Perelman's Ricci Flow in Topological Quantum Gravity

Abstract

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

Figures (1)

• Figure 1: A typical qualitative example of a solution of Perelman's Ricci flow equations in $3+1$ dimensions, which includes both topology-changing transitions at time instants $t_s$ and $t'_s$ and Perelman's extinction of positively curved spatial regions. (a): The evolution in the Perelman frame, and (b): in our frame.