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.
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Reshetikhin, N., Vertman, B. Combinatorial Quantum Field Theory and Gluing Formula for Determinants. Lett Math Phys 105, 309–340 (2015). https://doi.org/10.1007/s11005-015-0744-3
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DOI: https://doi.org/10.1007/s11005-015-0744-3