Notes On Higher Spin Symmetries

TL;DR

In the AdS/CFT framework, the leading large-$N$ limit of ${\cal N}=4$ SYM suggests a bulk theory with an infinite tower of higher spin fields. The paper constructs explicit boundary-to-bulk propagators for massless higher spin fields in AdS, showing bulk solutions correspond one-to-one with bilinear deformations of the free boundary action, and analyzes how higher spin symmetry fixes boundary correlators up to a finite set of permutations of doubleton states. It develops the oscillator realization of the higher spin algebra $hs(2,2)$, clarifying the bulk–boundary symmetry correspondence and the role of doubletons as the free boundary degrees of freedom. The results provide sufficient conditions for a local higher spin bulk theory to reproduce the free boundary correlators, illuminating how higher spin symmetry constrains correlation functions in the AdS/CFT context and organizing the symmetry structure via $hs(2,2)$ and its real/complex forms.

Abstract

The strong form of the AdS/CFT correspondence implies that the leading $N$ expressions for the connected correlation functions of the gauge invariant operators in the free ${\cal N}=4$ supersymmetric Yang-Mills theory with the gauge group SU(N) correspond to the boundary S matrix of the classical interacting theory in the Anti de Sitter space. It was conjectured recently that the theory in the bulk should be a local theory of infinitely many higher spin fields. In this paper we study the free higher spin fields ($N=\infty$) corresponding to the free scalar fields on the boundary. We explicitly construct the boundary to bulk propagator for the higher spin fields and show that the classical solutions in the bulk are in one to one correspondence with the deformations of the free action on the boundary by the bilinear operators. We also discuss the constraints on the correlation functions following from the higher spin symmetry. We show that the higher spin symmetries fix the correlation functions up to the finite number of parameters. We formulate sufficient conditions for the bulk theory to reproduce the free field correlation functions on the boundary.