[Paper Review] Holographic Duality for three-dimensional Super-conformal Field Theories

Benjamin Assel|arXiv (Cornell University)|Jul 16, 2013

Black Holes and Theoretical Physics108 references3 citations

TL;DR

This thesis establishes a large class of new holographic dualities between three-dimensional $σ=4$ superconformal field theories (SCFTs) and type IIB string theory on $AdS_4 \times K_6$ backgrounds. It constructs these dualities via infrared fixed points of 3D $σ=4$ quiver gauge theories, verifies the correspondence through precise matching of free energy computed via localization and supergravity actions, and provides a comprehensive holographic dictionary for linear, circular, and domain-wall quiver geometries.

ABSTRACT

This thesis is based on the publications arXiv:1106.4253, arXiv:1206.2920, arXiv:1210.2590. We construct warped "AdS_4 x S^2 x S^2 x Σ" type IIB supergravity solutions dual to the infrared fixed points of all 3d N=4 linear and circular quivers. We provide checks of the correspondence by evaluating the supergravity action in a large N limit and by matching it with the computation of the free energy obtained from matrix models. To complete previous work, we establish the holographic dictionary for the more general supergravity solutions corresponding to domain walls (non-compact Σ) and dual to 4d SYM CFT coupled to a 1/2 BPS 3d defect. We also use our evaluations of the free energy to show that the F-theorem is verified for RG-flow relating T^ρ_{\hatρ}(SU(N)) SCFTs.

Motivation & Objective

• To construct and study a broad class of holographic dualities between 3D $σ=4$ superconformal field theories and type IIB string theory on $AdS_4 \times K_6$ backgrounds.
• To provide a holographic description of strongly coupled 3D $σ=4$ quiver gauge theories, which are inaccessible via perturbative methods due to divergent Yang-Mills coupling in the infrared.
• To establish a complete holographic dictionary between the gauge theory side (via localization and matrix models) and the supergravity side (via $IIB$ solutions and domain wall geometries).
• To verify the duality through quantitative checks, particularly the matching of free energy between the gauge theory and supergravity calculations.
• To explore the emergence of 4D gravity in the large-$\gamma$ limit, analyzing the localization and mass spectrum of graviton modes in the dual geometry.

Proposed method

• Constructing 3D $σ=4$ SCFTs as infrared fixed points of quiver gauge theories with product unitary gauge groups and matter content including fundamental and bifundamental fields.
• Using the technique of localization to compute the $S^3$ partition function and free energy of the gauge theories, particularly for $T[SU(N)]$ and $T^\rho_{\hat{\rho}}[SU(N)]$ theories.
• Deriving and solving the linearized Einstein equations for graviton fluctuations in $IIB$ supergravity backgrounds with $OSp(4|4)$ symmetry, focusing on the strip and annulus geometries.
• Establishing the holographic dictionary by relating harmonic functions $h_1, h_2$ and the warp factor $W$ to the gauge theory parameters and computing the supergravity action.
• Analyzing the mass spectrum of graviton modes via variational methods and test functions, particularly for the lightest mode in the central region between $AdS_5$ wells.
• Studying the large-$\gamma$ limit to assess the emergence of 5D gravity and the breakdown of the Karch-Randall scenario due to decompactification of the central region.

Experimental results

Research questions

• RQ1 What is the precise holographic dual of a 3D $σ=4$ quiver gauge theory with linear or circular quiver structure?
• RQ2 How can the free energy of strongly coupled 3D $σ=4$ SCFTs be computed non-perturbatively and matched to the on-shell supergravity action?
• RQ3 What is the behavior of the lightest graviton mode in domain-wall $AdS_4$ solutions dual to 4D SYM coupled to a 3D defect theory?
• RQ4 Does the Karch-Randall scenario for localized gravity emerge in the large-$\gamma$ limit of the constructed supergravity solutions?
• RQ5 How do the geometry of the internal space $K_6$ and the distribution of 5-branes determine the dual field theory structure and its symmetries?

Key findings

• The free energy computed via localization in the gauge theory side matches precisely with the on-shell supergravity action, providing a strong quantitative check of the holographic duality.
• For the $T[SU(N)]$ theory, the free energy scales as $F \sim N^2 \log N$ in the large-$N$ limit, consistent with the supergravity prediction.
• The lightest graviton mode in the central region between two $AdS_5$ wells has a mass $m_0 \propto \alpha / \gamma$, which is smaller than the naive scaling in the Karch-Randall model.
• In the large-$\gamma$ limit, the central region decompactifies as $\sim \gamma^{1/2} \ln(\gamma/\alpha)$, leading to a delocalized graviton mode and a breakdown of the Karch-Randall scenario.
• The supergravity solutions for circular quivers are constructed on an annulus geometry, with D3-brane charges computed via integration over the internal space, confirming consistency with the field theory side.
• The holographic dictionary is fully established for linear and circular quivers, including the correspondence between 5-brane moves and dualities in the gauge theory.