Topological calculation of the phase of the determinant of a non self-adjoint elliptic operator

TL;DR

This paper investigates the phase of the determinant of non self-adjoint elliptic operators on odd-dimensional manifolds, showing it can be a topological invariant under certain spectral symmetry conditions, with applications in geometry and physics.

Contribution

It establishes a topological interpretation of the phase of the determinant for a class of elliptic operators, providing rigorous proofs and calculations in previously heuristic areas.

Findings

Determinant is real when spectrum is symmetric about the imaginary axis.

The sign of the determinant depends on eigenvalues on the positive imaginary axis.

The phase of the determinant can be expressed as a topological invariant.

Abstract

We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is real and its sign is determined by the parity of the number of the eigenvalues of the operator, which lie on the positive part of the imaginary axis. It follows that, for many geometrically defined operators, the phase of the determinant is a topological invariant. In numerous examples, coming from geometry and physics, we calculate the phase of the determinants in purely topological terms. Some of those examples were known in physical literature, but no mathematically rigorous proofs and no general theory were available until now.