Duality group actions on fermions

In this short paper we look at the action of T-duality and string duality groups on fermions, in maximally-supersymmetric theories and related theories. Briefly, we argue that typical duality groups such as SL( 2, Z) have sign ambiguities in their actions on fermions, and propose that pertinent duality groups be extended by Z 2, to groups such as the metaplectic group. Specifically, we look at duality groups arising from mapping class groups of tori in M theory compactifications, T-duality, ten-dimensional type IIB S-duality, and ( briefly) four-dimensional N = 4 super Yang-Mills, and in each case, propose that the full duality group is a nontrivial Z 2 extension of the duality group acting on bosonic degrees of freedom, to more accurately describe possible actions on fermions. We also walk through U-duality groups for toroidal compactifications to nine, eight, and seven dimensions, which enables us to perform cross-consistency tests of these proposals.