Gravitational anomalies on the Newton-Cartan background

We derive the trace and diffeomorphism anomalies of the Schr\"odinger 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 Newton-Cartan 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.