Heriot-Watt Mathematics Report Series
HWM10-22, 28 May 2010
Matrix models and stochastic growth in Donaldson-Thomas theory
R J Szabo and M Tierz
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 wall-crossing
phenomena 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 leads to an
interpretation of Donaldson-Thomas theory in terms of
non-intersecting paths in the lock-step model of vicious walkers. We also
show that these generating functions can be interpreted as normalization
constants of a corner growth/last-passage stochastic model.
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