Higher spin fluctuations on spinless 4D BTZ black hole

We construct linearized solutions to Vasiliev's four-dimensional higher spin gravity on warped $AdS_3 \times_\xi S^1$ which is an $Sp(2)\times U(1)$ invariant non-rotating BTZ-like black hole with $\mathbb{R}^2\times T^2$ topology. The background can be obtained from $AdS_4$ by means of identifications along a Killing boost $K$ in the region where $\xi^2\equiv K^2\geqslant 0$, or, equivalently, by gluing together two Ba\~nados--Gomberoff--Martinez eternal black holes along their past and future space-like singularities (where $\xi$ vanishes) as to create a periodic (non-Killing) time. The fluctuations are constructed from gauge functions and initial data obtained by quantizing inverted harmonic oscillators providing an oscillator realization of $K$ and of a commuting Killing boost $\widetilde K$. The resulting solution space has two main branches in which $K$ star commutes and anti-commutes, respectively, to Vasiliev's twisted-central closed two-form $J$. Each branch decomposes further into two subsectors generated from ground states with zero momentum on $S^1$. We examine the subsector in which $K$ anti-commutes to $J$ and the ground state is $U(1)_K\times U(1)_{\widetilde K}$-invariant of which $U(1)_K$ is broken by momenta on $S^1$ and $U(1)_{\widetilde K}$ by quasi-normal modes. We show that a set of $U(1)_{\widetilde K}$-invariant modes (with $n$ units of $S^1$ momenta) are singularity-free as master fields living on a total bundle space, although the individual Fronsdal fields have membrane-like singularities at $\widetilde K^2=1$. We interpret our findings as an example where Vasiliev's theory completes singular classical Lorentzian geometries into smooth higher spin geometries.


Higher spin resolution of gravitational singularities
An interesting problem in gravity is whether classical spacetime singularities can be resolved by switching on higher spin gauge fields. Indeed, the resulting non-abelian interactions are spacetime non-local already at the classical level, akin to those of a full quantum effective field theory.
Moreover, higher spin gravities contain infinite towers of massless fields at weak coupling that one may argue become massive due to quantum effects, hence associated to screened charges in weakly coupled asymptotic regions, while supporting moduli spaces of classical solutions interpolating between asymptotic regions and strongly coupled core regions with nontrivial topology. This motivates examining whether classical spacetime singularities can be completed into smooth higher spin geometries given by classical solutions to unbroken higher spin gravities with bounded field configurations and finite observables accessible to asymptotic observers, providing semi-classical realizations of geometrically entangled quantum states.
To concord with basic properties of the holographic correspondence between generally covariant theories with anti-de Sitter vacua and conformal field theories in the context of higher spin theory [1][2][3][4], we shall a) presume a higher spin symmetry breaking mechanism whereby weakly coupled gauge fields with spins greater than two (and possibly also some fields with spins less than or equal to two) acquire masses so as to leave a spectrum with massless subsector corresponding to mattercoupled gravity; and b) construct exact solutions to unbroken higher spin gravities that describe smooth higher spin geometries containing asymptotically locally anti-de Sitter (ALAdS) (or de Sitter) regions where the full theory can be approximated by (free) Fronsdal fields 1 .
Thus, in the broken phase, the asymptotic fall-off of the fields that have acquired mass is enhanced, ensuring that they do not affect the leading orders of the Fefferman-Graham expansion of an effective lower-spin theory containing gravity (even though the spectrum of the broken phase is not fully gapped). By this screening mechanism, we envisage weakly coupled asymptotic regions described by an effective gravity theory glued to strongly coupled core regions described by an unbroken higher spin gravity; that is, we trust the latter when its curvatures are large, and the former when its curvatures are small.
Moreover, drawing on recent progress in assigning entanglement entropy to topologically nontrivial spacetimes [19], our working hypothesis is that the emerging higher spin geometries are not only smooth but also entangled in the sense that c) the higher spin resolution of a gravitational singularity yields a set of topologies including manifolds with numerous boundaries; and d) manifolds with multiple boundaries are represented quantum mechanically by geometrically entangled states.
Combining the dynamical higher spin symmetry breaking mechanisms (a) and (b) with the geometric entangling mechanisms (c) and (d), we envisage asymptotic observers represented by operators acting in the Hilbert space of the broken phase (with asymptotically enhanced mass-gap), sandwiched between geometrically entangled states with one "external" leg in the broken state space and multiple "internal" legs in unbroken state spaces, represented semi-classically by ALAdS higher spin geometries with non-trivial core topology. In other words, we propose that gravitational singularities are resolved into moduli spaces of smooth higher spin geometries, whereby physical observables are given by sums over unbroken core states: the latter are organized into ensembles by geometrically entangled states [19] represented semi-classically by classical solutions to higher spin gravities.
In this paper, we shall focus on (b) by exploring classical solutions to Vasiliev's equations in four spacetime dimensions [20]. Vasiliev's theory has been conjectured [2][3][4]21] to undergo dynamical symmetry breaking due to mixing between (massless) one-particle states and multiparticle Goldstone modes in the presence of special boundary conditions in anti-de Sitter spacetime.
As this mechanism does not require any coupling to additional fields, the theory, possibly including Yang-Mills-like gauge fields and fermions [22,23], provides a relatively minimalistic framework for studying singularity resolutions already at the classical level in accordance with (a) and (b) 2 .
The following classical singularities of matter-coupled gravity will be of interest for this work: i) Degenerate metrics; 2 Stringy extensions [2,[24][25][26] by extra massive fields are likely required in order to admit flat space limits with significant mass-gaps.
ii) Analytic singularities 3 in generalized Weyl curvatures 4 ; iii) Delta function sources in the equations of motion.
At the center of the Schwarzschild black hole, all three types of singularities arise: the metric degenerates on trapped spheres (leading to geodesic incompleteness); the Weyl tensor blows up; and the linearized equations of motion have a delta function source. In order to disentangle these types singularities, it is useful to instead consider fluctuations over constantly curved black holes, as the background only exhibits degenerate metrics related to trapped submanifolds, while the fluctuations can be made to exhibit curvature singularities.
Constantly curved black holes were first constructed in three dimensions by Bañados, Teitelboim and Zanelli (BTZ) [27], and further studied by Bañados, Henneaux, Teitelboim and Zanelli (BHTZ) [28] within the context of a more general moduli space of three-dimensional constant curvature geometries including extremal black holes, conical singularities and proper three-dimensional antide Sitter spacetime itself. BHTZ-like geometries in four spacetime dimensions were first studied bẙ Aminneborg, Bengtsson, Holst and Peldan in [29,30], who observed that the uplift of the spinless BTZ black hole only has quasi-horizons that fail to trap any two-dimensional subspaces. The latter geometry was later revisited by Bañados, Gomberoff and Martinez (BGM) [31], who properly interpreted it (by representing it using a three-dimensional Penrose diagram) as a black hole that traps (one-dimensional) circles rather than any two-manifold.
In this paper, we shall examine fluctuations around the eternal spinless BGM black hole thought of as a classical solution of Vasiliev's bosonic higher spin theory. More precisely, we shall construct linearized massless Weyl tensors of arbitrary integer spin obeying Bargmann-Wigner equations of motion on the aforementioned background and subject to various boundary conditions corresponding to different representations of the background symmetry group, including modes with momenta around the trapped sphere and quasi-normal modes. Pending a fully non-linear construction, we shall verify our main hypothesis, namely that the linearized Vasiliev master fields admit analytic continuations across singularities as well as horizons, so as to create field configurations on extended manifolds with topologies that differ from that of the original BGM geometry. We shall focus on resolved geometries with a single asymptotic region, though the formalism readily produces resolutions with multiple asymptotic regions as well.
More broadly speaking, Vasiliev's higher spin gravity tempers (i) using differential algebras and (ii) and (iii) using non-commutative algebras. In particular, I) Degenerate metrics are handled by abandoning the Fronsdal formulation in favour of the unfolded formulation [32][33][34][35], in which the fundamental fields are differential forms obeying 3 We refer to a singularity in a real-analytic function as an analytic singularity. 4 In a matter-coupled gravity theory without fermions, the generalized Weyl curvatures consist of the spin-two Weyl curvature, the spin-one Faraday tensors and the scalar fields. covariant constancy conditions referring to backgrounds with differential Poisson structures rather than Lorentzian structures. As we shall see, this formalism permits the construction of fluctuation fields from objects defined in coordinate-free bases that remain well-defined as the frame field degenerates, and that hence admit continuation across singularities of type (i). To our best understanding, this mechanism for sending fluctuations through singularities associated to degenerate metrics has so far not been exhibited in the higher spin literature 5 , though degenerate background metrics have been considered within the context of "wormholes" of three-dimensional Chern-Simons higher spin gravity [38], and the first-order formulation of gravity in the context of topology change [39]. II) As for resolving analytic Weyl curvature singularities in higher spin gravity, the basic mechanism involves assembling infinite-dimensional towers of fields into horizontal forms on fibered spaces with non-commutative fibers, that we shall refer to as correspondence spaces. Locally, the horizontal forms, that we shall often refer to as master fields, are forms on the base manifold valued in algebras of quantum mechanical operators realized as various distributions on the fiber including real-analytic functions and non-real analytic objects such as delta functions and their derivatives. Above generic points of the base manifold, the master fields are realanalytic with Lorentz-covariant Taylor expansions in the fiber coordinates, the coefficients of which are bounded component fields. Closing in on special points, however, the master fields approach non-real analytic distributions in the fiber that nonetheless remain well-defined as symbols of quantum mechanical operators belonging to a star product algebra with a trace, though their naive interpretation in terms of Lorentz-covariant component fields clearly breaks down. So far, this mechanism has been shown to resolve Coulomb-like singularities (of codimension three) in the Weyl curvatures of four-dimensional higher spin black hole-like solutions of Vasiliev's theory [10]. In this paper, we shall extend this result to membrane-like singularities (of codimension one) in linearized fluctuation fields over BGM black holes as well as AdS 4 . More precisely, the mechanism at work trades the analytic spacetime singularities in the Weyl curvatures for delta function singularities in the fiber supported on fiber submanifolds of codimension two, which can be shown to be well-defined operators in the above sense upon using a certain regular presentations [8,10] to be outlined below.
III) Delta function sources in equations of motion typically accompany the singularities in (II), at least at the linearized level. As for the analytic singularities of odd codimension referred to above, the corresponding delta function sources in the equations of motion of the spacetime 5 Rather, in constructing unfolded systems of equations it is usually assumed that if the frame field is invertible then the system must admit a dual interpretation as a complex for an algebraic differential whose cohomology in different degrees consists of the dynamical Fronsdal fields, their gauge parameters, and equations of motion and Bianchi identities [34]; for analogous treatment of mixed symmetry fields, see [36,37]. gauge fields are transferred by the Vasiliev system to delta function sources of codimension two in the equations of motion for the gauge field on Vasiliev's internal noncommutative twistor space leaf of the base manifold (sometimes referred to as Z-space). 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. Moreover, the latter is interpretable as 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 [40,41], referred to as Frobenius-Chern-Simons (FCS) gauge theory. The FCS two-form can develop various expectation values in spacetime as well including codimension-two delta function sources, as these can be regularized by embedding spacetime as a Lagrangian submanifold into its non-commutative cotangent bundle (or phase-spacetime) [10]. This suggests the existence of fully nonlinear higher spin geometries serving as resolutions of conical singularities arising in BHTZ-like geometries interpretable as entanglement surfaces extended into the bulk [19,42,43].
The higher spin singularity resolution mechanisms introduced so far can be implemented off-as well as on-shell, using an adaptation of the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) formalism [44] to Cartan integrable systems on non-commutative manifolds [45]. In fact, as far as one is concerned with resolving degenerate metrics in ordinary gravity, the AKSZ formalism permits the inclusion of degenerate frame fields into the classical theory, though quantum corrections are more delicate as they require a balance between even and odd forms in order for topological anomalies [46] to combine into a finite one-loop partition function. On the other hand, the FCS model is manifestly topologically supersymmetric in the sense that its spectrum of even and odd forms is identical thus ensuring a finite one-loop normalization of the partition function on a given manifold [40,41,47].
As for resolving curvature singularities, the attendant non-commutative geometries are not visible in ordinary gravity nor in the perturbatively defined Fronsdal formulation of higher spin gravity.
Hence, to our best understanding, the resolution of classical singularities in gravity relies crucially not only on higher spin extension but also on Vasiliev's formulation of higher spin gravity using master fields.

