Communications in Mathematical Physics

, Volume 331, Issue 3, pp 1191–1235 | Cite as

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

  • C. Kalla
  • D. KorotkinEmail author


In this paper we study the 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 the Bergman tau-function (which was studied before in the context of Hurwitz spaces and spaces of holomorphic Abelian and quadratic differentials) and the 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.


Modulus Space Riemann Surface Line Bundle Modular Form Variational Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.UFR Sciences MAPMO-UMR 6628, Département de MathématiquesUniversité d’OrléansOrléans Cedex 2France
  2. 2.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada

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