<<

IC/81+/237

INTERNATIONAL CENTRE FOR

SUPERTWISTORS AUD SUPERSPACE

M. Kotrla

and

J. Niederle

INTERNATIONAL ATOMIC ENERGY AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1984 Ml RAM ARE-TRIESTE Mmm: •<•-•» 4 TC/8V23T I, INTRODUCTION

It is well known that tvo fundamental physical - the general of relativity and the Yang-Mills theory - can he clearly formulated International Atomic Energy Agency geometrically in the framework of the Riemannian spaces and fibre bundles. On and the other hand, the geometrical interpretation of the supersymmetric and of the United nations Educational Scientific and Cultural Organization supergravitational theories - the promising candidates for unified theories of all fundamental interactions of particles - is far from being clear and complete. INTERNATIONAL CENTRE FOE THEORETICAL PHYSICS Thus in order to clarify the geometrical structure of these theories a super- symmetric extension of the twistor approach will be developed. In what follows we shall restrict ourselves to the flat case since the space of twistors is particularly simply defined for 4-dimensional complex conformally flat space-time.

SUPERTWISTORS AHD SUFERSPACE * The twistor theory (for reviews see [l] and papers in [2]) gives an alternative description of space-time and of various objects defined on it be means of and holomorphic geometry. The description is particularly simple whenever one is dealing with a theory having the conformal symmetry.

M. Kotrla The aim of the so-called twistor programme [3] is to give a new description Institute of Physics, Czechoslovak Academy of Sciences, of rel&tivlstie as well as quantum physics in terms of twi3tors. So far, two Prague, Czechoslovakia big successes of the twistor theory have been reached: a nice description of

and free mas3less fields with an arbitrary [l],[2], [U] and a general method for solving Yang-Mills [2],[5],[6] as well as the non-linear Einstein [2],[7] J. Niederle ** equations in the self-dual case. International Centre for Theoretical Physics, Trieste, Italy In 19T8 Ferber [8] introduced the supersynmietrie generalization of twistors and International Bchool for Advanced Studies, Trieste, Italy. the supertwistors. They are defined as elements of a complex projective super- space on which the conformal SU(2,2|N) acts linearly. By using supersupertwistors, non-linear realizations of the conformal supersymmetry on superspace and representations of the conformal supersymmetry on the space of ABSTRACT chiral superfields have been constructed. Then it was observed "by Witten [9] that the N = 3 supersymmetric Yang-Mills equation can be interpreted in terms In the paper the usual correspondence between twistors and of supertwistors in the same way as the self-dual Yang-Mills theories were by geometrical objects in the is generalized to the twistors. An extension of the approach was recently discussed to Include also supertwistor case by means of flag supermanifolds. This super- the supersymmetric Yang-Mills theories for II = 1,2 and h [10]. twistor correspondence Is treated in detail. The chiral and non- chiral superspaces are constructed and their properties studied by In this article we shall define supertwistors a little bit more abstractly means of supertwistors. than in [8] and show first the correspondence betveen the supertwistor space and the complex superspace containing our space-time in detail and then that the results analogous to those of the twistor theory hold. MIRAHARE - TRIESTE November 19SU The correspondence is essential for formulations of supersymmetric theories in terms of supertwistors. However, before discussing it we shall recall some • To be submitted for publication. basic facts about the usual twistors and the correspondence between twistors and ** Permaner.t address: Institute of Physics, Czechoslovak Academy geometrical objects in the MinkowskI space. of Sciences,18040 Prague, Czechoslovakia. II. TWISTOKS AMD TWISIOR CORRESPONDENCE tine and T, respectively CM and P{T), is given by the basic relation TwistoiE can be introduced "by several essentially equivalent ways [lj. Thus for example in physics the twistors are usually defined as quantities U) = 1 X Tat li.Jj P describing the states of free massless particles whereas in mathematics as It which expresses the condition for Z £ P(T) given ny Z = C Y 2 1° * Y * C,Z = elements of the complex T "x. C in which the covering = ((A TI ,)£T } to be incident with x 6 CM. Namely each point Z of P(T) group SU(2,2) of the identity-connected component of the eonformal group of the (i.e. each projective twistor) can be interpreted as the set of points in CM Minkowski space acts according to the fundamental representation. incident with it, i.e. with the set of complex xa0' satisfying (2.3) with ^ a The coordinates of a twistor Z S T will be denoted by Z , and TtfiI fixed. Inversely, the set of solutions of (2.3) in Z with x Z=(Z)=(Z,Z,Z,Z). By choosing a proper basis twistor Z can be held fixed may be interpreted as the set of projective twistors Z in P(T) J a expressed as Z (u) , ID , TT t, TT .), where io , TT , are and cospinors,- incident with point x of CM. In more detail a null twistor Z (provided respectively of the Lorentz group contained in SU(2,2) . •- Moreover there exists •n , i 0) can be interpreted as a null-line in CM. A non-null twistor of P(T) a hermitean form (...) invariant with respect to SU(2,2), For twistor as a totally null plane in CM, i.e. as a complex 2-plane in CM all^tangent a (Z ) = ( and r= (Y11) , it is of the form vectors of which are null, mutually orthogonal and of the form p° na , where TI is fixed and p varies. This plane in CM is called an a-plane. (Similarly a a a' (Z,Y) = 5° Xa, T7 v a B-plane in CM is defined as that whose tangent vectors have the form p IT , where now it is the unprimed p that it fixed and TT that varies. The 6-plane in CM corresponds to a projective plane in P(T)J The points in CM where Z = (IT SO ) is the complex conjugate twistor to Z . corresponds to a projective line in P(T) etc.

