Axial anomalies of Lifshitz fermions

TL;DR

This paper calculates the axial anomaly for Lifshitz fermions with anisotropic scaling, showing it matches the relativistic case, and explores implications for chiral symmetry breaking in Horava-Lifshitz gravity.

Contribution

It demonstrates that the axial anomaly for Lifshitz fermions is identical to the relativistic case and applies spectral methods to verify the eta-invariant equality, extending anomaly analysis to non-relativistic gravity.

Findings

The axial anomaly matches the relativistic case for Lifshitz fermions.

The eta-invariant of Lifshitz and Dirac operators are equal in three dimensions.

Chiral symmetry breaking is possible in Horava-Lifshitz gravity under certain conditions.

Abstract

We compute the axial anomaly of a Lifshitz fermion theory with anisotropic scaling z=3 which is minimally coupled to geometry in 3+1 space-time dimensions. We find that the result is identical to the relativistic case using path integral methods. An independent verification is provided by showing with spectral methods that the eta-invariant of the Dirac and Lifshitz fermion operators in three dimensions are equal. Thus, by the integrated form of the anomaly, the index of the Dirac operator still accounts for the possible breakdown of chiral symmetry in non-relativistic theories of gravity. We apply this framework to the recently constructed gravitational instanton backgrounds of Horava-Lifshitz theory and find that the index is non-zero provided that the space-time foliation admits leaves with harmonic spinors. Using Hitchin's construction of harmonic spinors on Berger spheres, we obtain explicit results for the index of the fermion operator on all such gravitational instanton backgrounds with SU(2)xU(1) isometry. In contrast to the instantons of Einstein gravity, chiral symmetry breaking becomes possible in the unimodular phase of Horava-Lifshitz theory arising at lambda = 1/3 provided that the volume of space is bounded from below by the ratio of the Ricci to Cotton tensor couplings raised to the third power. Some other aspects of the anomalies in non-relativistic quantum field theories are also discussed.