Chern-Simons theory on L(p,q) lens spaces and Gopakumar-Vafa duality

TL;DR

This paper explores the relationship between Chern-Simons theory on lens spaces and topological string theory, revealing new matrix model representations and demonstrating the failure of Gopakumar-Vafa duality for certain cases.

Contribution

It introduces novel matrix integral representations for Chern-Simons partition functions on L(p,q) and analyzes their large N behavior, providing insights into dualities and geometric transitions.

Findings

Gopakumar-Vafa duality fails for q>1 in fixed vacua

New matrix integral representations for CS partition functions

Explicit construction of dual string backgrounds via conifold transition

Abstract

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, some novel matrix integral representations of the SU(N) CS 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, we perform in full detail the construction of a family of would-be dual closed string backgrounds via conifold geometric transition from T^*L(p,q). We can then explicitly prove 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.