Crystals, instantons and quantum toric geometry

TL;DR

This paper explores the connections between melting crystal models, instantons, and quantum toric geometry, revealing their roles in various physical theories and mathematical structures, including noncommutative geometry and gauge theories.

Contribution

It introduces a novel framework linking melting crystal models to noncommutative instantons and quantum geometry using advanced mathematical tools.

Findings

Partition function computed via noncommutative instantons

Relation between 3D melting crystals and 6D gauge theories

Mathematical construction of quantum geometry with toric and noncommutative methods

Abstract

We describe the statistical mechanics of a melting crystal in three dimensions and its relation to a diverse range of models arising in combinatorics, algebraic geometry, integrable systems, low-dimensional gauge theories, topological string theory and quantum gravity. Its partition function can be computed by enumerating the contributions from noncommutative instantons to a six-dimensional cohomological gauge theory, which yields a dynamical realization of the crystal as a discretization of spacetime at the Planck scale. We describe analogous relations between a melting crystal model in two dimensions and N=4 supersymmetric Yang-Mills theory in four dimensions. We elaborate on some mathematical details of the construction of the quantum geometry which combines methods from toric geometry, isospectral deformation theory and noncommutative geometry in braided monoidal categories. In particular, we relate the construction of noncommutative instantons to deformed ADHM data, torsion-free modules and a noncommutative twistor correspondence.