Duality group actions on fermions

TL;DR

We address sign ambiguities in fermion transformations under conventional duality groups in maximally supersymmetric theories. The main approach is to promote duality groups to ${\mathbb Z}_2$-central extensions, notably the metaplectic group $Mp(2,\mathbb{Z})$, to achieve a consistent action on fermions across mapping class groups, T-duality, and S-/U-duality. The key contributions include a detailed proposal of $Z_2$-extended duality groups in nine, eight, and seven dimensions, with cross-checks that show coherent structure between bosonic and fermionic sectors, and a discussion of moduli-stack and geometric Langlands implications. This framework clarifies how fermions can be consistently incorporated into duality symmetries and suggests new avenues for understanding moduli spaces in string theory and related mathematical structures.

Abstract

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.