The extended $Z_N$-Toda hierarchy

We construct the extended flow equations of a new $Z_N$-Toda hierarchy taking values in a commutative subalgebra $Z_N$ of $gl(N,\mathbb C)$. We give the Hirota bilinear equations and tau function of this new extended $Z_N$-Toda hierarchy. Taking the presence of logarithmic terms into account, we construct some extended vertex operators in generalized Hirota bilinear equations, which might be useful in topological field theory and the Gromov–Witten theory. We present the Darboux transformations and bi-Hamiltonian structure of this hierarchy. Using Hamiltonian tau-symmetry, we obtain another tau function of this hierarchy with some unknown mysterious relation to the tau function derived using the Sato theory.