History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of spinor/tensor fields on the compact Riemann surface $M$ (for simplicity in this question I'm assuming $n\geq0$). Their result reads: $$\det(\Delta_n^+)=\mathcal{Z}_{n-[n]}(n+1)\cdot e^{-c_n\mathcal{X}(M)}.$$

Where $\mathcal{Z}_{n-[n]}(s)$ are two Selberg zeta functions, and $c_n$ is a constant which is explicitly computed in the article above.

In their exposition there are for sure some trivial mistakes. One comes from an error in the reference paper "Fourier coefficients of the resolvent for a Fuchsian group" (1977) by Fay, and it has been considered in the review paper "Geometry of String Perturbation Theory" (1988) of D'Hoker and Phong. But it doesn't lead to any change in the above considered formula.

Another trivial mistake is a misuse of the volume formula for an hyperbolic Riemann surface, this has been first pointed out by Bolte and Steiner, in their (unpublished) paper: "Determinants of Laplace-like Operators on Riemann Surfaces" (1988). In this article the authors use a new way to compute the same quantity $\det(\Delta_n^+)$, but they arrive to a different result. Calling $\frac{1}{2}\Delta_n^+$ what D'Hoker and Phong called $\Delta_n^+$ (according to Bolte and Steiner), it reads: $$\det(\frac{1}{2}\Delta_n^+)=\mathcal{Z}_{n-[n]}(n+1)\cdot e^{-k_n\mathcal{X}(M)}.$$

Where the constant $k_n$ has been explicitly computed, and it turns out to be different from $c_n$.

Unfortunately the mistakes pointed out above are not enough to account for the difference between $c_n$ and $k_n$. Bolte and Steiner suspect that the error in D'Hoker and Phong could come from a missing factor $2$ in the definition of $\Delta_n^+$.

The end of the story seems to be the article "Notes on determinants of Laplace-type operators on Riemann surfaces" (1990) of Oshima. In this article the author recalls the correction of some mistakes in the original paper of D'Hoker and Phong, moreover he points out a conceptual mistake made by Bolte and Steiner. Finally, simply assembling previous results, he provides the following formula: $$\det(\frac{1}{2}\Delta_n^+)=\mathcal{Z}_{n-[n]}(n+1)\cdot e^{-l_n\mathcal{X}(M)}.$$

Again $l_n$ is given explicitly, and it turns out to be different from both $c_n$ and $k_n$.

Question: Is it really the end of the story? It seems to me that the whole situations is quite messy, for example no one seems to point out new specific mistakes in the original article of D'Hoker and Phong (there must be some, according to Oshima). In specific, do the experts agree on the validity of the formula provided by Oshima? Are there other successive articles on the same topic?

Note: I added the tags "Arithmetic Geometry" and "Analytic Number Theory" because the laplacians $\Delta_n^+$ are conjugated to the Maass Laplacians $D_n$ acting on automorphic forms of weight $n$ on $M$. So I suspect an answer could come from people in arithmetic as well.

Thank you very much for reading all this!

  • 1
    $\begingroup$ Relating the determinant of the Laplacian to the Selberg Zeta function in the article is almost by definition up to a constant, so some formula like that is no surprise. This involves comparing the trace formula with the logarithmic derivative of the Selberg Zeta function. So that part of the formula should be true. Are you asking whether $l_n$ are the right constants? $\endgroup$
    – Marc Palm
    Mar 25 '13 at 14:23
  • $\begingroup$ I mean, are you asking whether $e^{l_n \chi(M)}$ is the right constant? $\endgroup$
    – Marc Palm
    Mar 25 '13 at 14:34
  • $\begingroup$ Yes, this is what I'm asking for. In the sense that the other part of the formula is well understood (not only by experts, but by me as well! :) ). $\endgroup$ Mar 25 '13 at 15:40
  • $\begingroup$ Okay, then the remaining contributions should most likely come from the holomorphic functions on M of even (odd) weight between two and n for n even (odd). The corresponding eigenvalues are explicitly known and depend on the weight only. The can be computed in terms of cohomology, hence the Euler number.What remains is a constant coming from the trace formula. Standard reference is Hejhal LNM for this, compact things equals first volume. In principle, defining the Selberg zeta function for arbitrary weight should give a more direct expression. $\endgroup$
    – Marc Palm
    Mar 25 '13 at 18:56
  • $\begingroup$ I'll try to give a look to Hejhal. But the problem is not about grasping the meaning of the formula, there are at least three articles computing it. My problem is to know if the last formula, which seems to be the definitive one, is indeed definitive and reliable. Thank you for your interest! $\endgroup$ Mar 26 '13 at 16:48

I do not know if this is really the end of the story. You may be interested in the following paper by Jay Jorgenson.

Basically he extended the work by Ray-Singer by fixing all unknown invariants, but he was unable to extend their work to the full "Siegel half plane", which may yield him a closed form formula. Recently there has been more work being done on Riemann surfaces viewed as homogeneous spaces by Gerald Montplet and Anna von Pippich, but the analysis there are rather delicate using the rescaling of the Wolpert metric. Since this space has cusp singularities, there has been work done by people like Rafe Mazzeo and Xuwen Zhu to analyze the behavior of the Laplacian using microlocal analysis. But I think they are not working on it in an arithemetic context.

If you are not into manifold with singularities, then I think the work done by Jorgenson is reasonably complete and may be the best one available. His work relies heavily upon the "holomorphic factorization" technique, which I do not know whether has any scope of futher refinement.


A fairly complete treatment is given by Christian Grosche:

Grosche, C., Path integrals, hyperbolic spaces and Selberg trace formulae, Singapore: World Scientific. xi, 280 p. (1996). ZBL0883.58003.

(the citation tool gives a reference to the first edition, the second edition came out in 2013. See, in particular, Chapter 14 in the Second Edition (I don't have the first).


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