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Spectral Determinants on Mandelstam Diagrams

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Abstract

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric \({|\omega|^2}\), where \({\omega}\) is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic form \({\omega}\). As an important intermediate result we prove a decomposition formula of the type of Burghelea–Friedlander–Kappeler for the determinant of the Laplacian for flat surfaces with cylindrical ends and conical singularities.

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Authors and Affiliations

  1. MAPMO (UMR 7349 Université d’Orléans-CNRS) UFR Sciences, Bâtiment de mathématiques rue de Chartres, BP 6759, 45067, Orléans Cedex 02, France

    Luc Hillairet

  2. Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West, Montreal, H3G 1M8, Quebec, Canada

    Victor Kalvin & Alexey Kokotov

Authors
  1. Luc Hillairet
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  2. Victor Kalvin
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  3. Alexey Kokotov
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Correspondence to Alexey Kokotov.

Additional information

Communicated by S. Zelditch

With an appendix by A. Kokotov and D. Korotkin

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Hillairet, L., Kalvin, V. & Kokotov, A. Spectral Determinants on Mandelstam Diagrams. Commun. Math. Phys. 343, 563–600 (2016). https://doi.org/10.1007/s00220-015-2506-6

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