Abstract
We consider the vortex equations for a U(n) gauge field A coupled to a Higgs field \({\phi}\) with values on the n × n matrices. It is known that when these equations are defined on a compact Riemann surface Σ, their moduli space of solutions is closely related to a moduli space of τ-stable holomorphic n-pairs on that surface. Using this fact and a local factorization result for the matrix \({\phi}\) , we show that the vortex solutions are entirely characterized by the location in Σ of the zeros of det \({\phi}\) and by the choice of a vortex internal structure at each of these zeros. We describe explicitly the vortex internal spaces and show that they are compact and connected spaces.
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References
Abrikosov, A.: On the magnetic properties of superconductors of second group. Sov. Phys. JETP 5, 1174 (1957) [Zh. Eksp. Teor. Fiz. 32, 1442 (1957)]; Nielsen, H., Olesen, P.: Vortex-line models for dual strings. Nucl. Phys. B61, 45 (1973)
Auzzi R., Bolognesi S., Evslin J., Konishi K., Yung A.: Nonabelian Superconductors: Vortices and Confinement in \({\mathcal{N}=2}\) SQCD. Nucl. Phys. B673, 187–216 (2003)
Auzzi R., Shifman M., Yung A.: Composite non-abelian flux tubes in N = 2 SQCD. Phys. Rev. D73, 105012 (2006)
Baptista J.M.: Vortex equations in abelian gauged sigma-models. Commun. Math. Phys. 261, 161–194 (2006)
Bertram A., Daskalopoulos G., Wentworth R.: Gromov invariants for holomorphic maps from Riemann surfaces to grassmannians. J. Amer. Math. Soc. 9, 529–571 (1996)
Bradlow S.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135, 1–17 (1990)
Bradlow, S., Daskalopoulos, G., Garcí a-Prada, O., Wentworth, R.: Stable augmented bundles over Riemann surfaces. In: Vector bundles in algebraic geometry, London Math. Soc. Lecture Note Ser. 208, Cambridge: CUP, 1995, pp. 15–67
Eto, M., Evslin, J., Konishi, K., Marmorini, G., Nitta, M., Ohashi, K., Vinci, W., Yokoi, N.: On the moduli space of semilocal strings and lumps. Phys. Rev. D76, 105002 (2007) Shifman, M., Yung, A.: Non-abelian semilocal strings in N=2 supersymmetric QCD. Phys. Rev. D73, 125012 (2006) Popov, A.: Non-abelian vortices on Riemann surfaces: an integrable case. Lett. Math. Phys. 84, 139–148 (2008)
Eto, M., Isozumi, Y., Nitta, M., Ohashi, K., Sakai, N.: Moduli space of non-abelian vortices. Phys. Rev. Lett. 96, 161601 (2006) Eto, M., Isozumi, Y., Nitta, M., Ohashi, K., Sakai, N.: Solitons in the Higgs phase – the moduli matrix approach –. J. Phys. A39, R315–R392 (2006) Eto, M., Fujimori, T., Gudnason, S., Konishi, K., Nitta, M., Ohashi, K., Vinci, W.: Constructing non-abelian vortices with arbitrary gauge groups. Phys. Lett. B669, 98–101 (2008)
Eto M., Konishi K., Marmorini G., Nitta M., Ohashi K., Vinci W., Yokoi N.: Non-abelian vortices of higher winding numbers. Phys. Rev. D74, 065021 (2006)
Hanany A., Tong D.: Vortices, instantons and branes. JHEP 0307, 037 (2003)
Hashimoto K., Tong D.: Reconnection of non-abelian cosmic strings. JCAP 0509, 004 (2005)
Taubes C.H.: Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations. Commun. Math. Phys. 72, 277–292 (1980)
Tong, D.: Quantum vortex strings: a review. http://arXiv.org/abs/0809.5060v2[hep-th], 2008
Acknowledgement
It is a pleasure to thank Minoru Eto, Muneto Nitta, Keisuke Ohashi and Norisuke Sakai for kindly hosting me during a visit to TITech, Tokyo, three years ago. I am grateful to them and to David Tong for explaining me their work on non-abelian vortices. I am partially supported by the Netherlands Organisation for Scientific Research (NWO) through the Veni grant 639.031.616.
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Communicated by G. W. Gibbons
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Baptista, J.M. Non-Abelian Vortices on Compact Riemann Surfaces. Commun. Math. Phys. 291, 799–812 (2009). https://doi.org/10.1007/s00220-009-0838-9
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DOI: https://doi.org/10.1007/s00220-009-0838-9