Vasiliev versus Fronsdal formulations
While the deformed Fronsdal formulation of higher spin gravity refers to a Lorentzian spacetime background, Vasiliev's formalism [34,35,48] introduces a non-commutative background for a differential graded (homotopy) associative algebra (DGA) of differential forms. This algebra is a deformation of the classical algebra of differential forms (with its compatible wedge product and de Rham differential) along a differential Poisson structure so as to produce a space of symbols equipped with an associative star product and a mutually compatible differential.
The DGA operations can be realized together with compatible trace and hermitian conjugation operations by attaching differential forms as boundary vertex operators to an induced first-quantized differential Poisson sigma model [49], which is a two-dimensional topological field theory with an N = 1 supersymmetry (of degree one) generated by the de Rham differential 6 . The fibration of the correspondence space (giving rise to the horizontal forms) arises from additional supersymmetries (of degree minus one) generated by inner derivatives along vector fields that preserve the differential Poisson structure, which are hence special fundamental vector fields [47].
The DGA operations induce a class of star product local functionals given by traces of star products of horizontal forms and their exterior derivatives. This class remains closed under the Batalin-Vilkovisky (BV) bracket modulo a set of boundary conditions (usually solved by choosing a polarization and setting all momenta to zero at the boundary). Thus, the BV master equation poses a well-defined deformation problem for a gauge invariant BV path integral measure based on a star product local master action, leading to a notion of star product local (quasi-)topological non-commutative field theories of AKSZ-type [45].
Assuming the existence of a topological open string on T * Sp(4) × C 2 (with holomorphic symplectic structure on C 2 ) obtained from deformation quantization of a single conformal particle [24], we think of the FCS theory as a truncation that retains the zero-and winding modes, which thus coordinatize the correspondence space with non-commutative fibers arising from fermionic zeromodes on C 2 induced via the aforementioned special fundamental vector fields [47]. We then embed Vasiliev's theory into the FCS theory as an on-shell branch with "order parameter" given by the aforementioned two-forn vacuum expectation value 7 .
The Vasiliev branch contains ALAdS solutions, which are master field configurations (on the total non-commutative fibered space) subject to boundary conditions giving rise to asymptotically free Fronsdal fields [8,9,51] 8 . Our basic hypothesis is that the free energy, i.e. the on-shell action, of the FCS model is finite on these ALAdS configurations. The FCS free energy given by the on-shell value of a topological vertex operator (TVO) [40,41,53], i.e. a higher spin invariant star product local boundary functional whose total variation vanish on-shell (such that it can be added to the AKSZ bulk action without affecting the smoothness and nilpotency of the BRST operator). The FCS theory only admits a finite number of TVO's, given by Chern classes and Chern-Simons forms, that is, the FCS free energy functional contains only a finite number of free parameters. In a stark contrast, Vasiliev's theory admits an infinite number of TVO's, suggesting that these can be used 6 Within the context of higher spin gravity, one may think of the differential Poisson sigma model as describing first-quantized conformal particles making up the partons of a tensionless string [24]. 7 We expect that reductions of the FCS model in the presence of various vacuum expectation values create a moduli space of unfolded systems on the reduced correspondence spaces (with four-dimensional commuting base manifold and non-commutative C 2 fiber), containing the plethora of "formal" higher spin gravities [50] obtained by deformations of the fiber star product. 8 The ALAdS boundary conditions add non-trivial perturbative corrections to the gauge function already at the linearized level which steer the perturbative expansion away from the singular gauge found in [52].
as building blocks for the FCS free energy functional.
As for classical observables in the Vasiliev branch of the FCS theory, the simplest ones are zeroform charges [5,10,[53][54][55], which are integrals over the non-commutative twistor space of constructs formed out of spacetime curvatures and their derivatives evaluated at a single point in spacetime.
These observables have cluster decomposition properties characteristic of extensive variables [10,54], and hence serve as natural building blocks for higher spin amplitudes, referred to in [54] as quasiamplitudes. Indeed, their classical perturbative expansion around AdS 4 backgrounds reveal a direct correspondence between the first-quantized topological open string amplitudes and the correlation functions of holographically dual conformal field theories [55][56][57][58][59].
We expect the deformed Fronsdal theory to be perturbatively equivalent to the Vasiliev branch of FCS model at the level of amplitudes rather than at the level of spacetime vertices [52,[60][61][62] (or microscopic field configurations). In other words, we propose that the Fronsdal program set up on a Lorentzian spacetime manifold and the Vasiliev program set up on a non-commutative manifold are dual (at the level of free energy functionals) provided the two sides are supplemented with corresponding ALAdS boundary conditions.
The Fronsdal and Vasiliev formulations exhibit an important conceptual difference. The simplest TVO of the FCS model does not receive any quantum corrections, as it is built from forms in degree one; for details, see [40,63]. This simple result is qualitatively in agreement with the holographically dual conformal field theory 9 . As for the perturbatively defined Fronsdal action, on the other hand, its natural interpretation is as a quantum effective action fixed essentially by uplifting the conformal bootstrap approach into the bulk [64,65]. Moreover, from its 1/N -expansion it follows that it has no (non-trivial) classical limit. In other words, it appears that the deformed Fronsdal theory does not provide any path integral measure based on a classical (possibly quasi-local) action formulated directly on spacetime 10 .
In summary, in order to embed the results of this paper (which hold on their own) into the above broader physical context, we will assume that -Vasiliev's equations describe a quantum effective field theory including quantum effects from second as well as first quantization analogously to string field theory [67]; -there exists a free energy functional that makes Vasiliev's equations dual to perturbatively defined Fronsdal formulation on ALAdS backgrounds.
Thus, to the extent that one expects that quantum corrections are important in order to smoothen out 9 Non-trivial quantum corrections to the FCS free energy can be generated by adding TVO's that depend on forms in higher form degrees [63]. 10 Nonetheless, it has been demonstrated [66] that perturbative "re-quantization" of the deformed Fronsdal theory dual to the free theory can be interpreted sensibly at least at one-loop, suggesting that its realm of validity can be extended so as to include boundary conditions corresponding to non-trivial conformal field theories.
classical spacetime singularities, Vasiliev's equations provide a background-independent formulation for studying such effects within the context of higher spin gravity.

