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Register a new userSingular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface
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In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view.
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In this note we quantize the usual symplectic (K{a}hler) form on the vortex moduli space by modifying the Quillen metric of the Quillen determinant line bundle.
We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows that a disk with two corners admits a conformal metric with constant Gauss curvature and constant geodesic curvature on its boundary if and only if the two corners have the same angle. In fact, we can classify all the solutions in a more general situation, that of the 2-sphere cut by two planes.
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