lf

H’(M,

the equivalence

One

should

AkE) = Ofor

E).

k = 1, . . . . r, then the only graded

HL-bundle

up to

%(E) is the trivial one & AE.

that the condition H’(M, Ak E) # 0 for a certain for the existence of a non-trivial graded bundle Y(E). will be given in 2.6 A.’

be aware

k = 1, . . . . r is not necessary

A counterexample

Proof: 1” Let A be an n x n matrix consisting of zeros and [n2/4] units such that A2 = 0. An example is the matrix having units in block II and zeros in the remaining positions:

c-2+31 I ! II .. . . ...

) [“i--J

I.1 IIIiIV

Then I,+ AOH’

(M, E) can be naturally

identified

1 Proof. M. F. Atiyah has told me that the condition P. S. Green in his lectures in Oxford in Spring 1980.

with a subspace

of SC&,,

H'(M, E)# 0 had been considered

by

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212

through W&E whose topological dimension is 2 [n2/4] h’(M, E). If r = 1, then A may be replaced by the n x n matrix having all elements equal to one. 2“ The proof is obvious. 3” There exists a set of sections (s,) of HL(n, A?!,) such that -1 (1) = 0. = (s, 911ssg 1

s$

Then we can apply (s’,) such that $‘) = 0 and (.g gas z;

and continuation

of this procedure

l)C2)= 0

up to Y allows us to complete

the proof. QED

H’(M, HL(n, AE)) and bundles spaces of graded Remark. The H’ (M, GL(n, AE)) may be regarded as sets of classes of data for a generalized multidimensional mixed (additive-multiplicative-distributive) skew-symmetric Cousin problem. To see this one should write down the equivalence relation for matrix elements. Below we present several holomorphic graded bundles.

“individual”

PROPERTY1. The space H’ (M, GL(n,

and

“collective”

properties

of

AE)) admits an algebraic structure given

by (1) multiplication by multiplication of cocycles,

complex

(zgq?) (k) : = ZkgL$/,

(2) additions I+A@H’(M,

of

points

in

:4(E) + z%(E)

numbers

k =O,

subspaces

of

determined

by

the

1, . . . . r.

H’(M,

HL(n, AE))

of

the

type

E), where A2 = 0.

Recall that the only algebraic structure in the space of vector bJndles is a distinguished zero element. H’(M, GL(n, K)) Observe that the graded bundle O%‘(E) admits the cocycle (G,,) := (g$)) so that it is trivial. The vector n-bundle admitting the cocycle (G,@) will be denoted B,(E) or W. Thus we have O%(E) = %O(E)@/lE

= W@AE,

so that

The spaces S’GE,+,make a fibration of H’ (M, GL(n, indexed by points of H’(M, GL(n, C)) since H’ (M, GL(n,

AE))s g(E) --t OQ(E)EH’

AE)) and the fibres can be

(M, GL(n,

C))@AE.

ON GRADED

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213

The above mapping can be combined with the identity mapping by means of

Hence we have PROPERTY 2. The space SSL,w is always Hausdorthe point WQAE particular, S%e,, W@AE.

and C-astral with respect

to

(i.e. each point can be joined with WQAE by a C-like space). In is connected, contractible and attains its maximal dimension in

Both spaces H’(M, GL(n, AE)) and H’(M, GL(n, C’)) have the same homotopy type. Recall that the coincidence of the homotopy types means coincidence of all homotopy groups and all Linvariants, see [77]. Note that Ms,, is Hausdorff because the orbits (classes of equivalent graded bundles) are closed. That is so because if G,, is fixed than s, take values in an RPlike subspace of GL(n, AE,). But the topology of H’(M, GL(n, AE)) is usually cumbersome and not necessary Hausdorff. Thus it is convenient to consider instead of H’(M, GL(n, AE)) its subspace with a good topology, i.e. a moduli space of stable bundles. We must remember that the dependence E + S%:(‘E,Wis not always continuous by virtue of the discontinuity of E + H’(M, E). An example are the twisted complex instanton bundles E” OH’ in the Atiyah-Ward A(” Ansatz, see Section 3 and [7]. The examples A and C in the next section suggest that the dependence W+ S%,, is much more complicated. PROPERTY 3. Total spaces of the graded bundles z!/;(E), ZEC,

are holomorphic (i.e. the complex structures are isomorphic). Thus the total space of a graded bundle (G(E) is holomorphic to the total space of the trivial graded bundle :GO(E)@AE. Proof:

Consider the set

%c: = (U & A(6’ 1 where the equivalence relation - js

(61,)))x (a) x C)/,,

(e:, r, 4 ‘c (e,, B, z)-=& = (=gap)ex,

XE%~.

The functions in the above relations are holomorphic so that the quotient set %c inherits the structure of a complex manifold. It is clear that the condition z = zO determines a complex submanifold of %c which is the total space of z. 9(E). Every such submanifold can be mapped holomorphically into another one by means of a flow in !eC along the holomorphic vector field k?/az. This map is obviously desired holomorphism.

bijective so that it gives the

J. CZYil.

214

One can quickly check that the above map of total spaces is not compatible with the additive structures so that it is not a graded-bundles-isomorphism. QED One of the consequences

of Property

3 is

PROPERTY3a. If the base space is compact then S
9(/(E) -+ 0.
of graded

bundles determines

a stratification

of each

spuce S ‘GE.W.

One can assign to any graded bundle
ON GRADED

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215

SPACES

PROPERTY 5. 1” Let ti( .) be a topological ,first Chern class, the Euler M = P3 C). Then we have

charateristic

invariant of vector bundles (such as the or the Atiyah-Rees invariant in the case

ti(CGt,(E)) = ti(?YO(E)@AE). 2” Total spaces of S”(E) and !Go(E)@AE are holomorphic. 3” Every vector bundle !““(E) corresponding to a graded bundle Y(E) admits a sequence of subbundles 3”(E) : = W, 2 W, 2 . . . I W, such that rank of W, is n(c)+rT’)+

and

. . . +[))

W, = 30(E)@A’E.

4” The vector bundle
cI AB

detA,

CD’

where A, B, C, D are quadratic

Z2-structure,

i.e.

the

detD#O,

matrices.

holomorphic graded bundles 3(E) and So(E)@AE are Proof: 1” The equivalent as smooth graded bundles. Therefore ‘S”(E) and %o(E)@AE are equivalent as smooth vector bundles. 2“ The proof is similar to that of Property 3. 3” The subbundles W,, k = 0, 1, . . ., r can be obtained by the action of the transition

functions

on

& & AjE

and

then

by

restricting

to

the

linear

j=k

structures. n

n

4” It suffices to use the local decompositions

where

A,E:=

& 2i=

A”E,

A,E:=

0

&

$(E)lo,

Azi + ’ E and

= (0

A0 E,)@ (0

then to restrict

A, E,),

to linear

Zi+l=O

structures. Let us state the inverse problems: When does a given vector n2’-bundle I/ admit a non-trivial graded bundle S(E) such that I/ = 3”(E)? If it does, we ask whether 3(E) is unique or not? How can the auxiliary bundle E be reproduced if the vector bundle V = S”(E) is given? Property 5 enables us to do no more than answer negatively the inverse problem in particular cases. 2.5. Algebraic

graded

bundles

Recall that a graded bundle is called algebraic if all elements of the basic setup from Section 2.2 (i.e. manifolds, mappings and the auxiliary vector boundle) are

J. CZYt

216

algebraic objects. For brevity it suffices to assume that the auxiliary vector bundle E is algebraic and the projection zs is an algebraic mapping. (We recall that algebraic objects like algebraic manifolds (known also as smooth or non-singular algebraic varieties), mappings and bundles can be introduced by means of a class of local functions which are analytic and rational.) We can simply extend the definition of algebraic graded bundles to the case when the base space is any algebraic variety (not necessary smooth). Each complex algebraic graded bundle determines a holomorphic graded bundle. If the base space is compact, then non-equivalent algebraic graded bundles determine non-equivalent holomorphic ones. This is a special case of the Serre theorem, in which such relation is proved for algebraic group-sheaves over compact manifolds, see [76]. In the case of graded HL-bundles, a more exact relation is satisfied, namely, for and algebraic base space manifolds there is a 1 - 1 natural compact correspondence between holomorphic and algebraic graded bundles up to equivalence. This is a direct consequence of the result of Serre and the Rosenlicht theorem concerning algebraic solvable Lie groups [69]. I do not known whether the last fact remains true for all graded bundles (not necessary HL-ones). By virtue of the aforementioned result of Serre it does if there exists a rational section s: GL(2*n,

C)/GL(n,

AV) + GL(2’n,

C).

Let us note that Theorem /3 of the previous section remains true if we replace holomorphic graded bundles and sections in the Tech cohomologies by complex algebraic ones. 2.6. Examples A. Graded

bundles

over

M = P’ C

In this case we have by virtue of the Grothendieck E :=

W:= ‘II,(E) := 6

& HPk,

i=l

k=l

H is the hyperplane bundle. Thus W is associated with the cocycle Gall := y$’ =

6 i=l

z ,

HI’,

theorem pk, riEZ*

(G,,))

'i

()

(x, B, = (09 I), (1, 0).

L;1

Let us write in the “natural” coordinates of P’ C the k-th component matrix element of the transition function yO1 of a graded bundle %:(I( E)

of a

ON GRADED BUNDLES AND THEIR MODULI SPACES

217

where kc., denotes the projection in E onto H”. Let s,,, si be two sections of HL(n, LIE) lying over U0 and U, resp. defined as o! = 1, 2): follows (cl, : = z E P’ c 12, # 0, x: “$,‘,jj : = zF c af+rjZf 3 /=o

k

kSylj =

pk pk z. z

af+pk+ri Z* 7

where k ._ Oij e-

0,

Pk+li

tj-tiepk-1,

pk+ti+ 1 2 tje


(Al)

and s$=@=O

for 1=2,...,r.

