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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 438648, 47 pages
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.


We give a short new proof of large 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 -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 -shifted multifactorials.