U( N) framed links, three-manifold invariants, and topological strings

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 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.