Hirota quadratic equations for the extended Toda hierarchy

Abstract

The extended Toda hierarchy (ETH) was introduced by E. Getzler [Ge] and independently by Y. Zhang [Z] in order to describe an integrable hierarchy that governs the Gromov-Witten invariants of $\mathbb{C}{P}^1$. The Lax-type presentation of the ETH was given in [CDZ]. In this article, we give a description of the ETH in terms of tau functions and Hirota quadratic equations (HQEs), also known as Hirota bilinear equations (HBEs). 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