The pair so, si determines a manner of passing function go, for which the following holds: (0, kgbyij =

k(sogo~ ‘; ‘ti” =

pk+ri+l ‘j-

z;k

>

from go1 to the transition

tj,

1

c f=pk+ti+l

LXJZ", pk+ti+

1 < rj*

642)

One can quickly prove that the array of all coefficients al is determined by the whole equivalence class of T(E) so that the space S+& of all such graded bundles up to equivalence projects on the space of sets of coefficients a/. By summarizing “degrees of freedom” of al we obtain dim ~w~E=%.w

2 2 i

i

i,j=

1 k=

H(tj-ti-I)

=:u,

1

The same procedure can be repeated for the components of cJ~; belonging to /i2E, . . . . NE successively defined with respect to A2 E, . . ., NE in a similar way as those in (Al). Note that then the parts of the transition functions belonging to Grassmann spaces of lower degrees remain unaltered (e.g. in the second step we have i&rib= &,\)). In such a way we can confine the space of parameters of representatives of all graded bundles which were considered and compute an upper bound for the dimension of SC!&, at the point corresponding to the trivial bundle W@,4E. Thus we have

J. CZVi

218

dim

2 i,j=

1 1= 1 &I

<...

H(rj-ri-p,,-

... -~k~-l):=

We will show that both bounding values (that in (A3) attained by the studied dimension. Let us put E: = H”@K2 Let us take a graded bundle having a cocycle gap(z) have expansions in the region given by (A2). and gb: = ah/z. Let us take shl) : = (-b/2, = (0, a/z) # (0, 0) Then (s,g,,

20.

ck,

s;l)(2) = sb” A g$$+gb’: A s\l’-.$” = _;_;+o+ub

and that in (A4)) are so that EAE = Hd2. whose all elements Assume that gfj 0), s(ll) : = (-b/2,

0).

A $‘+g(o:’

= 0. ‘7

The second case is g g/ = 0. Then one can check that the part gb2i remains unaltered under transformations which maintain go1 in the region of (A2) and that there is no other equivalence in this region. Hence S9JHo~DH _ 2 Ho = HL(l, /i(H”@H-2)) is topologically equivalent to a “complex cross”, that is a pair of complex planes having one common point. Here we can see the stratification noted in Property 4. In this case the inequality (A3) turns out to be the equation. One can verify that the space HL(2, n(H”@,Hw2)) has components of different dimensions which intersect at the zero point. Now let us consider

bundles

E and

fj-fi

W such that the following

-pkl > 1

inequality (A4)

is satisfied for all i, j, 1, . . . , k, as before. Then by virtue of (A2) no non-identical equivalence transformation preserves any cocycle in the region given by (A2). That is so because the situation .sp) = 0, k = 1, . . ., I-1 and (Sosol+GoIs~‘)(“=O which has appeared in the previous cases in now impossible. Therefore in this case (A3) may be replaced by the equality. Now we prove the existence of graded bundles of 2nd degree over P’ C. Recall the cocycle formula restricted to A2 E gg)

= &a A g~“/+LJyb:+&;

A (S~‘)‘2’+S~‘(S;1)“‘+S~1’g~‘l+g~*~(S;1)’1’.

We see that g$:’ = 0 is possible only if each matrix element of r decomposes into a sum of at most 3n simple 2-vectors. Recall that a generic 2-vector in ,42 E2, *-E P’ C, decomposes into a sum of [r/2] simple 2-vectors. So let us assume that 3n c [r/2] and look for a section yb2j of /i2 Eluo, such that at a point of P’ C none of the sums

ON GRADED &’

BUNDLES +

AND THEIR

$1 + dj”

)

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C&W-(/l%

219

SPACES

(A3

I%,)

admits a decomposition into 3n simple 2-vectors. Such a section is an (i, j)-matrix element of cocycles of 2nd degree graded bundles. (2) by means of the following conditions: Let us define yoI 1” The rank of yb2j is greater than 6n generically. 2” For each monomial m(z) appearing in the expansion in z of yb: projected on any H

pkl

AH

pkz c A2 E we have (Pkl+Pk2+ri+1)~deg(m)~rj-1.

maX

kl fk.2

Then the rank of the sum in (A5) is generically also greater than 6n which makes decompositions into 3n simple 2-vectors impossible. The procedure described works also while searching for graded bundles of degrees 3, 4, . . . But then the procedure turns out to be ineffective because the linear algebra as yet cannot indicate for which dimensions the decompositions of k-vectors occurring in the equivalence formula of cocycles cannot be performed in general. In order to see another property of graded bundles over P1 C let us consider ~8:. The following decomposition can be easily proved for sufficiently large integer 4: gb”l = -(so where so = I,+I,@o,

cr := zg’ zqeT(n’ gb’: := (s,g,,

A s; l)(r), E) so that

s;l)(r) = sb”gol+g~](s;‘)(*-‘).

We can choose such a large q that the (i, j)-matrix have in its expansions any term aJzJ with pt+

. . . +pr+ti+

element

of gg\ does not

1 < f < tj-1.

Then there exist sb, s; such that (sb 901 s;- r)(r) = 0. Thus we have proved

that deg S(E) < r.

Another

fact which can be quickly

proved

for each 9(E)

over P1 C is

Yt,(E) = W&lE. That is so because classifications on P’ C are identical.

of smooth

and holomorphic

graded

bundles

220

J. CZYt

B. M=P”C,m>l auxiliary bundle E is trivial that then the graded bundle

In the first part of the example assume that the and the bundle W =
01

(l’

g’”12 --

(Gap

902.

is the trivia1 line bundle, i.e.

is :=

sh”s’).

Let us define a decomposition such that the components

( ’ ’ = &’ .cl,p satisfy (1)G

12

OYOl

032)

+ ,Kl:‘,’

=

1

od3:

1gb’; G12+Go, d,:’ = 0, G 01

2dl:’

=

The existence of the above decomposition the expansions of g$ in zo, zl, z2. The following equalities are consequences -0Qo1

(1)G

lo=

ogb:

-

033)

2.(/b:.

becomes

obvious

of the equations G20

GIo ,g&’ = -lg’r1:G21

=

if we put in (Bl) (B3):

: so,

=:sl.

(B4)

G20 z.y&’ = Gz 1 2y:‘2) = : s2. The transition functions gor,, admit singularities only in z, = 0 and zp = 0 and in z. f 0, z1 # 0, z2 # 0 resp. therefore so, sir s2 are non-singular The equations (B2), (B3) and (B4) imply g$’ = -soGo

+Gols,,

g&’ = -so Go2 -t- Go2 s2, g’&=

-s,

(B5)

G12+G12s2,

which means that 1, can be proved by a natural induction. For genera1 auxiliary bundles E the equation (Bl) makes sense if a choice of a coordinate chart containing Uolz is given Therefore fix the coordinates in E corresponding to the covering sets Ud, 0, 1, 2, 3, and denote by &’ the section g$’ in the Ud-coordinates. Then the additional transformation law is

ON GRADED BUNDLES

221

AND THEIR MODULI SPACES

(B6

where (y,)

is a cocycle

of E and the multiplication

matrix elements of $rr’ by yae. The equation (Bl) in U,-coordinates

“*” means

all

is

Observe that $r’J, &i, &/ admit poles at z. = 0, z1 = 0, in the previous case of the trivial E. Let us put P E:= HP so that yae= 2 . 0Zd Then the components

multiplying

z2

= 0 contrary

to g$/

(B7)

of the decomposition Q$ = L& + &$

defined

(B8)

in the same way as .g,@ in (B2) fulfil &)

= Yye*&I$) 7

&7’

= Y& . &I

(Note that the compatibility of (B9) and (B6) is a consequence are monomials.) Therefore we can define so : = -O&‘GIo

= -o;b:‘G20

s1 : = GIo ,&,‘: = - ,& s2 : =

We see that s, are regular

(B9)

*

of the fact that y.@

,

Gzr,

(BLO)

Gzo 2&rj = G2, ,jj\:,. on U, and that s, trivialize

S(E) as in (B5).

Thus, by passing from E = HP, M = P2 C to E = & HPk, M=P”C,

m>l

k=l

we can prove that all considered graded bundles are trivial. Note that this fact can be interpreted as a generalization of the Bott formula H’ (P” C, HP) = 0 if m > 1. A modification allows us to prove the same fact when M is a space of flags in Cm. Formally it suffices to consider multiindicides instead of the indices a. Observe that the above proof seems to work also for a class of algebraic manifolds which do not admit vector bundles having transition functions corresponding to a 2-set covering. One can also prove that each holomorphic graded bundle over PC is isomorphic to a graded algebraic bundle and that the moduli space S<+, consists

222

J. CZYi

of algebraic varieties as components, these varieties admit singularities.

cf. [23]. The question

C. Complex

is whether

and when

tori as base spaces

Let us consider as the base space a complex torus Cm/r, where r is a lattice, i.e. r c C and r z Z2”. Let E be an auxiliary holomorphic vector bundle of rank Y over the torus. Thus E is completely determined by the following equivalence relation:

where (z, e) w (z+y, periodical condition

4:(z)‘),

4::

C” -+ GL(r,

C) and

4:

satisfies

the

quasi-

4;+ ?’(z) = 4; (z + y’) 4; (z) = 4:s (z) 4; (z + 1”). The because

E -+ 4:

correspondence

is compatible

@‘(z) =f(z+~)@(z)f-l(z),

with

,f: C”-,GL(r,

the

(Cl)

equivalence

C),

Es

relation,

E’.