Outline of paper
The scope of the paper is to show that probing the spinless BGM black hole using linearized higher spin master field fluctuations leads to smooth linearized higher spin geometries.
In Section 2, we recall some of the basics of the unfolded formulation, and the resolution of the analytic part of the Coulomb-like singularity (in the Weyl curvature) at the level of linearized master fields, that we shall then generalize to membrane-like singularities in Section 5 and Appendix D.
In Section 3, we first recall the basics of the spinless BTZ black hole in three dimensions and its uplift to the spinless BGM black hole in four dimensions. We then show how the topologies of both of these black holes can be extended using the gauge function approach so as to cross over singularities as well as include additional boundaries. Finally, we discuss some generalities of embedding the spinless BGM black hole into Vasiliev's four-dimensional higher spin gravity.
In Section 4, we construct a space of building blocks for the integration constant for the Weyl zero-form that gives rise to fluctuation modes on the spinless BGM black hole. This building blocks are stargenfunctions of two number operators with complex eigenvalues that obey kinematical conditions as well as the quantization condition induced by the BHTZ-like identification on the BGM background. In particular, we find that quantizing the identification Killing vector implies that the spectrum of the dual Killing vector has imaginary parts, which we interpret as quasi-normal modes.
In Section 5, we unfold the initial datum and extract fluctuation fields in a simple case (when there are no quasi-normal modes) which allows us to examine the fiber real-analyticity properties of the Weyl zero-form in detail and exhibit the resolution of its membrane-like singularity.
In Section 6, we conclude, stressing the limitations in our approach visavi non-linear perturbative corrections, which we hope to present elsewhere.
In the Appendices, we spell out our conventions; collect various formulae that are used in the body of the paper; and analyze in detail the fiber distribution arising at the membrane-like singularity; and discard an apparent singularity of no physical importance.

Resolving of curvature singularities in ALAdS backgrounds
In this Section, we outline key features of the unfolded formulation of higher spin gravity of relevance for resolving singularities and generating vacua with nontrivial topology associated to geometrically entangled vacuum states. We exemplify the resolution of analytic Weyl curvature singularities in the context of codimension-three Coulomb-like singularities using an extension of the Weyl algebra by delta function distributions, referred to as the extended Weyl algebra [8,10,40].

