Higher Spin Gauge Theory and Holography: The Three-Point Functions (0912.3462v4)

Abstract: In this paper we calculate the tree level three-point functions of Vasiliev's higher spin gauge theory in AdS4 and find agreement with the correlators of the free field theory of N massless scalars in three dimensions in the O(N) singlet sector. This provides substantial evidence that Vasiliev theory is dual to the free field theory, thus verifying a conjecture of Klebanov and Polyakov. We also find agreement with the critical O(N) vector model, when the bulk scalar field is subject to the alternative boundary condition such that its dual operator has classical dimension 2.

• The paper develops a robust method for computing tree-level three-point functions using boundary-to-bulk propagators for scalar and higher spin currents.
• It confirms the duality between Vasiliev’s higher spin theory in AdS4 and free or critical O(N) vector models by matching correlation structures.
• The analysis distinguishes between Δ=1 and Δ=2 boundary conditions, providing insights for extending holographic approaches to higher-point and loop corrections.

Higher Spin Gauge Theory and Holography: Evaluating Three-Point Functions

The paper by Simone Giombi and Xi Yin explores the evaluation of three-point functions in the context of Vasiliev's higher spin gauge theory in four-dimensional Anti-de Sitter space ($AdS_4$). It seeks to establish the duality proposed by Klebanov and Polyakov, between Vasiliev's theory and free (or critical) $O(N)$ vector models at large $N$, a cornerstone of the AdS/CFT correspondence.

Background and Motivation

The AdS/CFT correspondence provides a profound link between gravitational theories in higher-dimensional AdS spaces and conformal field theories (CFTs) on their boundaries. This duality has predominantly been explored in scenarios involving superstring theories or strongly coupled gauge theories. However, Vasiliev's theory, which encompasses fields of all even integer spins, offers a compelling scenario in which a weakly coupled bulk theory corresponds to a free boundary theory, providing a novel angle to approach the holographic principle.

Theoretical Framework

Vasiliev's higher spin theory is a sophisticated framework that encapsulates gauge fields with increasing spins, extending beyond traditional finite spin theories. A key challenge in this setup is the lack of a conventional Lagrangian, with the theory defined through complex nonlinear equations of motion. These involve auxiliary fields and a star product algebra to structure interactions in the AdS background.

Key Contributions

The paper primarily focuses on calculating tree-level three-point functions for these higher spin fields. These computations are pivotal for testing the conjectured duality with the dual field theories on the boundary—specifically, the free $O(N)$ vector models and their critical counterparts. The objective is to see whether these correlation functions in the bulk theory manifest the same structures as those expected from a free field theory perspective.

1. Three-Point Function Calculations: The authors develop methodologies to compute these correlation functions via boundary-to-bulk propagators. They solve for propagators associated with scalar and higher spin currents, asserting a nontrivial structure consistent with CFT predictions.
2. Verification of Duality: By finding agreement with the free $O(N)$ vector model correlators, the paper substantiates the conjecture of duality. Notably, the agreement extends to scenarios involving two scalar operators and a spin-$s$ current and vice versa.
3. Boundary Conditions: The analysis distinguishes between $\Delta=1$ and $\Delta=2$ boundary condition cases, reflecting different dual operator scaling dimensions, and further verifies these scenarios by matching bulk computations with known boundary model results.

Implications and Future Directions

The verification of such dualities provides a robust initial framework for understanding theories that harbour infinite towers of massless fields potentially relevant to string theory and past high-energy phenomenological investigations. This also broadens the scope of the AdS/CFT correspondence by elucidating its applicability beyond traditional finite spin models.

For future directions, critical tasks include resolving ambiguities in quartic interactions within Vasiliev's framework and understanding the potential spontaneous symmetry breaking within the AdS space, particularly for the critical $O(N)$ vector model where exact higher spin symmetries do not hold at finite $N$. Moreover, extending this analysis to higher-point functions and loop level corrections would further solidify our understanding and application of higher spin theories in holography.

In conclusion, Giombi and Yin's paper advances our comprehension of higher spin gauge theories, providing concrete computational strategies to bridge bulk theories with their associated boundary CFTs, and opening avenues for richer holographic interpretations in quantum gravity research.