Chern–Simons theory on lens spaces and Gopakumar–Vafa duality

We consider aspects of Chern–Simons theory on L(p,q) lens spaces and its relation with matrix models and topological string theory on Calabi–Yau threefolds, searching for possible new large N dualities via geometric transition for non-SU(2) cyclic quotients of the conifold. To this aim we find, on one hand, a useful matrix integral representation of the partition function in a generic flat background for the whole L(p,q) family and provide a solution for its large N dynamics; on the other hand, we perform in full detail the construction of a family of would-be dual closed string backgrounds through conifold geometric transition from T∗L(p,q). We can then explicitly prove the claim in [5] that Gopakumar–Vafa duality in a fixed vacuum fails in the case q>1, and briefly discuss how it could be restored in a non-perturbative setting.