Ricci flows and infinite dimensional algebras

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.

arXiv comment10 pages, contribution to 36th Ahrenshoop meeting, Berlin

References in corpus (5)

• Don't Panic! Closed String Tachyons in ALE Spacetimes
• Critical phenomena in gravitational collapse (Physics Reports)
• Spacetime Energy Decreases under World-sheet RG Flow
• Spacetime decay of cones at strong coupling
• Renormalization group flows and continual Lie algebras

Cited by in corpus (8)

• The integrable harmonic map problem versus Ricci flow
• Irreversibility of World-sheet Renormalization Group Flow
• The algebraic structure of geometric flows in two dimensions
• Geometric flows and (some of) their physical applications
• The Generalized Ricci Flow for 3D Manifolds with One Killing Vector
• Symmetry Reduction of the Two-Dimensional Ricci Flow Equation
• On the integrability of spherical gravitational waves in vacuum
• Reduction of coupling constants in two-dimensional sigma models