Gradient flows and instantons at a Lifshitz point

TL;DR

This paper develops a framework for embedding gradient flow equations in non-relativistic models with anisotropic scaling, exploring instantons in Lifshitz gravity and their connections to non-equilibrium theories.

Contribution

It introduces a broad framework for gradient flows in Lifshitz-type models and reviews instanton solutions in Horava-Lifshitz gravity as geometric flows.

Findings

Instantons in Lifshitz gravity can be viewed as geometric flow solutions.

Chiral instantons occur at anisotropic scaling exponent z=3.

Lifshitz theories serve as toy models for vacuum selection in relativistic theories.

Abstract

I provide a broad framework to embed gradient flow equations in non-relativistic field theory models that exhibit anisotropic scaling. The prime example is the heat equation arising from a Lifshitz scalar field theory; other examples include the Allen-Cahn equation that models the evolution of phase boundaries. Then, I review recent results reported in arXiv:1002.0062 describing instantons of Horava-Lifshitz gravity as eternal solutions of certain geometric flow equations on 3-manifolds. These instanton solutions are in general chiral when the anisotropic scaling exponent is z=3. Some general connections with the Onsager-Machlup theory of non-equilibrium processes are also briefly discussed in this context. Thus, theories of Lifshitz type in d+1 dimensions can be used as off-shell toy models for dynamical vacuum selection of relativistic field theories in d dimensions.