A twistor Z (or Za) is said to be positive, negative or null The correspondence between the projective twistor space P{T) and the according as (Z,Z) = Z Za is positive, negative or zero. compactified complex Minkowski space CM mapping points of P(T) onto o-planes in CM and points in CM onto projective lines in P(T) may be described more Following Penrose [1], we now set the correspondence between twistors abstractly by using flag and manifolds (for definitions see e.g. 111]). and objects in the Minkowski space. In order to interpret this correspondence geometrically we need to pass to the projective twistor space P(T) and to the Let F be the following flag manifold compactified complex Minkowski space CM. Space P(T) is defined as space CP whose points are the equivalence classes of proportional twistors, i.e. F = V(T) := (£.2) (2.U) Space CM is a compact complex It-space the poins of which are composed not only of the points of the complex Minkowski space but in addition of the points of and G the Grassraannian manifold a complex 3-surface - a light-cone - defining the "points at infinity" of complex Minkowski space. Thus denoting "by xa and x the vectorial and spinori&l 1 (2.5) components of a point x in CM, the correspondence between the Minkowski space- (€) := { L2 \ L2 are subspaees in t" ,

which is isormorphic to the compactified complex Minkowski space CM. Let u *) However, the twistor is not a pair of spinors, since under translation with the parameters a we have HI™ -»• in +ia IT , n , -•• IT ,. and v be natural projections defined by

-3- -h- . F (2.6) en (X X) = X (3.1) H>* f2l (It (2)°fll (2)

Then the correspondence between CM and F(T) is given by which is invariant under transformations from SU(2,2|H). The infinitesimal transformations from EU(2,2|H) are of the form tf (2.7)

(for details see [12]).

The twistor correspondence is essential for finding solutions of massless (3.2) field'equations and (anti)self-dual Yang-Mills field equations by means of \ / tvistors [it], [5].

where a, b, 1, D, G, S £ (A)Q, E, p 6 (A)1> conjugation is defined via the involution on A, a and b are hermitian, S antihermitian and D and Q real.

III. SUFERTVISTORS AHD SUPEE5PACES In the twistor theory space CM is isomorphic to Grassmannian manifold In the paper [8] supertwistors were introduced analogously to twistors, i.e. 0 j ), i.e. to the set of 2-dimensional subspaces in T ^ t . A similar as quantities describing free massless particles, but in a superspace. In this situation happens for supertwistors. However, since in this case some extra odd section we shall use an abstract definition of supertvistors which provides to dimensions appear, we have more possibilities. introduce various superspaces containing the Minkowski space. Thus let us consider complex 2/k-dimensional subspaces (0^ k^ M) of the super- ,H Let us denote by a space of supertwistors for N-extended super- twistor space D % d . A 2/k-subspace of S is a 2/k-dimensional plane symmetry. I is a vector space isomorphic to I (for details see the going through the origin and thus determined by 2 even and (N-k) odd homogenous Appendix) on which tue fundamental representation of confonnal supergroup equations. The set of all S/k-subspaces in 1 forms the Grassmannian manifold SU(2,2|(T): acts. The coordinates of an element X of I W i.e. of the (A) (see the Appendix). It is covered by several maps,. In one of them, supertwistor X will be denoted by = (Xa, X1) a = 0,1,2,3, i = 1,...,N say , the 2/k-subspaces are determined by the equations and A = 0,...,3+H, where XL (respectively" X ) belongs to the even (respectively odd) part of a commutative B.anach superalgebra A over it, i.e. A e (A)ot (A) (for detail s^e the Appendix). In a suitable basis of ZN

