Chern-Simons theory on L (p, q) 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-S U (2) cyclic quotients of the conifold. To this aim we find, on one hand, a useful matrix integral representation of the S U (N) C S 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.