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

We consider the Gopakumar-Ooguri-Vafa correspondence, relating ${\rm U}(N)$ Chern-Simons theory at large $N$ to topological strings, in the context of spherical Seifert 3-manifolds. These are quotients $\mathbb{S}^{\Gamma} = \Gamma\backslash\mathbb{S}^3$ 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.

💡 Research Summary

The paper extends the celebrated Gopakumar‑Ooguri‑Vafa (GOV) correspondence, which relates large‑N U(N) Chern‑Simons theory on S³ to topological string theory on the resolved conifold, to the broader class of spherical Seifert 3‑manifolds (S^{\Gamma}=\Gamma\backslash S^{3}). Here (\Gamma) is a finite subgroup of SU(2) acting freely on S³; by the McKay correspondence each such (\Gamma) is labelled by an ADE Dynkin diagram. The authors propose a dual description of the 1/N expansion of the Chern‑Simons partition function (restricted to the contribution of the trivial flat connection) in terms of both A‑model and B‑model topological strings on non‑toric local Calabi‑Yau threefolds.

On the A‑model side they construct a family of local Calabi‑Yau threefolds (Y_{\Gamma}) by a natural Γ‑equivariant generalisation of the conifold transition. When (\Gamma) is non‑abelian the resulting geometry is non‑toric, but it still admits a well‑defined Gromov‑Witten theory. The A‑model free energy is interpreted as counting open and closed holomorphic curves in (Y_{\Gamma}), generalising the known results for lens spaces.

On the B‑model side the target space is identified with the spectral curve of a Toda‑type integrable system built on the double Bruhat cells of the affine co‑extended loop group (\widehat G) associated with the ADE label of (\Gamma). The curve has the form
\