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.)
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?
For example, if $M = S^1$, then $\zeta_{S^1}(s) = 2 \zeta(2s)$, for the usual Riemann zeta function, and $exp( - \zeta_{S^1}'(0) ) = 2 \pi$... which has a clear interpretation, though it's not clear that the clear interpretation is correct.
I asked a similar question in the comments of my previous question here: What is $e^{- \zeta_{\Delta} '(0)}$ for a $\Delta$ the Laplacian of a manifold?
