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Non-abelian vortices on compact Riemann surfaces · Around

2008

Non-abelian vortices on compact Riemann surfaces

Baptista, J. M.

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We consider the vortex equations for a U(n) gauge field coupled to a Higgs field with values on the n times n square matrices.

It is known that when these equations are defined on a compact Riemann surface, their moduli space of solutions is closely related to a moduli space of tau-stable holomorphic n-pairs on that surface.
Using this fact and a local factorization result for the Higgs matrix, we show that the vortex solutions are entirely characterized by (1) the location in the surface of the zeros of the determinant of the Higgs matrix and (2) 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.

Built on

A. Abrikosov : ‘ On the magnetic properties of superconductors of second group
1973
Earlier work this paper cites.
C.H. Taubes : ‘ Arbitrary N N -vortex solutions to the first order Ginzburg-Landau equations
1980
Earlier work this paper cites.
S. Bradlow : ‘ Vortices in holomorphic line bundles over closed Kähler manifolds
1990
Earlier work this paper cites.
S. Bradlow, G. Daskalopoulos, O. García-Prada, R. Wentworth : ‘ Stable augmented bundles over Riemann surfaces
1995
Earlier work this paper cites.
A. Bertram, G. Daskalopoulos and R. Wentworth : ‘ Gromov invariants for holomorphic maps from Riemann surfaces to grassmannians
1996
Earlier work this paper cites.

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