Topological quantum gravity of the Ricci flow

We present a family of topological quantum gravity theories asso-ciated with the geometric theory of the Ricci flow on Riemannian manifolds. First we use BRST quantization to construct a “prim-itive” topological Lifshitz-type theory for only the spatial metric, with spatial diffeomorphism invariance and no gauge symmetry, as-sociated with Hamilton’s Ricci flow: Hamilton’s flow equation ap-pears 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 anN = 2extendedBRSTsymmetry.First,wegaugespatialdiffeo-morphisms, and show that this gives us access to the mathematical technique known as the DeTurck trick. Finally, we gauge foliation-preserving time reparametrizations, +L3mtz9r7A9ATwj/cY/bCgeLSmBUboth with the projectable and nonprojectable lapse function. The path integral of the full theory is localized to the solutions of Ricci-type flow equations, general-izing those of Perelman. The role of Perelman’s dilaton is played by the nonprojectable lapse function. Perelman’s F-functional ap-pears 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.