Quantum Calabi-Yau and Classical Crystals

Abstract

We propose a new duality involving topological strings in the limit of large string coupling constant. The dual is described in terms of a classical statistical mechanical model of crystal melting, where the temperature is inverse of the string coupling constant. The crystal is a discretization of the toric base of the Calabi-Yau with lattice length g s . As a strong evidence for this duality we recover the topological vertex in terms of the statistical mechanical probability distribution for crystal melting. We also propose a more general duality involving the dimer problem on periodic lattices and topological A-model string on arbitrary local toric threefolds. The (p, q) 5-brane web, dual to Calabi-Yau, gets identified with the transition regions of rigid dimer configurations.

Universal character and large N factorization in topological gauge/string theory

“…expression in terms of specialization of skew Schur functions [14]. Finally we prove (1.2) in section 4 by using the formulas for the specialization of skew Schur function.…”

Topological strings and largeNphase transitions II: chiral expansion ofq-deformed Yang-Mills theory

“…This defines the cubic topological vertex [42]. It is proportional to the combinatorial quantity Y q |Y| , where the sum runs over plane partitions Y whose edges in the three directions correspond to Young tableaux with the shapesR (a) [45]. Explicitly, the topological vertex amplitude in the canonical framing (2.33) is given by…”

Crystal melting and black holes

“…In [3] it was shown that the target space formulation of the topological A-model on C 3 describes quantum foam on the same geometry. Remarkably, the partition sum of this model is identical to that of three dimensional crystal melting [4]. In two dimensions, crystal melting adds squares to a 2d Young tableau.…”

“…We caution that this material is more formal than other parts of this note. The A-model partition function on C 3 coincides with the crystal melting partition function [4]:…”

“…original motivation for this note was to give a physical interpretation of the infinite honeycomb lattice dimer model of crystal melting purely as a quiver gauge theory of the type shown in figure (4).…”