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Axial anomalies of Lifshitz fermions

Ioannis Bakas, Dieter Lust

TL;DR

We demonstrate that axial anomalies for massless Lifshitz fermions with z=3 in 3+1 dimensions reproduce the relativistic local anomaly forms: ∂_μ J_5^μ = -(1/8π^2) Tr(F ∧ F) and ∇_μ J_5^μ = -(1/192π^2) Tr(R ∧ R). An explicit Lifshitz–Dirac index theorem is established by matching the eta-invariant of the Lifshitz operator to the Dirac operator on spatial leaves, and the integrated anomaly equals the Dirac index on gravitational instanton backgrounds. In Hořava–Lifshitz gravity, instantons arising in the unimodular phase (λ = 1/3) can produce a nonzero fermion index when the Berger-sphere leaves develop sufficient harmonic spinors, i.e., δ>4, implying possible chiral symmetry breaking and baryon/lepton number violation in suitable regimes. The analysis includes explicit Berger-sphere spectra, eta-invariant calculations, and topological checks showing vanishing Euler and signature invariants for these instanton spaces, with comparisons to Taub–NUT and Eguchi–Hanson relativistic instantons. Overall, the results reveal that nonrelativistic gravity preserves the anomaly’s topological character, while enabling novel chiral phenomena under specific geometric and unimodular 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.

Axial anomalies of Lifshitz fermions

TL;DR

We demonstrate that axial anomalies for massless Lifshitz fermions with z=3 in 3+1 dimensions reproduce the relativistic local anomaly forms: ∂_μ J_5^μ = -(1/8π^2) Tr(F ∧ F) and ∇_μ J_5^μ = -(1/192π^2) Tr(R ∧ R). An explicit Lifshitz–Dirac index theorem is established by matching the eta-invariant of the Lifshitz operator to the Dirac operator on spatial leaves, and the integrated anomaly equals the Dirac index on gravitational instanton backgrounds. In Hořava–Lifshitz gravity, instantons arising in the unimodular phase (λ = 1/3) can produce a nonzero fermion index when the Berger-sphere leaves develop sufficient harmonic spinors, i.e., δ>4, implying possible chiral symmetry breaking and baryon/lepton number violation in suitable regimes. The analysis includes explicit Berger-sphere spectra, eta-invariant calculations, and topological checks showing vanishing Euler and signature invariants for these instanton spaces, with comparisons to Taub–NUT and Eguchi–Hanson relativistic instantons. Overall, the results reveal that nonrelativistic gravity preserves the anomaly’s topological character, while enabling novel chiral phenomena under specific geometric and unimodular 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.

Paper Structure

This paper contains 33 sections, 304 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Axial anomalies of Lifshitz fermions
  3. Lifshitz fermions and axial symmetry
  4. Gauge field contribution to the anomaly
  5. Metric field contribution to the anomaly
  6. Instantons of Hořava--Lifshitz gravity
  7. General considerations
  8. Unimodular phase at special values of $\lambda$
  9. Instantons with $SU(2) \times U(1)$ isometry
  10. Sectional Ricci curvature of instantons
  11. The index of Lifshitz fermion operator
  12. Atiyah--Patodi--Singer index theorem
  13. Lifshitz operator on Berger spheres
  14. Computation of the $\eta$--invariant
  15. Chiral symmetry breaking by instantons
  16. ...and 18 more sections

Figures (6)

  • Figure 1: Schematic picture of an interpolating instanton solution $\mathbb{R} \times \Sigma_3$
  • Figure 2: Effective potential barriers for instanton tunneling with varying $\omega$
  • Figure 3: Dependence of curvature of Berger spheres upon squashing and stretching
  • Figure 4: Level crossing of the modes $Z_-(\delta)$ from negative to positive values
  • Figure 5: Effective potential for $SU(2) \times U(1)$ mixmaster dynamics in Einstein gravity
  • ...and 1 more figures