Lower dimensional volumes and the Kastler-Kalau-Walze type theorem for Manifolds with Boundary

TL;DR

This paper introduces lower dimensional volumes for spin manifolds with boundary, computes specific cases in 5 and 6 dimensions, and establishes a Kastler-Kalau-Walze type theorem for these manifolds.

Contribution

It defines new lower dimensional volume invariants for manifolds with boundary and extends the Kastler-Kalau-Walze theorem to these cases.

Findings

Computed ${ m Vol}^{(2,2)}$ for 5- and 6-dimensional manifolds with boundary

Established a Kastler-Kalau-Walze type theorem for manifolds with boundary

Extended geometric analysis tools to manifolds with boundary

Abstract

In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume ${\rm Vol}^{(2,2)}$ for 5-dimensional and 6-dimensional spin manifolds with boundary and we also get the Kastler-Kalau-Walze type theorem in this case.