Orbifold melting crystal models and reductions of Toda hierarchy

Abstract

Orbifold generalizations of the ordinary and modified melting crystal models are introduced. They are labelled by a pair of positive integers, and geometrically related to orbifolds of local geometry of the and 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 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 . That of the orbifold version of the modified melting crystal model is the rational reduction of bi-degree . 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 orbifold of the resolved conifold.