U(N) Framed Links, Three-Manifold Invariants, and Topological Strings

TL;DR

This paper explores the connection between three-manifold invariants derived from U(N) Chern-Simons theory and topological string theory, revealing a duality that links surgery procedures, invariants, and string expansions.

Contribution

It establishes a correspondence between U(N) three-manifold invariants from framed links and topological string expansions, including the computation of Gopakumar-Vafa and Gromov-Witten coefficients.

Findings

U(N) three-manifold invariants relate to trivial connection contributions.

Large N expansion matches closed string A-model partition functions.

Explicit Gopakumar-Vafa and Gromov-Witten coefficients are determined.

Abstract

Three-manifolds can be obtained through surgery of framed links in $S^3$. We study the meaning of surgery procedures in the context of topological strings. We obtain U(N) three-manifold invariants from U(N) framed link invariants in Chern-Simons theory on $S^3$. These three-manifold invariants are proportional to the trivial connection contribution to the Chern-Simons partition function on the respective three-manifolds. Using the topological string duality conjecture, we show that the large $N$ expansion of U(N) Chern-Simons free energies on three-manifolds, obtained from some class of framed links, have a closed string expansion. These expansions resemble the closed string $A$-model partition functions on Calabi-Yau manifolds with one Kahler parameter. We also determine Gopakumar-Vafa integer coefficients and Gromov-Witten rational coefficients corresponding to Chern-Simons free energies on some three-manifolds.