Symmetries of supergravity backgrounds and supersymmetric field theory

TL;DR

This paper explores the geometric structures of $$ superspace in four-dimensional supergravity, focusing on (conformal) Killing tensor superfields and their role in symmetries of supersymmetric theories.

Contribution

It introduces and analyzes (conformal) Killing tensor superfields in $$ superspace, revealing their significance in generating symmetries and extended supersymmetry transformations.

Findings

Killing vector superfields generate superspace isometries.

Killing spinor superfields generate extended supersymmetries.

Higher-rank Killing tensor superfields generate all symmetries of the Wess-Zumino operator.

Abstract

In four spacetime dimensions, all ${\cal N} =1$ supergravity-matter systems can be formulated in the so-called $\mathsf{U}(1)$ superspace proposed by Howe in 1981. This paper is devoted to the study of those geometric structures which characterise a background $\mathsf{U}(1)$ superspace and are important in the context of supersymmetric field theory in curved space. We introduce (conformal) Killing tensor superfields $\ell_{(\alpha_1 \dots \alpha_m) ({\dot \alpha}_1 \dots {\dot \alpha}_n)}$, with $m$ and $n$ non-negative integers, $m+n>0$, and elaborate on their significance in the following cases: (i) $m=n=1$; (ii) $m-1=n=0$; and (iii) $m=n>1$. The (conformal) Killing vector superfields $\ell_{\alpha \dot \alpha}$ generate the (conformal) isometries of curved superspace, which are symmetries of every (conformal) supersymmetric field theory. The (conformal) Killing spinor superfields $\ell_{\alpha }$ generate extended (conformal) supersymmetry transformations. The (conformal) Killing tensor superfields with $m=n>1$ prove to generate all higher symmetries of the (massless) massive Wess-Zumino operator.