[Paper Review] Rigid Holography and Six-Dimensional N=(2,0) Theories on AdS_5 times S^1

Ofer Aharony, Micha Berkooz+1 more|arXiv (Cornell University)|Jan 13, 2015

Black Holes and Theoretical PhysicsPhysics and Astronomy52 references17 citations

TL;DR

This paper introduces 'rigid holography'—a duality framework linking six-dimensional $ scr{N}=(2,0)$ $A_{K-1}$ superconformal field theories on $\mathrm{AdS}_5 \times \mathbb{S}^1$ to subsectors of large-$N$ four-dimensional ${\cal N}=2$ quiver SCFTs. It shows that the $\mathrm{AdS}_5$ boundary theory hosts a weakly coupled ${\cal N}=2$ $SU(K)$ gauge theory coupled to moduli-dependent couplings, with non-trivial corrections to the moduli space metric due to holographic dynamics.

ABSTRACT

Field theories on anti-de Sitter (AdS) space can be studied by realizing them as low-energy limits of AdS vacua of string/M theory. In an appropriate limit, the field theories decouple from the rest of string/M theory. Since these vacua are dual to conformal field theories (CFTs), this relates some of the observables of these field theories on AdS to a subsector of the dual CFTs. We exemplify this `rigid holography' by studying in detail the 6d N=(2,0) A_{K-1} superconformal field theory (SCFT) on AdS_5xS^1, with equal radii for AdS_5 and for S^1. We choose specific boundary conditions preserving sixteen supercharges that arise when this theory is embedded into Type IIB string theory on AdS_5xS^5/Z_K. On R^{4,1}xS^1, this 6d theory has a 5(K-1)-dimensional moduli space, with unbroken 5d SU(K) gauge symmetry at (and only at) the origin. On AdS_5xS^1, the theory has a 2(K-1)-dimensional `moduli space' of supersymmetric configurations. We argue that in this case the SU(K) gauge symmetry is unbroken everywhere in the `moduli space' and that this 5d gauge theory is coupled to a 4d theory on the boundary of AdS_5 whose coupling constants depend on the `moduli'. This involves non-standard boundary conditions for the gauge fields on AdS_5. Near the origin of the `moduli space', the theory on the boundary contains a weakly coupled 4d N=2 supersymmetric SU(K) gauge theory. We show that this implies large corrections to the metric on the `moduli space'. The embedding in string theory implies that the 6d N=(2,0) theory on AdS_5xS^1 with sources on the boundary is a subsector of the large N limit of various 4d N=2 quiver SCFTs that remains non-trivial in the large N limit. The same subsector appears universally in many different 4d N=2 SCFTs. We also discuss a decoupling limit that leads to N=(2,0) `little string theories' on AdS_5xS^1.

Motivation & Objective

• To develop a new holographic duality—'rigid holography'—for non-gravitational field theories on $\mathrm{AdS}_5 \times \mathbb{S}^1$ by decoupling them from full string/M-theory backgrounds.
• To study the six-dimensional ${\cal N}=(2,0)$ $A_{K-1}$ theory on $\mathrm{AdS}_5 \times \mathbb{S}^1$ with equal radii, preserving 16 supercharges via embedding in Type IIB on $\mathrm{AdS}_5 \times \mathbb{S}^5/\mathbb{Z}_K$.
• To identify the boundary dual of this $\mathrm{AdS}_5$-based theory as a subsector of large-$N$ four-dimensional ${\cal N}=2$ quiver SCFTs with universal structure across different models.
• To analyze the spectrum of Kaluza-Klein modes and match them to BPS operators in the 4D quiver theory, confirming consistency of the duality.
• To explore the structure of the 'moduli space' on $\mathrm{AdS}_5 \times \mathbb{S}^1$, showing that $SU(K)$ gauge symmetry remains unbroken everywhere, unlike in flat space.

