Renormalization group flows and continual Lie algebras

TL;DR

This paper explores the renormalization group flows of two-dimensional sigma models, revealing their connection to an infinite-dimensional algebra and integrable Toda equations, with applications to geometric deformations and string theory phenomena.

Contribution

It introduces a continual Lie algebra framework for RG flows, providing explicit solutions, and applies this to models like the sausage and black hole geometries, linking to string theory.

Findings

RG flows form a continual Toda system based on G(d/dt;1)

Explicit solutions for RG flows using free fields and Bäcklund transformations

New interpretations of geometric models and string phenomena in the RG context

Abstract

We study the renormalization group flows of two-dimensional metrics in sigma models and demonstrate that they provide a continual analogue of the Toda field equations based on the infinite dimensional algebra G(d/dt;1). The resulting Toda field equation is a non-linear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time. We provide the general solution of the renormalization group flows in terms of free fields, via Backlund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten's two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormalization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. Finally, we revisit the transition of a flat cone C/Z_n to the plane, as another special solution, and note that tachyon condensation in closed string theory exhibits a hidden relation to the infinite dimensional algebra G(d/dt;1) in the regime of gravity. Its exponential growth holds the key for the construction of conserved currents and their systematic interpretation in string theory, but they still remain unknown.