The virtual intersection theory of isotropic Quot Schemes

TL;DR

This paper develops a virtual intersection theory for isotropic Quot schemes over curves, providing explicit formulas and comparing invariants with stable map approaches, thus extending known enumerative formulas.

Contribution

It introduces a virtual fundamental class for isotropic Quot schemes and derives explicit intersection formulas, generalizing the Vafa-Intriligator formula.

Findings

Explicit formulas for intersection numbers when r=2

Comparison between Quot scheme invariants and stable map invariants

Generalization of the Vafa-Intriligator formula

Abstract

Isotropic Quot schemes parameterize rank $r$ isotropic subsheaves of a vector bundle equipped with symplectic or symmetric quadratic form. We define a virtual fundamental class for isotropic Quot schemes over smooth projective curves. Using torus localization, we prescribe a way to calculate top intersection numbers of tautological classes, and obtain explicit formulas when $r=2$. These include and generalize the Vafa-Intriligator formula. In this setting, we compare the Quot scheme invariants with the invariants obtained via the stable map compactification.