Rigid 6D supersymmetry and localization

TL;DR

This paper develops rigid supersymmetric theories on six-dimensional manifolds, deriving geometric conditions for supersymmetry and demonstrating localization of super Yang-Mills path integrals on Calabi-Yau threefolds.

Contribution

It formulates necessary and sufficient geometric conditions for supersymmetry in 6D theories and applies localization techniques to compute path integrals on complex manifolds.

Findings

Supersymmetric theories constructed on various 6D manifolds.

Localization of super Yang-Mills on Calabi-Yau threefolds.

Geometric constraints for supersymmetry derived from Killing spinor equations.

Abstract

We construct rigid supersymmetric theories for interacting vector and tensor multiplets on six-dimensional Riemannian spin manifolds. Analyzing the Killing spinor equations, we derive the constraints on these theories. To this end, we reformulate the conditions for supersymmetry as a set of necessary and sufficient conditions on the geometry. The formalism is illustrated with a number of examples, including manifolds that are hermitian, strong Kaehler with torsion. As an application, we show that the path integral of pure super Yang-Mills theory defined on a Calabi-Yau threefold M_6 localizes on stable holomorphic bundles over M_6.