Complex Chern-Simons Theory at Level k via the 3d-3d Correspondence

We use the 3d-3d correspondence together with the DGG construction of theories T (n) [M] labelled by 3-manifolds M to define a non-perturbative state-integral model for Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k = 1 state-integral, as well as the 3d index, which is k = 0. The Chern-Simons partition functions correspond to partition functions of T (n) [M] on squashed lens spaces L(k, 1). At any k, they admit a holomorphic-antiholomorphic factorization, corresponding to the decomposition of L(k, 1) into two solid tori, and the associated holomorphic block decomposition of the partition functions of T (n) [M]. A generalization to L(k, p) is also presented. Convergence of the state integrals, for any k, requires triangulations to admit a positive angle structure; we propose that this is also necessary for the DGG gauge theory T (n) [M] to flow to a desired IR SCFT.