Emergent Calabi-Yau Geometry

TL;DR

This paper demonstrates how the smooth geometry of Calabi-Yau manifolds can be derived from a crystal melting statistical model, linking thermodynamics to complex geometry and topological string theory.

Contribution

It establishes a novel connection between crystal melting models and Calabi-Yau geometry through the thermodynamic limit and partition functions.

Findings

Thermodynamic limit yields Calabi-Yau geometry from crystal melting.

Partition function of molten crystals matches topological string theory.

Ronkin function relates to the holomorphic 3-form on Calabi-Yau manifolds.

Abstract

We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the partition function of the topological string theory by relating the Ronkin function of the characteristic polynomial of the crystal melting model to the holomorphic 3-form on the corresponding Calabi-Yau manifold.