Lifshitz Scale Anomalies

TL;DR

This paper develops a comprehensive framework to analyze Lifshitz scale anomalies across various dimensions and dynamical exponents, revealing that all such anomalies are trivial descents, with detailed cohomology classifications provided.

Contribution

It introduces a detailed cohomological framework for Lifshitz scale anomalies applicable in multiple dimensions and dynamical exponents, including complete classifications and selection rules.

Findings

All Lifshitz scale anomalies are trivial descents (B-type).

Complete cohomology classifications for 1, 2, and 3 dimensions.

Comparison between Lifshitz and conformal anomalies for dynamical exponent one.

Abstract

We analyse scale anomalies in Lifshitz field theories, formulated as the relative cohomology of the scaling operator with respect to foliation preserving diffeomorphisms. We construct a detailed framework that enables us to calculate the anomalies for any number of spatial dimensions, and for any value of the dynamical exponent. We derive selection rules, and establish the anomaly structure in diverse universal sectors. We present the complete cohomologies for various examples in one, two and three space dimensions for several values of the dynamical exponent. Our calculations indicate that all the Lifshitz scale anomalies are trivial descents, called B-type in the terminology of conformal anomalies. However, not all the trivial descents are cohomologically non-trivial. We compare the conformal anomalies to Lifshitz scale anomalies with a dynamical exponent equal to one.