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 $\Omega$, and $\Delta_{n}$ the Laplacian on \n differentials on \Ga\bk\Omega, we define a notion of a Bers dual basis $\phi_{1},...c,\phi_{2d}$ for $\ker\Delta_{n}$. We prove that $\det\Delta_{n}/\det$, 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 $\det\Delta_{n}=c_{g,n}Z(n)$, 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