Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 2

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

Pages: 305 – 394



Gaëtan Borot (Max Planck Institut für Mathematik, Bonn, Germany)

Andrea Brini (UMR 5149 du CNRS, Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, France)


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 $\mathbb{S}^{\Gamma} = \Gamma \setminus \mathbb{S}^3$ 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 $\Gamma$. Its mirror A-model theory is realized as the local Gromov–Witten theory of suitable ALE fibrations on $\mathbb{P}^1$, 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 $\mathrm{sl}_{N+1}$ LMO invariant of $\mathbb{S}^{\Gamma}$ 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 $\mathbb{S}^{\Gamma}$, is computed by the Eynard–Orantin topological recursion.

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