Spectral theoretic characterization of the massless Dirac operator

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.