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In this paper we study Baker-Akhiezer spinor kernel on moduli spaces ofmeromorphic differentials on Riemann surfaces. We introduce the Baker-Akhiezertau-function which is related to both Bergman tau-function (which was studiedbefore in the context of Hurwitz spaces and spaces of holomorphic and quadraticdifferentials) and KP tau-function on such spaces. In particular, we derivevariational formulas of Rauch-Ahlfors type on moduli spaces of meromorphicdifferentials with prescribed singularities: we use the system of homologicalcoordinates, consisting of absolute and relative periods of the meromorphicdifferential, and show how to vary the fundamental objects associated to aRiemann surface (the matrix of $b$-periods, normalized Abelian differentials,the Bergman bidifferential, the Szeg\"o kernel and the Baker-Akhiezer spinorkernel) with respect to these coordinates. The variational formulas encodedependence both on the moduli of the Riemann surface and on the choice ofmeromorphic differential (variation of the meromorphic differential whilekeeping the Riemann surface fixed corresponds to flows of KP type). Analyzingthe global properties of the Bergman and Baker-Akhiezer tau-functions weestablish relationships between various divisor classes on the moduli spaces.

Baker-Akhiezer spinor kernel and tau-functions on moduli spaces of meromorphic differentials