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15 June 2016 Classification of affine vortices
S. Venugopalan, C. Woodward
Duke Math. J. 165(9): 1695-1751 (15 June 2016). DOI: 10.1215/00127094-3450315

Abstract

We prove a Hitchin–Kobayashi correspondence for vortices on the complex affine line with Kähler target, which generalizes a result of Taubes for the case of a line target. More precisely, suppose that K is a compact Lie group and that the target X is either a compact Kähler K-Hamiltonian manifold or X is a symplectic vector space with linear K-action and a proper moment map. Suppose that the action of the complexified Lie group G satisfies stable = semistable. Then, for some sufficiently divisible integer n, there is a bijection between gauge equivalence classes of K-vortices with target X and isomorphism classes of maps from the weighted projective line P(1,n) to X/G that map the stacky point at infinity P(n) to the semistable locus of X. The results allow the construction and partial computation of the quantum Kirwan map from Woodward and play a role in the conjectures of Dimofte, Gukov, and Hollands relating vortex counts to knot invariants.

Citation

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S. Venugopalan. C. Woodward. "Classification of affine vortices." Duke Math. J. 165 (9) 1695 - 1751, 15 June 2016. https://doi.org/10.1215/00127094-3450315

Information

Received: 17 July 2013; Revised: 7 July 2015; Published: 15 June 2016
First available in Project Euclid: 22 April 2016

zbMATH: 1344.53019
MathSciNet: MR3513572
Digital Object Identifier: 10.1215/00127094-3450315

Subjects:
Primary: 53D06

Keywords: Hitchin–Kobayashi , Kirwan map , vortices

Rights: Copyright © 2016 Duke University Press

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Vol.165 • No. 9 • 15 June 2016
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