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Anomalies for Galilean fields

Kristan Jensen

Abstract

We initiate a systematic study of `t Hooft anomalies in Galilean field theories, focusing on two questions therein. In the first, we consider the non-relativistic theories obtained from a discrete light-cone quantization (DLCQ) of a relativistic theory with flavor or gravitational anomalies. We find that these anomalies survive the DLCQ, becoming mixed flavor/boost or gravitational/boost anomalies. We also classify the pure Weyl anomalies of Schrödinger theories, which are Galilean conformal field theories (CFTs) with $z=2$. There are no pure Weyl anomalies in even spacetime dimension, and the lowest-derivative anomalies in odd dimension are in one-to-one correspondence with those of a relativistic CFT in one dimension higher. These results classify many of the anomalies that arise in the field theories dual to string theory on Schrödinger spacetimes.

Anomalies for Galilean fields

Abstract

We initiate a systematic study of `t Hooft anomalies in Galilean field theories, focusing on two questions therein. In the first, we consider the non-relativistic theories obtained from a discrete light-cone quantization (DLCQ) of a relativistic theory with flavor or gravitational anomalies. We find that these anomalies survive the DLCQ, becoming mixed flavor/boost or gravitational/boost anomalies. We also classify the pure Weyl anomalies of Schrödinger theories, which are Galilean conformal field theories (CFTs) with . There are no pure Weyl anomalies in even spacetime dimension, and the lowest-derivative anomalies in odd dimension are in one-to-one correspondence with those of a relativistic CFT in one dimension higher. These results classify many of the anomalies that arise in the field theories dual to string theory on Schrödinger spacetimes.

Paper Structure

This paper contains 12 sections, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Newton-Cartan geometry
  4. Null reduction
  5. Flavor and/or gravitational anomalies from DLCQ
  6. Anomaly inflow
  7. Constructing the counterterm
  8. The anomalous boost Ward identity
  9. Weyl anomalies for $z=2$
  10. The basic idea
  11. Two spatial dimensions
  12. The general result

Figures (1)

  • Figure 1: The short sequence which relates the anomalies of a relativistic theory to those of the $d$-dimensional NR theory realized by DLCQ. The relativistic parent lives on $\mathcal{M}_{d+1}$, and its null reduction leads to the NR theory on $\mathcal{M}_d$. The anomalies of the parent are described via inflow by letting $\mathcal{M}_{d+1}$ be the boundary of a $d+2$-manifold $\mathcal{M}_{d+2}$ which we equip with a Chern-Simons form $I$. The anomaly polynomial $\mathcal{P}$ is a formal $d+3$ form which may be thought of as living on a formal $\mathcal{M}_{d+3}$.