Index Theory on Incomplete Cusp Edge Spaces

TL;DR

This paper develops index theory for Dirac operators on incomplete cusp edge spaces, constructing heat kernels and proving self-adjointness and Fredholm properties, leading to an index formula including a signature formula.

Contribution

It introduces a new index formula for Dirac operators on incomplete cusp edge spaces, extending previous theories to these singular geometries.

Findings

Constructed heat kernel for Laplace-type operators.

Proved Dirac operators are self-adjoint and Fredholm.

Derived an index formula including a signature formula.

Abstract

We study Dirac-type operators on incomplete cusp edge spaces with invertible boundary families. In particular, we construct the heat kernel for the associated Laplace-type operator and prove that the Dirac operators are essentially self-adjoint and Fredholm on their unique self adjoint domain. Using the asymptotics of the heat kernel and a generalisation of Getzler's rescaling argument we establish an index formula for these operators including a signature formula for the Hodge-de Rham operator on Witt incomplete cusp edge spaces.