Horizontal forms and quasi-topological noncommutative field theories
The fundamental field of the FCS model is a flat horizontal odd multi-form, or Quillen superconnection, on a fibered non-commutative manifold, or correspondence space, valued in an internal 3-graded Frobenius algebra. This master field decomposes under the internal algebra into a set of differential forms of different degrees, including zero, on the total space, all of which are horizontal, that is, given locally by differential forms on the base space valued in a space of zero-forms on the fiber space forming an associative operator algebra. Finally, the flatness condition on the Quillen superconnection implies that all its horizontal components obey Cartan integrable covariant constancy conditions on the correspondence space.
The appearance of horizontal forms has two immediate consequences for resolving singularities: a) A finite set of covariantly constant master fields contains an infinite set of covariantly constant differential forms on the base manifold, capable of capturing ordinary local degrees of freedom propagating on commutative spacetime leafs of the base manifold; b) The associative fiber algebra contains various higher spin representations including delta functions as well as real analytic functions, capable of capturing spacetime singularities as well as regular, possibly ALAdS, configurations.
Further below, we shall exemplify how (a) and (b) play a crucial role in resolving classical singularities associated with degenerate frame fields and analytic Weyl curvature singularities, respectively.
The FCS model provides an example of a quasi-topological field theory, i.e. a functorial map [68,69] into a category of infinite-dimensional tensors, which one may think of as a set of generalized representation spaces, from a category of topological manifolds with geometrical decorations, which one may think of as a generalized group; in the case of the FCS model, differential Poisson manifolds (with conformal infinities and other defects) are encoded into differential graded star product algebras (with vacuum gauge functions and other cohomologically nontrivial elements).
These maps provide a natural generalization of the representations used in ordinary quantum mechanics, whereby manifolds with boundaries and other defects are mapped to geometrically entangled "vacuum" states on which locally defined quantum fields act modulo overlap conditions encoding transition functions and other boundary conditions.
The simplest example of a quasi-topological field theory is a quantum mechanical system with symplectic manifold S on which acts a group G, which maps open oriented intervals to a group element g ∈ G and a corresponding quantum mechanical evolution operator in H ⊗ H * .
In the case of a quantum field theory, its quasi-topological re-formulation amounts to an unfolded formulation and a corresponding AKSZ model in one higher dimension [37,45] with a non-trivial TVO dual that is dual to the original generating function, thereby facilitating the prospect of dual computations of S-matrices or holographically dual conformal field theories.
In two dimensions, the group theoretic approach to string scattering amplitudes [70][71][72] provides a functorial operator formulation of two-dimensional (matter-coupled) gravity, whereby surfaces with multiple boundaries are mapped to geometrically entangled tensors. Thus, the dual quasitopological field theory is a three-dimensional AKSZ model with suitable TVO [73].
As for gravity in higher dimensions and string field theory, it is certainly natural to ask for AKSZ formulations as well; in this vein, it has been proposed that the Gibbons-Hawking entropy of the (proper) de Sitter vacuum is due to geometrical entanglement [19]. In the case of higher spin gravity, we may treat (proper) AdS background as a closed manifold with a circle defect representing the conformal infinity where the frame fields blow up [40]. Thus, the resulting compact spacetime can be taken to be the boundary of an AKSZ bulk manifold in one higher dimension, to which suitable TVO's can be attached. In this sense, higher spin gravities (and other gravity-like theories) are treated as two-category [74] gauge theories mapping bulk manifolds to morhisms of boundary In what follows, we shall detach ourselves from the above larger picture and limit ourselves to the construction of semi-classical boundary states, that is, classical solutions to higher spin gravity with multiple conformal infinities and curvature singularities giving rise to finite free energies.

Extended Weyl algebra
To exhibit this resolution mechanism, we take the fiber to be the non-commutative holomorphic symplectic C 2 with canonical coordinates Y α = (y α ,ȳα) subject to the canonical star product commutation rules The chiral star product where each auxiliary doublet is integrated over R 2 , is equivalent to the Moyal product for the space P of Weyl ordered polynomials. It admits the following compatible hermitian conjugation operation: Realizing the Lie algebras sp(4) and sl(2; C) as Weyl-ordered bilinears, Y α form a real sp(4)-quartet, and y α a complex sl(2; C)-doublet.

Vacuum gauge functions and topology change via degenerate metrics
The base manifold is taken to be (2.10) We let Ω be an sp(4)-valued vacuum connection obeying 12) where N ∈ S 3 , and take the π-odd component of Ω to obey ALAdS boundary conditions at S 1 ×{N }, which requires it to be a frame field in a tubular neighbourhood of S 1 × {N }. Thus, in any simple where L : U 4 → Sp(4) is a gauge function. For the boundary conditions imposed, we may extend L to a function on M ′ 4 ; for example, taking (r = 1, 2, 3) L = exp ⋆ (iET ) ⋆ exp ⋆ (iP r n r arcsinhf (ρ)) , T ∈ [0, 2π) , n r n r = 1 , (2.14) where E is the energy operator and P r the spatial transvections on AdS 4 (for conventions, see Appendix A), and choosing yields the AdS 4 metric in standard global coordinates 11 .
Two gauge functions are considered equivalent as long as they are homotopic in the interior of M ′ 4 and obey the boundary conditions; for example, keeping the boundary condition at S 1 × {N } fixed there is nothing preventing the gauge function from collapsing in the interior of M ′ 4 so as to create a degenerate frame field as long as no new singularities arise in Ω.
Modifying the boundary conditions, one may create additional ALAdS regions in M 4 ; for example, taking where a is a positive real constant, gives a connection on 17) where N, S ∈ S 3 . The resulting gauge function describes a constantly curved manifold with two ALAdS regions in the tubular neighbourhoods of S 1 × {N, S} with conformal infinities given by Lorentzian S 1 × S 2 . These regions are connected by a cylinder with a degenerate metric which is Lorentzian except at ρ = 0. More generally, one may create any given number of conformal infinities given by Lorentzian S 1 × S 2 by adding singularities to f at further points on S 3 .

Linearized Weyl zero-form
The higher spin fluctuations are contained in a zero-form Φ, referred to as the Weyl zero-form, valued in the extended twisted-adjoint representation The linearized Weyl zero-form obeys whose general solution is given by where Ψ is a constant in W and L is a gauge function. The constant Φ ′ contains all spacetime derivatives of the physical fields evaluated at a spacetime point, which we shall refer to as the unfolding point, that are invariant under linearized (or abelian) higher spin gauge transformations.
The gauge function L "spreads", or "unfolds" this local datum, which we shall also refer to as initial datum, on the spacetime chart U 4 where L is defined [34,35]. Using the gauge function in (2.14), which is globally defined on M ′ 4 with one or two conformal infinities (depending on the choice of f ), the periodicity of the linearized Weyl zero-form under T → T + 2π follows from exp ⋆ (2πiE) = κ y ⋆κȳ , (2.25) and the fact that ππ(Φ ′ ) = Φ ′ .

Particle and black hole states in AdS 4
Families of (exact) biaxially symmetric, generalized Petrov-type D solutions to Vasiliev's equations have been constructed in [10,11,13] using gauge functions and Weyl zero-form integration constants.
These integration constants are expanded in special bases such that each distinct (micro)state consists of an infinite tower of Fronsdal fields. The corresponding master fields are valued in a fiber algebras spanned by delta functions as well as real-analytic functions [8-10, 13, 14].
In particular, there are two branches with two compact Killing symmetries, of which one consists of black-hole states with ALAdS regions, including the linearized fields of the charged Kerr-AdS black hole of the Einstein-Maxwell theory (which we think of as a broken phase of the higher spin gravity theory).
At the linearized level, the black hole states arise naturally together with particles states by taking Ψ ∈ End(F) , where F + and F − consist of the direct product of two Fock and anti-Fock spaces, respectively, as Ψ P with initial data Ψ P ∈ A P := Hom(F + , F − ) ⊕ Hom(F − , F + ) . (2.28) From κ y ⋆ F ± = F ± , it follows that these two types of modes are exchanged by the duality trans- Presenting the initial data using regular presentations [8,10,14], yields the orthogonality relations where σ, σ ′ , σ ′′ , σ ′′′ = ±, that is which turn out to dictate the self-interactions among particle and black hole states governed by the quadratic terms in Vasiliev's equations [10].

