A Note on the Einstein-Hilbert action and the Dirac operator on R^n

TL;DR

This paper extends the relation between Einstein-Hilbert action and the Dirac operator from closed spin manifolds to R^n with a suitable metric, using complex powers and heat kernel techniques.

Contribution

It introduces a method to define the regularised Wodzicki Residue on R^n and generalizes the known relation to non-compact settings.

Findings

Established the relation on R^n with a suitable metric.

Defined regularised Wodzicki Residue on R^n.

Used heat kernel properties to derive the result.

Abstract

We prove an extension to R^n, endowed with a suitable metric, of the relation between the Einstein-Hilbert action and the Dirac operator which holds on closed spin manifolds. By means of complex powers, we first define the regularised Wodzicki Residue for a class of operators globally defined on R^n. The result is then obtained by using the properties of heat kernels and generalised Laplacians.