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d=3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories

Ofer Aharony, Guy Gur-Ari, Ran Yacoby

TL;DR

The paper analyzes 3d O(N)_k and U(N)_k Chern-Simons theories coupled to scalars in the large-N limit, uncovering a parity-violating deformation controlled by λ that preserves a line (or surface) of conformal fixed points when λ_6 is also marginal. Through explicit two-loop calculations and an all-orders argument leveraging supersymmetric embeddings, it shows that at infinite N both λ and λ_6 are exactly marginal, while at finite N they generate nontrivial fixed points and anomalous dimensions remain suppressed in 1/N. The operator spectrum at large N mirrors the free theory, consisting of an infinite tower of higher-spin currents and a scalar of Δ=1, yet certain correlators, such as ⟨T J J⟩, depend on λ, signaling a continuous family of holographic duals that deform Vasiliev higher-spin gravity. These results illuminate the AdS/CFT map for weakly coupled duals and suggest rich structures for parity-violating higher-spin theories in AdS4, with multiple avenues for further exploration including holographic construction, finite-N corrections, and generalizations to fermions or supersymmetry.

Abstract

We study three dimensional O(N)_k and U(N)_k Chern-Simons theories coupled to a scalar field in the fundamental representation, in the large N limit. For infinite k this is just the singlet sector of the O(N) (U(N)) vector model, which is conjectured to be dual to Vasiliev's higher spin gravity theory on AdS_4. For large k and N we obtain a parity-breaking deformation of this theory, controlled by the 't Hooft coupling lambda = 4 πN / k. For infinite N we argue (and show explicitly at two-loop order) that the theories with finite lambda are conformally invariant, and also have an exactly marginal (φ^2)^3 deformation. For large but finite N and small 't Hooft coupling lambda, we show that there is still a line of fixed points parameterized by the 't Hooft coupling lambda. We show that, at infinite N, the interacting non-parity-invariant theory with finite lambda has the same spectrum of primary operators as the free theory, consisting of an infinite tower of conserved higher-spin currents and a scalar operator with scaling dimension Δ=1; however, the correlation functions of these operators do depend on lambda. Our results suggest that there should exist a family of higher spin gravity theories, parameterized by lambda, and continuously connected to Vasiliev's theory. For finite N the higher spin currents are not conserved.

d=3 Bosonic Vector Models Coupled to Chern-Simons Gauge Theories

TL;DR

The paper analyzes 3d O(N)_k and U(N)_k Chern-Simons theories coupled to scalars in the large-N limit, uncovering a parity-violating deformation controlled by λ that preserves a line (or surface) of conformal fixed points when λ_6 is also marginal. Through explicit two-loop calculations and an all-orders argument leveraging supersymmetric embeddings, it shows that at infinite N both λ and λ_6 are exactly marginal, while at finite N they generate nontrivial fixed points and anomalous dimensions remain suppressed in 1/N. The operator spectrum at large N mirrors the free theory, consisting of an infinite tower of higher-spin currents and a scalar of Δ=1, yet certain correlators, such as ⟨T J J⟩, depend on λ, signaling a continuous family of holographic duals that deform Vasiliev higher-spin gravity. These results illuminate the AdS/CFT map for weakly coupled duals and suggest rich structures for parity-violating higher-spin theories in AdS4, with multiple avenues for further exploration including holographic construction, finite-N corrections, and generalizations to fermions or supersymmetry.

Abstract

We study three dimensional O(N)_k and U(N)_k Chern-Simons theories coupled to a scalar field in the fundamental representation, in the large N limit. For infinite k this is just the singlet sector of the O(N) (U(N)) vector model, which is conjectured to be dual to Vasiliev's higher spin gravity theory on AdS_4. For large k and N we obtain a parity-breaking deformation of this theory, controlled by the 't Hooft coupling lambda = 4 πN / k. For infinite N we argue (and show explicitly at two-loop order) that the theories with finite lambda are conformally invariant, and also have an exactly marginal (φ^2)^3 deformation. For large but finite N and small 't Hooft coupling lambda, we show that there is still a line of fixed points parameterized by the 't Hooft coupling lambda. We show that, at infinite N, the interacting non-parity-invariant theory with finite lambda has the same spectrum of primary operators as the free theory, consisting of an infinite tower of conserved higher-spin currents and a scalar operator with scaling dimension Δ=1; however, the correlation functions of these operators do depend on lambda. Our results suggest that there should exist a family of higher spin gravity theories, parameterized by lambda, and continuously connected to Vasiliev's theory. For finite N the higher spin currents are not conserved.

Paper Structure

This paper contains 14 sections, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The $O(N)$ Model with Chern-Simons Interactions
  3. Conformal Symmetry
  4. The Beta Function $\beta_{\lambda_6}$ at Two Loops
  5. The Large $N$ Beta Function $\beta_{\lambda_6}$ to All Orders
  6. Spontaneous Breakdown of Conformal Symmetry
  7. Higher-Spin Currents
  8. Correlation Functions
  9. Computation of $\left<J_2 J_1 J_1\right>$
  10. Summary and Future Directions
  11. Conventions
  12. 2-Loop Diagram Results
  13. Conformal Transformations
  14. Anomalous Dimension of $\phi^i\phi^i$

Figures (1)

  • Figure 1: The space of coupling constants and the renormalization group flow towards the IR for large but finite $N$, based on the two-loop beta function. There are two lines of fixed points. Since $\beta_{\lambda}=0$, the flow lines are all in the $\lambda_6$ direction.