Lattice calculations on the spectrum of Dirac and Dirac-K\"ahler operators

TL;DR

This paper introduces a matrix-based method for calculating the spectrum and analytical index of elliptic operators like Dirac and Dirac-Kähler on compact Riemannian manifolds, providing exact results and insights.

Contribution

It presents a novel matrix technique to exactly compute spectra and indices of elliptic operators, including Dirac and Dirac-Kähler, on specific manifolds.

Findings

Exact spectrum of Dirac operator on torus and sphere obtained

Analytical index of Dirac operator calculated on these manifolds

Spectrum of Dirac-Kähler operator on the sphere explored

Abstract

We present a matrix technique to obtain the spectrum and the analytical index of some elliptic operators defined on compact Riemannian manifolds. The method uses matrix representations of the derivative which yield exact values for the derivative of a trigonometric polynomial. These matrices can be used to find the exact spectrum of an elliptic operator in particular cases and in general, to give insight into the properties of the solution of the spectral problem. As examples, the analytical index and the eigenvalues of the Dirac operator on the torus and on the sphere are obtained and as an application of this technique, the spectrum of the Dirac-Kahler operator on the sphere is explored.