Resolving Coulomb-like singularities
The linearized black hole geometries contain Coulomb-like singularities, which consist of analytic singularities in the Weyl curvatures and delta function sources for the Fronsdal curvatures. They arise by first expanding the horizontal forms into Lorentz tensors in the ALAdS region, and then following these fields towards the origin. On the other hand, the horizontal forms remain well-defined as symbols of operator algebra elements defined on the entire correspondence space.
To exhibit this resolution mechanism, we start by observing that if Φ is real-analytic on all fibers above a region U 4 ⊂ M 4 , then it follows from the master field equations of motion thať where the spacetime one-formŤ Φ ∈Ť is given by a distribution on M 4 with support on M 4 \ U 4 , which we refer to as the Bargmann-Wigner source ofΦ.
Taking L to be the global gauge function on M ′ 4 with a single conformal boundary, the particle modes Φ The higher spin resolution of these Coulomb-like singularities amounts to the fact that from the integrability condition it follows thatŤ = D (0) χ locally, such that the extended Weyl zero-form obeys a source free equation with initial data Ψ ∈ W that is regular in the sense that Ψ ∈ A and hence Ψ ⋆ Ψ is well-defined (by the assumption that A has a well-defined star product).
We expect that the Fronsdal fields carrying the black hole modes have delta function sources on M 4 [75]. The Fronsdal fields are assembled together with distributions in Y -space into a spacetime one-form master field valued in the extended higher spin algebra hs(4). Vasiliev's equations maps this spacetime one-form to a horizontal twistor space one-form field with a source in noncommutative twistor space of codimension two. Remarkably, in the FCS model, the latter source has a finite free energy, given by the on-shell value of a TVO given by the second Chern class on twistor space. Thus, Vasiliev's formalism replaces the ill-defined free energy for a Coulomb-like configurations, computed from singular sources in spacetime using the Fronsdal on-shell action, by a well-defined finite free energy, computed from the regular source in non-commutative twistor space.
An interesting problem is to extend the black hole solutions of [10], which were constructed in trivial topology, to higher spin eternal black holes (with topology R × S 2 × S 1 ) by using gauge functions of the form (2.14) with f (ρ) given by (2.16). As f (ρ) is bounded from below, it follows that the resulting solutions will consist of infinite towers of Lorentz tensors that are bounded. We leave this for future work.

BTZ-like geometries in gravity and higher spin gravity
In this section, we spell out the key differences between the ambient (metric) and intrinsic (gauge function) approaches to BTZ-like geometries in the case of the spinless BTZ black hole in three dimensions and its direct BGM uplift to four dimensions. We then discuss generalities of embedding the BGM geometries into Vasiliev's system together with linearized higher spin fluctuations fields.

Generalities
The BTZ black hole [28] has contributed in many respects to our understanding of gravity. This geometry is a remarkable toy model comprehending many crucial aspects of black holes in higher dimensions: mass and angular momentum; area law for entropy; and a causal structure making it a proper background geometry for the study of properties of quantum fields in curved spacetime.
The BTZ black hole highlights the fact that Einstein gravity in three dimensions can be thought of as a topological theory on a non-compact manifold M ′ 3 with connection one-form Ω valued in the Lie algebra of G = SO(3, 1), SO(2, 2) or ISO(2, 1) depending on whether the cosmological constant is positive, negative or null, respectively. The connection is defined locally in charts subject to boundary conditions such that M ′ 3 can be obtained from a closed three-manifold M 3 by removing conformal infinities.
Ambient approach. Three-dimensional Einstein spaces with non-trivial topologies can be ob- is a Lorentzian manifold with isometry group G and Γ a discrete subgroup of G. These procedures present three problems: a) Identification of points in M ′ 3 connected by time-like curves gives rise to closed time-like curves. If Γ is generated from a Killing vector field − → K on M ′ 3 , referred to as the identification Killing vector, then these curves can be removed by excising the regions of M ′ 3 in which In the BTZ geometry, the closed time-like curves are excised by taking ξ ⩾ 0, and the causal singularity at ξ = 0 is surrounded by a horizon at ξ = √ M .
b) The fixed points of Γ in M ′ 3 may turn out to be conical delta function singularities in the Riemann curvature of (M ′ 3 , ds 2 )/Γ. In the BTZ black hole, however, the Riemann curvature of (M ′ 3 , ds 2 )/Γ remains bounded.
c) The identifications induce a topology as well as metric structure, which may turn out to be non-Hausdorff, as is indeed the case for the spinless BTZ black holes, though, as we shall see, this type of singular topology is not an intrinsic property of the spinless BTZ black hole.
Intrinsic approach. Following this approach, one constructs gauge functions defined locally on a non-compact manifold M ′ 3 obtained from a closed manifold M 3 by removing conformal infinities. Locally, it follows that Ω = L −1 dL where L : U 3 → G is a gauge function, which can be trivialized inside U 3 . Thus, imposing non-trivial boundary conditions on Ω, and hence G, at boundaries or other defects yields boundary states [76][77][78][79] and localizable bulk degrees of freedom [43,80].

Ambient vs. intrinsic approaches to 3D spinless BTZ black holes
In what follows, we first describe the the eternal spinless BTZ black hole obtained using identifications in the ambient space. We then construct a topologically extended version using an intrinsic gauge function.
Ambient approach. The eternal BTZ black hole geometry arises for negative cosmological con- ⊂ AdS 3 consists of all the points in AdS 3 where ξ 2 ⩾ 0; and The identification Killing vector belongs to a specific conjugacy class of so(2, 2), referred to in the literature [28] as I b , which can be taken to be spanned by Taking α 2 (or α 1 ) to vanish, yields the non-rotating BTZ black hole with mass M = (α 1 ) 2 and identification Killing vector To exhibit this geometry, one may embed (proper) AdS 3 into a flat four-dimensional ambient space with signature (−1, −1, 1, 1) as the quadratic form whose isometries are generated by the Killing vectors 3 /Γ can be embedded into AdS 3 by taking This orbispacetime consists of two smooth regions with ξ > 0 and ξ < 0, respectively, glued along a singularity at ξ = 0.
Restricting to ξ > 0, the induced geometry is the spinless eternal BTZ black hole given by the warped product 12 with metric where ds 2 is the metric on one of the two stereographic coordinate charts of AdS 2 . Kruskal-Szekeres-like coordinates can be introduced via the embedding (m = 0, 2) for which the two-dimensional line element and warp factor, respectively, take the form Thus, the black hole has topology The black hole can be restricted further to a Schwarzschild-like spinless BTZ black hole, with line element where r ⩾ 0 and t ∈ R can be introduced by taking and embedding The Killing vectors are now given by The orbispacetime AdS 3 /Γ thus consists of two eternal black holes with ξ > 0 and ξ < 0, respectively, glued together across their singularities into a single topologically extended eternal spinless BTZ black hole and singularities of R × S 1 topology at ξ = 0 hidden behind future and past horizons at ξ = ± √ M .
The resulting closed time-like curves can be removed by going to the covering space with topology Intrinsic approach. Starting instead from the topological space the globally defined gauge function (r = 1, 2) 13 is the so(2, 2) generator corresponding to the identification Killing vector, and yields a frame field e a and Lorentz connection ω ab that are bounded and constantly curved in the interior of M ′ 3 . The resulting intrinsically defined quasi-Lorentzian metric is a topological extension of the spinless BTZ black hole, wherein CMink 2 has been extended to (proper) AdS 2 with Lorentzian metric (3.28) The three-geometry has two conformal infinities with conformal quasi-Lorentzian metric (3.29) 13 A gauge gauge function adapted to the stereographic coordinate system on CMink2 is given by where ξ m = 4Υ(x 2 )x m with Υ given in Appendix C.1 of [14].
The warp factor is valued in R, and it follows that the warped circle shrinks to zero size at T = π/2 and 3π/2.
The topologically extended black hole is 2π-periodic in T , as exp ⋆ (2πiE) is a central element in SO(2, 2). Indeed, restricting T to [π/2, 3π/2] yields the spinless eternal BTZ black hole. As this restriction respects the flow lines of the globally defined Killing vectors of the topologically extended BTZ black hole (along the warped S 1 and the surfaces of constant ξ), the restriction of these vector fields remain globally defined on the BTZ black hole.

