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arXiv · 2304.04957

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

Yiannis Loizides
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Abstract

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.

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Yiannis Loizides. 2023-04-11. Families of elliptic boundary problems and index theory of the Atiyah-Bott classes. https://arxiv.org/abs/2304.04957

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