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Letters in Mathematical Physics

, Volume 105, Issue 3, pp 309–340 | Cite as

Combinatorial Quantum Field Theory and Gluing Formula for Determinants

  • Nicolai Reshetikhin
  • Boris Vertman
Article

Abstract

We define the combinatorial Dirichlet-to-Neumann operator and establish a gluing formula for determinants of discrete Laplacians using a combinatorial Gaussian quantum field theory. In case of a diagonal inner product on cochains we provide an explicit local expression for the discrete Dirichlet-to-Neumann operator. We relate the gluing formula to the corresponding Mayer–Vietoris formula by Burghelea, Friedlander and Kappeler for zeta-determinants of analytic Laplacians, using the approximation theory of Dodziuk. Our argument motivates existence of gluing formulas as a consequence of a gluing principle on the discrete level.

Keywords

zeta-determinant gluing formula Gaussian quantum field theory asymptotic determinant 

Mathematics Subject Classification

58J52 81T27 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  3. 3.ITMO UniversitySaint PetersburgRussia
  4. 4.Mathematisches InstitutUniversität MünsterMünsterGermany

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