Ambient vs. intrinsic approaches to spinless 4D BGM black holes
In what follows, we first describe the the eternal spinless BTZ black hole obtained using identifications in the ambient space. We then construct two topologically extended versions using intrinsic gauge functions.
Ambient approach. Higher-dimensional orbispacetimes AdS n /Γ with n > 3 are more complex than their three dimensional counterparts, as the identification Killig vector leaves more than one ambient plane invariant, which may lead to non-abelian (residual) Killing symmetries. An identification Killing vector in a conjugacy class of Type I preserves the foliation defined by its norm, whose leaves are constantly curved manifolds whose signature as well as radius may vary along the foliation.
Four-dimensional constantly curved orbispacetimes were studied in [29,30]. The direct uplift of the three-dimensional eternal spinless BTZ black hole to four dimensions is the eternal spinless BGM black hole [31] ( Its warp factor ξ ∈ R + , and it does not contain any closed time-like curves. The black hole contains past and future singularities at ξ = 0, which arise behind past and future horizons at ξ = √ M .
More precisely, expressing AdS 4 as the quadratic form the uplifted eternal spinless BTZ black hole can be obtained by an identification along Thus, the black hole can be embedded into AdS 4 via the map A Schwarzschild-like patch with coordinates {r, t, ϕ, θ} can be obtained by taking ξ = r , (3.35) leading to the eternal spinless BGM black hole for which the three-dimensional line element and warp factor, respectively, take the form The resulting spacetime manifold and singularities of R 2 ×S 1 topology at ξ = 0 hidden behind future and past horizons at ξ = ± √ M .
The resulting closed time-like curves can be removed by going to the covering space with topology Intrinsic approach. The topologically extended eternal spinless BGM black hole has a globally defined gauge function given by (r = 1, 2, 3) Indeed, this gauge function yields the line-element for AdS 3 × ξ S 1 with In the oscillator realization, from which it follows that under T → T +2π, one has L → L⋆κ y ⋆κȳ; that is, besides the periodicity in ϕ (which is imposed a fortiori by quantizing P ), there is a periodicity in the non-Killing time T .
Taking instead leads to a geometry with two asymptotic regions, that one may view as a semi-classical geometry for an entangled state [19].

Higher spin extension of the 4D BGM black hole
The three-dimensional BHTZ geometries admit generalizations to higher dimensions as well as various signatures. As these BHTZ-like geometries are described by connections that are flat, they are vacuum solutions of corresponding Vasiliev-like higher spin gravities. In three dimensions, linearized higher spin fluctuations around the spinning BTZ black hole have been constructed in [16] from gauge functions defined on Schwarzschild patches and zero-form initial data for conformally coupled scalar and spinor fields.
In what follows, we shall consider higher spin fluctuations around the four-dimensional extended eternal spinless BGM black hole, i.e. on AdS 3 × ξ S 1 . The linearized master fields are thus doubly periodic on the time-like S 1 and as well as on the warped S 1 .
Periodic boundary conditions on warped S 1 and group algebra. The boundary conditions on the warped circle require the Weyl zero-form integration constant Ψ to be expanded over a basis of fiber functions Ψ n [ν], n ∈ Z, where ν is a set of amplitudes, that diagonalizes the adjoint star product action of the boost K ∈ so(2, 3) and forms a basis of an amplitude dependent generalization of the group algebra C[Z], viz. 14 where • n,n ′ a composition rule obeying the co-cycle condition Real and chiral density matrices. The boost K, which is a non-compact operator, is realized in the fiber as the direct product of two inverted harmonic oscillators, also known as Hubble Hamiltonians, viz.
The spectrum of a single (normalized) Hubble Hamiltonian H := p 2 − x 2 has been determined in [82]. The point spectrum of H, i.e. the eigenstates |λ⟩ with complex eigenvalue λ, belong to the Banach space L p iff p > 2 and |Imλ| < 1/2 − 1/p; one can show that the Hölder dual (L p ) * ∼ = Lp, where 1 p + 1 p = 1, is also in the spectrum, i.e. it is not possible to invert H − λ in Lp for λ in the strip |Imλ| < 1/2 − 1/p. Taking the limit p = 2, it follows that H − λ remains non-invertible in L 2 for real λ, i.e. the Hubble Hamiltonian has a continuous real spectrum, for which we can use the Thus, one has However, the reality condition on Ψ requires κ y ⋆κȳ to have a well-defined one-sided action on Ψ n . To this end, we shall assume the existence of complexified stargenfunctions f C λ 1 ,λ 2 |µ 1 ,µ 2 obeying where λ i , µ i ∈ C, and for λ, µ ∈ C and l, m ∈ Z, and Thus, using such a chiral direct product of two Hubble Hamiltoniams, one has and ν n (λ, l; µ, m) ≡ 1 2 (1 + (−1) l+m )ν n (λ, l; µ, m) in order to impose the ππ-projection.

Regular prescription.
In what follows, we shall a) write the group algebra elements, which are special functions in Y , as integral transforms with contours in the S and T planes corresponding to Mellin and Laplace transforms; b) take the unfolding point, where the zero-form initial data is defined, to be an intersection point between a future and a past horizon of the topological black hole background.
As we shall see, this yields a Weyl zero-form that is real-analytic in the fiber over generic spacetime points (hence liftable to a master field configuration solving the linearized Vasiliev system), provided that all star products (between the initial data and gauge functions) are performed prior to reading off Lorentz tensorial component fields. In particular, this procedure yields a Weyl zero-form that is real-analytic in the fiber above the original unfolding point, whereas Ψ ⋆ κ y is a non-real-analytic function involving complex powers of oscillators 15 .
The above prescription is part of a broader scheme [8,10,14] for perturbative computations in The scheme facilitates perturbative computations in Vasiliev's theory using the gauge function method [8,10,14,51], 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 [51].
In what follows, the scheme will only be used to extract the linearized Weyl zero-form (which is the first object of the full set of Vasiliev fields to be encountered at every order of perturbations), 15 A similar computational method, based on displacing the unfolding point away from the horizon, was employed in [16]. 16 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).
though this nonetheless constitutes a nontrivial application of the formalism as it removes the aforementioned unphysical singularity at the unfolding point, and, moreover, lifts an apparent ambiguity in the choice of regular presentation of the initial data of the Weyl zero-form; for details, see Appendix E.
Singularity structure. As we shall see, the fiber real-analyticity of the Weyl zero-form only breaks down on two codimension-one submanifolds: -at ξ = 0, i.e. at the singularity of the BGM background, where the Weyl zero-form approaches a fiber distribution with a regular presentation given in Section 5.1; -at membrane-like singularities, where the Weyl zero-form approaches fiber delta functions with regular presentations, as will be shown in Appendix D in a special case, namely when Ψ is an operator in a complexified Fock space, which will be the topic of the next Section.
In particular, this means the Weyl zero-form remains real-analytic in the fiber above the entire horizons at ξ = ± √ M including the unfolding point (except at possible intersections between the horizons and the membrane-like singularity).

