Laplacians in spinor bundles over translation surfaces: self-adjoint extentions and regularized determinants

TL;DR

The paper addresses the challenge of defining and comparing determinants of Dolbeault Laplacians acting on spinor bundles over translation surfaces with flat conical metrics, focusing on self-adjoint extensions (Friedrichs, Szegö, holomorphic) and their moduli dependence.Its core approach combines a novel regularization of the Quillen metric with precise heat-kernel and resolvent analyses on singular surfaces, enabling explicit determinant formulas without requiring full spectral data for the singular operators.Key contributions include a Szegö-extension determinant expression in terms of Bergman tau-functions and theta-constants, a moduli-dependent comparison formula between det ΔF and det ΔS via the scattering matrix, and a spinor bosonization-style result linking determinants to theta data in the non-smooth setting.The work establishes almost-isospectrality between certain extensions, provides a rigorous framework for zeta-regularization on conical surfaces, and extends bosonization-type identities to flat singular metrics, offering tools and formulas of value for mathematical physics and spectral geometry.

Abstract

We study the regularized determinants ${\rm det}\, Δ$ of various self-adjoint extensions of symmetric Laplacians acting in spinor bundles over compact Riemann surfaces with flat singular metrics $|ω|^2$, where $ω$ is a holomorphic one form on the Riemann surface. We find an explicit expression for ${\rm det}\, Δ$ for the so-called self-adjoint Szegö extension through the Bergman tau-function on the moduli space of Abelian differentials and the theta-constants (corresponding to the spinor bundle). This expression can be considered as a version of the well-known spin-$1/2$ bosonization formula of Bost-Nelson for the case of flat conformal metrics with conical singularities and a higher genus generalization of the Ray-Singer formula for flat elliptic curves. We establish comparison formulas for the determinants of two different extensions (e. g., the Szegö extension and the Friedrichs one). The paper answers a question raised by D'Hoker and Phong \cite{DH-P} more than thirty years ago. We also reconsider the results from \cite{DH-P} on the regularization of diverging determinant ratio for Mandelstam metrics (for any spin) proposing (and computing) a new regularization of this ratio.

Figures (3)

• Figure 1: A genus 3 example of a surface representing coordinates (\ref{['moduli space coordinates']}). Here the opposite side of parallelograms are identified.
• Figure 2: Deformation of the torus $\mathbb{T}_j$: a) the case $\alpha\le 0$ ($V$ is removed, the double-sided arrow denotes the identification of the points), b) the case $\alpha>0$ ($V$ is glued in). For simplicity, we assumed that $e_j=0$.
• Figure 3: Deformation of the neighborhood of $P_k$ corresponding to the variation $\delta C_k$.

Theorems & Definitions (28)

• Remark 1.1
• Lemma 2.1
• Definition 2.2
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Theorem 3.6
• Theorem 3.7