  We study a natural family of non-local elliptic boundary problems on a
compact oriented surface $\Sigma$ parametrized by the moduli space
$\mathcal{M}_\Sigma$ of flat $G$-connections with framing along $\partial
\Sigma$. 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 $\mathcal{M}_\Sigma$. 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 $\mathcal{M}_\Sigma$; 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.
