Higher spin gauge theory and holography: the three-point functions

Abstract

In this paper we calculate the tree level three-point functions of Vasiliev's higher spin gauge theory in AdS 4 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.

New classes of bi-axially symmetric solutions to four-dimensional Vasiliev higher spin gravity

“…This suggests a holographic relationship to three-dimensional conformal field theories [3][4][5]; see also [6][7][8]. In [9,10] this relation was examined under the assumption that the Gubser-Klebanov-Polyakov-Witten (GKPW) prescription [11,12] for on-shell computations of Witten diagrams can be applied to classical field configurations obtained from Vasiliev's equations. However, the Fronsdal fields embedded into Vasiliev's master fields have non-local interactions [13][14][15] 1 that belong to a functional class widely separated [17] from that of the quasi-local Fronsdal theory [18], which is built by applying the canonical Noether approach to Fronsdal fields in anti-de Sitter spacetime.…”

“…The differential star product algebra is assumed to be trivial in strictly positive degrees, in the sense that dΞ M := (dx µ , dz α , dzα) are taken to be graded anti-commuting elements obeying 9) which are consistent with associativity. The algebra Ω(M) ⊗ A is also assumed to be equipped with an anti-linear anti-automorphism †, for which we use the convention…”

“…We shall refer to Y 4 × Z 4 as the full twistor space, and Y 4 and Z 4 , respectively, as the internal and external twistor spaces. 9 The Sp(4;R) quartets are split into SL(2;C) doublets, viz. 10…”

“…The equations given so far provide a formal definition of the basic bosonic model. 9 Taking the master fields to be smooth functions of Y4 yields an anti-de Sitter analog of the PenroseNewman transformation; to our best understanding, the precise relation between Y4 × Z4 and the original (commuting) twistor space of Penrose remains to be spelled out in detail. 10 The doublet indices are raised and lowered using f α = ε αβ f β and f β = f α ε αβ idem fα.…”

“…Beyond the linearized approximation, it has been used in amplitude computations [9,10,28,29] and related recent works [38,40]. Most exact solutions found so far, however, have been given in other gauges argued to be equivalent to Vasiliev gauge; for example, see [26,37,41].…”