The chiral torsional anomaly and the Nieh-Yan invariant with and without boundaries

TL;DR

This paper investigates the torsional contribution to the 4d chiral anomaly, clarifying the role of the Nieh-Yan invariant, boundary effects, and renormalization, with implications for torsional anomalous transport.

Contribution

The study clarifies the conditions under which the Nieh-Yan invariant contributes to the chiral anomaly and introduces explicit counterterms to cancel these contributions, highlighting boundary effects.

Findings

Nieh-Yan invariant vanishes on boundaryless spacetimes with well-defined Dirac index.

Heat kernel expansion breakdown correlates with non-vanishing Nieh-Yan on manifolds without boundaries.

Boundary terms lead to a torsional anomalous Hall effect and can be canceled by explicit counterterms.

Abstract

There exists a long-standing debate regarding the torsion contribution to the 4d chiral anomaly of a Dirac fermion. Central to this debate is the Nieh-Yan anomaly, which has been considered ill-defined and a regularization artifact. Using a heat-kernel approach, we examine the relationship between the Dirac operator index, the Nieh-Yan invariant and the torsional anomaly. We show the Nieh-Yan invariant vanishes on spacetimes without boundaries, if the Dirac index is well-defined. In the known examples of non-vanishing Nieh--Yan invariant on manifolds without boundaries, the heat kernel expansion breaks down, making the index ill-defined. Finally, for finite boundaries we identify several finite bulk and boundary anomaly terms, alongside bulk and boundary Nieh-Yan terms. We construct explicit counterterms that cancel the Nieh-Yan terms and argue that the boundary terms give rise to a torsional anomalous Hall effect. Our results emphasize the importance of renormalization conditions, as these can affect the non-thermal Nieh-Yan anomaly coefficients. In addition, we demonstrate that anomalous torsional transport may arise even without relying on the Nieh-Yan invariant.