Lifshitz Weyl Anomalies in Two and Three Dimensions

Our modern understanding of the forces of nature as described by quantum field theories is fundamentally based on symmetries and their associated conservation laws. Quantum anomalies occur when a symmetry of a classical field theory is violated upon quantization. Gravitational anomalies of one-loop quantum effective actions arise after coupling classical field theories to external background geometry and integrating out all dynamical matter fields in the partition function. A gravitational Weyl anomaly of a relativistic field theory is the statement that the quantum effective action is not invariant under local rescaling of the background geometry.

In this work, we study Weyl anomalies in non-relativistic Lifshitz field theories in (1+1) and (2+1) dimensions. Lifshitz field theories introduce a degree of scaling anisotropy between space and time measured by the dynamical scaling exponent z. In 1+1 dimensions, we analyze and study the physical and mathematical nature of a particular z=1 and z=2 Lifshitz Weyl anomaly. We then use the Fujikawa method to derive the z=1 Lifshitz Weyl anomaly from a two-dimensional massless chiral field theory. We also derive the (1+1)-dimensional z=2 Lifshitz Weyl anomaly from a (2+1)-dimensional non-relativistic Chern-Simons action on a manifold with a boundary. We evaluate the z=1 Lifshitz Weyl anomaly on the Mobius strip and relate it to a topological invariant that counts the parity of its number of half-twists.

In 2+1 dimensions, we extend a background metric optimization procedure for Euclidean path integrals, first introduced for a two-dimensional conformal field theory, to a z=2 anisotropically scale-invariant (2+1)-dimensional Lifshitz field theory of a free massless scalar field. We find optimal geometries for static and dynamic correlation functions. For the static correlation functions, the optimal background metric is equivalent to an AdS metric on a Poincare patch, while for dynamical correlation functions, we find a Lifshitz-like metric.