Families of elliptic boundary problems and index theory of the Atiyah-Bott classes

We study a natural family of non-local elliptic boundary problems on a compact oriented surface $\mathrm{\Sigma }$ parametrized by the moduli space ${\mathcal{M}}_{\mathrm{\Sigma }}$ of flat $G$-connections with framing along $\mathrm{\partial }\mathrm{\Sigma }$. This family generalizes one introduced by Atiyah and Bott for closed surfaces. In earlier work we constructed an analytic index morphism out of a subring of the K-theory of ${\mathcal{M}}_{\mathrm{\Sigma }}$. In this article we apply that morphism to the K-class of the Fredholm family and derive cohomological formulas. The main application is to calculate K-theory intersection pairings on symplectic quotients of ${\mathcal{M}}_{\mathrm{\Sigma }}$; the latter are compact moduli spaces of flat connections on surfaces with boundary, where the boundary holonomies lie in prescribed conjugacy classes. The results provide a gauge theory analogue of the Teleman-Woodward index formula.