Orbifold melting crystal models and reductions of Toda hierarchy

Orbifold generalizations of the ordinary and modified melting crystal models are introduced. They are labelled by a pair $a,b$ of positive integers, and geometrically related to ${\mathbf{Z}}_a\times {\mathbf{Z}}_b$ orbifolds of local ${\mathbf{C}\mathbf{P}}^1$ geometry of the $\mathcal{O}(0)\oplus \mathcal{O}(-2)$ and $\mathcal{O}(-1)\oplus \mathcal{O}(-1)$ types. The partition functions have a fermionic expression in terms of charged free fermions. With the aid of shift symmetries in a fermionic realization of the quantum torus algebra, one can convert these partition functions to tau functions of the 2D Toda hierarchy. The powers $L^a,{\bar{L}}^{-b}$ of the associated Lax operators turn out to take a special factorized form that defines a reduction of the 2D Toda hierarchy. The reduced integrable hierarchy for the orbifold version of the ordinary melting crystal model is the bi-graded Toda hierarchy of bi-degree $(a,b)$. That of the orbifold version of the modified melting crystal model is the rational reduction of bi-degree $(a,b)$. This result seems to be in accord with recent work of Brini et al. on a mirror description of the genus-zero Gromov-Witten theory on a ${\mathbf{Z}}_a\times {\mathbf{Z}}_b$ orbifold of the resolved conifold.