The Higher Spin/Vector Model Duality (1208.4036v2)

Abstract: This paper is mainly a review of the dualities between Vasiliev's higher spin gauge theories in AdS4 and three dimensional large N vector models, with focus on the holographic calculation of correlation functions of higher spin currents. We also present some new results in the computation of parity odd structures in the three point functions in parity violating Vasiliev theories.

• The paper demonstrates that holographic duality enables matching three-point functions of higher spin currents between Vasiliev's AdS4 theories and 3D vector models.
• It rigorously constructs Vasiliev’s nonlinear equations using master fields and investigates both parity even and odd contributions in boundary conditions.
• The study identifies key challenges in quantizing higher spin interactions and proposes future research directions to explore non-perturbative aspects of quantum gravity.

Analysis of the Holographic Duality: The Higher Spin/Vector Model Duality

Simone Giombi and Xi Yin present a comprehensive paper of the dualities between Vasiliev's higher spin gauge theories in $AdS_4$ and three-dimensional large $N$ vector models. These dualities are framed within the broader context of the AdS/CFT correspondence, which posits a profound connection between gravitational theories in anti-de Sitter (AdS) space and conformal field theories (CFTs) defined in fewer dimensions. The paper focuses on correlation functions of higher spin currents, emphasizing their holographic calculations and exploring parity odd structures in three-point functions, particularly in parity violating Vasiliev theories.

Overview of Vasiliev's Higher Spin Gauge Theory

Vasiliev's higher spin theories are characterized by their infinite towers of gauge fields of increasing spin, formulated in a fully covariant and gauge-invariant manner. The paper begins with a detailed construction of Vasiliev's equations using master fields: the one-form $W$, scalar $B$, and additional auxiliary fields depending on twistor-like variables $Y$ and $Z$. These master fields satisfy nonlinear equations that are simplified in the frame-like formalism, analogous to Cartan's formalism in general relativity.

A crucial aspect of Vasiliev's theory is the presence of a holographic dual description in terms of vector models, specifically $O(N)$ symmetric theories for the bosonic case. For the bosonic and supersymmetric generalizations, the gauge theory in $AdS_4$ encompasses gravity and higher spin fields, and by imposing appropriate boundary conditions, we establish a connection with the dual vector models in three dimensions.

Correlation Functions and Boundary Conditions

The duality between higher spin theories and vector models manifests prominently in the computational agreement of correlation functions. The paper meticulously derives holographic results for the three-point functions of boundaries currents. This involves treating two of the boundary currents as sources and computing the boundary values of the dual fields in the bulk. The authors present an intricate analysis of parity even and parity odd contributions, determined by a parity breaking phase $\theta_0$, and validate their results against known structures from field theory.

Parity Violation and Holographic Implications

For parity-invariant theories, significant parallels are drawn with free and critical $O(N)$ models. However, parity violation introduces additional complexity and interest due to potential Chern-Simons vector model duals. These vector models, parameterized by a ’t Hooft-like coupling $\lambda = N/k$, indicate a richer structure where duality extends to parity violating interactions in Vasiliev theory, modulated by $\theta_0 = \frac{\pi}{2}\lambda$ in the scalar case.

The exploration of parity violation is particularly robust, examining its impact on conformal dimensions and symmetry-breaking patterns of higher spin currents. The theoretical framework predicts consistency with Chern-Simons-matter theories, guiding the understanding of holography beyond perturbative regimes.

Challenges and Future Directions

The complexity of Vasiliev's equations, lack of a conventional action from which they derive, and subtleties in gauge fixing pose challenges for quantization within these models. The paper outlines an ongoing need to explore non-perturbative aspects and exact results to gain coherent insights into both the Vasiliev theory and its dual vector models. Moreover, exploring these holographic dualities further might provide critical insight into quantum gravity and the role of higher spin symmetries in string theory and field theory analogs.

Overall, Giombi and Yin's paper is a substantial contribution to the understanding of higher spin theory dualities, offering an in-depth exploration of theoretical constructs poised to illuminate intricate aspects of holography. This research underscores the speculative yet promising avenues in understanding gravitational interactions in AdS space and their dual conformal field theories.