Orbifold melting crystal models and reductions of Toda hierarchy

Author: Takasaki Kanehisa  

Publisher: IOP Publishing

E-ISSN: 1751-8121|48|21|215201-215234

ISSN: 1751-8121

Source: Journal of Physics A: Mathematical and Theoretical, Vol.48, Iss.21, 2015-05, pp. : 215201-215234

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Abstract

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 &${{mathbb{Z}}_{a}}times {{mathbb{Z}}_{b}}$; orbifolds of local &$mathbb{C}{{mathbb{P}}^{1}}$; 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 &${{mathbb{Z}}_{a}}times {{mathbb{Z}}_{b}}$; orbifold of the resolved conifold.