Introduction to the Gopakumar–Vafa Large N Duality

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 S 3 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.

Chern–Simons theory on L(p,q) lens spaces and Gopakumar–Vafa duality

“…This duality is realised through a particular kind of geometric transition, called conifold transition, which plays a relevant rôle in the study of the moduli space of Calabi-Yau three-folds (CY 3) (see e.g. [13,2,22] for reviews). Let us recall the basic features of Gopakumar-Vafa (GV ) duality.…”

Torus Knots in Lens Spaces and Topological Strings

“…which reduces to (29) when q = 1. From the point of view of CS theory, the relationship with topological string theory and therefore the algebraic curve description is not expected to hold in this case [6]. Hence we restrict to L(p, 1) in the sequel, and perform explicit computations for L(2, 1) only.…”

Quantum Barnes Function as the Partition Functionof the Resolved Conifold

“…. , γ n [8,30]. In general, the enumerative interpretation fails and γ 1 · · · γ n g,α are only rational numbers, this is always the case for Calabi-Yau manifolds.…”

“…Indeed, the complex degree of the integrand in (12) is 1 2 (deg γ 1 + · · · + deg γ n ) and for the integral to be non-zero it should be equal to the virtual dimension (11). There are other natural classes on M g,n (X, α) that lead to more general Gromov-Witten invariants, gravitational descendants and Hodge integrals [8,14,19], but we need not concern ourselves with them here.…”

“…Since repackaging involves transcendental substitutions, analytic continuation and asymptotic expansions -things one does with functions and not with formal series -it makes sense to identify the underlying holomorphic functions to establish a duality. This is the strategy of this paper and it distinguishes it from previous approaches [8,21,41] that use double expansions in genus and degree. This makes for a cleaner comparison of partition functions with a clear view of what matches and what does not match in them (Theorem 3).…”