Ricci flows and infinite dimensional algebras

TL;DR

This paper explores Ricci flows in two-dimensional sigma models, revealing their description as Toda systems via a novel infinite-dimensional Lie algebra, and provides explicit solutions with applications to string theory.

Contribution

It introduces a new infinite-dimensional Lie algebra framework for Ricci flows in 2D, enabling systematic analysis and explicit solutions, with potential extensions to higher dimensions.

Findings

Ricci flows in 2D sigma models can be formulated as Toda systems.

A novel infinite-dimensional Lie algebra with exponential growth is constructed.

Explicit solutions include the sausage model and conical singularity decay.

Abstract

The renormalization group equations of two-dimensional sigma models describe geometric deformations of their target space when the world-sheet length changes scale from the ultra-violet to the infra-red. These equations, which are also known in the mathematics literature as Ricci flows, are analyzed for the particular case of two-dimensional target spaces, where they are found to admit a systematic description as Toda system. Their zero curvature formulation is made possible with the aid of a novel infinite dimensional Lie algebra, which has anti-symmetric Cartan kernel and exhibits exponential growth. The general solution is obtained in closed form using Backlund transformations, and special examples include the sausage model and the decay process of conical singularities to the plane. Thus, Ricci flows provide a non-linear generalization of the heat equation in two dimensions with the same dissipative properties. Various applications to dynamical problems of string theory are also briefly discussed. Finally, we outline generalizations to higher dimensional target spaces that exhibit sufficient number of Killing symmetries.