A Vafa-Intriligator formula for semi-positive quotients of linear spaces
Riccardo Ontani
TL;DR
This work develops a comprehensive framework for genus zero quasimap invariants of targets of the form $V/\!/G$. It proves an abelianisation principle reducing nonabelian invariants to torus data on $V/\!/T$, and derives Jeffrey–Kirwan residue formulae that express invariants as explicit residues after abelianisation. Under a semi-positivity assumption, it extends these ideas to a Vafa–Intriligator-type formula for the generating series, written as a finite sum of explicit contributions via an involution on the dual torus and a Jacobian determinant. The approach combines fixed-locus localisation, the Chern–Weil morphism on fixed loci, and known toric results (Szenes–Vergne) to achieve closed expressions and convergence results, thus providing practical computational tools for quasimap invariants beyond toric and Grassmannian cases. This bridges nonabelian GIT quotients with toric techniques, enabling residue- and generating-function computations across a broad class of reductive quotients.
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.