Let $M$ be a Riemann surface and $\mu$ a metric on it, which could be non-compact. Moreover let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its $\zeta$-regularized determinant. The structure constant is defined to be: $$c_{\Delta,\,\mu}(M) = \log \left(\frac{\mathrm{det}^*(\Delta_{\mu,\,M})}{\mathrm{vol}_\mu(M)} \right).$$

An important example where this constant appears is the Polyakov formula, which expresses the modification of the structure constant as the metric on the surface is conformally perturbed.

Although I didn't find such a statement in the literature, it seems quite natural to me that the structure constant should be additive on connected components. To be precise, if $(C,\mu_C) = (A\sqcup B,\mu_A + \mu_B)$ is the disjoint union of the metrized Riemann surfaces $(A,\mu_A)$ and $(B,\mu_B)$, then $$c_{\Delta,\,\mu_C}(C) = c_{\Delta,\,\mu_A}(A) + c_{\Delta,\,\mu_B}(B).$$

Trying to prove it I observed that $\log(\mathrm{det}^*(\Delta_{\mu,\,M}))$ is already additive on connected components, but the log of the volume is clearly not! I'm quite confused about something which is supposed to be fairly simple, in specific where does the "need" of the volume come from.

Summing up I have two questions:

1. Is the structure constant additive on connected components?

2. What is the role of the volume in the definition of the structure constant?

Thanks in advance!

This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would be most grateful for clarifications!

Definition of the Structure Constant Let $M$ be a possibly non-compact Riemann surface and $\mu$ a smooth metric on it; let $\Delta_{\mu,\,M}$ be the Laplacian on $M$ induced by $\mu$ and $\mathrm{det}^*(\Delta_{\mu,\,M})$ its $\zeta$-regularized determinant. The structure constant is defined by $$c_{\Delta,\,\mu}(M) := \log \left(\frac{\mathrm{det}^*(\Delta_{\mu,\,M})}{\mathrm{vol}_\mu(M)} \right).$$

A precise statement for the Additivity on Connected Components is the following. Let $(C,\mu_C) = (A\sqcup B,\mu_A + \mu_B)$ be the disjoint union of the metrized Riemann surfaces $(A,\mu_A)$ and $(B,\mu_B)$, then $$c_{\Delta,\,\mu_C}(C) = c_{\Delta,\,\mu_A}(A) + c_{\Delta,\,\mu_B}(B).$$

Does the additivity on connected components hold?

Here follow some observations:

1. By direct examination of its construction it seems to me that the logarithm of the regularized determinant is already additive on connected components.

2. As pointed out by Carlo Beenakker in his answer, if we scale the metric by a positive factor $\gamma^2$ then the regularized determinant scales as $$\mathrm{det}^*(\Delta_{\gamma^2\cdot\mu,\,M}) = \gamma^{-\mathcal{X}(M)/3} \mathrm{det}^*(\Delta_{\mu,\,M}),$$ where $\mathcal{X}$ is the Euler characteristic. Given the additivity on connected components of the Euler characteristic, this is not in contradiction with the observation above.

3. Also in his answer, Carlo Beenakker mentions that the role of the volume in the definition of the structure constant is to "normalize to unit area". My lack of understanding of this point is twofold: on one hand, since $$c_{\Delta,\,(\mathrm{vol}_\mu(M))^{-1}\cdot\mu}(M) = \log \left(\frac{\mathrm{det}^*(\Delta_{(\mathrm{vol}_\mu(M))^{-1}\cdot\mu,\,M})}{\mathrm{vol}_{(\mathrm{vol}_\mu(M))^{-1}\cdot\mu}(M)} \right) = \log \left(\frac{\mathrm{det}^*(\Delta_{\mu,\,M})}{\mathrm{vol}_{\mu}(M)^{-\mathcal{X}(M)/6}} \right),$$ I do not get the idea why this is a meaningful renormalization for the regularized determinant. On the other hand I do not see why this renormalization should ensure the additivity on connected components rather than spoiling it.

All in all it appears I'm pretty confused about the whole business. Thanks for your support!