Holography of 3d-3d correspondence at Large N

TL;DR

This paper explores the holographic duality of M5-branes on hyperbolic 3-manifolds, proposing a large N conjecture for the perturbative free energy of associated Chern-Simons theories, supported by partial proofs and numerical tests.

Contribution

It introduces a conjecture relating large N free energy terms to 3-manifold volume, combining holography and 3d-3d correspondence, with partial proofs and numerical validation.

Findings

Tree and one-loop free energy terms scale as N^3 and are proportional to the 3-manifold volume.

Two-loop free energy matches the conjectured N^3 scaling and volume dependence.

Higher loop terms are suppressed at large N, confirmed numerically in examples.

Abstract

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 $\mathcal{N}=2$ 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.