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On a polygon version of Wiegmann-Zabrodin formula

Alexey Kokotov, Dmitrii Korikov

TL;DR

The paper develops a Hadamard-type variational formula for the ζ-regularized determinant of the Dirichlet Laplacian on a convex polygon under infinitesimal polygonal deformations that preserve the vertex count. It leverages the polygon's Schottky double with a flat conical metric to handle corners and derives a main variation formula $\delta \log \det Δ_D = \frac{1}{6\pi}\Im \mathscr{H}\int_{∂P}\{z,x\}(A\cdot\nu)\hat{ν}(x)\,dx + \frac{\gamma-\log 2}{12}\sum_{i=1}^n\left(\frac{\pi}{α_i}-\frac{α_i}{π}\right)\frac{\delta α_i}{α_i}$, including explicit corner- and rotation-perturbation analyses that ensure polygon-preserving deformations obey the formula. The authors provide rigorous justification for polygon-preserving deformations and connect their result to the smooth-domain Wiegmann–Zabrodin framework via a detailed appendix. By treating non-smooth polygonal domains with conical geometry, the work extends spectral-geometry insights to polyhedral domains and clarifies how corner angles and Schwarzian data govern determinant variations. Overall, it offers a robust, geometry-driven method to relate spectral determinants to polygonal shape changes with potential implications for mathematical physics and geometry.

Abstract

Let $P$ be a convex polygon in ${\mathbb C}$ and let $Δ_{D, P}$ be the operator of the Dirichlet boundary value problem for the Lapalcian $Δ=-4\partial_z\partial_{\bar z}$ in $P$. We derive a variational formula for the logarithm of the $ζ$-regularized determinant of $Δ_{D, P}$ for arbitrary infinitesimal deformations of the polygon $P$ in the class of polygons (with the same number of vertices). For a simply connected domain with smooth boundary such a formula was recently discovered by Wiegmann and Zabrodin as a non obvious corollary of the Alvarez variational formula, for domains with corners this approach is unavailable (at least for those deformations that do not preserve the corner angles) and we have to develop another one.

On a polygon version of Wiegmann-Zabrodin formula

TL;DR

The paper develops a Hadamard-type variational formula for the ζ-regularized determinant of the Dirichlet Laplacian on a convex polygon under infinitesimal polygonal deformations that preserve the vertex count. It leverages the polygon's Schottky double with a flat conical metric to handle corners and derives a main variation formula , including explicit corner- and rotation-perturbation analyses that ensure polygon-preserving deformations obey the formula. The authors provide rigorous justification for polygon-preserving deformations and connect their result to the smooth-domain Wiegmann–Zabrodin framework via a detailed appendix. By treating non-smooth polygonal domains with conical geometry, the work extends spectral-geometry insights to polyhedral domains and clarifies how corner angles and Schwarzian data govern determinant variations. Overall, it offers a robust, geometry-driven method to relate spectral determinants to polygonal shape changes with potential implications for mathematical physics and geometry.

Abstract

Let be a convex polygon in and let be the operator of the Dirichlet boundary value problem for the Lapalcian in . We derive a variational formula for the logarithm of the -regularized determinant of for arbitrary infinitesimal deformations of the polygon in the class of polygons (with the same number of vertices). For a simply connected domain with smooth boundary such a formula was recently discovered by Wiegmann and Zabrodin as a non obvious corollary of the Alvarez variational formula, for domains with corners this approach is unavailable (at least for those deformations that do not preserve the corner angles) and we have to develop another one.

Paper Structure

This paper contains 7 sections, 2 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Hadamard formula and formulation of the main result
  3. Bending the side far from vertices
  4. Parallel infinitesimal shifts of sides near vertices
  5. Infinitesimal rotations of sides near vertices
  6. Justifying formula (\ref{['Main formula']}) for polygon-preserving deformations
  7. Appendix: on Wiegmann-Zabrodin formula

Key Result

Lemma 3.1

Let $F$ be a function holomorphic in some neighborhood of the negative semi-axis containing the curve $\Gamma$ admitting the asymptotics as $\Re\mu\to -\infty$, where $r_k\in\mathbb{R}$, $F_k,\tilde{F_k}\in\mathbb{C}$, and the remainder obeys for some $\kappa<0$. Denote by $F(\infty)$ and $\tilde{F}(\infty)$ the constant term and the coefficient at ${\rm log}(-\mu)$ in (F asymp). Let $\widehat{F

Theorems & Definitions (5)

  • Lemma 3.1
  • proof
  • Remark 6.1
  • Theorem 7.1
  • Remark 7.2