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International Centre for Theoretical Physics IC/81+/237 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS 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 theories - the general theory 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 complex analysis 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 spin [l],[2], [U] and a general method for solving Yang-Mills [2],[5],[6] as well as the non-linear Einstein [2],[7] field 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 supersymmetry 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 Minkowski space 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 twistor space 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 vector space 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 spinors 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 spinor 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 Grassmannian 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" , <Li<Lim 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.
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