Ricci flows and infinite dimensional algebras. (English) Zbl 1052.81061
Summary: 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 Bäcklund 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.
MSC:
81T10 | Model quantum field theories |
83C47 | Methods of quantum field theory in general relativity and gravitational theory |
81T17 | Renormalization group methods applied to problems in quantum field theory |
22E70 | Applications of Lie groups to the sciences; explicit representations |
81T20 | Quantum field theory on curved space or space-time backgrounds |
83C75 | Space-time singularities, cosmic censorship, etc. |
81R15 | Operator algebra methods applied to problems in quantum theory |
81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |