A Vafa-Intriligator formula for semi-positive quotients of linear spaces

TL;DR

This paper develops a residue formula for genus zero quasimap invariants of semi-positive GIT quotients, expressing them in terms of invariants of torus quotients, and proves a Vafa-Intriligator formula under positivity assumptions.

Contribution

It introduces a Vafa-Intriligator formula for semi-positive quotients and relates invariants of $V/ G$ to those of $V/T$, providing explicit residue and sum formulas.

Findings

Residue formulae for quasimap invariants of $V/ G$

Expression of invariants in terms of torus quotients

Vafa-Intriligator formula for generating series

Abstract

We consider genus zero quasimap invariants of smooth projective targets of the form $V/\!/G$, where $V$ is a representation of a reductive group $G$. In particular we consider integrals of cohomology classes arising as characteristic classes of the universal quasimap. In this setting, we provide a way to express the invariants of $V/\!/G$ in terms of invariants of $V/\!/T$, where $T$ is a maximal subtorus of $G$. Using this, we obtain residue formulae for such invariants as conjectured by Kim, Oh, Yoshida and Ueda. Finally, under some positivity assumptions on $V/\!/G$, we prove a Vafa-Intriligator formula for the generating series of such invariants, expressing them as finite sums of explicit contributions.