Working within Vasiliev's theory, we construct an equivalence class of linearized gauge functions that yields free Fronsdal fields in accordance with Vasiliev's central on mass shell theorem. Using this construction, we map linearized solutions of Vasiliev's equations obtained in Weyl order from zero-form integration constants and vacuum gauge functions to generating functions for Weyl tensors and Fronsdal fields obtained in normal order. We exemplify this map for massless particle and higher spin black hole modes. We also show that the required gauge condition on the linearized twistor space connection is weaker than the one used in Vasiliev's original analysis. We incorporate this relaxed gauge condition into a Fefferman-Graham-like scheme for imposing asymptotically locally anti-de Sitter boundary conditions on the full master fields in which the latter approach free master fields asymptotically. Our results support the embedding of Vasiliev's theory as a branch of a Frobenius-Chern-Simons theory.1 The solutions that we refer to as higher spin black hole states are so called essentially because they possess a tower of Weyl tensors of all integer spins that include and generalize the spin-2 Weyl tensor of an AdS Schwarzschild black hole. However, at present there is no known higher-spin invariant quantity ensuring that the singularity of the individual Weyl tensors is physical, and whether there exists any invariant notion of an event horizon -as well as an entropy attached to it -remains an open problem. On the other hand, the fact that each such solution has identical black-hole asymptotics but is possibly non-singular and horizon-free may suggest an interpretation in terms of black-hole microstates, similar to fuzzballs [23][24][25][26]. In that sense, the name black-hole states may turn out to be appropriate in an even deeper sense. See [19] for more details on this proposal and on our usage of the terminology. 14 Examples of non-trivial flat twistor space connections are given in [12]. with classical closed path in Z 4 given by L(M ) ∪ L(M ), which is invariant under large Cartan gauge transformations and independent of Proj X 4 (p 0 ), provided that it can be evaluated using the regular scheme. In other words, W L(M )∪L(M ) is the trace of the holonomy resulting from parallel transporting an object from Z to Z + M along L(M ) using V , and then back along the same path using V # . The non-triviality of the closed Wilson loop follows from H Lp 0 →p 1 (M ) [V ] ⋆ H Lp 1 →p 0 (M ) [V # ] = H Lp 0 →p 1 (M ) [V ] ⋆ H Lp 0 →p 1 (M ) [V # ] −1 = e iM α Sα(p 0 )/2 ⋆ ⋆ e −iM α Zα/2 ⋆ ⋆ e iM α S # α (p 0 )/2 ⋆ ⋆ e −iM α Zα/2 ⋆ −1

Abstract. We first prove that, in Vasiliev's theory, the zero-form charges studied in 1103. 2360 and 1208.3880 are twisted open Wilson lines in the noncommutative Z space. This is shown by mapping Vasiliev's higher-spin model on noncommutative Yang-Mills theory. We then prove that, prior to Bose-symmetrising, the cyclically-symmetric higherspin invariants given by the leading order of these n-point zero-form charges are equal to corresponding cyclically-invariant building blocks of n-point correlation functions of bilinear operators in free conformal field theories (CFT) in three dimensions. On the higher spin gravity side, our computation reproduces the results of 1210.7963 using an alternative method amenable to the computation of subleading corrections obtained by perturbation theory in normal order. On the free CFT side, our proof involves the explicit computation of the separate cyclic building blocks of the correlation functions of n conserved currents in arbitrary dimension d > 2 using polarization vectors, which is an original result. It is shown to agree, for d = 3 , with the results obtained in 1301.3123 in various dimensions and where polarization spinors were used.

We investigate the formulation of Vasiliev’s four-dimensional higher-spin gravity in operator form, without making reference to one specific ordering. More precisely, we make use of the one-to-one mapping between operators and symbols thereof for a family of ordering prescriptions that interpolate between and go beyond Weyl and normal orderings. This correspondence allows us to perturbatively integrate the Vasiliev system in operator form and in a variety of gauges. Expanding the master fields in inhomogenous symplectic group elements, and letting products be controlled only by the group, we specify a family of factorized gauges in which we are able to integrate the system to all orders, producing exact solutions, including but not restricted to ones presented previously in the literature; and then connect, at first order, to a family of rotated Vasiliev gauges in which the solutions can be represented in terms of Fronsdal fields. The gauge function responsible for the latter transformation is explicitly constructed at first order. The analysis of the system in various orderings is facilitated by an analytic continuation of Gaussian symbols, by means of which one can distinguish and connect the two branches of the metaplectic double cover and give a rationale to the properties of the inner Klein operators as Gaussian delta sequences defining analytic delta densities. As an application of some of the techniques here developed, we evaluate twistor space Wilson line observables on our exact solutions and show their independence from auxiliary constructs up to the few first subleading orders in perturbation theory.

In this note, we review the results of 1705.03928 on a special class of observables of Vasiliev's four-dimensional higher spin gravity theory, referred to as zero-form charges. These objects were shown to be twisted open Wilson lines in the noncommutative twistor Z space. For boundary conditions corresponding to higher spin fields in asymptically anti-de Sitter spacetime, their classical perturbative expansion was then proven to start with cyclically-invariant building blocks of n-point correlation functions of bilinear operators in free conformal field theories (CFT) in three dimensions. On the CFT side, the proof involves the explicit computation of the aforementioned building blocks in arbitrary dimension d > 2, which generalizes known results on 3-point functions for any d > 2, and on n-point functions for d = 3.