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Fay-like identities of the Toda Lattice Hierarchy and its dispersionless limit
In this paper, we derive the Fay-like identities of tau function for the Toda
lattice hierarchy from the bilinear identity. We prove that the Fay-like
identities are equivalent to the hierarchy. We also show that the
dispersionless limit of the Fay-like identities are the dispersionless Hirota
equations of the dispersionless Toda hierarchy.Comment: 20 page
Holomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization
For a quasi-Fuchsian group \Ga with ordinary set , and
the Laplacian on \n differentials on \Ga\bk\Omega, we define a notion of a
Bers dual basis for . We prove that
, is, up to an anomaly computed by
Takhtajan and the second author in \cite{TT1}, the modulus squared of a
holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the
Selberg zeta Z(n). This generalizes the D'Hoker-Phong formula
, and is a quasi-Fuchsian counterpart of the result
for Schottky groups proved by Takhtajan and the first author in \cite{MT}.Comment: 15 page
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