Proposed method

• Utilizes the AdS/CFT correspondence in a 'rigid' limit where gravity is decoupled, treating the $\mathrm{AdS}_5$ background as fixed, and focusing on a subsector of the full CFT dual.
• Employs Gaiotto's class ${\cal S}$ construction to describe the 4D ${\cal N}=2$ quiver SCFTs as compactifications of the 6D ${\cal N}=(2,0)$ theory on a Riemann surface with punctures.
• Performs Kaluza-Klein reduction of the 6D $\mathrm{AdS}_5 \times \mathbb{S}^1$ theory, classifying modes by $U(1)_R$ charge and $SU(2)_R$ representation, and matching their quantum numbers to operators in the 4D quiver.
• Analyzes boundary conditions for gauge fields on $\mathrm{AdS}_5$, introducing non-standard conditions that allow the emergence of a weakly coupled 4D ${\cal N}=2$ $SU(K)$ gauge theory on the boundary.
• Matches the Kaluza-Klein spectrum of 6D fields (vector, tensor, scalar) to BPS operators in the 4D quiver, including those in twisted sectors labeled by $\mathbb{Z}_K$ quantum numbers.
• Derives the scaling dimensions and $U(1)_R$ charges of the 4D operators from the bulk modes, confirming exact match with the 6D Kaluza-Klein tower.

Experimental results

Research questions

• RQ1 How can a non-gravitational 6D ${\cal N}=(2,0)$ theory on $\mathrm{AdS}_5 \times \mathbb{S}^1$ be consistently described as a subsector of a 4D ${\cal N}=2$ SCFT in the large-$N$ limit?
• RQ2 What is the nature of the 'moduli space' of supersymmetric configurations on $\mathrm{AdS}_5 \times \mathbb{S}^1$, and how does it differ from the flat-space moduli space?
• RQ3 How do the boundary conditions for gauge fields on $\mathrm{AdS}_5$ affect the emergence of a 4D ${\cal N}=2$ gauge theory, and what is the role of the $U(1)_R$ charge in organizing the Kaluza-Klein modes?
• RQ4 Why does the $SU(K)$ gauge symmetry remain unbroken across the entire 2$(K-1)$-dimensional 'moduli space' on $\mathrm{AdS}_5 \times \mathbb{S}^1$, unlike in flat space?
• RQ5 What is the origin of large corrections to the metric on the 'moduli space' in the $\mathrm{AdS}_5 \times \mathbb{S}^1$ background, and how are they related to the holographic dual?

Key findings

• The 6D ${\cal N}=(2,0)$ $A_{K-1}$ theory on $\mathrm{AdS}_5 \times \mathbb{S}^1$ with equal radii and 16 supercharges is dual to a subsector of large-$N$ 4D ${\cal N}=2$ quiver SCFTs, with the duality being universal across different 4D models.
• On $\mathrm{AdS}_5 \times \mathbb{S}^1$, the theory has a 2$(K-1)$-dimensional 'moduli space' where $SU(K)$ gauge symmetry is unbroken everywhere, in contrast to the 5$(K-1)$-dimensional moduli space on $\mathbb{R}^{4,1} \times \mathbb{S}^1$ where it is broken except at the origin.
• The boundary of $\mathrm{AdS}_5$ hosts a weakly coupled 4D ${\cal N}=2$ $SU(K)$ gauge theory whose coupling constants depend on the 'moduli' of the 6D theory, arising from non-standard boundary conditions for the gauge fields.
• The Kaluza-Klein spectrum of 6D fields—vector, tensor, and scalar modes—precisely matches the BPS operators in the 4D quiver SCFT, with matching $SU(2)_R$ and $U(1)_R$ quantum numbers and scaling dimensions.
• The missing operators in the 4D quiver (e.g., $n=0$ scalar, $n=1$ fermionic operator) are accounted for by hypermultiplet bilinears and $U(1)^{K-1}$ currents, consistent with the $SU(N)^K$ gauge group structure.
• Large corrections to the metric on the 'moduli space' arise due to the holographic coupling of the 4D boundary theory to the 6D bulk, indicating non-trivial quantum corrections not captured by classical geometry.