Let $(M,g)$ be a Riemannian manifold, with Laplacian $\Delta$. If $\lambda_i$ are the nonzero eigenvalues of $\Delta$, we can define the zeta function $\zeta(s) = \Sigma \lambda_i^{-s}$. By analytic continuation using the Mellin transform and the heat kernel, we can define $e^{ - \zeta'(0)}$, which is the 'zeta regularized determinant.'  (See, for example, [this paper][1].)

For a finite graph $G$, if $L$ is the combinatorial laplacian, we can make similar definitions. Then $e^{- \zeta_L'(0)} / |G|$ counts the number of spanning trees. (Essentially a reformulation of Kirkoffs tree-matrix theorem.)

Question: Is there a parameter / moduli space of geometric objects associated to $(M,g)$ whose volume $e^{- \zeta'(0)} / Vol(M)$ computes?

I asked a similar question in the comments of my previous question here: https://mathoverflow.net/questions/304274/what-is-e-zeta-delta-0-for-a-delta-the-laplacian-of-a-manifold/304286?noredirect=1#comment758955_304286


  [1]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.486.558&rep=rep1&type=pdf