Dirac Functional Determinants in Terms of the Eta Invariant and the Noncommutative Residue

TL;DR

This paper rigorously constructs zeta and eta-functions for Dirac operators on manifolds, clarifies ambiguities in fermion determinants, and expresses results using noncommutative residues, connecting to recent mathematical findings.

Contribution

It provides a rigorous analysis of Dirac functional determinants, resolving ambiguities and expressing results via noncommutative residues, with explicit formulas and connections to recent literature.

Findings

Ambiguity in massless case determinants is resolved in the massive case.

Results express determinants in terms of noncommutative residues.

Explicit formulas are derived after resummation of mass series expansions.

Abstract

The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrisic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the noncommutative residue. The ambiguity is here seen to disappear in the massive case, giving rise to a phase of the Dirac determinant - that agrees with very recent calculations appeared in the mathematical literature - and to a multiplicative anomaly - also in agreement with other calculations, in the coinciding situations. After explicit, nontrivial resummation of the mass series expansions involving zeta and eta functions, the results are expressed in terms of quite simple formulas.