### The $L^2$ geometry of vortex moduli spaces

**Speight, M ***(Leeds)*

Thursday 24 February 2011, 15:30-16:30

Seminar Room 1, Newton Institute

#### Abstract

Let L be a hermitian line bundle over a Riemann surface X. A vortex is a pair consisting of a section of and a connexion on L satisfying a certain pair of coupled differential equations similar to the Hitchin equations. The moduli space of vortices is topologically rather simple. The interesting point is that it has a canonical kaehler structure, geodesics of which are conjectured to approximate the low energy dynamics of vortices. In this talk I will review what is known about this kaehler geometry, focussing mainly on the cases where X is the plane, sphere or hyperbolic plane.

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