Venugopalan, S.; Woodward, C. Classification of affine vortices. (English) Zbl 1344.53019 Duke Math. J. 165, No. 9, 1695-1751 (2016). Summary: 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 \(\mathbb P(1,n)\) to \(X/G\) that map the stacky point at infinity \(\mathbb P(n)\) 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 T. Dimofte et al. [Lett. Math. Phys. 98, No. 3, 225–287 (2011; Zbl 1239.81057)] relating vortex counts to knot invariants. Cited in 11 Documents MSC: 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 57R57 Applications of global analysis to structures on manifolds 35Q99 Partial differential equations of mathematical physics and other areas of application 37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) Keywords:Hitchin-Kobayashi; vortices; Kirwan map; moment map Citations:Zbl 1239.81057 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] R. A. Adams, Sobolev Spaces , Pure Appl. Math. 65 , Academic Press, New York, 1975. [2] A. Adem, J. Leida, and Y. Ruan, Orbifolds and Stringy Topology , Cambridge Tracts in Math. 171 , Cambridge Univ. 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