Quantum Calabi-Yau and Classical Crystals
Andrei Okounkov, Nikolai Reshetikhin, Cumrun Vafa
TL;DR
The paper introduces a duality between topological strings on toric Calabi–Yau threefolds in the large Kahler/large-$g_s$ regime and a discrete 3D crystal melting model on the toric base, with the crystal lattice spacing $g_s$ and temperature $T=1/g_s$. It derives the topological vertex from crystal-melting statistics using transfer-matrix methods and skew-Schur function techniques, and then extends the framework to periodic dimers and $(p,q)$-brane webs, connecting dimer spectral curves to mirror geometry. The key technical contributions include a precise correspondence between 3D partitions and A-model amplitudes, an explicit perpendicular partition function that yields the vertex up to framing, and a dimer-based interpretation of toric local CYs via Ronkin-function limit shapes. This work provides a concrete combinatorial and statistical-mechanical realization of local Calabi–Yau topological strings, linking mirror symmetry, limit shapes, and crystal/dimer models with brane-web constructions and Gromov–Witten/Gopakumar–Vafa-type invariants.
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.