On the Uniqueness of String Superalgebra Derived from Light-Like Integrability

Abstract

Ten dimensional physics attracts much attention recently in connection with the developments of superstring theory, which may provide a consistent unified theory including gravity. It is one of interesting aspects of these approaches that supersymmetric Y ang-Mills theory in ten dimensions has a twistorial interpretation as in four. dimensions, or in other words that in superspace formulation the constraint equations of the theory are equivalent to integrability on a light-like line_I) It is also shown that N = 1 supergravity in ten. dimensions has a similar property. I) These facts suggest that the underlying superstring theory may have such a feature as well. In fact, Freund and Mezincescu) have shown that the integrability generalized to superstrings yields Siegel's superalgebra) which describes the classical dynamics of superstrings. Freund and Mezincescu, however, obtained another superalgebra, which is similar to Siegel's, and the ones extended with central charges. In this paper, we would like to point out that only Siegel's superalgebra without central charges satisfies the whole set of Jacobi identities. We believe that the present observation strengthens the physical merits of the work by Freund and Mezincescu.) Let Da(a) and Da(a) be fundamental operators (or covariant derivatives) for the superstring with the parameter representing a position along the string. Here and below Lorentz vector and spinor indices are denoted as a, b, ---( =0, ---,9) and a,/3, ---(=1, ---, 16), respectively. Lorentz group O(l, 9) has two spinor representations, the 16 and 16, which we distinguish by indices. up and down respectively, so that we have two. gamma matrices raap and raap. Those matrices are symmetric and obey the Clifford algebra raapr/' + rbapraP' = 27JabO/. To describe the dynamics of the superstring, Siegel has proposed a superal. gebra of covariant derivatives) such that