Abstract
We consider the Gopakumar-Ooguri-Vafa correspondence, relating U(N) Chern-Simons theory at large N to topological strings, in the context of spherical Seifert 3-manifolds. These are quotients $S^\Gamma=S^3/\Gamma$ of the three-sphere by the free action of a finite isometry group. Guided by string theory dualities, we propose a large N dual description in terms of both A- and B-twisted topological strings on (in general non-toric) local Calabi-Yau threefolds. The target space of the B-model theory is obtained from the spectral curve of Toda-type integrable systems constructed on the double Bruhat cells of the simply-laced group identified by the ADE label of $\Gamma$. Its mirror A-model theory is realized as the local Gromov-Witten theory of suitable ALE fibrations on $CP^1$ generalizing the results known for lens spaces. We propose an explicit construction of the family of target manifolds relevant for the correspondence, which we verify through a large N analysis of the matrix model that expresses the contribution of the trivial flat connection to the Chern-Simons partition function. Mathematically, our results put forward an identification between the 1/N expansion of the $sl_{N+1}$ LMO invariant of $S^\Gamma$ and a suitably restricted Gromov-Witten/Donaldson-Thomas partition function on the A-model dual Calabi-Yau. This 1/N expansion, as well as that of suitable generating series of perturbative quantum invariants of fiber knots in $S^\Gamma$, is computed by the Eynard-Orantin topological recursion.