Gravitational anomalies on the Newton-Cartan background

Abstract

We derive the trace and diffeomorphism anomalies of the Schrödinger field minimally coupled to the Newton-Cartan background using Fujikawa's path integral approach. This approach in particular enables us to calculate the one-loop contributions due to all the fields of the NewtonCartan structure. We determine the coefficients and demonstrate that gravitational anomalies for this theory always arise in odd dimensions. Due to the gauge field contribution of the background we find that in 2 + 1 dimensions the trace anomaly contains terms which have a form similar to that of the 1 + 1 and 3 + 1 dimensional relativistic trace anomalies. This result reveals that the Newton-Cartan background which satisfies the Frobenius condition possesses a Type A trace anomaly in contrast with the result of Lishitz spacetimes. As an application we demonstrate that the coefficient of the term similar to the 1 + 1 dimensional relativistic trace anomaly satisfies a c-theorem condition.

Trace anomaly for non-relativistic fermions

“…The first explicit calculation of anomalies for a physical system was performed in [27] with the Heat Kernel (HK) method, for the case of a free scalar. Later this result was confirmed in [28] using Fujikawa approach. 2 On the other hand, both the calculations in [27,28] disagree with [29].…”

“…Later this result was confirmed in [28] using Fujikawa approach. 2 On the other hand, both the calculations in [27,28] disagree with [29].…”

“…First of all, we show that it is possible to couple the fermion to the 1 An interesting subtlety is that, in order to get a type-A anomaly, gravity backgrounds which do not satisfy Frobenius condition must be considered [21,25]. 2 In [28] an extra term is present, which could not be detected with the background chosen in [27]; this issue deserves further study which goes beyond the purpose of the present paper.…”

“…In its essence, this regularization is not different from the one normally used in the relativistic case to adapt the heat kernel procedure to fermions: to make elliptic the Dirac operator, one first considers its square, perform the heat kernel, and then takes the square root of the resulting determinant. An alternative way to compute the trace anomaly is by the Fujikawa's method: this was recently performed in [28], obtaining the same numerical result for the scalar case. An alternative treatment of the heat kernel procedure, applied to parabolic operators, was developed in [29], obtaining a results which disagrees with both [27] and [28].…”

“…An alternative way to compute the trace anomaly is by the Fujikawa's method: this was recently performed in [28], obtaining the same numerical result for the scalar case. An alternative treatment of the heat kernel procedure, applied to parabolic operators, was developed in [29], obtaining a results which disagrees with both [27] and [28].…”