Non-Relativistic Supersymmetry on Curved Three-Manifolds

TL;DR

This paper constructs explicit non-relativistic supersymmetric field theories on curved three-manifolds by reducing four-dimensional theories, deriving Killing spinor equations, and identifying conditions for supersymmetry on Newton-Cartan backgrounds.

Contribution

It provides a systematic method to realize non-relativistic supersymmetry on curved spaces via null reduction and characterizes the geometric conditions for supersymmetry.

Findings

Derived algebraic and differential Killing spinor equations.

Identified conditions for Newton-Cartan backgrounds to admit supersymmetry.

Presented examples including twistless torsional and contact geometries.

Abstract

We construct explicit examples of non-relativistic supersymmetric field theories on curved Newton-Cartan three-manifolds. These results are obtained by performing a null reduction of four-dimensional supersymmetric field theories on Lorentzian manifolds and the Killing spinor equations that their supersymmetry parameters obey. This gives rise to a set of algebraic and differential Killing spinor equations that are obeyed by the supersymmetry parameters of the resulting three-dimensional non-relativistic field theories. We derive necessary and sufficient conditions that determine whether a Newton-Cartan background admits non-trivial solutions of these Killing spinor equations. Two classes of examples of Newton-Cartan backgrounds that obey these conditions are discussed. The first class is characterised by an integrable foliation, corresponding to so-called twistless torsional geometries, and includes manifolds whose spatial slices are isomorphic to the Poincar\'e disc. The second class of examples has a non-integrable foliation structure and corresponds to contact manifolds.