Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces
type: Item
instance of: scholarly article
title: Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces
zbMATH Open document ID: 0629.58029
DOI: 10.1007/BF01217814
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: The author gives an estimate for the Selberg zeta function Z(R,s) of a compact Riemann surface of genus \(g\geq 2\) valid for s in a neighborhood, depending only on the genus, of \(s=1\). The author uses a formula of McKean relating Z'/Z to the heat kernel K of R and determines then the behaviour of K when R tends to infinity in the moduli space \(M_ g\) of stable curves. From this he deduces the behaviour of the spectral measure, and the functional determinant, as R tends to infinity in \(M_ g\). Using this and the expression of the holomorphic parameter for the transverse direction to the compactification divisor of \(M_ g\) in terms of Fenchel-Nielsen coordinates (that he etablishes), the author proves that the bosonic Polyakov integral \((d=26)\) is infinite. Similar results are obtained by Hejhal who uses the B-groups as model for the degeneration of Riemann surfaces [\textit{D. A. Hejhal}, Preprint, Chalmers Univ. Göteborg (1987)].
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 4024359
zbMATH Keywords: Laplacian
zbMATH Keywords: functional determinant
zbMATH Keywords: string theory
zbMATH Keywords: Selberg zeta function
author: Scott A. Wolpert
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4024359