(C2)

We are looking for a lattice-formula for graded bundles over C”‘/T. Let p: c” -, c”/T be the projection and p* be the induced mapping of graded bundles. The graded bundle p* (T(E)) IS ’ t rivial as every graded bundle over C”. Thus p* (T(E)) is isomorphic

to 6 p*(AE)

and T(E) satisfies

se(E)=

(&P* t-W)/-,

where

(~3P*(U)) - (z+Y,~*('J'~(z)a)), Using (Cl) for matrix Y,(z):=

Cz, F(z,

elements

Y&kGW,

ME,,,,),

a~&A&,,:,.

of Y?(z), we obtain

Y)I= CZ+Y,&%)W,

191, F(z, Y)EGL(~,AC),

where &? is applied to all matrix elements of F(z, y). We can see that F (z, y) acts on a fibre in z+y. Hence Cl applied to Y gives us the following quasi-periodical condition for F: F(z, Y+Y’) = F(z+y’, In order to get the equivalence

YN-&~(z+VPF(~,

relation

Y’(z) = B(Z+Y) Yy,(z)g-’ (4,

(C3)

for F observe that, similarly as in (C2), g(z)EGL(n,

so that g(z) := cz,

~‘11.

cp(41= cz+y, 4.?(4~P41

A&z,)

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and the equivalence

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relation

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223

SPACES

is

F’(z, y) = q(z+‘1’)F(z, Let us assume

THEIR

Y)[+y(Z).cp-l(Z)].

that E is a line bundle

and

W = & w, i=

where

w are line

1

bundles too. Then functions @ of (Cl) corresponding to these bundles up to the equivalence are &functions 0” and @ resp. Recall that O,.(:) is given by 0.) (-_)

=

c(

tl/)

enW=.;9 +

x(;‘, +I!*) =

'~H(Y.Y),

H(Z, u) =

pmfw.~2$

ZQU,-_),

(y,)x,(y,)

ImH(rxf)

and

~2,

[x(:1)/ = 1.

By virtue of (C3) we have F!!‘(I, ‘) l!+J”) = F!?‘(-fq” IJ - I,/ “1)04, (Z)+ 0; (1 + 7’) 0;’ (Z + ;v’)Fif I(=, “r’) = F$’ (z + y’, y) fl-$ (-_)+ “{, (2 + 7’) 0;’ (z + 11’)F$‘(z,

y’),

where E’ = Eons Wj*. By forgetting i, j we obtain the equation in the l-dimensional case E = E’, W to Fij has the complex = Wj. Hence the subspace of S%E,W corresponding dimension equal to h’ (M, El). Recall that if M is algebraic (i.e. an Abelian variety) and E is ample, i.e. the corresponding form H is non-degenerate, then we have the following formula for first cohomologies (see [62]): h’(M,

E) =

f?(H) # 1,

0,

.PfImH:=, where n is the number D. M is a compact

of negative

Riemann

n(H) = 1,

detImH,

eigenvalues.

surface

In this case we can and will use only finite coverings ASSERTION1. The ,following

estimution

(Q,) such that QegY= 8.

is satisfied:

dim ,,,EH’ (M, HL(n, LIE)) < 2n2 i k=

h’(M,

Ak E).

1

Proqf: It suffices to point out such a transformation of any graded HL-bundle which maps all components of all matrix elements of the given cocycle elements into a fixed representation of H’(M, Ak E), k = 1, . . . . r. We can realize this by acting on (gXa) with sets of sections (~2) such that

(s$Z,)~~EI-(M,

and all remaining

matrix

elements

of sijk -I,

AkEn)

vanish. We are obliged

to use the

J. CZYi

224

following order: for each k taking subsequently the values 1, . . . . r each of the indices i, ,j runs over whole the set (1, . . ., n). q.e.d. Recall that to each vector n-bundle W on a Riemann surface one can relate a sequence of line bundles L,, . . . , L, such that W I L, in the sense of a subbundle, W/L, 13 L,, (W/L,)/L, I L, and so on, cf. [6], [45]. ASSERTION 2. Let

graded bundle S(E) Proof: In the there is a cocycle form a cocycle correspond to a

L1 = L, and H’(M, E) # 0. Then there exists a non-trivial such that CGO(E)= Mi.

language of cocycles the “filtration” by L, , . . . , L, means that (G,,) of W consisting of upper-diagonal matrices such that GzBvii of Li and in our case Gaps,1 = Gap,,,. Let the cocycle (h,,) non-zero element of H’ (M, E). Put y$’ : = G,,,

Y$!~I : = GE,,, 1 k,,

gh~~!ij: = 0

q$ := 0 Then

if

(i,.iJ # (4 11,

if I> 1.

we have (s,g,&‘)b?

The above

formula

= (~~‘~,+h,~-sg,nll)“‘Gll~,t1

leads us to the following dim w3,,ES$,W

# 0.

estimation:

3 h’ (M, 0,

q.e.d. ASSERTION 3. If The proof

W = & L, then dim,,,S!//,,W k= 1

> nh’(M,

E).

is clear if we put g$‘,ij : = Gag,ii h,, di,, g$ = 0

if I>

ASSERTION 4. There exist graded HL-bundles Proof: Let let us define

n = r = 1 and

H’(M,

1.

for which Cqu(E) # 6 AE.

E) # 0. Choose

0 #(c,@)EH~(M,

E) and

where (;I,~) corresponds to W, be a cocycle of a vector bundle I/ which is the extension of E by W determined by (c,,), cf. [7]. Then the cocycle gab : = 1@~,~ ;‘$’ corresponds to a graded bundle Y(E) such that 9”(E) = I/. Since

ON GRADED

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I/ is a non-trivial extension then the correspondence lished in this manner for general n, r. E.

The triviality

of graded

bundles

SPACES

cap c*gmB cannot

225

be estab-

over every Stein manifold

Recall that Stein manifolds are those holomorphic manifolds which satisfy two postulates: 1” Each of their points is an isolated zero of a finite system of global holomorphic functions. 2” They are holomorphically convex, i.e. holomorphic hulls of compact subsets are compact. A space satisfying both these conditions but admitting singularities is called a Stein space. An important fact is that Stein manifolds are not compact. Furthermore, each non-compact Riemann surface is a Stein manifold. By virtue of one of Grauert theorems, see Satz 2a in [42], for each Stein manifold M (more generally, for each Stein space) there exists an isomorphism between H’(M, GL(n, M)) and H’(M, GL(FI, C)) which trivializes each graded bundle over M. Observe that the triviality of graded HL-bundles over Stein manifolds is a consequence of a more elementary Cartan-Serre vanishing theorem, see [44], which says that H’ (M, V) = 0 for 4 > 1 and each holomorphic vector bundle V over M. F. 9Jo(E) is flat Suppose that we have 1. A flat vector bundle W determined by a cocycle (G,& of constant transition functions associated to a 3-set covering (Qa), SI= 0, 1, 2 (for the main properties of flat vector bundles see [31]). 2. An arbitrary auxiliary vector bundle E. 3. An (!&)-cocycle (hap) of a graded HL-bundle such that matrix elements of (hap -I,) take values in a fixed Grassmann bundle nk E and !I$! Gzo hiki = 0. Then the formulae g$’ : = G,,, g$ : = 0,

I=

gd”; : = htj Gzl,

1, . . . . r and l#li, g’:: : = G,, hiki,

g’oi : = hf;

uniquely determine a graded bundle %(E) such that !qO(E) = W The above assigment is preserved by transformations of the type figp = .s, h,, s; ‘3 (s, -I,)‘”

&,j = s; g,,j s;-

= (s; -I,)‘”

’,

= 0,

where 1 = 0, 1, . . ., r and 1 f k, 4, = so G,,,

4 = G,osl

Gzlr

s; = Gzosz.

J. CZYi

226

Hence in particular

we obtain,

cf. Theorem

\j

dim W.IC S’//F,.w 3 21:ln. r) h’ ((Q,)f_ (,, AE), where r = 1. r:(n, r) :=

G. A review of N’(M, Gl.

r > 1.

M and

E) for particular

in Cm, E is trivial In this case we have the or-iginal additive

E

M is open

= @ If’ (M, I’) $- 0 (resp. = 0) means

that

Cousin

problem

this problem

and H’ (M, E)

cannot

(resp. can) be

solved for each compatible data. In particular, if M is an open holomorphy domain in C2 then H’ (M, C!) f 0 iff ,Zf is not holomorphically convex. This is the case when e.g. C2 - M contains compact components or M is not convex in the sense of Levi. Rougly speaking, the Levi convexity means that it is possible to “touch” i7M from outside with an open subset of the zero-set of a holomorphic function at each point of JM. In particular, a region in C2 having zero-point and admitting boundary equations of the type 1~~1= Y(~z,~), which is a Reinhardt region, is not Levi convex if in a point. where r12In ‘I’

-~~~> 0

(ln I=, I)” and. more generally, which is a Hartogs

such a region determined by boundary region, is not Levi convex if

equations

1~~1= Y(;,),

A In Y > 0. The above

conditions

are proved

in [74].

G2. M is a Riemann surface Let E be trivial and rank E = 1. Then we have /I’ (M. E) = h’ (M, C) = genus M (recall that genus M = 0 iff M = P’ C). The second natural choice of E is E :-

0, h’(M,

7-M) =

1, i 3(Y- f),

TM.

genus genus genus

Then

we have, cf. [SS],

M = 0, M = 1,

M=s>l.

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G3.

M is primary Hopf surface, E is trivial Recall that a primary Hopf surface can be defined to be complex manifold which is homeomorphic to S’ xS3. An example is the original Hopf surface UO (C2- [0),/r, where r is generated by o LI , 0 < /u/ < 1. 1 1 K. Kodaira proved that H’(M, I’) = C, see [54].

G4. Compact KPhler manifolds with dim, M = 2, c1 (M) = 0, H”(M, C) = 0 In algebraic geometry one proves that the class of such manifolds includes a 20-dimensional family of algebraic manifolds. One of them is the famous Kummer surface. The calculations carried out in [72] yield h’(M,

G5. E = TM

and

TM) = 20.

