Quantum Witten localization and abelianization for qde solutions

TL;DR

This paper develops quantum localization and abelianization formulas for GIT quotients, providing new tools for computing quantum invariants and solutions to quantum differential equations in geometric representation theory.

Contribution

It introduces quantum versions of Witten localization and abelianization formulas, extending classical results to quantum invariants and applying them to moduli spaces and Nakajima quiver varieties.

Findings

Proved a quantum localization formula relating GIT quotient invariants to equivariant invariants.

Established a quantum abelianization formula for GIT quotients by a group and its maximal torus.

Derived a quantum Lefschetz principle for holomorphic symplectic reductions.

Abstract

We prove a quantum version of the localization formula of Witten that relates invariants of a git quotient with the equivariant invariants of the action. Using the formula we prove a quantum version of an abelianization formula of S. Martin relating invariants of geometric invariant theory quotients by a group and its maximal torus, conjectured by Bertram, Ciocan-Fontanine, and Kim. By similar techniques we prove a quantum Lefschetz principle for holomorphic symplectic reductions. As an application, we give a formula for the fundamental solution to the quantum differential equation (qde) for the moduli space of points on the projective line and for the smoothed moduli space of framed sheaves on the projective plane (a Nakajima quiver variety).