Rigid supersymmetric theories in 4d Riemannian space

TL;DR

This paper develops a formalism for constructing and analyzing rigid supersymmetric theories on four-dimensional Riemannian manifolds, emphasizing boundary terms and geometric structures to identify supersymmetric backgrounds.

Contribution

It introduces a method to formulate supersymmetric theories directly in Euclidean signature using G-structures, expanding the understanding of supersymmetric backgrounds with non-trivial curvature.

Findings

Reformulation of supersymmetry conditions via torsion classes

Construction of Lagrangians in Euclidean signature with boundary considerations

Examples of supersymmetric backgrounds with non-zero Weyl tensor

Abstract

We consider rigid supersymmetric theories in four-dimensional Riemannian spin manifolds. We build the Lagrangian directly in Euclidean signature from the outset, keeping track of potential boundary terms. We reformulate the conditions for supersymmetry as a set of conditions on the torsion classes of a suitable SU(2) or trivial G-structure. We illustrate the formalism with a number of examples including supersymmetric backgrounds with non-vanishing Weyl tensor.