a the coordinates of the supertwistor X are A = (oi , ), where to, ir and (3.3) ^-transform under •'6h{2,1} C SU(2,2|N) as spinors, cospinors and scalars, respectively and in ftddtlion ^}y 's form the fundamental representation of the internal symmetry group KTJ(M) C LiU(2,2 ju). Using complex conjugate where v,X 6. (A)n, p,n £ (A) are constants characterizing the given subspace. dual'supertwistors 'X.) - (^ u°, - $• • j to A 's we can introduce the These objects were considered in flit ] for H = 1 and 2, Tfie 2/k-subspace hermit ian form on £ x T'\ is determined "by columns of the supermatrix of the type lt/N x 2/k which can be N expressed in map t ? , in the form K

•) Algebra A is with the involution *: = (iiJ

-5- paper, we shall not use the superspaces with^ k + 0,H, but they might be important haE the dimension + Supermanifold §2i\ li/B^ ^ k(N-k)]/2H. Insisting for formulations of supersymmetric theories that the even dimension should be It we have two possibilities: either to put k = 0 or k = H which corresponds to the supermanifold G : = G , , .,T(A) and In order to get a non-chiral superspaces we have to consider more complicates

^T. - ^o/w )•/HWW » respec"&-i.vei.y, On superspaces Gn and GT we introduce the manifolds than Grassmannian, namely the flag supermanifolds with the length of the a map C f and C fP~ , respectively so that the corresponding 2/k-subspaces are flag equal to 2 (see the Appendix}. Then there are more possibilities to obtain generated by columns of the supermatrices a l»/!)N-dimensional superspace [16]. The usual transformation properties are obtained if we choose ,-V respectively . (3.5) M: = F(2/0, 2/H, c"*/IJ) Then taking the map C & on the set of 2/0-subspaces and the map C iP on the We shall denote the element from C J"_ , i.e. a 2/0-subspace vith parameters set of 2/H-subspaces we can choose on M the map - (v,n), by R(w,n) and analogously the element from C j/>L "by L(u,p). They are specified by the relations R(*r,f) : In order that R(w,n) cL(u,p) the following condition must he satisfied V (3.6) (3.9)

The element Z = (R,L) S C/ can be described, e.g. by using the coordinates and (w, D, ii). However, it is more convenient to choose another parametrization (3.7) namely (z, p, n), in which

Since under SU(2,2|jl) transformations on T. , any 2/k-subspace is mapped into (3.10) a A - For in:rin:ites:i ma another 2/k-subspace, SU(2,2[H) acts on 2/i 1»/N^ ' - l trans format ions (3.2) the action can "be written by using the coordinates (v, X, p, n) (see (3.1*))- Thus we can find that v"8', A1 pa and r\la< Then, in the same was as previously, we obtain the usual transformations under transform under the Poincare' group as a vector, scalar, spinor and cospinor, SU(2,2|lT). For instance, for super symmetry transformations we get respectively and that X with respect to the internal symmetry group SU(H) transforms non-linearly. In the special case k = 0 and k = N the situation is simpler since there are no scalar coordinates X. The realizations of the conformal supersymmetry are then obtained on chiral superspaces which for N = 1 (3.11) were introduced in [15]- In particular, for the supersymmetry transformation with parameter £ we obtain

C9 '{I and (3.8)

*) Let us note in this context that in the particular case H = 2 we get a superspace of dimension 5A and the scalar coordinate X tranforais under SU(2)r. SU(2,2|ll) as a projective coordinate on P(C)2. We can also construct Thus the complex superspaces C and C ¥ of dimension 1*/2N derived from R j more complicated flag supermanifolds (as we see later on) and thus probably find supertwistors are generalizations of chiral superspaces for H > 1. In this :. coiii.ec+ion to the recent approach to N = 2 cupel-symmetric theories [l6]. -7- The point Z with coordinates (s, p, is determined by the system Here the mappings p, v>, TT. IT- are projections defined lay of equation:;

(3.12a) - «,'o •

- :. i • :• v (3.12b)

EQ.-(3.12a) defines a 2/N-subspace and together with (3.12b) specifies the '•_ ' "2/4 • '2/.'/' '"/ 2/0-subspace contained in it.