Q: M--t R is a line bundle

where

R is a Riemann

surface,

RfP’C

Below let me outline the proof due to K. Dabrowski of the h’ (M, TM) = cxj . It will be given in our joint paper. Let us observe that it suffices to prove that h’(M, Q* TR) = ^x:. We have the following exact sequence of sheaves of sections: O-+ Q* TR + g*(TR@[p])

fact

that

+ c’@_ ltp, + 0,

where PER and [p] is a line bundle given by p as a divisor. The corresponding exact sequence of col~omologies is

0 - H’(M,

e*TR) - H”(M, e*(TR -tH’(M,

It is obvious

0 [PI, -

Ho&’

(~1,

I’@-I(,,,)

Q* TR)-.

that h’(~- ’ (p), Cp_ lcp,) = cc. Thus if a term in the middle is of a

finite dimension then we have h’ (M, Q* TM) = x. Therefore the main point is to prove that h”(M, e*(TR@[p]) < x . This may be proved by using the Nagata compactification &!l of M, considering suitable divisors and their exact sequences and, finally, by proving the existence of a surjective map on this cohomology from the finite dimensional space HO(M, L). G6. Another

example

of h’(M,

E) = cc

A class of complex manifolds M, such that for every holomorphic vector bundle E we have h’(M, E) = cc, can be constructed by applying the results of Andreotti and Norguet, see [2]. A particular case is the space of positive (negative) twistors, i.e. elements of P3 C which satisfy Im(z’ 33 -z’?) > 0 (< 0), cf. [2]. Below we present two cases of vanishing of first cohomologies.

228

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J. CZVi

M is a compact quadric in PC Then for each vector bundle V over M we have W(M,

V) = 0,

if q > 0, see e.g. [44]. In particular, the first cohomologies of auxiliary Grassmann bundles H’ (M, Ak E), k = 1, . . , Y vanish. Thus this example is a model for Theorem /I. G8. The Kodaira vanishing theorem is satisfied Let M be a compact Kahler manifold, dim,M = m. It follows from the Kodaira vanishing theorems that Hi (M, E) = 0 if E is a holomorphic line bundle such that either m # 1 and -ci (E) is the class of a Kahler form or ci (EOTM) is the class of a KPhler form, see [SO], [8 1). The previous examples concerning P” C and tori allow us to suppose that in this case all graded bundles may be trivial. 2.7. Graded bundles ouer G-supermanifolds I

realize that graded bundles may seem to be unfit for mathematical gourmands because of an inhomogeneity in their structure. Namely, fibres of graded bundles are provided with Grassmann moduli structures while their bases are ordinary (not “super”) manifolds. But we will get rid of this disharmony soon by using functions of G-classes and by replacing in the basic setup of graded bundles all manifolds with G-supermanifolds in the sense of Alice Rogers, see [68]. Let us recall the notions of G-functions and G-supermanifolds (G is like Grassmann).

In this section

52 c (2 & V)@(% /1, V), where

/1,(,,

is the even

(odd) part of Al’, is an open set in the usual Euclidean topology of the vector space. We say that a function f’: Sz -+ nP’ is of G’ class if there exist functions and a function E: s2 -+ nF’ such that if (a, b), fs: Q-t/lV, q = 1, . . . . m,+m, (a+/?, h+k)~U,

where a, u+h~&,

.f‘(a+k b+l) =.f(a, h)+ (II.Ij is the Euclidean

z

q=1

V’, b, b+k~%

4J&l,

w+

A, V, then

m’r2 k&f& q=nt,

b)+ll(k W&v& 4

+1

norm) and

Jlr:(h, k)(l -+ 0 as Il(h, k)l! -+ 0. Note that the functions jb, q = m, + 1, . ., m, +m, are not uniquely defined contrast to the traditional differential calculus, a problem that is discussed in original paper [68]. The functions of GP class are defined inductively; the class G” is defined to the intersection of all GP. The functions of G’“-class (i.e. analytic G-functions) can be defined locally means of power series of the type

in the be by

‘ON GRADED BUNDLES AND THEIR MODULI SPACES

f(x) =

c

LQ=o.....p,,

aP1’ +m2=

. ..’

Pm,+m2

229

(sl-JgP1 . . . (x,1+m2-J)“1+~2)Pml+m2.

0

Furthermore, an analytic G-function is algebraic if its restriction to Q n Km1 (K = R, C is the scalar part of /iv) is an algebraic n I/-valued function. Thus G-functions of (m,, m&-variables, when restricted to s2 n Km1 are /iV valued functions of m,-variables of the same class as before. This is so because each G”-function in the sector of “pure Grassmann” variables looks like a polynomial of degree at most (r- l)(m, + m,). In the next step we should define G-supermanifolds of dimension (m,, m,) and of the above classes. But their definition is parallel to that of differentiable manifolds. It suffices to use as an atlas a set of G-functions of suitable class taking values in (m&/1, V)@(mG ,4, V) and to assume compatibility

on intersecting

sets in

the sense of the above Grassmanndifferentiability. The definition of G-mappings between G-supermanifolds is also simple and natural. The next notion is a line G-bundle. This is a G-bundle (i.e. the base space and the total space are G-supermanifolds and the projection is a G-mapping) which locally looks like the product Q x AK Note that any line G-bundle is determined by a cocycle of transition G-functions. Now we are ready to introduce graded G-bundles. One way of defining them is to establish the quadruplet (SG, xG, M,, E,), where the total space ‘4(; and base space M, are G-supermanifolds, the projection Q: ‘//(; 2 M, is a G-mapping and the auxiliary bundle E, is a line G-bundle. Furthermore, we assume a local triviality of the graded G-bundle, which means that the decomposition

holds for an open covering (a,) of M,. Note that in this case we need not “grassmannize” the auxiliary bundle. As a second way of introduction of graded G-bundles we can use the I- 1 correspondence between the set of classes of equivalent graded G-bundles and the cohomology space H’ (MG, GL(n, E,)). The above proposition of graded G-bundles does not allow us to put local coordinates in base spaces and local coordinates in fibres on quite an equal footing. This is so because there is no general procedure which separates locally 11, even and n2 odd variables in fibres, in contrast to base spaces. In order to obtain a class of graded G-bundles satisfying this separability postulate we can assume: Ga. The auxiliary line G-bundle E, splits E, = Eco@E,, , where fibres of EGO(EG 1) consist of even (odd) vectors for every admissible local map of E,; . Gb. The transition

functions

take values in groups

GL((n,,

n,), (EGOxr EciLs))

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230

of .4E,,,-linear transformations of IV, : = ($ E,,,,,)O($ decomposition of IV into the even and odd part.

E,;,,v)

preserving

the

Instead of giving examples of graded G-bundles here, we refer to [16], where certain graded G-bundles are explicitly constructed. They are in their even sectors direct continuations of linear bundles by the rule K + /iO I/ and are flat in their odd sectors. Similarly as in Section 2.2, we say that a graded G-bundle is trivial if its corresponding cohomology class contains a cocycle whose transition functions take values in GL(n, K). Recall that derivatives of G”-functions restricted to “purely Grassmann” sectors are polynomials so that G”-functions are analytic in “Grassmann directions”. Hence we can simply prove that K-valued G”-functions are constant. (The analogous statement in the case of analytic functions says that each real-valued holomorphic function is constant.) Thus each trivial graded Gbundle admits a cocycle of constant K-functions. Recall that in the case of C”linear bundles constant cocycles determine a class of flat bundles. Now we point out a non-trivial graded G-bundle over the ordinary torus T2 : = R2/f. Assume that the lattice r is spanned by (0, 1) and (1, 0). In the aforementioned paper [68] one can find a construction of the structure of a G”manifold of the (1, l)-dimension on T’. This structure is naturally induced from R2 provided with the Grassmann algebra structure RZ = R’@R’ = R’@ V = AT/. Let us take a covering (Sz,, Q,) of T2 such that Q1 n 0, has components 52’ and 0” with non-interesecting projections on V/Z. Consider a transition function hiz; sZi2 +GL(l, /iI’) such that

(ii) “h,,,,.(x, v), XE R’/Z, VE V/Z is affine but not constant in L’, where ‘(.) denotes the projection on I/: Suppose that there exist G r-functions s, on 52, and s2 on.QR, taking values in 1(1, ~‘)IL’EV) such that hi, =.s,+hiz-sz is constant in I: on Q2,,. Then in particular (s, -,s~),~?. = c(x). Hence the following equalities hold on Q”

5, (x,

c) -

“J’2 ( x3 V)=

“SI(X,

C)-

“S1(X,

V +

I)

--Cl

(x) = const

for .x = const

since admissible G z-functions si are affine in v. As a consequence h,, is not constant on Q”. Thus the graded G-bundle determined by hi2 is not trivial. 2.8. Graded Z,-bundles In order rewrite the main formulae of supersymmetry in terms of graded bundles certain reductions of general graded bundles to smaller transformation groups are necessary. Below we outline a number of such reductions.

