Virtual invariants of critical loci in GIT quotients of linear spaces

TL;DR

This paper develops a localization formula to compute virtual invariants of critical loci in GIT quotients, confirming predictions in quiver moduli spaces from physics.

Contribution

It introduces an equivariant localization approach for virtual invariants in GIT quotients, connecting mathematical formulas with physical predictions.

Findings

Derived formulas for DT, χ_y, and Ell invariants of critical loci

Recovered physicists' predictions for quiver moduli spaces

Established a new computational framework for virtual invariants

Abstract

We use an equivariant version of the localization formula of Jeffrey and Kirwan to prove a formula for virtual invariants $(\text{DT}$, $\chi_y$, $\text{Ell})$ of critical loci in quotients of linear spaces by actions of reductive algebraic groups. In particular we recover formulae for the invariants of critical loci of potentials in moduli spaces of quiver representations predicted by physicists.