Holography of 3d-3d correspondence at large N

We study the physics of multiple M5-branes compactified on a hyperbolic 3-manifold. On the one hand, it leads to the 3d-3d correspondence which maps an N = 2 superconformal field theory to a pure Chern-Simons theory on the 3-manifold. On the other hand, it leads to a warped AdS(4) geometry in M-theory holographically dual to the superconformal field theory. Combining the holographic duality and the 3d-3d correspondence, we propose a conjecture for the large N limit of the perturbative free energy of a Chern-Simons theory on hyperbolic 3-manifold. The conjecture claims that the tree, one-loop and two-loop terms all share the same N-3 scaling behavior and are proportional to the volume of the 3-manifold, while the three-loop and higher terms are suppressed at large N. Under mild assumptions, we prove the tree and one-loop parts of the conjecture. For the two-loop part, we test the conjecture numerically in a number of examples and find precise agreement. We also confirm the suppression of higher loop terms in a few examples.