Abstract
This is a review of recent results on the integrable structure of the ordinary and modified melting crystal models. When deformed by special external potentials, the partition function of the ordinary melting crystal model is known to become essentially a tau function of the 1D Toda hierarchy. In the same sense, the modified model turns out to be related to the Ablowitz-Ladik hierarchy. These facts are explained with the aid of a free fermion system, fermionic expressions of the partition functions, algebraic relations among fermion bilinears and vertex operators, and infinite matrix representations of those operators.
Highlights
This is a sequel of our previous work [1, 2] on the integrable structure of the “melting crystal model” of topological string theory [3] and 5D N = 1 supersymmetric U (1) gauge theory [4]
We here address the same issue for a modified melting crystal model that is related to topological string theory on the resolved conifold
Local Gromov-Witten invariants of these manifolds are studied by the localization technique [7]. Generating functions of these topological invariants coincide with the topological string amplitudes obtained by the method of topological vertex
Summary
Speaking more fairly, the relativistic Toda hierarchy [16] We expect that this explains an origin of the integrable structure that Brini [17] observed in the generating function of local Gromov-Witten invariants of the resolved conifold by a genus-by-genus analysis. This is a place where matrix-valued quantum dilogarithmic functions show up. Lax operators are thereby shown to take a factorized form as expected
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