Fronsdal fields from gauge functions in Vasiliev’s higher spin gravity

Abstract

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

Higher spin fluctuations on spinless 4D BTZ black hole

“…2) the holonomies of which can be combined with open Wilson lines in twistor space [32][33][34][35] so as to provide a set of classical observables;…”

“…As for the analytic curvature singularities of odd codimension referred to above, we expect that the linearized Vasiliev system [41,47] provides a map that transfers corresponding delta function sources in the Fronsdal field equations to delta function sources of codimension two for the noncommutative twistor space connection. In fact, independently of whether the spacetime fields are singular or not, any Vasiliev higher spin geometry exhibits a twistor space delta function source of codimensions two; for a recent treatment of these singularity structures, see also [35]. Moreover, the latter arises via a vacuum expectation value of a dynamical two-form of an extension of Vasiliev's theory off-shell based on an internal 3-graded Frobenius algebra [48,49], referred to as Frobenius-Chern-Simons (FCS) gauge theory.…”

“…The BTZ-like black hole instead arises by taking Vasiliev's higher spin gravity, on the other hand, contains a flat one-form valued in a higher spin algebra (even in the presence of a non-trivial Weyl zero-form). Thus, the theory maps closed curves in spacetime to holonomies valued in the higher spin group; for a recent review, see Section 6 of [35]. Indeed, these holonomies reduce to those of the unfolded BHTZ-like geometries upon embedding the latter into higher spin gravity as vacua.…”

“…20 Assumption (a) follows (i), and the order of operations that we apply to evaluate the Weyl zero-form at the original unfolding point is in accordance with (ii). The scheme facilitates perturbative computations in Vasiliev's theory using the gauge function method [8,10,14,35], since the initial data (and other twistor space constructs arising in Vasiliev's Z-space) indeed admit regular presentations and Gaussian kernels can be star multiplied and traced straightforwardly. Thus, at every order of perturbation theory condition (iii) serves as an arbitrator among otherwise potentially ambiguous choices of (complex) contours for parametric integrals, thereby removing potential ambiguities from the scheme, though the scheme may clearly break down (provided that there exists either no or multiple consistent nestings of parametric integrals).…”

“…In the case of particle and black hole states in AdS 4 , the scheme has been implemented to all orders in [8,10] albeit in a (holomorphic) gauge which does not respect ALAdS boundary conditions, as stipulated by the central on mass-shell theorem, though ALAdS configurations can be reached by perturbative modifications of the gauge function at least at the linearized level [35].…”