Shifts of prepotentials (with an appendix by Michele Vergne)

TL;DR

This paper investigates the geometric and physical properties of supersymmetric theories in five dimensions derived from M-theory compactified on Calabi-Yau threefolds, focusing on the structure of equivariant volumes and their applications.

Contribution

It introduces finite-difference equations for equivariant volumes and explores their implications in supersymmetric gauge theories, extending previous results with new mathematical insights.

Findings

Derived shift equations for equivariant volumes.

Connected geometric structures to physical gauge theories.

Provided an alternative proof of key shift equations.

Abstract

We study the dynamics of supersymmetric theories in five dimensions obtained by compactifications of M-theory on a Calabi-Yau threefold X. For a compact X, this is determined by the geometry of X, in particular the Kahler class dependence of the volume of X determines the effective couplings of vector multiplets. Rigid supersymmetry emerges in the limit of divergent volume, prompting the study of the structure of Duistermaat-Heckman formula and its generalizations for non-compact toric Kahler manifolds. Our main tool is the set of finite-difference equations obeyed by equivariant volumes and their quantum versions. We also discuss a physical application of these equations in the context of seven-dimensional gauge theories, extending and clarifying our previous results. The appendix by M. Vergne provides an alternative local proof of the shift equation.