Quasimaps to quivers with potentials

TL;DR

This paper develops a framework for counting quasimaps to the critical locus of potentials in non-compact GIT quotients, with applications to quivers, vertex functions, and Bethe equations in geometric representation theory.

Contribution

It introduces a new method for defining virtual counts of quasimaps to critical loci, extending quasimaps theory to quivers with potentials and related geometric structures.

Findings

Defined virtual counts of quasimaps to critical loci.

Proved a gluing formula within cohomological field theories.

Computed Bethe equations for various examples.

Abstract

This paper is concerned with a non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential, and prove a gluing formula in the formalism of cohomological field theories. The main examples studied in this paper is when the above setting arises from quivers with potentials, where the above construction gives quantum correction to the equivariant Chow homology of the critical locus. Following similar ideas as in quasimaps to Nakajima quiver varieties studied by the Okounkov school, we analyse vertex functions in several examples, including Hilbert schemes of points on $\mathbb{C}^3$, moduli spaces of perverse coherent systems on the resolved conifold, and a quiver which defines higher $\mathfrak{sl}_2$-spin chains. Bethe equations are calculated in these cases. The construction in the present paper is based on the theory of gauged linear sigma models as well as shifted symplectic geometry of Pantev, To\"en, Vaquie and Vezzosi, and uses the virtual pullback formalism of symmetric obstruction theory of Park, which arises from the recent development of Donaldson-Thomas theory of Calabi-Yau 4-folds.