Duality group actions on fermions

TL;DR

This paper investigates how duality groups act on fermions in supersymmetric theories, proposing Z_2 extensions to account for sign ambiguities and testing these ideas across various compactifications.

Contribution

It introduces Z_2 extensions of duality groups to properly describe their action on fermions in string and M-theory contexts.

Findings

Duality groups like SL(2,Z) have sign ambiguities in fermionic actions.

Proposed Z_2 extensions, such as the metaplectic group, resolve these ambiguities.

Cross-consistency tests support the proposed group extensions.

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.