[Paper Review] Toward a Higher-Spin Dual of Interacting Field Theories

Leopoldo A. Pando Zayas, Cheng Peng|arXiv (Cornell University)|Mar 26, 2013

Black Holes and Theoretical Physics51 references4 citations

TL;DR

This paper extends the duality between free vector models and higher-spin gauge theories to interacting field theories by mapping the exact renormalization group (ERG) equations of interacting vector models in the large N limit to equations of motion in a higher-spin theory. Using a multiparticle extension of the higher-spin algebra, the authors show that interactions in the boundary theory correspond to curvature terms in the bulk, explicitly breaking higher-spin symmetry in a 1/N-suppressed manner, thus providing a covariant, holographic realization of RG flow as higher-spin gauge dynamics.

ABSTRACT

We show explicitly how the exact renormalization group equation of interacting vector models in the large N limit can be mapped into certain higher-spin equations of motion. The equations of motion are generalized to incorporate a multiparticle extension of the higher-spin algebra, which reflects the "multitrace" nature of the interactions in the dual field theory from the holographic point of view.

Motivation & Objective

• To extend the known duality between free vector models and higher-spin theories to interacting field theories in the large N limit.
• To map the exact renormalization group (ERG) equations of interacting vector models to higher-spin equations of motion in the bulk.
• To incorporate multi-trace interactions from the boundary field theory into the bulk via a multiparticle extension of the higher-spin algebra.
• To demonstrate that interactions in the boundary theory break higher-spin symmetry in the bulk, consistent with known constraints.
• To establish a covariant formulation of RG flow as higher-spin gauge dynamics, bridging ERG and gravity-like equations.

Proposed method

• The authors use the exact renormalization group (ERG) formalism to describe the RG flow of interacting vector models in D ≥ 3 dimensions, focusing on irrelevant deformations that preserve proximity to the free theory fixed point.
• They introduce a multiparticle extension of the higher-spin algebra to encode the multitrace nature of interactions in the boundary field theory.
• The mapping from the ERG equation to higher-spin equations of motion is achieved via a projection operation analogous to that used in Vasiliev’s higher-spin theory.
• The higher-spin equations are generalized to include curvature terms proportional to the interaction strength and suppressed by powers of 1/N, reflecting symmetry breaking.
• The construction relies on a star product formalism and momentum-space representation of the ERG kernel, enabling a direct comparison with higher-spin field equations.
• The analysis is performed in the large N limit, ensuring that the duality remains well-defined and that corrections are systematically controlled by 1/N.

Experimental results

Research questions

• RQ1 How can the exact renormalization group (ERG) equations of interacting vector models be mapped to equations of motion in a higher-spin theory?
• RQ2 What role does the multiparticle extension of the higher-spin algebra play in encoding multi-trace interactions in the boundary field theory?
• RQ3 How is higher-spin symmetry broken in the bulk when interactions are introduced on the boundary?
• RQ4 In what way does the RG flow near the free theory fixed point manifest as a higher-spin gauge theory in the bulk?
• RQ5 Can the RG=GR correspondence be extended beyond the free theory to include interacting field theories via a covariant higher-spin formulation?

Key findings

• The ERG equations of interacting vector models in the large N limit can be mapped to higher-spin equations of motion in the bulk, with the mapping being exact only in the free limit.
• Multi-trace interactions in the boundary theory are naturally encoded in a multiparticle extension of the higher-spin algebra, which generalizes the standard algebra to include composite states.
• The higher-spin equations in the bulk contain curvature terms proportional to the interaction strength and suppressed by powers of 1/N, signaling explicit breaking of higher-spin symmetry.
• This symmetry breaking is consistent with recent results by Maldacena and Zhiboedov, which constrain the existence of exact higher-spin symmetries in quantum field theories.
• The construction provides a covariant realization of the RG=GR correspondence, showing that first-order ERG equations are equivalent to first-order higher-spin gauge equations in the connection formulation.
• Dimensionality plays a crucial role in the mapping, with dimension-dependent structures emerging in the interacting case, unlike in the free theory where dimension was secondary.