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

Abstract

In this paper we study Baker-Akhiezer spinor kernel on moduli spaces of meromorphic differentials on Riemann surfaces. We introduce the Baker-Akhiezer tau-function which is related to both Bergman tau-function (which was studied before in the context of Hurwitz spaces and spaces of holomorphic and quadratic differentials) and KP tau-function on such spaces.In particular, we derive variational formulas of Rauch-Ahlfors type on moduli spaces of meromorphic differentials with prescribed singularities: we use the system of homological coordinates, consisting of absolute and relative periods of the meromorphic differential, and show how to vary the fundamental objects associated to a Riemann surface (the matrix of b-periods, normalized Abelian differentials, the Bergman bidifferential, the Szegö kernel and the Baker-Akhiezer spinor kernel) with respect to these coordinates. The variational formulas encode dependence both on the moduli of the Riemann surface and on the choice of meromorphic differential (variation of the meromorphic differential while keeping the Riemann surface fixed corresponds to flows of KP type).Analyzing the global properties of the Bergman and Baker-Akhiezer tau-functions we establish relationships between various divisor classes on the moduli spaces.DK was partially supported by NSERC, FQRNT and CURC.

Spaces of Abelian Differentials and Hitchin’s Spectral Covers

“…We refer to [14,12] for explicit formula for τ B and to [19,12] for its properties and applications.…”

Spectral Determinants on Mandelstam Diagrams

“…Note that it is a straightforward generalization of the Bergman tau-function on the moduli space of Abelian differentials [23] (i. e. the moduli space of pairs (X, ω), where X is a compact Riemann surface, and ω is a holomorphic one-form on X ) to the case of a meromorphic one-form ω with pure imaginary periods and simple poles with (fixed) real residues. This generalization (along with many others) was also recently discussed in [19].…”

“…The authors thank D. Korotkin for extremely useful discussions. We thank also C. Kalla and D. Korotkin for communicating the results from [19] long before their publication.…”

“…Relying on the results obtained in [23,24], it is not hard to propose an explicit formula for the solution of this system (its main ingredient, the Bergman tau-function on the space of meromorphic differentials of the third kind, was recently introduced by Kalla and Korotkin in [19]). …”

Variational formulas on spaces of SL(2) Hitchin's spectral covers

“…The Bergman tau-function on moduli spaces of differentials was originally defined as a higher genus generalization of the Dedekind eta function on elliptic surface. Beginning from the moduli space of holomorphic Abelian differentials [14] it was extended to the case of Abelian differential with arbitrary divisor [13]; further generalizations cover moduli spaces of holomorphic quadratic [6] and N-differentials [20]. Recently in [1] Bergman tau-function was applied to describe the discriminant class of Sp(2n) Hitchin's spectral covers.…”