Hirota equations for the extended bigraded Toda hierarchy and the total descendent potential of CP^1 orbifolds

Abstract: We prove that the Hirota quadratic equations of Milanov and Tseng define an integrable hierarchy which is equivalent to the extended bigraded Toda hierarchy. In particular this proves a conjecture of Milanov-Tseng that relates the total descendent potential of the orbifold $C_{k,m}$ with a tau function of the bigraded Toda hierarchy.