The Extended Toda Hierarchy (shortly ETH) was introduce by E. Getzler \cite{ Ge} and independently by Y. Zhang \cite{ Z} in order to describe an integrable hierarchy which governs the Gromov--Witten invariants of $\C P^1$. The { \em Lax type} presentation of the ETH was given in \cite{ CDZ}. In this paper we give a description of the ETH in terms of { \em tau-functions} and Hirota Quadratic Equations (known also as Hirota Bilinear Equations). A new feature here is that the Hirota equations are given in terms of vertex operators taking values in the algebra of differential operators on the affine line.
Categories: math.AG