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Quantum Calabi-Yau and classical crystals. (English) Zbl 1129.81080
Etingof, Pavel (ed.) et al., The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand. Papers from the conference held in Cambridge, MA, USA, August 31–September 4, 2003. Boston, MA: Birkhäuser (ISBN 0-8176-4076-2/hbk). Progress in Mathematics 244, 597-618 (2006).
Summary: 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.
For the entire collection see [Zbl 1083.00015].

MSC:
81T45 Topological field theories in quantum mechanics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
82B23 Exactly solvable models; Bethe ansatz
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