International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 438648, 47 pages
doi:10.1155/2008/438648
Research Article
Quantum Barnes Function as the Partition Function
of the Resolved Conifold
Department of Mathematics, Northwestern University, Evanston, IL 60208, USA
Received 3 July 2008; Accepted 15 December 2008
Academic Editor: Alberto Cavicchioli
Copyright © 2008 Sergiy Koshkin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We give a short new proof of large N duality between the Chern-Simons invariants
of the 3-sphere and the Gromov-Witten/Donaldson-Thomas invariants of
the resolved conifold. Our strategy applies to more general situations, and it is
to interpret the Gromov-Witten, the Donaldson-Thomas, and the Chern-Simons
invariants as different characterizations of the same holomorphic function. For the
resolved conifold, this function turns out to be the quantum Barnes function, a
natural q-deformation of the classical one that in its turn generalizes the Euler
gamma function. Our reasoning is based on a new formula for this function that
expresses it as a graded product of q-shifted multifactorials.