From equivariant volumes to equivariant periods

TL;DR

This paper extends the concept of equivariant volumes to partition functions of supersymmetric GLSMs on various manifolds, revealing their relation to equivariant quantum cohomology and Gromov-Witten invariants.

Contribution

It introduces generalized equivariant volumes as partition functions for GLSMs on different manifolds and explores their differential/difference equations and connection to quantum cohomology.

Findings

Partition functions satisfy shift equations analogous to equivariant volumes.

These functions are annihilated by operators encoding quantum cohomology/K-theory relations.

Expansion in equivariant parameters encodes genus-zero Gromov-Witten invariants.

Abstract

We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d supersymmetric GLSM on $S^1$, $D^2$ and $D^2 \times S^1$, respectively. We define these objects and study their dependence on equivariant parameters for non-compact toric K\"ahler quotients. We generalize the finite-difference equations (shift equations) obeyed by equivariant volumes to these partition functions. The partition functions are annihilated by differential/difference operators that represent equivariant quantum cohomology/K-theory relations of the target and the appearance of compact divisors in these relations plays a crucial role in the analysis of the non-equivariant limit. We show that the expansion in equivariant parameters contains information about genus-zero Gromov-Witten invariants of the target.