THE SIGNATURE TRIALITY OF MAJORANA–WEYL SPACE–TIMES

Abstract

In this paper, a general property of field theories with Majorana–Weyl spinors (such as ten-dimensional SYM theories, superstrings, five-branes, etc.) is presented. It is shown that these theories, in a given dimension, admit different formulations in the allowed space–times signatures which are all interconnected by transformations which close the S 3 permutation group (S 3 duality). Explicit formulas to "translate" a theory from a given signature into another signature, are given. They are conveniently chosen to realize the S 3 group. The above construction applies for any dimension d which admit Majorana–Weyl spinors in more than one signature, that is for d ≥ 8. We prove that the S 3 duality (valid for any even d ≥ 8) is in consequence of the famous triality property of the 8 dimensions, together with (for d > 8) the mod 8 property of Clifford Γ matrices.