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Determinants of Laplacians. (English) Zbl 0618.10023

Let \(\Gamma <\text{PSL}(2,\mathbb R)\) be a discrete subgroup such that \(\Gamma\setminus\mathfrak H\) is a compact Riemann surface of genus \(\geq 2\), where \(\mathfrak H\) denotes the upper half-plane. The author considers \[ D_{2n}=-y^ 2(\partial^ 2/\partial x^ 2+\partial^ 2/\partial y^ 2)+2iny(\partial /\partial x)\qquad (n\in {\mathbb Z}) \] as a self-adjoint linear operator on an appropriate space of automorphic forms with transformation behaviour governed by \(\chi^{2n}\), where \(\chi\) is an odd character on \(\Gamma\).
The main result of the work under review is Theorem 1 which says that \(\det (D_{\nu}+s(s-1))\) is equal to the Selberg zeta-function \(Z_{\nu}(s)\) for \(\chi^{\nu}\) \((\nu =0,1)\) times an elementary factor involving the Barnes double gamma function. The proof is based on a regularization procedure for determinants of Laplacians.
The connections with recent work on string theory are indicated.

MSC:

11F12 Automorphic forms, one variable
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
35P99 Spectral theory and eigenvalue problems for partial differential equations
58J52 Determinants and determinant bundles, analytic torsion
47F05 General theory of partial differential operators
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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References:

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