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

Abstract

General $$ \mathcal{N} $$ 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) 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 ϵα, which proves to generate extended superconformal transformations. Among its cousins are the conformal Killing vector ξa and tensor ζa(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 a limited class of backgrounds, including all conformally flat ones.

Superconformal geometries and local twistors

“…The (1, 0) theory in D = 6 has been discussed in SU(2) superspace in [25] and in conformal superspace [20][21][22]. Recently the geometric symmetries of (1, 0) supergravity backgrounds was treated in [48]. The theory was also discussed earlier in harmonic superspace in [24] and, some years ago, in projective superspace [25].…”

Killing-Yano Cotton currents

“…There are applications in General Relativity (GR) [6,7] to G-structures [8,9], to WZW models [10], to classical mechanics [11] and to symmetries of the Dirac operator and super Laplacians [12,13]. Supersymmetric conformal KTs and KYTs are discussed in [14], in [15] and [16]. Finally KTs arise in the context of hyperKähler geometry [17].…”

Uses of Killing and Killing-Yano Tensors

“…Killing-Yano tensors square to Killing tensors and characterise the symmetries of the Dirac equation [10] and are also related to novel supersymmetries in sigma models and strings [11,12]. Supersymmetric Killing-Yano tensors [13] characterise the symmetries of super Laplacians [14][15][16]. Finally, KTs arise in the context of hyperKähler geometry [17].…”