The Hirota equation for the descendent potential of orbifold $CP^1$ and the Lax formulation of bigraded Toda hierarchy

The total descendent potential $D$ associated, by the Givental formula, with the calibrated Frobenius manifold of Laurent polynomials is conjectured to coincide with the Gromov-Witten potential of a $\C P^1$ orbifold. Milanov and Tseng proved that the potential $D$ is a solution of an Hirota quadratic equation, defined in terms of vertex operators whose coefficients are obtained from singularity theory, and conjectured that such Hirota equation is equivalent to the Lax formulation of the bigraded Toda hierarchy, introduced previously. After briefly reviewing the statement of Milanov-Tseng and the Givental formula, and we show how to obtain the bigraded Toda Lax equations, therefore proving the Milanov-Tseng conjecture, and we point out analogies and differences between the bigraded Toda hierarchies and the well-known Gelfand-Dickey reductions of KP. Based on joint work with J. van de Leur.