231

ON GRADED BUNDLES AND THEIR MODULI SPACES

III’), q +s = n, dim, I/ = r consist

c GL(n,

Let the group GL(nlq,slnV) invertible matrices of the type

4

of

b

-i-

(1) LJ=

where

aij,

hijE ‘40 I!

cij,

dij~/lr

I/

(Recall

that

/1, I/: =

&

/iZk X

A, V

2k=O

:=

&

A2k+1 V)

2k+ 1

This group can be considered U : = c$ AV preserving cl := UOOUl,

Then the Z,-gradation

as the group of all AI/-linear transformations

the following u, :=

of

Z,-gradation:

(& A, v)&

n, v), u, := (6 now6 Al VI.

of U agrees with the Z,-gradation grad vu = grad v + grad u,

of /iV by

VE/iV, UEU,

where “+” is mod 2 and the grades of all elements are assumed to make sense. More physically, the group elements are proper transformations of a superspace which is spanned by q generators of bosonic and s generators of fermionic vector fields. (Recall that the bosonic generators can be defined as “usual” vectors but the fermionic ones are rigorously defined as contractions in ny cf. [37], [SS].) A slightly more special group consists of only those transformations g of (1) which preserve a measure. Let us recall that similarly to a change of a volume element, induced by a usual vector transformation given by its determinant, the Berezinian of any transformation like (1) B(g) := det(A-BD-’

C)det D-’

is the object corresponding to a change of “supervolume”. Thus each of the conditions (a) B(g)E K and IS(g)( = 1 (K = R or C), (b) B(g) = 1 determines a K-analytic subgroup of GL(n I s, q I AL’) denoted by BGL( .) and SBGL( .) resp. Furthermore, these three groups, when intersected with HL(n, AV), give smaller groups HL(nls, ql AV), BHL(.) and SBHL( .) resp. consisting of suitable “purely Grassmann” Z,-transformations. By a natural transition from a vector space I/ to families of hbres E, of

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J.CZYi

auxiliary bundles we obtain classes of group-bundles whose fibres are isomorphic to the above groups. Elements of first cohomologies formed of sections of these group-bundles correspond to graded Z,-bundles assigned to the considered groups like graded BHL-bundles. Such graded bundles can be also considered as reductions of graded bundles to the corresponding subsheaves of the sheaf of sections of GL(n, .4E). The theory of any above class of graded Z,-bundles seems to have much in common with that of general graded bundles and general graded HL-bundles so that we close this section by briefly recalling existence theorems for the mentioned graded Z,-bundles. In the case of smooth graded Z,-bundles the proof of Theorem r from section 2.3 can be repeated with only slight changes due to the restrictions to transformations of a given type. Thus we can formulate STATEMENT 'x.Each smooth graded Z,-bundle in the sense of any of the above groups over any paracompact base manifold M is trivial in the sense of its groupbundle (i.e. trivialization making sections are those of the group-bundle).

In the holomorphic 5(E)

and algebraic

case we have

STATEMENT p. !f H’ (M, E) # 0, then there exist non-trivial graded in the sense of all group-bundles previously mentioned. If H’ (M, AkE)‘= 0, k = 1, . . . . r, then each graded Z,-HL-bundle

Direct preceding analogous

is trivial.

algebraical calculations allow us to improve in the Z,-cases the estimates for dimensions of spaces of suitable graded Z,-bundles to S’G,,w. This is so in Theorem p, (A3) and (A4) and others.

2.9.Projective

By

Z,-bundles

graded

bundles

projective graded bundle of a class (holomorphic, algebraic or of the cohomology space G”) up to equivalence we mean an element such that N(E,) H’(M, GL(n, AE)/N(E)), where N(E) + M is a group-bundle is a normal subgroup of GL(n, AE,). We can replace GL(n, IIE,) by a bundle of their subgroups which are not contained in GL(n, K) being a subgroup of all GL(n, AE,). Then we obtain a class of reduced projective graded bundles. This notion seems to be useful because some of these subgroups admit more normal subgroups. Assume that all groups N(E,) are Abelian. Then the following sequence can be proved : --t H’(M,

a

N(E))

+ H’(M,

GL(n,

AE)) + H’(M,

GL(n,

AE)/N(E))

4 H’(M,

N(E)),

where the extreme terms are topological groups and the terms in the middle are topological spaces with distinguished zero-elements.

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Let us put N(E,) : = Z,+Ak ,??,@I,, where k = r or (k is even and r/2 < k < r). Then N (E,) are Abelian. We can simply prove a stronger fact, namely that N(E,) is a subgroup of the centre of GL(n, AE,). In this case 6 takes values in H’(M, AkE). This is one of the analogies between 6 and the first Chern class of complex vector bundles. 2.10. Local geometry

in graded

bundles

In this section we do not need to assume analicity of graded bundles, though the holomorphic case remains more interesting. K. Gawedzki in [37], cf. also [56], worked out a superspace tensor calculus which can be simply adapted for the needs of graded bundles. As a superspace we mean here the base manifold M together with an algebra r,,,(AE) of local sections of the Grassmann bundle AE ---tM (this algebra replaces the algebra C,zc(M) from differential geometry). The algebra r,,,(AE) possesses a natural Z,-gradation given by AE = A,,E@At

E,

where &E:=

ar/21

@

,4’“E,

2k=O 2[(r-

/i,E:=

u/21+

@

1

AZk+lE,

r = dim, E,,

XE M.

2k+l=l

We consider as a (graded) vector any K-linear mapping X of r,,,(AE) which can be decomposed as X = X,+X,, where X, (the even part) and X0 (the odd part) satisfy X, (r,,, (4 E)) c rlo, (4 4, X,(@)

a, BE~,~~(M,

= X,(cOP+aX,(/% XO (r,m (Ai X06@)

=

E))

c

rl”c (-4i 01 El,

XO@)B+~(a)Xo(PL

where ZlnkE= ( - l)k. One can easily check that for even vector fields the commutator is well defined and the anticommutator makes no sense but for odd vector fields it is just anticommutator that makes sense whereas the commutator does not. Any vector X has the following expansion, cf. [37]: m = dim M, r = dim, E,,

where coefficients a’(x), j?(x) belong to the Grassmann algebras AE,, ~?/c?x’ are “ordinary” vectors which constitute bases of tangent spaces 7; M and c3/afP are contractions in the Grassmann algebras AE, with local sections Oj of E forming bases in the fibres E,. The vectors of type a(8/lo’.x), where a(x) e/i0 E, (resp. r (X)E Ai E,) and /?(8/%), where p(x) E A, E, (resp. P(x)E no E,) are even (resp. odd).

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This notion of a vector is a starting point to a complete graded tensor calculus, see [37], [56], [SS]. In this calculus both symmetric and antisymmetric graded tensors consist of two parts and one of them is symmetric while the second one is antisymmetric in the usual sense. The exterior derivative of antisymmetric graded tensor fields satisfies the equality d2 = 0 which determines cohomologies of superspaces, see [56]. We will make a usual geometric calculations as in the Chern-Weil construction, see [20], so that only special differential graded forms will be used, namely

Note that if cx(x)~/l~E,,

P(x)cAjE,,

then

q A $ = (- l)PSfij(C/ A cp, By a connection in the graded bundle -+ Tloc(‘~(E)@T* M) such that

S(E) we mean a mapping

V: f,,,(%(~))

V(.s, +s,) = vs, +vsz, V(rs) = rlr@s+xVs,

sd-,&4(E)),

cr~I-,,,<(AE)).

For a given system s = (s, , . . , s,), si E f,,,(%(E)), of local sections which form a basis of the A&module r,,,(S(E)) one gets the connections matrix vs, = ejsj. Note that 02 are graded

one-forms,

where ~$3~= u{i”(x) dx,. If .$ = sjsj

is another

basis then w transforms

as follows:

ro’y = v”g+lY$JJ, where V” is a connection in AE@K”‘. Note that in order to describe a connection in ‘G(E) a connection in the auxiliary bundle must be given. Similarly as in geometry of fibre bundles we may introduce a connection by means of a connection graded form 1’ on the space of the principal graded bundle which fulfils w = s* y. It is natural to define a curvature matrix Q by

ON GRADED

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235

that the skew product of x, dxi, fil dxj, where slI (.Y), The transformation rule is Q’ = gQg_1.

There

is no problem

with the Bianchi

identity

vQ+s2Aw-oAQ=o. But from this point the traditional geometry and the graded geometry seem to move in different ways. In order to see this let us try to carry out the Chern-Weil construction (then K = 0. One can quickly verify that dtrS2, =O,

dtrSZ, = dtro,

A o1

(2)

(Q, : = q.I*E? where the cut is made in the set of values) so that d tr Q is a graded 3-form depending on (J.I,. Let us recall that the Chern-Weil construction is a consequence of the fact: there exist j-homogeneous polynomial functions in matrix elements defined in the set of all complex n x n matrices pj such that pj(GAG-

‘) = pj(A),

GcGL(n,

C),

npj(a) = 0.

j = degpj < n,

(3) (4)

Let us observe that none of the above properties can be maintained in the case of graded bundles. In order to prove the non-existence of any polynomial-function satisfying (3) it is sufficient to check for a certain matrix A with elements in /1V that eigenvalues of matrices gAg_’ , geGL(n, /iv) may change independently of each other. Since p1 (0) = tr Q, the formula (1) makes (4) not correct for graded bundles. But the above obstructions do not exclude any possibility of defining “graded Chern classes” in this way for special graded bundles. For instance, if 5(E) is a graded K (II, NZ)-bundle such that the group K (n, .4E),.,o, is Abelian and if we assume that dtro, A o, = 0, then

~$1 := trf+(,,0+.41,E defines graded The troubles As copying

an element of H’(M, Z@)E) which looks like a first Chern class of such bundles, cf. Section 2.9. above obstructions in the Chern-Weil construction seem to be relevant to in integration of odd graded forms, cf. [25]. a conclusion of this section I propose to pay maximal attention when differential geometry formulae in the super or graded geometries.

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2.11. How to gmeralize

the idea of graded

bundles?

At first we will depict correlations between graded bundles and supermanifolds in the sense of Kostant and Berezin, see [14], [56], [SS] and Grassmann moduli over them which are given in recent Manin’s papers [84], [SS], [86]. Such a supermanifold is defined to be a Z2-graded sheaf of Grassmann algebras (M, I’ = Co + Cl ) over a base manifold M (only Z,-graded structures of this type seem to be relevant to the physical situation). Roughly speaking, a sheaf is most abstract global structure in which local sections are well defined. Therefore the notion of a sheaf of vector spaces is a generalization of that of a vector bundle. For definitions of sheaves see [40], [SO]. In practice the supermanifold may be locally regarded as a Grassmann bundle /iE but, as transition functions, invertible functions of the type y(x, e) --t (g(x, e), h(x, e)),

1

g(x, 4 = go(-9+ il

h(x, e) = h,(x)+

<...

