The signature triality of Majorana-Weyl space-times

In this paper, a general property of field theories with Majorana-Weyl spin ors (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 signatu re into another signature, axe given. They are conveniently chosen to reali ze the S-3 group. The above construction applies for any dimension d which admit Majorana-Wey l spinors in more than one signature, that is for d greater than or equal t o 8. We prove that the S-3 duality (valid for any even d greater than or eq ual to 8) is in consequence of the famous triality property of the 8 dimens ions, together with (for d > 8) the mod 8 property of Clifford Gamma matric es.