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

2023/04/11 by Loizides, Yiannis
#Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2304.04957

Abstract

We study a natural family of non-local elliptic boundary problems on a compact oriented surface Σ parametrized by the moduli space M_Σ of flat G-connections with framing along ∂ Σ. 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 M_Σ. 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 M_Σ; 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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