Title: Topological Quantum Gravity of the Ricci Flow

Abstract: Using the techniques of cohomological quantum field theories, we construct a topological nonrelativistic quantum gravity, designed around the mathematical theory of the Ricci flow on Riemannian manifolds, as used by Perelman in his proof of the Poincare conjecture and Thurston's geometrization theorem. This quantum field theory brings together three, previously not directly related areas: Topological field theory of the cohomological type, nonrelativistic quantum gravity of the Lifshitz type, and the mathematics of geometric flows on manifolds. We identify the precise theory whose path integral localizes to the solutions of the Perelman's version of the Ricci flow equations, and identify various features of Perelman's construction in the physics language, such as Perelman's entropy functional (which plays the role of our super potential), Perelman's dilaton field (which maps to the lapse function of nonrelativistic gravity), and others. With this embedding of Perelman's Ricci flow to topological quantum gravity, many intriguing results about the structure of the flow accumulated on the mathematical side in the past two decades can now be imported into the physical picture in the path integral formulation of quantum gravity.

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