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We present a class of topological quantum gravity theories associated with the geometric theory of the Ricci flow on Riemannian manifolds. The most primitive theory is a nonrelativistic topological gravity of the spatial metric, whose path integral is localized to the solutions of Hamilton’s Ricci flow. After gauging foliation-preserving diffeomorphisms of spacetime, we establish connection to Perelman’s Ricci flow: The role of Perelman’s dilaton is played by the nonprojectable lapse function, and the F and W-functionals appear as our superpotential.

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