Index formulae for mixed boundary conditions on manifolds with corners

TL;DR

This paper develops a new index formula for Dirac operators on manifolds with corners, incorporating mixed boundary conditions using a Lie groupoid approach and heat kernel methods.

Contribution

It introduces a novel glueing construction and Lie groupoid framework to compute the index of Dirac operators with mixed boundary conditions on complex manifolds.

Findings

Derived a general Atiyah-Singer type index formula

Established a Lie groupoid and heat kernel approach for manifolds with corners

Extended index theory to include mixed boundary conditions

Abstract

We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calculate the index we introduce a glueing construction and a corresponding Lie groupoid. We describe the Dirac operator subject to mixed boundary conditions via an equivariant family of Dirac operators on the fibers of the Lie groupoid. Using a heat kernel method with rescaling we derive a general index formula of the Atiyah-Singer type.