Symmetries of $\mathcal{N} = (1,0)$ supergravity backgrounds in six dimensions

TL;DR

This paper investigates the geometric symmetries of six-dimensional $ abla=(1,0)$ supergravity backgrounds, introducing conformal Killing spinor superfields and analyzing their role in superconformal transformations and higher symmetries.

Contribution

It develops a framework for understanding conformal Killing spinor superfields and their associated symmetries in 6D supergravity backgrounds, including higher symmetries and their geometric implications.

Findings

Conformal Killing vector and tensor superfields generate superconformal symmetries.

Higher symmetries of the hypermultiplet are linked to conformal Killing tensors.

Higher symmetries of the conformal d'Alembertian are only defined on conformally flat backgrounds.

Abstract

General $\mathcal{N}=(1,0)$ supergravity-matter systems in six dimensions may be described using one of the two fully fledged superspace formulations for conformal supergravity: (i) $\mathsf{SU}(2)$ superspace; and (ii) conformal superspace. With motivation to develop rigid supersymmetric field theories in curved space, this paper is devoted to the study of the geometric symmetries of supergravity backgrounds. In particular, we introduce the notion of a conformal Killing spinor superfield $\epsilon^\alpha$, which proves to generate extended superconformal transformations. Among its cousins are the conformal Killing vector $\xi^a$ and tensor $\zeta^{a(n)}$ superfields. The former parametrise conformal isometries of supergravity backgrounds, which in turn yield symmetries of every superconformal field theory. Meanwhile, the conformal Killing tensors of a given background are associated with higher symmetries of the hypermultiplet. By studying the higher symmetries of a non-conformal vector multiplet we introduce the concept of a Killing tensor superfield. We also analyse the problem of computing higher symmetries for the conformal d'Alembertian in curved space and demonstrate that, beyond the first-order case, these operators are defined only on conformally flat backgrounds.