Spectral theoretic characterization of the massless Dirac operator

TL;DR

This paper provides a spectral characterization of the massless Dirac operator on 3D manifolds, deriving an explicit asymptotic formula for the spectral function and identifying conditions that uniquely identify the operator as a massless Dirac operator.

Contribution

It introduces a new spectral criterion involving asymptotic coefficients to determine when an elliptic operator is a massless Dirac operator on half-densities.

Findings

Derived explicit two-term asymptotic formula for the spectral function.

Identified conditions on the subprincipal symbol and spectral asymptotics for the operator to be a massless Dirac operator.

Connected geometric objects like metric, torsion, and topological charge to spectral properties.

Abstract

We consider an elliptic self-adjoint first order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive lambda. We derive an explicit two-term asymptotic formula for the spectral function as lambda tends to plus infinity, expressing the second asymptotic coefficient via the trace of the subprincipal symbol and the geometric objects encoded within the principal symbol - metric, torsion of the teleparallel connection and topological charge. We then address the question: is our operator a massless Dirac operator on half-densities? We prove that it is a massless Dirac operator on half-densities if and only if the following two conditions are satisfied at every point of the manifold: a) the subprincipal symbol is proportional to the identity matrix and b) the second asymptotic coefficient of the spectral function is zero.