Supersymmetry algebras in arbitrary signature and their R-symmetry groups

TL;DR

This paper develops a unified framework for supersymmetry algebras in arbitrary dimensions and signatures, providing comprehensive tables of R-symmetry groups and applications to string theory and supergravity.

Contribution

It introduces a general construction of supersymmetry algebras in any signature, extending symplectic Majorana spinors and clarifying R-symmetry actions across signatures.

Findings

Tables of R-symmetry groups for all signatures

Identification of twisted supersymmetry algebras with non-compact R-symmetry

Application to relations between string theories and supergravity geometries

Abstract

String theory, specifically type-II superstring theory, can be formulated in any ten-dimensional signature. To facilitate the study of supergravity and superstring theories in this setting, we present a uniform construction of supersymmetry algebras in arbitrary dimension and signature, which generalizes the ideas underlying symplectic Majorana spinors. In our formalism R-symmetry acts on an auxiliary multiplicity space which makes its action manifest. This allows us to provide extensive tables which list the R-symmetry groups of extended supersymmetry algebras for all signatures together with other useful information. Twisted (`type-*') supersymmetry algebras in Lorentz signature with non-compact R-symmetry groups are shown to be part of a general pattern resulting from the interplay between complex superbrackets and reality conditions. As an application we show how the relations between type-II string theories in ten and nine dimensions can be extracted from their supersymmetry algebras. We also use our results to determine the special geometry of vector and hypermultiplet scalar manifolds of four-dimensional $\mathcal{N}=2$ and three-dimensional $\mathcal{N}=4$ supergravity theories for all signatures.