1 il

Yil...i,k(X)~il

*

...

*

eizk7

eizk

~___ hiI,.,i2k+1(x)ei,

_____ A ... A ei2k+l

C...ii~k+l

are admitted. The formal definition says that the supermanifold is a fibred space having as fibres (stalks) sets of germs of functions of the above type and provided with such a topology which is compatible with the operation of joining germs in local functions. Thus the notion of a supermanifold includes the additional dependence of points of the base space on points of fibres which is determined by the underlined terms. If the sheaf corresponding to a supermanifold is generated by the set of sections of /1E (i.e. the sheaf may be determined without the underlined terms) then the supermanifold is callled simple. By a supergraded bundle (Cc-module in Manin’s terminology) 5 we will mean a sheaf of moduli of Grassmann algebras such that the sheaf of Grassmann algebras If’ determines a supermanifold (M, CC’). Hence this supermanifold is in fact a generalized auxiliary Grassmann bundle /iE. Thus the category of supergraded bundles includes that of graded bundles, that of supermanifolds and the tensor product of both those categories. A supergraded bundle is called trivial if the underlying supermanifold is simple and it is trivial in sense of graded bundles. Let .Y’: = C 1@ Pf be the sheaf of ideas of a supermanifold (M, c'= Co@ I", ) and c be a super graded bundle. Then under suitable conditions (in particular Cocoherence

of [a,) the sheaf Gr 5 : = & 5 .Yi/.Yi+ ’ corresponds

to a trivial graded

i=O

bundle

W’@nE. Moreover,

i is trivial iff < = Gr < in sense of equivalent

sheaves.

ON GRADED BUNDLES AND THEIR MODULI SPACES

237

By combining the proof of the simplicity of smooth supermanifolds [14] with that in Section 2.3 one can prove that if M is paracompact, then each smooth super graded bundle is trivial. The same property is also shared by Stein manifolds as one can prove by means of Grauert’s theorems [42]. condition for the simplicity of supermanifolds is The necessary By virtue of Theorem p is H’(M, TM@dE)=O, k = 1, . . . . r, E = Y”Y’2. implicates the following necessary condition for the triviality of supergraded bundles: H’(M, A’E) =O, H’(M, TM@AkE) =O, k = 1, . . . . r and (Gr5)’ : = O/c4p= 6

W’, where

rank W’ = 1.

If rank E = 1, then the only supermanifold is the simple one. If rank E = 2, then the moduli space of supermanifolds is parametrized by H’(M, TM@AE). In other cases the description of this space seems to be complicated so that the general problem of moduli spaces of supergraded bundles is badly non-linear. Note that analytic supermanifolds depend explicitly on first order jets in contrast to graded bundles. If we consider a bundle of n- 1 order jets over M as a base space instead of the arbitrary manifold M, then we can define such super-manifolds and supergraded bundles which depend explicitly on jets up to n-th order. Let us observe that all the above structures turn out to be spaces which locally look like C” x(/iC)“. Furthermore, graded bundles are compatible with an intuition of manifold like objects which make of them sets of points with classes of local coordinates. In cases of the remaining sheaf structures such a compatibility is not so conspicuous, because coordinate charts are not distinguished in definitions of sheaves. The next direction of the generalization may be to define and to study Grothendieck-like rings generated by sets of graded bundles. It is possible to define such a ring X(M) in which one can add and tensor-multiply arbitrary graded bundles Y(E) and %(E’) over M (E and E’ may be different), see [23]. Then the resulting graded bundles have as the auxiliary bundle E v E’ (dimE, v Ei = dimE,+dimE:-dim E, n E:). The ring .X(M) may be obtained by means of the extension procedure of any semi-group S to a group G given by G:=SxS/,, where (sl, s2) h (si +s, s2 +s) which is applied to the additive semi-group of graded bundles over M. Note that the ring .X(M) includes graded bundles of an arbitrary integer “dimension” having auxiliary bundles of an arbitrary integer “dimension” too. Moreover, O-dimensional graded bundles having non-isomorphic auxiliary bundles are not identified. Observe that in the smooth case the subring of .X(M) made of graded bundles

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having a fixed auxiliary bundle E is a special Clifford module. The K-theory of the latter was elaborated by Atiyah et al. in [6]. Note that the tensor product in X(M) is non-commutative but it is graded commutative (a simple example in the case M = P’ C is given in [23]). This noncommutativity seems to be the main obstruction in attempts of constructing functors like “ch” and “td” in the analytic case. The next possible generalization is to let all the considered spaces be infinitedimensional and to assume that E, are copies of a linear topological space (Hilbert, Banach, locally convex and so on). This situation is convenient for physicists because if dim E, is finite, then the number of independent Faddeev Popov fields appears in relevant formulae though it is an unphysical parameter, cf. [16]. Observe that in the infinite-dimensional case some novelties occur on the level of trivial graded bundles (we have a few topological tensor products W@/1E) and smooth graded bundles because the group GL(n, AE), dimE = Z. n = 1, . . ., 3t does not decompose into a semi-simple and a nilpotent part. From the viewpoint of Lie groups it would be desirable to look for such generalizations which would more closely the approach whole “spectrum” in the Levi decomposition (any Lie group splits into a semi-simple and a solvable part) as it is in our case where the structural group splits into a special semi-simple and a special n’ilpotent one. To conclude the presentation of graded bundles from a mathematical viewpoint let us observe that all what we did was in fact a separation of and a look at “trivial” cases in the domain of graded bundles which enabled us to perform various transformations in the theory of vector bundles. Of course, the non-linearizable part of the domain of graded bundles seems to be more interesting. It still looks mysterious. 3. Applications of graded bundles to physical models A non-trivial graded bundle c/(E) of rank IZ can be interpreted as a system composed of n identical interacting or interferring or superposing objects represented by the auxiliary bundle E. The vector bundle ‘ho(E) can be understood as a system of weights for particular objects. In this chapter we limit ourselves to more simple cases when
ON GRADED BUNDLES AND THEIR MODULI SPACES

239

rigorous models. Then the base space is a manifold corresponding to space-time and provided with a structure adequate for general relativity (pseudoeuclidean, a complex one giving rise to gravitational twistors and so on) the auxiliary bundle is the bundle of 3/2-spinors (sometimes 5/2) in a spin structure in a tangent bundle and the rank of the graded bundle is the number of fundamental (basic) fermionic fields in a given model (this number in physical papers is denoted with n or N). Thus n = 1 corresponds to the simple supergravity and n > 1 to the extended supergravity, cf. [63]. If IZ> 1 then the non-triviality of the graded bundle is proposed to be associated with interactions between the fermionic fields. In the first, traditional approach to supergravity the base space is assumed to be a pseudoriemannian and orientable smooth real 4-manifold (the orientability gives the spin structure in the tangent bundle, se [65]). In this case Theorem CIsays that all admissible graded bundles are trivial. This means that the fermionic fields can be separated globally from each other so that the postulate which states that “the interactions between fermionic fields are measured by an unremovable curvature” cannot be realized in this model. Note that T. Ross [70] elaborated a geometric covariant formalism for the simple supergravity but his attempts in the case n > 1 gave no results. Perhaps the triviality of smooth graded bundles was the reason of his difficulties. A certain model of supergravity with a complex manifold as a base space had been proposed by Ogievetsky-Sokatchev [64] and then was developed by A. S. Schwarz and others in papers concerning the “space-field democracy”, see [38], [74], [75]. In this model the space of the parameters is provided with the structure of a complex flat superspace of variables (x, 8, 8) where x E C4 and 0 (resp. 8) are left (resp. right) 2-spinors and 0, B belong to the anticommutation sector. Let us extend this model by admitting as a space of the variable x any complex 4-manifold M together with a real structure rr: M---f M (i.e. o2 = id, and cr induces anti-linear transformations of the tangent spaces) such that fix-points of cr form an oriented real 4-manfild M,. Furthermore, we introduce the spin-structure SM, in TM, and denote by 8 (f?) the left (right) 2-spinors. The dynamics and field equations are given by the following three objects: an open set U c M, a mapping H: SMoILr --f M, n U, known as a fundamental superfield, and a group G of holomorphic transformations of TM preserving TM,, such that Sl. The coordinates.

set

U

S2. The mapping

is

provided

with

H in the above

+(x,-Q

= H&(x,-.Q,

a

system

coordinates 0, 8),

of

holomorphic

satisfies the equation a = 1, 2, 3, 4.

and

affine

240

J. CZYi

S3. The group G induces a group of supervolume preserving over Lr of the type x’, = &(x,, 8), 0, = &(.Y) l9,), x = 1, 2

transformations

preserving the equation in S2. Examples of mappings H( *) are given in [38]. If graded HL-bundles over M of rank n > 1 will be considered as global structures of supergravity, then the non-trivial ones will correspond to essentially non-simple supergravity including interactions of the fermionic fields. In this way we have obtained for such interactions a geometric approach using analytic compactifications (or complements) of the domain CT to a complex manifold M such that H’ (M, A” TM) # 0 for some u = 1, 2, 3, 4. However, we are far away from having an effective correspondence between lagrangians of supergravities and such compactilications (if such a correspondence exists). Recall that the extended supergravity in the A. S. Schwarz formulation has not been completed as yet. Recall that H’ (M, AT’ M) = 0 holds if e.g. M = P C, M, = p R and by virtue of G7 if M is a quadric in P5 C, M is a compactified Minkowski space. H’ (M, TM) # 0 is true when M = R, x R2 x R, x R,, where Ri are compact Riemann surfaces, genus R, # 0, or M = L, @L2@L,, where Li -+ R are line bundles, R is as previously, c,(L,) = 2, cl (L,) > 0, cf. G5 (the latter base space was considered by Hitchin in [Sl]). Thus by virtue of Theorem l3 there exist non-trivial graded bundles determined by certain space-time 4-manifolds and spinor bundles. These non-trivial graded bundles in the case n 3 1 correspond to parameter spaces of “non-trivial” extended supergravities because they admit no global decomposition into n components corresponding to a parameter space of R; = 1 supergravity. In the recent Manin’s approach to supergravity [86] the N = 1 lagrangian is defined by means of a transformation of a supermanifold. On the other hand such a transformation determines a graded bundle, cf. my paper [S7]. Hence graded bundles are present in N = 1 supergravity too. 3.2. The extension

of the class of instantons

by graded

bundles

We shall deal with instantons in the sense of SU(Y) gauge groups, r 2 2, and the Euclidean space-time compactified to S4. There are known several equivalent and a few possibly equivalent definitions of instantons, see [4], [29], [30]. Historically the first one, which is relevant to the geometric formalism of Yang-Mills fields says that the SU(r)-k-instanton is any self-dual hermitian connection V’ in the hermitian smooth vector bundle E’ of rank r over S4 such that the second Chern class is c2 (E’) = li > 0. The connection is understood up to equivalence. Note that the number k determines the bundle E’ uniquely up to equivalence.

