The Functional Determinant and the Partition Function in Geometric Flows

Abstract

We propose the use of the functional determinant of geometric operators in constructing an entropy functional associated to geometric flows. Our approach is based on the direct computation of the partition function, with a well-defined set of microstates and macrostates in the canonical ensemble. The approach is motivated by a fundamental enigma in Perelman's derivation of his famous $\mathcal{W}$-entropy. The defining feature of our entropy is that the energy of each microstate in the partition function is invariant along the associated geometric flow - a clue that could be inferred from Perelman's work. Moreover, the monotonicity of our entropy along the associated geometric flow is then a natural result in the statistical mechanics framework. While we will not argue in a completely rigorous manner, we will use the formalism to derive an explicit formula for an entropy associated to conformal flows on a closed surface based on the Polyakov formula for the determinant of the Laplacian. We also discuss possible extensions of our results to more general operators and manifolds.