Construction of zero-form initial data using Fock spaces
In this Section, we shall provide simple building blocks for the Weyl zero-form integration constant Φ ′ = Ψ ⋆ κ y that diagonalize the twisted adjoint action of the oscillator realization K of the identification Killing vector field − → K used to construct the four-dimensional BGM black hole background, as discussed in Section 3.4.
To this end, we shall start in Section 4.1 by recalling the construction in [8,10] of linearized Weyl zero-forms on AdS 4 by expanding Ψ over stargenfunctions obtained by dressing Fock space projectors and twisted projectors by polynomials in corresponding (complexified) creation and annihilation operators introduced so as to create integer left and right eigenvalues for Cartan generators in so(2, 3) whose Sp(4) matrices square to −1, namely, E, J, iP and iB.
To obtain linearized Weyl zero-forms on AdS 4 on BGM backgrounds with identification Killing iii) constrain the eigenvalues so as to implement the bosonic projection; the reality conditions; and the BTZ-like identification, as discussed in Sections 4.4 and 4.6.
iv) rewrite the stargenfunction on Sp(4, R)-covariant form, which will be particularly useful in analysing the singularity structure of the Weyl zero-form, which is the topic of Section 4.5.
We stress that the above construction provides a particular type of building blocks for Ψ that diagonalize ad ⋆ K . The fact that these elements do not span any associative algebra on their own does not pose any problem as long as we limit ourselves to a strictly linearized analysis.

Fock spaces associated to different Cartan subalgebras
The basic idea is thus to expand the initial datum Ψ (or, equivalently, Φ ′ of (2.24)) in operators that span specific representations of the complexified AdS 4 isometry algebra sp(4, C), and then subject them to the identification condition that characterizes the four-dimensional BGM background. To this end, we shall modify and extend the method developed in [8,10,11], which we shall briefly review in what follows for the reader's convenience.
Given a pair (K (+) , K (−) ) of mutually commuting and normalized generators of (the complexified) sp(4, C) with oscillator realization they can be written in terms of two number operators as where the creation and annihilation operators αβ . An extension of the Weyl algebra by delta functions contains operators P n L ,n R (Y ) obeying and with n L,R = (n 1 , n 2 ) L,R ∈ (Z + 1/2) × (Z + 1/2), idem m L,R , being half-integer eigenvalues under the left or right star-product action of number operators w i , Clearly, the P n L ,n R also diagonalize the adjoint as well as twisted-adjoint actions of K (±) , viz.
The diagonal elements P n,n ≡ P n = P n 1 ,n 2 are projectors and belong to the enveloping algebra of the number operators, and hence factorize as P n 1 ,n 2 (w 1 , w 2 ) = P n 1 (w 1 ) ⋆ P n 2 (w 2 ). In particular, the projectors onto the lowest-weight state of the Fock space (+) and the highest-weight of the anti-Fock space (−) correspond to In star-product form, the generic projector reads where ϵ i := sign(n i ).
gives rise to a Weyl zero-form that is real-analytic in Y at the unfolding point 19 . Thus, the twisted sector is nontrivial iff the principal Killing vector is taken to be E or iP , in which case we expand 20 where ν n L ,n R and µ n L ,n R are independent deformation parameters, while in the remaining families we set the µ-parameters to zero, viz.
In the latter case, once a regular presentation R [P n L ,n R (Y )] has been chosen, there remains an apparent ambiguity whether to expand Ψ in terms of R(P n L ,n R (Y )) or R(P n L ,n R (Y )) ⋆ κ y , as both choices lead to Weyl zero-forms on-shell whose component fields obey the same boundary conditions in spacetime. However, a closer inspection of how their regular presentations vary over spacetime (see Appendix E) reveals that only former choice is compatible with condition (iii) in Section 3.4.

Diagonalizing the adjoint actions of P and B
In what follows, we shall oscillator realize stargenfunctions f λ L ,λ R with general complex left and right eigenvalues λ L = (λ 1L , λ 2L ) and λ R = (λ 1R , λ 2R ) of the number operators 19 The one-sided star multiplication by κy exchanges a symbol by a dual symbol obtained by chiral Fourier transformation in y-space (but notȳ-space) followed by replacing the Fourier dual variable by y. This duality transformation, that need not be a symmetry of the symbols of a generic quantum mechanical system, leaves the solution spaces found in [8] invariant; whether it is a symmetry of higher spin gravity, possibly related to a GSO-like projection of an underlying topological open string, is an interesting open problem. 20 The (diagonal) projectors in the regular sector of M(E; J) gives rise to massless scalar particle modes in AdS4, while the twisted counterpart yields spherically symmetric higher spin black holes [8]. 21 As discussed in Section 3.4, we expect that additional states must be added to Ψ in order for the linearized solutions to admit completions into perturbatively defined nonlinear solutions, as this requires Ψ to belong to an associative algebra.  related to the Cartan pair (iB, iP ). To this end, we choose 17) and use the realization of the Dirac matrices given in Appendix A, to arrive at These operators can be projected out from Y α using a spin-frame idem their complex conjugates, as where To proceed, we use Thus, the stargenvalue problem is equivalent to Adding and subtracting these equations, one finds The solutions to (4.34) can be written equivalently as where 2wg (+) λ L ,λ R can be obtained from (4.37) by exchanging λ L ↔ λ R . For generic eigenvalues 22 , the solution to (4.37) can be given as where C andC are integration constants and g (+,1) Substituting an Ansatz of the form g , the latter is turned into the standard Kummer equation zg ′′ λ L ,λ R (z) + (b − z)g ′ λ L ,λ R (z) − ag λ L ,λ R = 0 with z = 4w, a = 1 2 − λR and b = λL − λR + 1 for g λ L ,λ R (w), so the usual criteria for the construction of the two independent solutions apply. In particular, for λL = λR, the two terms in the solution (4.40) degenerate into one. In this situation, the second term should be replaced with where U is the Tricomi confluent hypergeometric function. The latter can be expressed as the linear combination U (a, b, z) = π sin(πb) when b is not an integer, but can be extended to any b ∈ Z [83]. Combinations of one the two solutions in (4.40) and U enable one to write a complete solution to (4.37) also in the cases when λL − λR = ±1, ±2, ... in which one of the two terms in (4.40) has simple poles. For the special case λL = λR = 1 2 , 1F1 (0, 1, 4w) = U (0, 1, 4w) = 1 and a second independent solution is given by the exponential integralg 1/2,1/2 (w) = − ∞ −4w The corresponding solution for g (−) λ L ,λ R is obtained from (4.40) by performing the aforementioned exchange, whose action on the above basis elements is given by In what follows, we shall restrict the eigenvalues to as this will lead to regular prescriptions that simplify the analysis of the spacetime dependence of the Weyl zero-form. If λ R + 1 2 ∈ Z + , then the first confluent hypergeometric function in (4.40) reduces to a generalized Laguerre polynomial, These generalized polynomials capture g (+,1) Thus, for generic eigenvalues, we may take and generate the other branch by the exchange (4.38).
Solving the eigenvalue equations without any assumption of real-analyticity in Y implies in particular that the eigenvalues alone do not fully specify the function f λ L ,λ R , nor its algebraic properties. In fact, the total space of elements of the form (4.36), that satisfy the eigenvalue equations, can be described as the overlap of different solution subspaces, whose precise form goes beyond the scope of the present paper, and that we shall study systematically in a future publication.
Essentially, as we have seen Eq. (4.37) admits two independent solutions that are functions of w (plus two more independent solutions if we also admit distributions in w). The two independent λ L ,λ R (w), i = 1, 2, account for this degeneracy, and are distinct by the fact that f (1) admits a closed contour integral presentation while f (2) needs an open contour presentation 23 [84]. 23 For instance, admitting inverse powers of the oscillators in the realization of f λ L ,λ R implies that an element like f 1/2,1/2 can be realized both by the Fock space lowest-weight state projector f 1/2,1/2 , generated by means of non-analytic functions of the oscillators, can be shown to be equal to the second, independent and non-elementary solution.
In the following, we shall focus only on the simplest type of solutions that enable us to satisfy the periodicity condition in a non-trivial way and to study the possible resolution of singularities of the fluctuation fields in the higher-spin gravity setup. Such solutions admit a closed contour integral presentation for their "diagonal" factor g λ L ,λ R (w), corresponding to a Laplace-like transform. We shall now turn to describing this integral transform, specifying the eigenfunctions that it can encode.