ON GRADED

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SPACES

Belavin-Zakharov [12] and Ward [SO] observed that by choosing the only analytic connection form of the connection. V’ and then, after the analytic continuation on C4, by integrating it along anti-self-dual planes in C4 (the considered forms are integrable there) one obtains a cocycle of a holomorphic vector bundle E” of rank r over P3 C. This bundle can be understood as the Penrose transformation of (E’, V’). In the Atiyah-Ward paper [7] the “instanton” bundles E” have been distinguished in the class of holomorphic vector bundles over P3 C. This result and the Serre theorem [75], which allows us to use here the term “algebraic” instead of imply the following definition of instantons: “holomorphic”, DEFINITION. An algebraic vector bundle E” of rank r 3 2 over P3 C is called an SU (r)-k-instanton if (1) the Chern classes of E” are cl (E”) = c3 (E”) = 0 and c2 (E”) = k > 0, (2) the bundle E” is trivial over real lines in P3 C, i.e. over the Q-fibres, where r2,z3r z41 4 [z, +jz,,

P3C3CZ’,

z3 +jz,]

E P’ H

(5)

(H denotes quatermious and [ .] are equivalence classes of right multiplication by scalars.), (3) there exists and anti-linear analytic transformation J: E” -+ E”* (called a symplecric structure) lying over the involution D o(Cz1, z2, z3,241) = C-F*,

FI -24,

z31

that is J: Eg - E$‘,, where I‘*” means “dual”. Note that J is determined up to multiplicative constants on irreducible components of E”. In the theory of instantons the twisted instanton bundles E”(f) : = E”@H’ are important objects. The following dimension equalities: h1(P3C,

E”(I))=O,

h’(P3C,

E”(-1))

if 1 < -1,

(6)

=k

are essential in the Atiyah-Hitchin-Drinfeld-Manin (AHDM) construction instantons, which will be sketched at the end of this section. At this moment scrutinize the case 1 = 0, cf. [lo], [22].

of we

LEMMA.h” (P3 C, E”) = 2k - r. Proof: Consider sequence is exact:

the

case

of

SU(2)-instantons,

i.e. r = 2. The

following

0-E”(-l)-E”+Ef;2C+0. Since zeroth

cohomologies

in the above bundles

and the second cohomology

J. CZYi

242

HZ (P” c, E” ( - 1)) vanish, cohomologies

cf. [4],

therefore

the

standard

exact

sequence

of

gives us h’(P3CE”)

= h’(P3 C,

E”(-l))+h’(P3C,E~~2c).

The first term is equal to k, as we have seen before, and the second one is k - 2 as it follows from the Riemann-Roth-Hirzebruch theorem, see [49]. Thus we have h’ (P c, E”) = 2k - 2, The above equality remains true for Now consider the general case of characteristic x(P3 C, E”) is a topological r. The dimension of the first cohomology to the direct sum of bundles. In the case to signs as we shall prove below. Hence function of k and r, additive in r and a function is

li = 2, 3, . ..)

r=2

k = 1 as it is proved in [S]. SU(r) gauge group, r > 2. The Euler invariant of E” depending only on k and h’ (P3 C, E”) is additive in r with respect of instantons both numbers are equal up the dimension h’ ( .) is an integer-valued equal to 2k-2 if r = 2. The only such

h’ (P3 C, E”) = 2k-r. We have to prove that x(P3C, E”) = h0(~)-h1(~)+h2(.)-~3(.) = --h’(s). and then We can assume that E ” does not admit any trivial summand h”(P3 C, E”) = 0, cf. [4]. The triviality of E” over real lines implies a more general equality, namely h”(P3 C, E” (0) = 0

By virtue

of the Serre duality

if

I 6 0.

[SO] one has

h3 (P” C, E”) = h”(P3 C, E” (-4))

Furthermore,

the Serre duality

together

with (6) implies

h2 (P3 C, E”) = h’ (P” C, E” (-4))

which completes Remark.

= 0.

= 0,

the proof, q.e.d.

The equality proved states that h’ (P3 C, E”) is a topological invariant Nevertheless, connections between H’ (P3 C, E”) and the topology the general relations between remain obscure. In particular, h1 (P3 C, E”) and topological and geometrical characteristics of SU (r)-k-instantons have not been recognized and explained. E.g. the following coincidence takes place: h1(P3 C’, E”) = 0 iff k = 2r that is when the moduli space of E” is known to be contractible. The condition c1 (El’) = 0 makes nr E” trivial. As a consequnece, E” 2 nr- ’ E”. Hence and by virtue of Theorem B the following relations can be immediately proved : of instantons. of instantons

ON GRADED BUNDLES AND THEIR MODULI SPACES H’(P3

C, HL(n, ML’)) z R4n2(k-1)

H’ (P” C, HL(n, /1E”)) 2 R2n2(2k-r)

243

if r = 2, if Y > 2.

Any graded HL-bundle Z(E”), where E” is an instanton holomorphic vector bundle, will be called a graded instanton bundle. We will try to say what interpretation of such graded bundles is adequate. Recall that k-instantons are models of systems which look like configurations of k interacting pseudoparticles in the Euclidean space-time. Furthermore, a graded bundle of rank n can be understood as a system of n interferring copies of the auxiliary bundle. Therefore, a graded instanton bundle Y(E”) corresponds to the following abstract situation: There is a system of k objects. Each object consists of n pseudoparticles which are ordered linearly so that we will index them by 1,. _., II. The “horizontal” interactions in each level, i.e. in the group of k pseudoparticles having identical indices, are the same and such that their configuration and interactions correspond to the instanton given by the holomorphic vector bundle E”. (Recall that the correspondence between k-instantons and systems of k pseudoparticles in the case k > 2 is not quite clear, cf. [29].) The pseudoparticles and their interactions are sketched below

k objects

Furthermore, if 2k > I, then our pseudoparticles can also interact in “oblique” and “vertical” directions, as we see in the above diagram, and these interactions change the analytic geometry attached to t’he whole system making of the trivial graded

bundle

& AE” a non-trivial

one.

In quantum field theory in dilute gas approximations, see [17], [32] one integrates on the space of instantons and anti-instantons (in the considered models r = 2 or I = 3). Let us imagine that the pair instanton-anti-instanton creates a set of states. Then one of more simple computing procedures would require to join the set of spaces H’(P3 C,HL(Z, AE”)) of graded instanton bundles to the integration domain. Since of 1-instantons such spaces are one-point sets, this method increase the contributions of higher-order instantons. Recall that in certain recent models, e.g. in [32], the contributions of higher-order instantons also seem to be considerable.

244

J. CZYT

We wonder whether graded instanton bundles in models having S” as the base space are possible. I cannot realize this purpose directly. Nevertheless, the S4version of the AHDM construction allows us to see a role for H’ (P3 C, E”) in the geometry of S4. On the other hand, H1(P3 C, E”) generates the whole H’(P” C, HL(n, A,!?‘)) if r = 2 and its subspace if r > 2. In the considered version of the AHDM construction, see for details [4], [29], we take the trivial vector bundle T:= S4x c?“ and embed in it the bundle & Q-‘,

where

Q-’

denotes

the

complex

vector

2-bundle

over

S4 given

by

cl (Q- ‘) = - 1. Note that Q-’ can be also obtained from the projective quaternionic line bundle over P'H 2 S4 by passing from quaternionic to complex coordinates as it is done in (5). Moreover, a hermitian form (*, . ) is given in C?. A degeneration of ( *, . ) is admitted but (. , . ) restricted to all the subspaces & Q;‘, x E S4, is assumed non-degenerate. In this formulation instantons correspond hermitian form (.: ). Observe that the bundle (.;)

as follows:

E’ :=

to equivalence classes of the E’ can be defined by means of

where D is the degeneracy T/(&Q-l)/,,

and then the self-dual connection trivial connection in T.

V’ can be obtained

space of ( *, * )

by projection

on E’ at the

It is easy to check that dim,D = 2k-r, so that H’(P3 C, E”) 4 D. Thus the considered cohomology classes can be identified with points of the degeneracy space. However, I cannot deduce the whole hermitian structure from cohomologies of E". Acknowledgement I would like to thank my teachers and my colleagues who helped me to write this paper. In particular, I am deep indebted to K. Gawedzki, whose paper [37] had inspired me to study supergeometry, to G. Cieciura, who helped me to prepare the paper and to K. Maurin, who taught me mathematics for more than ten years. Furthermore, I thank Cz. Olech, S. Rolewicz and T. Traczyk for the help during the work and all people, whose advices and. suggestions were used in the paper, especially W. Barth, Mrs. A. Bojanowska, K. Dabrowski, N. Hitchin, S. Jackowski, and Yu. Manin. Let me not forget R. Picken and J. Diestel who helped me with the language proof. REFERENCES [l] Akulov, V. P. and Volkov, D. V.: Phys. Left. 46B (1973), 109-110. [2] Andreotti, A. and Norquet, F.: Ann. &da Norm. Sup. Piss 21 (1967), 31-82. [3] Arnowitt. R.. Nath P. and Zumino, B.: Phys. hr. 56B (1975), 81-84.

