Chiral Anomaly in Contorted Spacetimes

TL;DR

This paper reexamines the chiral anomaly in spacetimes with torsion, clarifying that only the Pontrjagin four-form appears in the anomaly, challenging recent claims about the Nieh-Yan term, with implications for quantum gravity and cosmic strings.

Contribution

It demonstrates that only the Pontrjagin four-form contributes to the chiral anomaly in Riemann-Cartan spacetimes, refuting the recent claim about the Nieh-Yan term's involvement.

Findings

Only the Pontrjagin four-form appears in the anomaly.

The Nieh-Yan term does not contribute to the chiral anomaly.

Implications for cosmic strings and quantum gravity are discussed.

Abstract

The Dirac equation in Riemann-Cartan spacetimes with torsion is reconsidered. As is well-known, only the axial covector torsion $A$, a one-form, couples to massive Dirac fields. Using diagrammatic techniques, we show that besides the familiar Riemannian term only the Pontrjagin type four-form $dA\wedge dA$ does arise additionally in the chiral anomaly, but not the Nieh-Yan term $d ^* A$, as has been claimed recently. Implications for cosmic strings in Einstein-Cartan theory as well as for Ashtekar's canonical approach to quantum gravity are discussed.