L )L £ Fl hen where L^ denotes a k/i-dimensional subspace, (\iQ, 2/O £/N' * the supertwistor correspondences are given by the pairs of the mappings.

IV. SUPERTWISTOR CORRESPONDENCE

Let us now describe the supertwistor correspondence, i.e. the correspondence (It.2) between the projection supertvistor space and the superspace. The first part of it was already discussed in the previous section. Since we have more super- spaces at our disposal, we can speak about the correspondence between P(Z ) The first part of the correspondence is determined by mapping 1(1. For instance, and M or between P(E )' and Q_ or between P(E ) and GT. to any point from GR (i.e. to any 2/0-subspace I^/Q) there corresponds the set L The supertwistor correspondence is given by means of natural projections of of all 1/0-subspaces LJ^/Q (projection twistors) for which L. /nC ?/n, i.e. flag supermanifoldspermE . Besides the introduced spaces GT, G_ and M we define the 1/0-dimensional projective straight line in P. the spaces: The images of the points from the other superspaces can be found analogouslyj N F = , 2/v ,Z ) By introducing the homogeneous coordinates {ID, TT,I? ) on P and the coordinates

(w,n), (u,p) and (a.p.ff) on GR, G^ and M, respectively as in the previous (for details see the Appendix). section, the condition LWQC 1>2IQ Can be expressed in the form Then the following diagram holds

and the condition 1^ ,Q <= L£ ,Q as

(h.k)

which are just £qs.(3-6) and (3-Y). In these coordinates the projections and IT are given "by

, f,

-9- -10- 1 Here the homogeneous coordinates on P a , anti The second part of tht- supertwistor correspondence is described by are (y , p .) and the 2/0-subspace B(u, p) is given by the system mapping to and specifies which object in the considered superspace corresponds to a protective supertvistor. It turns out that these objects are generalizations of tx-planes in CM. In superspaces JL, 0 and G we call them a-, N satisfying the system of Eqs.(3.12). On c 5 we have the coordinates (w , n ° ) and 2/N-subspace L(w, Thus the a-superplane can be parametrized in the following way is given "by the equation

(U = -1

(4.6) Analogously on C

Here ' (z, p, n) Is a fixed point satisfying (3.12) and \a e U) , ?"., , 9 o o nu I e € (A) are parameters. By using the projections TT and IT we obtain the a.,- and the a -superplanes from the a-superplane. In the coordinates these From the point of view of the transformations from SU(2,2/N) we have superplanes are defined as the set of solutions of Eqs.(3-5) and (3.6) for a given G_, M. The projections TT_ and if. are defined "by supertvistor {at, n,$ ). They can be parametrized as follows: h H

= V" where v, w are the same as in (4.5). The supertvistor correspondence is f l K" described in coordinates by Eqs.(U,9), C+.10) and (4,11). The sets corresponding

to the projective dual supertwistor In superspaces H, GT and §„ are generalizations

of fS-planes and are accordingly called B-, BL- and B -superplanea. In the lt ' coordinates they are defined as the sets of solutions of Eqs, (li.ll), (4.10) and + a r< = f^- (It.6) (4.9) for a given dual Supertwlstor (aa, u , - Z,). They are 2/3H-, 2/2M- and 2/H-dimensional sets whose points can be parametrized by At (A)Q> e, 5 e (A)-, A similar correspondence exists vhen ve are dealing with the space of dual super- in the following way twistors, E . We have the diagram

a-superplane F = F(l/O, 2/0, 2/U;

P = F(l/0, EN)

M = F(2/0, 2/S, l") (4.12)

H 3R = F(2/0, E )

-12-

" « * v* m then they lie on one and the same a-plane and the corresponding protective lines

