Matrix models and stochastic growth in Donaldson-Thomas theory

TL;DR

This paper connects Donaldson-Thomas invariants of local toric Calabi-Yau threefolds to symmetric functions, matrix models, integrable hierarchies, and stochastic growth models, revealing deep mathematical structures and physical interpretations.

Contribution

It introduces a matrix model representation for Donaldson-Thomas invariants and links them to integrable systems and stochastic processes, providing new computational and conceptual tools.

Findings

Partition functions expressed via Schur, Hall-Littlewood, and Jack measures.

Relation of generating functions to tau-functions of Toda and Toeplitz hierarchies.

Interpretation of invariants through non-intersecting paths and stochastic models.

Abstract

We show that the partition functions which enumerate Donaldson-Thomas invariants of local toric Calabi-Yau threefolds without compact divisors can be expressed in terms of specializations of the Schur measure. We also discuss the relevance of the Hall-Littlewood and Jack measures in the context of BPS state counting and study the partition functions at arbitrary points of the Kaehler moduli space. This rewriting in terms of symmetric functions leads to a unitary one-matrix model representation for Donaldson-Thomas theory. We describe explicitly how this result is related to the unitary matrix model description of Chern-Simons gauge theory. This representation is used to show that the generating functions for Donaldson-Thomas invariants are related to tau-functions of the integrable Toda and Toeplitz lattice hierarchies. The matrix model also leads to an interpretation of Donaldson-Thomas theory in terms of non-intersecting paths in the lock-step model of vicious walkers. We further show that these generating functions can be interpreted as normalization constants of a corner growth/last-passage stochastic model.