Spectral Determinants on Mandelstam Diagrams

Abstract

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric |ω| 2 , where ω is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic form ω. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplacian for flat surfaces with cylindrical ends and conical singularities.

Moduli spaces of meromorphic functions and determinant of the Laplacian

“…Using this BFK formula, we prove (as it was done in similar situations in [22], [16]) that the variations of the determinant of the Laplacian with respect to the moduli parameters z k remain the same if we replace the metric m = |df | 2 of infinite volume by a metricm of finite volume, wherem coincides with m outside vicinities of the poles of f and with some standard nonsingular metric of finite volume inside these vicinities. The aim of the second part of the paper is thus to study the zeta-regularized determinant of this new metricm and its variation with respect to moduli parameters.…”

“…The way to regularize such determinants is, in principle, also well-known (see, e. g., [31]): considering the Laplacian ∆ as a perturbation of some properly chosen "free" operator∆, one introduces a relative determinant det(∆,∆) in terms of the relative ζ-function ζ(s; ∆,∆) = 1 Γ(s) ∞ 0 Tr(e −∆t − e −∆t )t s−1 dt, (1.3) where a suitable regularization of the integral is made (being understood in the conventional sense the integral is usually divergent for any value of s). Following this approach, in [16] we studied the regularized determinants det(∆,∆) = e −ζ ′ (0;∆,∆) of the Laplacians on the so-called Mandelstam diagrams -the flat surfaces with cylindrical ends (more precisely, Riemann surfaces X with the metric |ω| 2 , where ω is a meromorphic one-form on X having only simple poles and such that all the periods of ω are pure imaginary and all the residues of ω at the poles are real).…”

“…can be obtained in a usual way, see e.g. [16,Lemma 4]. Thus for λ 2 < 0 the relative zeta function given by…”

“…In this case, the definition of det ζ (∆,∆) is rather straightforward, since, for K = 1, a conventional regularization (see, e.g., [16] and references therein) for the relative zeta function makes sense. Indeed, the first integral in the representation ζ(s; ∆,∆) = For the second integral we need the long time heat trace behaviour given by the following lemma.…”

Baker–Akhiezer Spinor Kernel and Tau-functions on Moduli Spaces of Meromorphic Differentials

“…Namely, let us introduce the following holomorphic Abelian differential: 18) and the following meromorphic quadratic differential:…”