Volume 8 (2006)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
MSP Books and Monographs
About this Series
Editorial Board
Ethics Statement
Author Index
Submission Guidelines
Author Copyright Form
Purchases
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
Other MSP Publications

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

Abstract

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.

Keywords

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

References
Publication

Received: 19 May 2006
Published: 21 September 2007

Authors
Dave Auckly
Department of Mathematics
Kansas State University
Manhattan KS 66506
USA
http://www.math.ksu.edu/~dav/
Sergiy Koshkin
Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston IL 60208-2730
USA