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A Deformation Approach to the BFK Formula

Romain Speciel

TL;DR

This work presents a deformation-based proof of the BFK gluing formula for zeta-regularized determinants of elliptic boundary problems by interpolating between boundary conditions B0 and B1 along a path Bt. The authors derive a fundamental log det Q(z) identity via resolvent and Poisson operator analysis, and regularize non-trace-class contributions through high powers of Q(z) within a Seeley calculus framework. By relating the zeta functions ζi(s) through a contour integral and applying boundary-symbol local data from log det Q, they obtain the relation det A1/det A0 = c det Q with a computable local factor c. The method extends to settings with geometric singularities such as corners and complements eta-invariant techniques, offering a robust boundary-condition deformation paradigm for spectral determinant localization.

Abstract

Understanding how spectral quantities localize on manifolds is a central theme in geometric spectral theory and index theory. Within this framework, the BFK formula, obtained by Burghelea, Friedlander and Kappeler in 1992, describes how the zeta-regularized determinant of an elliptic operator decomposes as the underlying manifold is cut into pieces. In this paper, we present a novel proof of this result. Inspired by work of Brüning and Lesch on the eta invariant of Dirac operators, we derive the BFK formula by interpolating continuously between boundary conditions and understanding the variation of the determinant along this deformation.

A Deformation Approach to the BFK Formula

TL;DR

This work presents a deformation-based proof of the BFK gluing formula for zeta-regularized determinants of elliptic boundary problems by interpolating between boundary conditions B0 and B1 along a path Bt. The authors derive a fundamental log det Q(z) identity via resolvent and Poisson operator analysis, and regularize non-trace-class contributions through high powers of Q(z) within a Seeley calculus framework. By relating the zeta functions ζi(s) through a contour integral and applying boundary-symbol local data from log det Q, they obtain the relation det A1/det A0 = c det Q with a computable local factor c. The method extends to settings with geometric singularities such as corners and complements eta-invariant techniques, offering a robust boundary-condition deformation paradigm for spectral determinant localization.

Abstract

Understanding how spectral quantities localize on manifolds is a central theme in geometric spectral theory and index theory. Within this framework, the BFK formula, obtained by Burghelea, Friedlander and Kappeler in 1992, describes how the zeta-regularized determinant of an elliptic operator decomposes as the underlying manifold is cut into pieces. In this paper, we present a novel proof of this result. Inspired by work of Brüning and Lesch on the eta invariant of Dirac operators, we derive the BFK formula by interpolating continuously between boundary conditions and understanding the variation of the determinant along this deformation.

Paper Structure

This paper contains 4 sections, 6 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup
  3. Properties of the operators $P$, $Q$ and $R$
  4. The BFK Formula

Key Result

Theorem 1.1

The zeta-regularized determinants of $A_0$ and $A_1$ are related by where $c$ is a local quantity which can be computed in terms of the symbols of $A$, $B_0$ and $B_1$ along $N$.

Figures (1)

  • Figure :

Theorems & Definitions (22)

  • Theorem 1.1: The BFK Formula BFK92
  • Remark 1.3
  • Example 2.3: The Laplacian on a manifold with boundary
  • Example 2.7: The Dirichlet-to-Neumann map
  • Remark 2.10
  • Definition 2.11
  • Example 2.17: The determinant of the Laplacian
  • Definition 2.19
  • Remark 2.20
  • Remark 2.21: Decomposing the determinant along a cut
  • ...and 12 more