A Derivation of AdS/CFT for Vector Models

Abstract

We explicitly rewrite the path integral for the free or critical $O(N)$ (or $U(N)$) bosonic vector models in $d$ space-time dimensions as a path integral over fields (including massless high-spin fields) living on ($d+1$)-dimensional anti-de Sitter space. Inspired by de Mello Koch, Jevicki, Suzuki and Yoon and earlier work, we first rewrite the vector models in terms of bi-local fields, then expand these fields in eigenmodes of the conformal group, and finally map these eigenmodes to those of fields on anti-de Sitter space. Our results provide an explicit (non-local) action for a high-spin theory on anti-de Sitter space, which is presumably equivalent in the large $N$ limit to Vasiliev's classical high-spin gravity theory (with some specific gauge-fixing to a fixed background), but which can be used also for loop computations. Our mapping is explicit within the $1/N$ expansion, but in principle can be extended also to finite $N$ theories, where extra constraints on products of bulk fields need to be taken into account.

arXiv comment71 pages + appendices (2 figures); v2: minor changes and added references; v3: minor changes; v4: corrected the discussion of the mass deformation when $d\ne 3$

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• Minimal model of Chiral Higher Spin Gravity
• Chiral Higher Spin Gravity and Convex Geometry
• Holography of Information in AdS/CFT
• Planar solutions of higher-spin theory. Nonlinear corrections
• Slightly Broken Higher Spin Symmetry: General Structure of Correlators
• Casimir Energy and Modularity in Higher-dimensional Conformal Field Theories
• ABJ Correlators with Weakly Broken Higher Spin Symmetry
• Disordered vector models: from higher spins to incipient strings
• The F-theorem in the melonic limit
• Four point functions in CFT's with slightly broken higher spin symmetry
• Anyonic correlation functions in Chern-Simons matter theories
• Matrix Entanglement
• Pure anti-de Sitter supergravity and the conformal bootstrap
• On $z$-dominance, shift symmetry and spin locality in higher-spin theory
• Instability of complex CFTs with operators in the principal series
• Mapping Large N Slightly Broken Higher Spin (SBHS) theory correlators to Free theory correlators
• On (spinor)-helicity and bosonization in $Ad{S}_4/CF{T}_3$
• Explicit holography for vector models at finite $N$, volume and temperature
• Emergent dual holographic description as a non-perturbative generalization of the Wilsonian renormalization group
• Hidden sectors of Chern-Simons Matter theories and Exact Holography
• Infinite distances in multicritical CFTs and higher-spin holography
• AdS from CFT for scalar QED
• Nonequilibrium thermodynamics perspectives for the monotonicity of the renormalization group flow
• On correlation functions as higher-spin invariants
• Holographic RG from ERG: Locality and General Coordinate Invariance in the Bulk
• Disordered $\mathcal{N}=(2,2)$ Supersymmetric Field Theories
• Renormalization group flow to effective quantum mechanics at IR in an emergent dual holographic description for spontaneous chiral symmetry breaking
• 3d Large $N$ Vector Models at the Boundary
• From Feynman graphs to Witten diagrams
• AdS/CFT correspondence for the $O(N)$ invariant critical ${\phi }^4$ model in 3-dimensions by the conformal smearing