Gauss-Bonnet for matrix conformally rescaled Dirac

TL;DR

This paper derives an explicit scalar curvature formula for a two-torus with a conformally rescaled Dirac operator, confirms the Gauss-Bonnet theorem in this setting, and extends the results to all Riemann surfaces.

Contribution

It provides a new explicit formula for scalar curvature under matrix conformal rescaling of Dirac operators and generalizes the Gauss-Bonnet theorem to this context.

Findings

Gauss-Bonnet theorem holds for the conformally rescaled Dirac operator on a two-torus.

Explicit scalar curvature formula derived for the conformally rescaled Dirac operator.

Extension of the results to all Riemann surfaces.

Abstract

We derive an explicit formula for the scalar curvature over a two-torus with a Dirac operator conformally rescaled by a globally diagonalizable matrix. We show that the Gauss-Bonnet theorem holds and extend the result to all Riemann surfaces with Dirac operators modified in the same way.