On the Heat Kernel and Weyl Anomaly of Schr\"odinger invariant theory

Understand

We propose a method inspired from discrete light cone quantization (DLCQ) to determine the heat kernel for a Schr\"odinger field theory (Galilean boost invariant with $z=2$ anisotropic scaling symmetry) living in $d+1$ dimensions, coupled to a curved Newton-Cartan background starting from a heat kernel of a relativistic conformal field theory ($z=1$) living in $d+2$ dimensions.

• We use this method to show the Schr\"odinger field theory of a complex scalar field cannot have any Weyl anomalies.
• To be precise, we show that the Weyl anomaly $\mathcal{A}^{G}_{d+1}$ for Schr\"odinger theory is related to the Weyl anomaly of a free relativistic scalar CFT $\mathcal{A}^{R}_{d+2}$ via $\mathcal{A}^{G}_{d+1}= 2\pi \delta (m) \mathcal{A}^{R}_{d+2}$ where $m$ is the charge of the scalar field under particle number symmetry.
• We provide further evidence of vanishing anomaly by evaluating Feynman diagrams in all orders of perturbation theory.