The Geometry of Time in Topological Quantum Gravity of the Ricci Flow

TL;DR

This work clarifies the gauge structure of topological nonrelativistic gravity tied to Ricci flow by showing it is equivalent to a standard one-step BRST gauge fixing of ADM variables with nonredundant symmetries, where the supercharge ${Q}$ acts as the BRST operator. A nonprojectable lapse plays the role of Perelman’s dilaton, while time reparametrizations and foliation-preserving diffeomorphisms are organized into a shifted, minimally redundant symmetry algebra that eliminates local propagating degrees of freedom. The analysis resolves the ghost-for-ghost question, provides a concrete BRST multiplet interpretation for all fields, and demonstrates how to realize Ricci-flow-type dynamics (via Perelman-like functionals) within a hybrid gauge-fixing framework. The results also reveal a dual interpretation in terms of a dual diffeomorphism copy, connecting the formal BRST construction to broader nonrelativistic quantum gravity structures and the mathematics of Ricci flow, with potential implications for nonrelativistic holography and mathematical physics.

Abstract

We continue the study of topological nonrelativistic quantum gravity associated with a family of Ricci flow equations on Riemannian manifolds. This topological gravity is of the cohomological type, and it exhibits an ${\cal N}=2$ extended BRST symmetry. In our previous work, we constructed this theory in a two-step procedure in the appropriate nonrelativistic ${\cal N}=2$ superspace, first presenting a topological theory of the spatial metric $g_{ij}$, and then adding the superspace versions of the lapse and shift variables $n$ and $n^i$ while gauging the symmetries of foliation-preserving spacetime diffeomorphisms. In the relation to Perelman's theory of the Ricci flow, the role of Perelman's dilaton is played by our nonprojectable lapse. Here we demonstrate that this construction is equivalent to a standard one-step BRST gauge-fixing of a theory whose fields are $g_{ij}$, $n^i$ and $n$, and whose gauge symmetries consist of (i) the topological deformations of $g_{ij}$, and (ii) the ultralocal nonrelativistic limit of spacetime diffeomorphisms. The supercharge $Q$ of our superspace construction plays the role of the BRST charge. The spacetime diffeomorphism symmetries appear in an interestingly "shifted" form, which may be of broader interest for nonrelativistic quantum gravity outside of the present topological context. In contrast to the foliation-preserving spacetime diffeomorphisms, the gauge symmetries identified in this paper act nonprojectably on time, making it clear that this theory has no local propagating degrees of freedom. We point out an intriguing dual interpretation of the same theory, as a gauge fixing of a dual copy of ultralocal spacetime diffeomorphisms, with the role of ghosts and antighosts interchanged and the second supercharge $\bar Q$ of the ${\cal N}=2$ superalgebra playing the role of the BRST charge in the dual picture.