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.
Highlights
We study the physics of multiple M5-branes compactified on a hyperbolic 3manifold. 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. It leads to a warped AdS4 geometry in M-theory holographically dual to the superconformal field theory
The geometric interpretation of S-dualities led to the celebrated AGT conjecture [7, 8], which states that some supersymmetric quantities such as the partition function on a squashed sphere can be identified with the ptn for some bosonic theory on M
We have studied the large N behavior of 3d TN [M ] theory by computing the free energy on a squashed 3-sphere Sb3
Summary
We review the gravity dual of TN [M ] and calculate the gravitational free energy F gravity in the supergravity approximation. With subleading corrections in 1/N. with subleading corrections in 1/N This result can be computed from an effective D = 4 gauged supergravity theory relevant to the current setup of M5-branes wrapped on a special Lagrangian 3-cycle in a Calabi-Yau three-fold. The N 3 dependence on the number of M5-branes generically appears when we relate the overall length scale with N using the four-form flux quantization condition in M-theory. In terms of M-theory, such solutions are interpreted as M5-branes wrapped on supersymmetric cycles [22, 39]. In this paper we aim to identify the field theory dual of the supersymmetric solution. The last step in the gravity computation is to use the general formula for holographic free energy for AdS4 derived in [32].
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