Tau Function and Hirota Bilinear Equations for the Extended Bigraded Toda Hierarchy

The Toda lattice equation is a nonlinear evolutionary differential-difference equation introduced by Toda [1] describing an infinite system of masses on a line that interact through an exponential force which is used to explain the well-known FermiPasta-Ulam phenomenon. It was soon realized that this equation is completely integrable, i.e. admits infinite conserved quantities. It has important applications in many different fields such as classical and quantum fields theory. Considering its application to 2D topological fields theory and the theory of Gromov-Witten invariants ([2], [3], [4], [5]), one replaced the discrete variables with continuous one. After continuous “ interpolation” ([6]) to the whole Toda lattice hierarchy, it was found the flow of spatial translations was missing. In order to get a complete families of flows ([7]), the interpolated Toda lattice hierarchy was extended into the socalled extended Toda hierarchy ([6]). It was firstly conjectured and then shown ([2], [7], [8]) that the extended Toda hierarchy is the hierarchy describing the GromovWitten invariants of CP 1 by matrix models ([9]) which describe in the large N limit of the CP 1 topological sigma model. The extended bigraded Toda hierarchy was introduced by Gudio Carlet ([10]) who hoped that EBTH might also be relevant for some applications in 2D topological fields theory and in the theory of GromovWitten invariants. Specifically, Carlet ([10]) generalized the Toda lattice hierarchy