2024/02/19 by Alexey Kokotov, Dmitrii Korikov, Kokotov, Alexey +1
Mathematics · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2402.12529
openalex publication_date 2024/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeDH-P more than thirty years ago. We also reconsider the results from \citeDH-P on the regularization of diverging determinant ratio for Mandelstam metrics (for any spin) proposing (and computing) a new regularization of this ratio.