0T-superplane Li ty(x) and ty(x + A x) intersect. A similar statement holds in the supertvistor theory. - f (it.13) First, on superspance C'f a metric can be introduced which is invariant with respect to the Poincare supergroup, namely

fij,-superplane = u (5.1) *. = p'i + £; (I*.lit) We say that two points Z = (R, L) and z" * (R*, L) from Of are null separates whenever R intersects T and R intersects L. Since R(w, n) and L(u,.p) a a 1 intersects if and only if v - u + o . fi is a null (non-zero} vector we can easily check that whenever 5? differs itifinitesimallj- from Z by aB tt 1 2 (S z ', (5 o t, S n " ) then according to our definition ds = 0. Form the definition it also follows that any two points both belonging either to an Let us also clarify the relations between a- and 6-superplanes. Two a-super- a-superplane or to a 6-superplane have the zero distance. planes intersect themselves in a 0/2N-dimensional set. The same is true for two B-superplanes. The a- and 6-superplanes intersect if and only if the following Analogously as in the twistor theory we shall use form (3-1) to define first relation holds for the corresponding supertwistor X and Y: Y X4 = 0. If conjugation and then real superspaees. Thus a £/N-subspace R is said to be a- and B-superplanes intersect then they intersect in the complex l/2N-dimensional conjugate to the 2/0-subspace R if R is orthogonal to R, i.e. if set which is a generalization of the complex light ray in CM. If the coordinates of the twistors JT, ?A are Xs = (a, ir, * ), Y^ = (

In other words, R = R(w , r|' then R+ =L(vaB', Thus conjugation 7 V3' gives one-to-one mapping between G.p and G as well as a mapping of M onto M : Z = (R,L) •+ Z+ = (L+, R+). Points of M for which Z+ = Z are called real. The set of all real points from M form a real superspace invariant under 10 •f '«'= f ".'«' SU(2,2/N). In " the real points have the coordinates (x 6" I '), B6' _ -aB' la ia = (e )* and form a real superspace denoted by f By using projection IT and rr we can find real chiral superspaees. They are where (z, p, ii) is a fixed point and a

(5.3)

V. METRIC AMD REAL STRUCTURE The real Minkowski space M C CM in the twistor theory is associated (via the twistor correspondence) with the projection space P(M) c p(T) of null twistors. The fact that two points fro™ C1-" have distance equal to zero can be Uamely, a real point x n M corresponds to a line in P(T) which lies whole in described In the twistor theory in tho following manner. If two points x and P(H) (and not meeting a special line corresponding to the vertex of a complex x + A x of CM are null separated in the sense of the metric ds"~ = A x. A x liftht cone at infinity of CM) and a null-twistor from P(lO corresponds to a

-13- -\k- null straight-line in M, I.e. to a geodesic. It is defined as an intersection the case Tor the Minkowski apace) but from invariance with respect to an anti- It Li £ of the corresponding a-plane in CM with the real space M. The o-plane linear mapping: a : C •+ C , a = —1. By introducing a quarternionic structure corresponding to twistor 1 Intersects M if and only If Z £ II, i.e. it is null. on C ^ H, this mapping represents the multiplication by quaternionic unit The saae situation happens for supertwistors. If we denote by fl c j j [5],[6], It is clear that space S is not invariant under SU(2,2). In

6 the set of all null supertwistors (i.e. those for which XA X = 0) and by R a this case we have to take the covering group of complexified SU(2,2) group, 2/0-suhspace then the necessary and sufficient condition that RC Tl Is that i.e. SLC^.C ) which covers SO(6,C) and S is then invariant with respect R C. R . Thus to a real point in M there corresponds a projective line lying to another its real form namely to SL(2,H) which covers S0(5,l). in E(Yt}. The a-superplane corresponding to a supertwistor contains real points In the supertwistor space E there exists a mapping analogous to the if and only if the corresponding supertwistor is null. The set of' real points mapping a only,in the case, when H is even. Thus assuming N = 2k the 2k ,2/k lying in the a-superplane represents a generalization of a null line. In the quaternionic structure E ">• a can then be Introduced as follows (for coordinates (x, e, §) it can be parametrized In the following way definition H*"",,2/fc1 see the Appendix). The first quaternionic unit In H is Identified with the imaginary unit In C so that any q t H can be written as 1 •= 1 + J 1 , I-,,"!,e c (J being the 3econd quaternionic unit). The isomorphism (5.!*) ^2k . ,.2/k H is then given by

k. (6.1) where (x, 6, 8) are the coordinates of a fixed real point and k e e, E e (A), are parameters satisfying The mapping c is equal to multiplication by quaternionic unit j from the right. k* = k, 1 U )* In what follows let indices p, q, w, s run from 1 to k (H = 2k) and let

