Newton–Cartan trace anomalies and renormalization group flows

I will discuss trace anomalies for non-relativistic Schrödinger theories in 2+1 dimensions coupled to a Newton–Cartan gravity background, which is used as a source of the energy-momentum tensor. The motivation is to identify candidates for a possible non-relativistic version of the a-theorem for theories with RG flows interpolating between an UV and an IR Schroedinger-invariant non-relativistic conformal fixed points. I will first discuss the general structure of the anomaly, which is determined by the Wess–Zumino consistency condition. Then I will present an explicit calculation for the anomaly in the case of a free scalar and of a free fermion, using heat kernel. There is a type A anomaly which is proportional to 1/m, where m is the mass of the particle. In analogy with the relativistic case, the irreversibility properties of the renormalization group can also be investigated by studying the Wess–Zumino consistency conditions for the trace anomaly of the theory in a Newton Cartan background with space-time dependent couplings.