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We find the regime of our recently constructed topological nonrelativisticquantum gravity, in which Perelman's Ricci flow equations on Riemannianmanifolds appear precisely as the localization equations in the path integral.In this mapping between physics and mathematics, the role of Perelman's dilatonis played by our lapse function. Perelman's local fixed volume conditionemerges dynamically as the $\lambda$ parameter in our kinetic term approaches$\lambda\to-\infty$. The DeTurck trick that decouples the metric flow from thedilaton flow is simply a gauge-fixing condition for the gauge symmetry ofspatial diffeomorphisms. We show how Perelman's ${\cal F}$ and ${\cal W}$entropy functionals are related to our superpotential. We explain the origin ofPerelman's $\tau$ function, which appears in the ${\cal W}$ entropy functionalfor shrinking solitons, as the Goldstone mode associated with time translationsand spatial rescalings: In fact, in our quantum gravity, Perelman's $\tau$turns out to play the role of a dilaton for anisotropic scale transformations.The map between Perelman's flow and the localization equations in ourtopological quantum gravity requires an interesting redefinition of fields,which includes a reframing of the metric. With this embedding of Perelman'sequations into topological quantum gravity, a wealth of mathematical results onthe Ricci flow can now be imported into physics and reformulated in thelanguage of quantum field theory.

Perelman's Ricci Flow in Topological Quantum Gravity