Abstract
On a smooth line bundle L over a compact Kähler Riemann surface Σ, we study the family of vortex equations with a parameter s. For each \({s \in [1,\infty]}\) , we invoke techniques in Bradlow (Commun Math Phys 135:1–17, 1990) by turning the s-vortex equation into an s-dependent elliptic partial differential equation, studied in Kazdan and Warner (Ann Math 2:14–47, 1978), providing an explicit moduli space description of the space of gauge classes of solutions. We are particularly interested in the bijective correspondence between the open subset of vortices without common zeros and the space of holomorphic maps. For each s, the correspondence is uniquely determined by a smooth function u s on Σ, and we confirm its convergent behaviors as \({s \to \infty}\) . Our results prove a conjecture posed by Baptista in Baptista (Nucl Phys B 844:308–333, 2010), stating that the s-dependent correspondence is an isometry between the open subsets when s = ∞, with L 2 metrics appropriately defined.
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Baptista J.M.: On the L 2 metrics of vortex moduli spaces. Nucl. Phys. B 844, 308–333 (2010)
Baptista, J.M.: Moduli spaces of Abelian vortices on Kähler manifolds. (2013). arXiv: 1211.0012
Bradlow S.B.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135, 1–17 (1990)
Bradlow S.B.: Special Metrics and Stability for Holomorphic Bundles with Global Sections. J. Differ. Geom. 33, 169–213 (1991)
Bertram A., Daskalopoulos G., Wentworth R.: Gromov Invariants for Holomorphis Maps from Riemann Surfaces to Grassmannians. J. Am. Math. Soc. 9, 529–571 (1996)
Cieliebak K., Gaio A.R., Mundet i Riera I., Salamon D.A.: The symplectic vortex equaions and invariants of Hamiltonian group actions. J. Symplectic Geom. 1(3), 543–645 (2002)
Donaldson, S.: Kähler metrics with cone singularities along a divisor, arxiv:1102.1196
Donaldson S.K., Kronheimer P.B.: The Geometry of Four-Manifolds. Oxford Science Publications, New York, NY (1990)
Garcia-Prada O.: A Direct Existence Proof for the Vortex Equations over a Riemann Surface. Bull. London Math. Soc. 26(1), 88–96 (1994)
Grirriths P.A., Harris J.: Principles of Algebraic Geometry. Wiley, Hoboken, NJ (1978)
Gaio A., Salamon D.: Gromov-Witten Invariants of Symplectic Quotients and Adiabatic Limits. J. Symplectic Geom. 3(1), 55–159 (2005)
Griffiths, P.A.: Introduction to Algebraic Curves, vol. 76. American Mathematical Society, Providence, RI (1983)
Jaffe A., Taubes C.: Vortices and Monopoles. Birkhäuser, Boston, MA (1981)
Kobayashi, S.: Differential Geometry of Complex Vector Bundles, Iwanami Shoten. Publishes and Princeton University Press, Princeton, NJ (1987)
Kallel S., Milgram J.: Space of holomorphic maps. J. Differ. Geom. 47, 321–375 (1997)
Kazdan J., Warner F.W.: Curvature functions for compact 2-manifolds. Ann. Math. 2(99), 14–47 (1978)
Manton N.S.: A remark on the scattering of BPS monopoles. Phys. Lett. 110B, 54–56 (1982)
Miranda, R.: Algebraic Curves and Riemann Surfaces, vol. 5. American Mathematical Society, Providence, RI (1995)
Morrison D., Plesser M.: Summing the instantons: quantum cohomology and mirror symmetry in toric varieties. Nucl. Phys. B 440, 279–354 (1995)
Ott, A.: Removal of singularities and Gromov compactness for symplectic vortices. (2009). arXiv: 0912.2500
Romao N.: Gauged vortices in a background. J. Phys. A: Math. Gen. 38, 9127 (2005)
Samols M.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135, 1–17 (1990)
Speight J.M.: The volume of the space of holomorphic maps from \({\mathbb{S}^2}\) to \({\mathbb{CP}^{k - 1}}\) . J. Geom. Phys. 61, 77–84 (2011)
Viaclovsky, J.A.: Math 865, Topics in Riemannian Geometry, Fall 2007 Class Notes in University of Wisconsin. Madison
Witten E.: Phases of n = 2 theories in two dimensions. Nucl. Phys. B 403, 159–222 (1993)
Wodward, C.: Quantum Kirwan morphism and Gromov–Witten invarants of quotients. arXiv: 1204.1765, April 2012
Xu, G.: U(1)- vortices and quantum Kirwan map. (2012). arXiv: 1211.0217
Ziltener, F.: A Quantum Kirwan Map: Bubbling and Fredholm Theory. Memoirs of the American Mathematical Society, Providence, RI (2012)
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Communicated by N. A. Nekrasov
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Liu, CC. Dynamics of Abelian Vortices Without Common Zeros in the Adiabatic Limit. Commun. Math. Phys. 329, 169–206 (2014). https://doi.org/10.1007/s00220-014-2016-y
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DOI: https://doi.org/10.1007/s00220-014-2016-y