On the Riemannian geometry of Seiberg–Witten moduli spaces

We construct a natural L2L2-metric on the perturbed Seiberg–Witten moduli spaces Mμ+Mμ+ of a compact 4-manifold MM, and we study the resulting Riemannian geometry of Mμ+Mμ+. We derive a formula which expresses the sectional curvature of Mμ+Mμ+ in terms of the Green operators of the deformation complex of the Seiberg–Witten equations. In case MM is simply connected, we construct a Riemannian metric on the Seiberg–Witten principal U(1)U(1) bundle P→Mμ+P→Mμ+ such that the bundle projection becomes a Riemannian submersion. On a Kähler surface MM, the L2L2-metric on Mμ+Mμ+ coincides with the natural Kähler metric on moduli spaces of vortices.