Chern-Simons theory on spherical Seifert manifolds, topological strings and integrable systems

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^Γ=Γ ∖ S^3 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 Γ. Its mirror A-model theory is realized as the local Gromov–Witten theory of suitable ALE fibrations on P^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 slN+1 LMO invariant of S^Γ 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^Γ, is computed by the Eynard–Orantin topological recursion