On the Riemannian geometry of Seiberg–Witten moduli spaces

Abstract

We construct a natural L 2 -metric on the perturbed Seiberg–Witten moduli spaces M μ + of a compact 4-manifold M , and we study the resulting Riemannian geometry of M μ + . We derive a formula which expresses the sectional curvature of M μ + in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case M is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U ( 1 ) bundle P → M μ + such that the bundle projection becomes a Riemannian submersion. On a Kähler surface M , the L 2 -metric on M μ + coincides with the natural Kähler metric on moduli spaces of vortices.