  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 $\lambda$ parameter in our kinetic term approaches
$\lambda\to-\infty$. 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 $\tau$ 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 $\tau$
turns out to play the role of a dilaton for anisotropic scale transformations.
The map between Perelman's flow and the localization equations in 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.
