Quantum Witten localization and abelianization for qde solutions

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).

arXiv comment41 pages. A previous version was called "Area-dependence in gauged Gromov-Witten theory". Some of that material was moved into "Wall-crossing for Gromov-Witten invariants under variation of git quotient", while some new material was added

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• The Abelian/Nonabelian Correspondence and Frobenius Manifolds
• Gromov-Witten invariants of blow-ups
• Quantum D-modules and generalized mirror transformations
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Cited by in corpus (7)

• Vortex invariants and toric manifolds
• The Abelian-Nonabelian Correspondence for $I$-functions
• Morphisms of CohFT algebras and quantization of the Kirwan map
• The Moduli Space in the Gauged Linear Sigma Model
• Quantum K-theory of flag varieties via non-abelian localization
• Gauged Gromov-Witten theory for small spheres
• Deformations of symplectic vortices