An AdS_3 Dual for Minimal Model CFTs (1011.2986v4)

Abstract: We propose a duality between the 2d W_N minimal models in the large N 't Hooft limit, and a family of higher spin theories on AdS_3. The 2d CFTs can be described as WZW coset models, and include, for N=2, the usual Virasoro unitary series. The dual bulk theory contains, in addition to the massless higher spin fields, two complex scalars (of equal mass). The mass is directly related to the 't Hooft coupling constant of the dual CFT. We give convincing evidence that the spectra of the two theories match precisely for all values of the 't Hooft coupling. We also show that the RG flows in the 2d CFT agree exactly with the usual AdS/CFT prediction of the gravity theory. Our proposal is in many ways analogous to the Klebanov-Polyakov conjecture for an AdS_4 dual for the singlet sector of large N vector models.

• The paper proposes a duality between 2D W_N minimal models in the large N ’t Hooft limit and higher spin theories on AdS3, with spectra matching across all coupling values.
• The paper employs detailed analyses of conformal dimensions, fusion rules, and RG flows to verify consistency with holographic predictions.
• The paper highlights the crucial role of two complex scalars in bridging the CFT and AdS formulations, advancing the understanding of higher spin/AdS duality.

An Analysis of AdS Duals for Minimal Model CFTs

The paper moves forward with a proposal for a duality between 2-dimensional conformal field theories (CFTs) and a class of higher spin interactions on 3-dimensional anti-de Sitter (AdS) spaces. Specifically, the authors claim a dual relationship between 2D W$_N$ minimal models in the large N 't Hooft limit and a family of higher spin theories on AdS$_3$. The 2D CFTs are characterized as WZW coset models and, for N=2, align with conventional Virasoro unitary series. The corresponding dual framework on the bulk side incorporates massless higher spin fields plus two complex scalars, whose masses are intricately linked to the 't Hooft coupling constant of the dual CFT.

The authors present stringent evidence supporting the conjecture that the spectra of the two theories are precisely matched across all 't Hooft coupling values. Furthermore, they illustrate a strong correlation between the renormalization group (RG) flows in the 2D CFTs and the anticipated AdS/CFT predictions from gravitational theory, drawing an analogy to the Klebanov-Polyakov conjecture for an AdS dual for the singlet sector of large N vector models.

Methodology and Results

1. Minimal Model Conformal Field Theories:
   • Detailed exploration of minimal W$_N$ models as WZW coset models with specific relationships defined by W-algebra symmetry.
   • Conformal dimensions, fusion rules, and RG flows are explicitly derived and explored in the context of these CFTs.
2. The 't Hooft Limit:
   • Consideration of a large N limit where both N and k approach infinity, keeping λ, the 't Hooft coupling, fixed. This delineates the CFT behavior scaling as N.
3. Higher Spin AdS Dual:
   • Description of corresponding higher spin theories in AdS$_3$ space characterized by a W-algebra symmetry.
   • Emphasis is placed on a match between the central charges of the 2D CFTs and this symmetry within the bulk theories.
   • Two massive scalars in the higher spin theories play a pivotal role in connecting spectra, whose masses conform to $M^2 = - (1 - λ^2)$.
4. Verification of Proposed Duality:
   • Verification of the proposed duality through comparisons of operator spectra and partition functions at one-loop levels.
   • Adherence to theoretical expectations for RG flows substantiates the claim, displaying concrete consistency with the CFT predictions.
5. Theoretical Implications:
   • The connection highlights a significant stride in elucidating higher spin theories with a controlled limit in 2-dimensional CFTs, shedding light on non-supersymmetric theories with potential statistical mechanics applications.
   • The work augments understanding of higher-spin/AdS correlatives, standing as a generalization of models to a complete one-parameter family of fixed points, alongside its ties to statistical and condensed matter systems.

Implications and Future Research

This research offers a significant insight into gauge-gravity duality devoid of superfluous elements often associated with string theory. Higher-spin interactions represent a non-trivial extension to the established frameworks and raise concerns about the limit and nature of Einstein gravity regimes. Further explorations into this proposal may illuminate understanding of realistic systems with intermediate coupling strengths.

Future endeavors could ensure the thorough substantiation of spectra matching and implement calculations for correlation functions, extending the findings towards holographic interpretations within strongly interacting frameworks. The implications explore generalized models, thus prompting exploration into theories encompassing both fermionic and supersymmetric considerations, potentially broadening the implications in other domains like condensed matter physics.