UC Berkeley

In this thesis we construct a family of topological quantum gravity theories with the goal of finding a regime in which these theories' localization equations are Perelman's celebrated Ricci flow equations. The natural setting for these theories is a nonrelativistic N = 2 superspace in which the spacetime is foliated by leaves of constant time. The basis for this construction is a "primitive" topological theory involving the spatial metric and its corresponding spatial diffeomorphism invariance, with N = 2 BRST symmetry. The algebra of the BRST charges is chosen so that the localization equations derived from a BRST-invariant action are flow equations and the couplings in the action are chosen so that these flow equations take the exact form of Hamilton's Ricci flow equations. Gauging spatial diffeomorphisms and foliation-preserving time reparametrizations (initially in two separate steps, but later in a single sweeping step) leads us to a set of geometric constraints that produce three distinct classes of field content in the BRST multiplets involved. Notably, the superpotential of the gauged theory is precisely Perelman's F-functional and the role of his dilaton is played by our nonrelativistic lapse function. Perelman's Ricci flow equations are then obtained as localization equations from the gauged theory in the same way as Hamilton's were obtained from the primitive theory (up to an interesting reframing), satisfying our goal.