  We present a family of topological quantum gravity theories associated with
the geometric theory of the Ricci flow on Riemannian manifolds. First we use
BRST quantization to construct a "primitive" topological Lifshitz-type theory
for only the spatial metric, with spatial diffeomorphism invariance and no
gauge symmetry, associated with Hamilton's Ricci flow: Hamilton's flow equation
appears as the localization equation of the primitive theory. Then we extend
the primitive theory by gauging foliation-preserving spacetime symmetries.
Crucially, all our theories are required to exhibit an ${\cal N}=2$ extended
BRST symmetry. First, we gauge spatial diffeomorphisms, and show that this
gives us access to the mathematical technique known as the DeTurck trick.
Finally, we gauge foliation-preserving time reparametrizations, both with the
projectable and nonprojectable lapse function. The path integral of the full
theory is localized to the solutions of Ricci-type flow equations, generalizing
those of Perelman. The role of Perelman's dilaton is played by the
nonprojectable lapse function. Perelman's ${\cal F}$-functional appears as the
superpotential of our theory. Since there is no spin-statistics theorem in
nonrelativistic quantum field theory, the two supercharges of our gravity
theory do not have to be interpreted as BRST charges and, after the
continuation to real time, the theory can be studied as a candidate for
nonrelativistic quantum gravity with propagating bosonic and fermionic degrees
of freedom.
