Equivariant orbifold structures on the projective line and integrable hierarchies

Let CP1k,m be the orbifold structure on CP1 obtained via uniformizing the neighborhoods of 0 and ∞ respectively by z↦zk and w↦wm. The diagonal action of the torus T=(S1)2 on CP1 induces naturally an action on the orbifold CP1k,m. In this paper we prove that if k and m are co-prime then Givental's prediction of the equivariant total descendent Gromov-Witten potential of CP1k,m satisfies certain Hirota Quadratic Equations (HQE for short). We also show that after an appropriate change of the variables, similar to Getzler's change in the equivariant Gromov-Witten theory of CP1, the putative Gromov-Witten potential turns into tau-function of a new integrable hierarchy, which we call the equivariant bi-graded Toda hierarchy. The Hamiltonian description of this hierarchy will be investigated in a future article.