Relations {5.M represent the real form of (It. 15). Thus the real light- each of their values correspond to two successive values of the indices of the ray in a real superspace has the real dimension 1/2N and since it is the part of internal symmetry so that for instant s*-*(£s - 1, 2s). Instead of quantities

twistor (ui, it, * ) goer\ *-"•*-••- s ^- ^-t. o^ ^^fsupertwistor (« = s u', TF intersect if and only if the corresponding supertwistors X. , X satisfy [0 -1] where e is the matrj.ix Ii. „! () (> A (if (2) In order to find out the Euclidean superspace we shall be interested in 2/0- and 2/N-subspaces, i.e. In elements from G_ and G invariant under a. In the coordinates the equations determining 2/0-subspace are now suitable to write in the form: VI. SUPERTWISTORE AND THE EUCLIDEAN SUPERSPACES

In the twistor theory we work with the complex space CM i G • (i) and (6.2) then by using the correspondence with null twistors we separate from it the •real Minkowskl space M invariant with respect to the conformal group. However, In order that the 2/0-suhspace will be invariant with respect to the conjugation the following relation must be satisfied: the compactified complex space CM alao contains the real Euclidean space E or 1) more precisely its compactiflc^tlon E . For this space the twistor lj correspondence can also be established. However, reality of space S does not follows from the requirement that some form on twistors vanishes (as was

-16- -15- : 4 *) (6-3) •' . * '2 P. ' The nejrt step is to find out transformations under which the Euclidean

auperspaces ET, E R end E T are invariant. We shall obtain them by means of supertwistors. Let us consider the complexiflcation of the supergroup These conditions determine a subset of C/ of the real dimension it/£N, Let SU(2,2|2k) namely, the supergroup SL(l+,c|2k) and let us take its real form us denote it by E R. which preserves a-reality, i.e. which maps the given Euclidean superspaee onto The equations for a 2/N-euispace are written in the form itself. The infinitesimal transformation from SL{U,c[2k) can be written in the form Us - U.H - i ps (6.1.)

The 2/N-subspace will be a-invariant whenever (6.8) (6.5)

The set STL of all cr-invariant 2/N-subspaces in Cj° . has again the real Here, 1tJl = Trk=TrS = 0. Due to the supertvistor correspondence these Jt N M dimension !(/2N, transformations are related vith ttose on C jr or on C ff _ or on C f L x\ Finally consider the Euclidean subspace in M, A point (H,L)'

(,.* ft (6.6) Thus transformations (6.8) restricted by (6.9) give the desired invariant in which transformations of superspacej ETR , IT and a via the supertwistor correspondence • In particular for E _ they are of the form (6.T) due to a-invariance.

(6.10)

•) Let us note that Eg.. (6.2) can then be rewritten in terms of quaternions into the form: q « v p, r^ = n p {q, p and r sure quaternions corresponding to u, IT and -J" as in (6.1) and w and rj3 are quaternions obtained by means of the correspondence (a + i b)*-»la " „]}. Thus we get the quaternionic superspace

n with the coordinates w = q p , n = r p which can be considered as S • are 2x2 matrices forming the elements of a (kik) matrix. the projective coordinates on P(H ). If the supergroup SL(2/k,H) acts on K , SL(2/lc,H)Si S0(5il) then we obtain the transformations of the coordinates -18- of the quaternionic superspaee including the Euclidean conforml transformations.

17- The/.1 car: '„<:. •••• i'r:. L .on ',r, - <_-r_- •.; of ',•

• W - .1 - yj ' ;/ PROJECTJVE , :;RASSMATIFIIA:I AND FLAG SUPER:#.UIFOLDS

(6.11) The notions of projective, Grassmannian and flag manifolds are assumed to be familiar (for definitions see e.g. [ll]). Here, we extend these Similarly, for supersp&ce E T ve find concepts to supermanifolds with an arbitrary commuting Banach superalge"bra A Li playing the role of the field of numbers. We use the definition of supermanifolds from [12] which is analogous to that considered in [13]. (6.12) Let R, C and M be the fields of real, complex and quaternionic numbers and let K denote one of these fields.

