From Rigid Supersymmetry to Twisted Holomorphic Theories

TL;DR

This paper explores how N=1 supersymmetric theories on complex four-manifolds can be analyzed using twisted fields, providing new formulas for partition functions and extending the framework to three dimensions.

Contribution

It introduces a twisted field approach to study supersymmetric theories on complex manifolds and derives explicit formulas for partition functions in various geometries.

Findings

Partition function Z_M depends on complex structure and background fields.

New explicit formulas for Z_M on S^3 x S^1 and squashed three-spheres.

Extension of analysis to three-dimensional N=2 theories with transversely holomorphic foliations.

Abstract

We study N=1 field theories with a U(1)_R symmetry on compact four-manifolds M. Supersymmetry requires M to be a complex manifold. The supersymmetric theory on M can be described in terms of conventional fields coupled to background supergravity, or in terms of twisted fields adapted to the complex geometry of M. Many properties of the theory that are difficult to see in one formulation are simpler in the other one. We use the twisted description to study the dependence of the partition function Z_M on the geometry of M, as well as coupling constants and background gauge fields, recovering and extending previous results. We also indicate how to generalize our analysis to three-dimensional N=2 theories with a U(1)_R symmetry. In this case supersymmetry requires M to carry a transversely holomorphic foliation, which endows it with a near-perfect analogue of complex geometry. Finally, we present new explicit formulas for the dependence of Z_M on the choice of U(1)_R symmetry in four and three dimensions, and illustrate them for complex manifolds diffeomorphic to S^3 x S^1, as well as general squashed three-spheres.