Abstract
In this paper, we revisit a number of issues in Vasiliev’s theory related to gauge functions, ordering schemes, and the embedding of Fronsdal fields into master fields. First, we parametrize a broad equivalence class of linearized solutions using gauge functions and integration constants, and show explicitly how Fronsdal fields and their Weyl tensors arise from these data in accordance with Vasiliev’s central on mass shell theorem. We then gauge transform the linearized piece of exact solutions, obtained in a convenient gauge in Weyl order, to the aforementioned class, where we land in normal order. We spell out this map for massless particle and higher spin black hole modes. Our results show that Vasiliev’s equations admit the correct free-field limit for master field configurations that relax the original regularity and gauge conditions in twistor space. Moreover, they support the off-shell Frobenius-Chern-Simons formulation of higher spin gravity for which Weyl order plays a crucial role. Finally, we propose a Fefferman-Graham-like scheme for computing asymptotically anti-de Sitter master field configurations, based on the assumption that gauge function and integration constant can be adjusted perturbatively so that the full master fields approach free master fields asymptotically.
Highlights
1.1 Motivations and summary of our main resultsAn outstanding problem in higher spin gravity (HSG) is the quest for a geometric formulation suitable for computing physical observables
The master fields are functions of fiber coordinates Y, and the base manifold is an extension of spacetime with additional coordinates Z, with Y and Z each making up a non-commutative twistor space
This construction was proposed as a method for encoding an infinite tower of Fronsdal fields together with highly non-local interactions into a remarkably simple set of constraints, but at the price of introducing ambiguities, entering via the resolution scheme for the Z-dependence, that affect the reading of the dynamics of physical fields from spacetime vertices and Witten diagrams
Summary
An outstanding problem in higher spin gravity (HSG) is the quest for a geometric formulation suitable for computing physical observables. We propose a Fefferman-Graham-like scheme for the perturbative construction of AAdS solutions to Vasiliev’s equations whose on-shell action yields physically meaningful holographic two-point functions in the leading order of classical perturbation theory This scheme involves fixing boundary conditions in both spacetime and twistor space; in particular, it is natural to expect that such a procedure will help in resolving the ambiguities that arise in integrating the Z-dependence perturbatively, and that it is instrumental to properly singling out the superselection sector that may be captured by the dual CFT. The analysis carried out in this paper addresses a number of open issues in the literature on Vasiliev’s theory concerning exact solutions, perturbative schemes, admissible classes of symbols, and choices of gauge and ordering prescriptions It provides the starting block for an iterative procedure for constructing ALAdS higher spin geometries with non-trivial topology both in spacetime and twistor space and related on-shell actions. We show that they are protected for Weyl-ordered solutions based on uncorrected initial data, in the sense that, there are sub-leading perturbative corrections to the observables, they are all proportional to the leading contribution
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