A commutative Banach superalgebra A over K is defined to he a and finally for superspace E Banach: STiperspace A over K (i.e. a Banach space (A, || • || ) with a <*> f (l +• tk + eta specified decomposition A = AQ

property ||abj < J a || || b ] for arbitrary a,b 6 A. A veil-known example of a commutative Banach superalgebra is the Grassmann VII. CONCLUDING REMAEKS algebra A over K. Every element a £ A can be written in the form In the paper the extension of the twistor correspondence to the super- a = o(a) + n(a), where a(a) 6 K and n(a) is nilpotent. It can be shown tvlstor case was given in full detail ty using flag supermanifolds. The the that a s A is invertible iff o{a) is invertible, i.e. a(a} ? 0. chlral and non-chiral superspaces were obtained and representations of SU(£,2|H) We may now define a A-module A for a fixed n,m« N as an (m+n) on these superspaces contruoted hy means of supertwistors. It will he shovn copies A x A < ... X A and decomposes' it into an even and odd part elsewhere that representations of in the space of superfields found for "the case of right-chiral superspace in [8] can also ~oe extended to other superspaces. On the other hand, it seems that the supertvistors can also he used as a suifbale framework for description of super-Yang-Mills theories = A' © A (see [9], [10], [17]) and, after a modification,of theories (see [18]).

The even part A is a An-module which we denote by IT (K is again the field over which the commutative Banach superalgebra A is defined) and call ACKNOWLEDGMENTS a A-superspace of dimension m/n. Before explaining the notion of dimension m/n let us mention that fi plays the role of the vector space !i in the usual On the authors (J.B.) would like to thank Professor P. Budinich and analysis and is a trivial example of a supermanifold in the sense of [12], the International School for Advanced Studies, Trieste. He would also like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. -20- -19- (.1) m/n m/n. We say that r vectors A (respectively e where z,X € KQ, P, n e A1- Thus ve have obtained the chart on G , (A). arfr-mutually "inTigpendent if the same property holds for vectors Of course, choosing another submatrix k/i x Xfl to be regular ve get another chart. In terms of these charts it can be proved that G, , , (A) is a o(A }, ,.., o(A ) (respectively «. YT). Thus there are maximally m k/ £ »xv £ m/n {respectively n) independent vectors In A-""11 (respectively Let us A-supermanI fold. choose them in both spacesand denote them by E , S. = l,.,.,m and 0^,, A = 1,... ,n. Finally ve generalize the notion of the Grassmannian supermanifold to super- ,m+n ,, , They form ef basis in can be written in the form A so that any manifolds vhose points are flags, i.e. sequences of submodules V-C V C c V of superspace K™ n. Namely a flag supermanifold ,k /i ,.. .k ft , F^/n) Is a STiperaanifold ^^ Vr)' V±c ^ is a aubmodiae of dimension X /t , i l,...,r, ± L /i.x < m/n and v where 6X <= A . l<= ••• C V'

ia a s ¥e say that AQ-submodule U of It haB dimension p[q if there exist Flag supermanifold «P«rmanifold Q p independent vectors E fi A_ r = l,...,p and q independent vectors of the dimension r" m/n 0 e A , p = 1,... ,

,m+n ,m+n The set of invertible A-linear mappings f ; A " •+ A""" forms a group and a A-supermanifold that (t^,Tj) •*• f f,~ is smooth (in the sense of D-smooth [12]), i.e. a- Lie- sufle^group GL(m/n,A). The elements of GL(m/n,A) transform independent We see that if r = 1 the flag supermanifold is Just the Grassmannian super- vectors to independent ones. manifold G^^U).

We now define a protective superspace. Let us consider

jjjn/n^j^^j Km/n> Q(X) = 0} and let us introduce on equlvalence relation: (x,9) ^ (y,n) if there exists ae A., o(a) 4 0 such that (x,9) = a(y,n). The set of all equivalence classes forms the pro^-ective- super-space P(it ). It ean he proved that Pdt™ ) is a compact analytic A-supermanifold of dimension

We may also generalize the motion of Grassmannian manifolds to the graded case. A Grassmannian supennanifold G, , , (A) is a set of all A,,-sutmodules *• k/l.m/n 0 , of dimension k/Jt < m/n fi.e. k < m, i < n or k < m, S. < n) of superspace ft . It has the dimension, equal to [(m-ls)k + (n-Jl}Jl] | [(m-k)jt + (n-Jt)k]. A Grassmannian supermanifolds can he equipped with a syctem of charts analogously as a Grassmannian manifold. 'Namely, suppose a submodule of dimension k/i < m/n is spanned by vectors E and 0 . We can arrange than to an even supenaatrix in x n|k x SL. With the help of transformations from GIj(k|i,A) we can put the supermatrix into the form

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