Pfaff $\tau$-functions

Abstract.

Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus $s=-t$ of the space of time variables. Restricting the solution to $s=-t$, we obtain another hierarchy called Pfaff lattice, which has its own tau function, being equal to the square root of the restriction of 2D-Toda tau function. We study its bilinear and Fay identities, W and Virasoro symmetries, relation to symmetric and symplectic matrix integrals and quasiperiodic solutions.