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.

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

References in corpus (11)

• Holographic representation of local bulk operators
• Spinning AdS Propagators
• AdS_4/CFT_3 Construction from Collective Fields
• Holographic dual of free field theory
• Quantum W-symmetry in AdS_3
• Double-Trace Deformations, Mixed Boundary Conditions and Functional Determinants in AdS/CFT
• Free field world-sheet correlators for ${\mathrm{AdS}}_3$
• Graviton 1-loop partition function for 3-dimensional massive gravity
• Real Forms of Complex Higher Spin Field Equations and New Exact Solutions
• The Exact Renormalization Group and Higher-spin Holography
• Holography as a Gauge Phenomenon in Higher Spin Duality

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