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Introduction to the Gopakumar–Vafa Large N Duality

Dave Auckly and Sergiy Koshkin

Geometry & Topology Monographs 8 (2006) 195–456

DOI: 10.2140/gtm.2006.8.195

arXiv: math.GT/0701568


Gopakumar–Vafa Large N Duality is a correspondence between Chern–Simons invariants of a link in a 3–manifold and relative Gromov–Witten invariants of a 6–dimensional symplectic manifold relative to a Lagrangian submanifold. We address the correspondence between the Chern–Simons free energy of S3 with no link and the Gromov–Witten invariant of the resolved conifold in great detail. This case avoids mathematical difficulties in formulating a definition of relative Gromov–Witten invariants, but includes all of the important ideas.

There is a vast amount of background material related to this duality. We make a point of collecting all of the background material required to check this duality in the case of the 3–sphere, and we have tried to present the material in a way complementary to the existing literature. This paper contains a large section on Gromov–Witten theory and a large section on quantum invariants of 3–manifolds. It also includes some physical motivation, but for the most part it avoids physical terminology.


Gromov–Witten invariants, Chern–Simons invariants, Reshetikhin–Turaev invariants, Gopakumar–Vafa conjecture, Large N Duality, 3–manifold, symplectic manifold, quantum invariants

Mathematical Subject Classification

Primary: 81T45

Secondary: 14N35, 17B37, 57M27, 81T30


Received: 19 May 2006
Published: 21 September 2007

Dave Auckly
Department of Mathematics
Kansas State University
Manhattan KS 66506
Sergiy Koshkin
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston IL 60208-2730