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[4] Atiyah, M. F.: Geometry of Yang-Mills Fields, Lezioni Fermiane, Piza, 1979. [S] -: K-Thmry, Harvard Univeristy Press, 1965. [6] Atiyah, M. F. Bott, R. and Shapiro, A.: Topology, 3-supplement (1964-65) 3-38, also in R. Bott, L..ectures on K(X), Benjamin, New York, 1969, pp. 14%178. [7] Atiyah, M. F. and Ward, R. S.: Commun. Mu/h. Phys. 55 (l977), 117-124. [8] Aupetit, H.: AsfPrisque 71-72 (1980), 35.-77. [9] Barth, W.: Math. Ann. 226 (1977) 125150. [lo] Barth, W. and Hulek, K.: Manuscripta Math. 25 (1978), 323-347. [ll] Batherol, M.: Trans. Amer. Math. Sot. 258 (1980), 257-270. [12] Belavin, A. and Zakharov, V.: JEPT Letters 25 (1977), 603-607 (in Russian). Nauka, Moscow, 1965 (in Russian). [l3] Berezin, F. A.: “Merod vtoriceskogo kvuntovanija”, peremennymi, MGU. 1983 (in Russian). [14] - : Kedenije v ulgehru i analiz s ontikommuripuyuscimi [l5] Bohr, N.: Atomic Physics and Humunn Know/edge, Wiley, New York, 1958. [16] Bonora, L., Pasta, P. and Tonin, M.: J. Math. Phys. 23 (1982), 839-845. [17] Colleman, S.: in: The whys of subnucleur physics (Ed. A. Zichichi), Plenum Press, New York, 1979, p. 805. ,’ ! q [18] Corvin, L., e’eman, Y. and Ste nberg, S.: Rerl. Mod. Phys. 47 (1975), 573-604. [19] Cremer, E.: r N = 8 Supe$ravity,~ f m: Unification of the Fun 1 hmenral Inreracrions, I Europhysics Co& jMarch, 1980. [20] Chern, S. S.: Complex Manifolds wirhour Potentiul Theory, D. van Nostrand, Princeton, 1967. [21] Czyi, J.: Lecture Nores in Marh. 838 (1981) 110-124. [22] -: Graded Bundles and 7heir Apphcutions in Supergravity nnd Instanrons, preprint LuminyMarseille CPT 8l/PE 1328. [23] - : to appeari? Banach Center ,Publications. L .? [24] -: preprint IM PAN 277, June I9831 Methods in Marh. Phys.! Stat&I/ 1982. i [25] Dell, J. and Smolin, L.: Commun. Murh. Phys. 66 (1979), 197-2 tr 1. [26] Deser, S.: L.ecrure Notes in Math. 676 (1978), 573-597. [27] - : Lecrure Nores in Phys. 94 (1979), 468477. [28] Deser, S. and Zumino, B.: Phys. Lprr. 658 (1976). 369-373. [29] Drinfeld, V. G. and Manin, Yu. I: Commun. Math. Phys. 63 (19781, 1777192. [30] - : Sot. J. Nucl. Phys. 29 (l979), 16461654 (in Russian). [31] Dupont, J.: Lecrure Notes in Moth. 640 (1978). [32] Fateev, V. A., Frolov, J. V. and Schwarz, A. S.: Nucl. Phys. 154B (1979) l-20. [33] Fayet, P. and Ferrara, S.: P/~Jx. Rcpr. 32C (1977). 249. [34] Feynman, R., Leighton, R. B. and Sands, M. L: The Feynmon Lectures on Physics. Vol. 3, Addison, Reading, 1965. [35] Freedman, D. Z., van Nieuvenhuizen, P. and Ferrara, S.: Phys. Rev. D13 (1976). 3214. [36] Frenkel, J.: Bull. Sot. Moth. France 85 (1957), 135-221. [37] Gawqdzki, K: Ann. Inst. Henri Poincure 27 (1977) 3555366. [38] Gayduk. A. V., Romanov, V. N. and Schwarz, A. S.: Commun. Mttrh. Phys. 79 (1981). 507-528. [39] Gindikin, S. G. and Henkin, G. M.: in: ltogi nnuki i rechniki, Vol. 17, Viniti, Moscow, 1981. pp. 57-109 (in Russian). [40] Godeman, R.: Topologie algibraique et thdorie des fuisceaux, Hermann, 1958. [41] Grauert, H.: Math. Ann. 133 (1957), 450-472. [42]

-

:

Math.

Ann.

135 (1958)

263-278.

Grauert, H. and Fritzsche, K.: Serertrl [44] Grauert, H. and Remmert, R.: Theorie [45] Gunning, R. and Rossi, H.: Anulyric Englewood, 1965. [43]

Compkx der

Funcrions

Springer, Berlin. 1976. Springer, Berlin, 1976. Several Complex Vtrritrb/e.s. Prentice-Hall,

Vtrritrbles.

Steinscher of

Riiume,

246 [46] [47] [48] [49] [SO] [Sl] 1523 [53] [54] [SS] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] 1733 [74] [75] [76] [77] [78] [79] [X0] [al] [82] [83] 1847 [SS] [86] [87]

J. CZVP Haag, R., Lopuszanski, J. T. and Sohnius, M.: Nucl. Phys. 8SB (1975), 257. Hartshorne, R.: Algebraic Geomerry, Springer, Berlin, 1977. -: Commun. Math. Phys. 59 (1977), t-15. - : Math. Anti. 238 (1978), 229-280. Hirzebruch, F.: Topological Methods in Algebruic Geometry, Springer, Berlin. 1968. Hitchin, N.: Murh. Proc. Cambridge Phi/. Sot. 85 (1979), 465476. Kac, V. G.: Adr. Marh. 30 (1978), 8-96. Kaplansky, I.: Gruded Lie Algebras I, II, preprints University of Chicago, 1975. Kodaira, I(.: Amer. J. Math. 88 (1966), 682-721. Kodaira. K. and Morrow, J.: Complex Mantftifds, Hott, New York, 1971. Kostant, B.: Lecture Notes in Math. 570 (1977), 177-306. Kotecky, R.: Rept. Math. Phys. 7 (1975), 457469. Leytes, D. A.: UMN 35, No 1 (1980) (in Russian). Manin, Yu. I.:/Dualir~ in: /Enciclopediu, K Diuin - Fume, Einaudi, Torino, 1978 (in Italian). / -. -Maurin, K: Analysis, i art II, PWN, Warszawa 1980. Milnor, J. W.: and Moore, J. L.: Ann. Muth. 81 (1965). 211-264. Mumford, D.: Abelian Vurieres, Oxford University Press, 1970. van Nieuvenhuizen, P.: Phys. Repr. 68, No 4 (1981), 189-398. Ogievetsky, V. and Sokatchev, E.: Phys. l,eft. 79B (1978), 222-224. Palais, R.: Seminur on the Atiyuh-Singer Index Theorem, Ch. IV, Princeton Universtiy Press, 1965. Penrose, R.: in: Quantun~ Graoiry, un Oxford Symposium, Clarendon Press, Oxford, 1975, p. 268. Rawnsley J.: Dijjerential Geometry of‘ lnstcmtons, preprint DIAS 25, Dublin, 1978. Rogers, A.: J. Marh. Phys. 21 (1980), 1352--1365. Rosenlicht, M.: Amer. J. Math. 78 (1956), 401443. Ross, T.: in: Supergrauity, Proc. of Stonybrook Conf. (Ed. P. van Nieuvenhuizen), North-Holland, Amsterdam, 1979, p. 155. Rabat, B. V.: Vvedenije 1’ kompleksnyi ctncrliz, Nauka, Moscow, 1969 (in Russian). Safarevitch, 1. R., (Ed.): .4lgebraiceskije poverchnosti, Trudy Inst. Steklov, 75 (1965); German translation: AIgebraische F&hen, Leipzig, 1968. Salam. A. and Strathdee, J.: Nucl. Phys. 76B (1974), 477482. Schwarz, A. S.: Commun. Marh. Phys. 87 (1982). 37-64. -: NW/. Phy.t. 171B (1980), 154-166. Serre, J.-P.: Ann. Inst. Fourier 6 (1956), 142. Spanier, E.: .4Igebruic Topology, MC Graw-Hill, New York, 1966. Steenrod, N.: Thr Topology, ef Fibre Bund/e.s. Princeton University Press, 1961. Townsend, P. K.: Topology und Supersymmetry Breaking in Ten-dimensional Super-Yav-Mills Theor!,, preprint LPTENS X3/2. Ward, R. S.: Phys. Lert. 61A (1977), 81-82. Witten, E.: Phys. Left. 77B (1978), 393-400. Wess, J. and Zumino, B.: Nut/. Phgs. 70B (1974), 39-50. de Wit. B.: in: Superyrtrcity, Proc. of Stonybrook Conf. (Ed. P. van Nieuvenhuizen) NorthHolland, Amsterdam. 1979. p. 113. Manin. Yu. l.:‘F/trg Superspuces trnd Supersymmetric Yang-Mills Equcrtions, in: Arithmetics and Geome/r)rC vol. II, Birklluser, 1983. - : Golomorfnu’u supergeomefri’tr i poPa Yungu-M&a, preprint 1983 (in Russian). -: Geometri’u supergruvitacii i superkrotki Subecta, preprint 1983 (in Russian). Czyi, J.: to appear in LL’tt. Math. PhJ,s.