Regular presentation of the stargenfunctions
The stargenvalue equations give rise to non-polynomial functions g(w) as well as complex powers of the oscillators. It is therefore important to specify a functional presentation for the eigenfunctions, both to make sense of the complex powers and because different presentation of the same nonpolynomial function may have different star-product properties. For instance, it was shown in [10] that in order to ensure that both Fock-space and anti-Fock-space elements (that are in general both required by reality conditions on the master fields) form an associative algebra, it is crucial to work with an integral presentation, with the prescription that all star products be worked out before evaluating the auxiliary integrals. The specific regular presentation that was used in that paper and its follow-ups (see [8,14] for the use of this integral presentation for solutions of various physical interpretation), involving an integral around a "small" contour, is technically the simplest one (see [85] for more general ones), and for this reason will be employed in the present paper to represent the factor g λ L ,λ R (w). We shall moreover use a Mellin transform to account for the complex powers. Fixing this presentation will resolve the degeneracy in f λ L ,λ R that we commented on above by limiting the choice of eigenfunction to a simple class. We postpone the study of more general regular presentations to a future publication [84].
Let us now show how one can solve the eigenvalue equation (4.37) by means of a closed-contour Laplace-like transform where the factor of 2 at the exponent has been inserted for future convenience and C(ς 0 ) is a closed contour encircling the point ς 0 to be determined later, subject to the condition that the integrand must be single-valued along the integration path. Eq. (4.37) is then converted into Expressing w in the square brackets in terms of derivatives of e −2ςw w.r.t. ς and integrating by parts one turns the condition (4.37) into a first-order differential condition on the transformg λ L ,λ R (ς), which is solved byg 49) where N is a constant. Thus we obtain 50) where N λ L ,λ R is a normalization constant, to be fixed by requiring closure of the associative algebra of f λ L ,λ R elements 24 . Imposing that the contour encircle the point ς 0 = 1, (4.50) gives an integral realization of the g(w) function for elements (4.36) with λ L ∈ C and λ R + 1 2 ∈ Z + , provided the contour is small enough as not to cross the branch cut from −1 to −∞ that arises from the numerator of the integrand when λ L − 1 2 / ∈ Z. Analogously, choosing a small contour that encircles the point ς 0 = −1 and does not cross the branch cut from 1 to +∞, (4.50) gives an integral realization of the g(w) function for elements with λ L − 1 2 ∈ Z − and λ R ∈ C 25 . We can therefore conclude that gives an integral presentation of the function of w accounting for the w-dependent factor of f λ L ,λ R in (4.36), with the limitation that We choose standard phase conventions around the branching points, with Arg(ς) ∈ (−π, π] for the integrand when λ R + 1 2 ∈ Z + and λ L ∈ C, and Arg(ς) ∈ [0, 2π) when λ L − 1 2 ∈ Z − and λ R ∈ C. The integral presentation (4.51) with (4.52) indeed covers the cases (4.43) with (4.42), as anticipated.
The factor (a + ) λ L −λ R of (4.36) also can be given an integral representation, which is in fact crucial to encode complex left eigenvalues. One way of doing that is via a Mellin transform,

53)
24 While for half-integer eigenvalues it is concretely possible to fix the normalization constants by requiring that working with complex eigenvalues makes this issue subtler, and the simple integral realization that we use in this paper, while good enough for a linearized analysis, is not a satisfactory choice for such purpose. Indeed, such "small contour" integral presentation can only capture discrete eigenvalues, according to (4.52), and these cover only a portion of the full spectrum, as discussed in Section 3.4. We expect in fact that the linearized solutions discussed in this paper can only be dressed into full solutions by starting from an enlarged set of states, with left and right complex eigenvalues. For this reason, we shall not fix the normalization constants in this paper, leaving this issue, as well as any other question related to the non-linear completion of these solutions, to a future work [84] where we shall use a different contour-integral presentation which evades such restriction. 25 Actually, when taking the product of two elements f λ L ,λ R and f λ ′ L ,λ ′ R for half-integer eigenvalues, the condition that they form an associative algebra in general requires that one can always deform one of the two closed contour to be infinitesimally close to ς0 = 1 or ς0 = −1, in such a way that, even after the star-product is evaluated, ς0 = ±1 is still the only pole encircled by the contour (for details, see [10,14]). Thus, in practice, we shall always assume the contour in (4.50) to be "sufficiently small" and to encircle ς0 = ±1. where Γ stands for the gamma function. The above integral only makes sense for Re (λ L − λ R ) < 0 and Re (a + ) > 0. In order to extend it to the rest of the parameter space of interest, we can analytically continue (4.53) with where γ is a contour of Hankel type, represented in Figure 1.
Such integral presentation is valid for any λ L − λ R ̸ = −1, −2, ... and Re(a + ) > 0. 26 In practice, as we shall see, when evaluating the spacetime-dependent master field it will be possible to formally use the simpler presentation (4.53) in the relevant computations, and then analytically continue λ L − λ R beyond the region Re (λ L − λ R ) < 0 after all star-products have been evaluated.
Thus the solutions to the eigenvalue problem (4.45) that we shall focus on can be rewritten as with a (possibly redefined) normalization constant N λ L ,λ R , λ R + 1 2 ∈ Z + or λ L − 1 2 ∈ Z − according to (4.52), and the proviso that for a proper analytic continuation one should use the Hankel contour integral (4.54). See Appendix C for further details on the elements f λ L ,λ R in the regular presentation.
As in this paper we are mainly concerned with the application of this formalism to the study of fluctuations over a BTZ-like background, for simplicity we shall limit ourselves to elucidating the main features of our construction by expanding the master fields on eigenfunctions of the form will require (see Sections 4.4 and 4.7). Elements admitting this type of "small-contour" integral transform correspond to eigenfunctions f λ L ,λ R belonging to the subset of the f (1) λ L ,λ R that can be obtained as (a + ) λ L − 1 2 ⋆ f (1) where λ L is at this level unconstrained and can have an imaginary part, while λ R is a positive half-integer. Constraints on λ L will arise from algebraic conditions and from imposing periodicity along the direction of identification. From now on we shall restrict our consideration to this class of eigenfunctions, and omit any of the superscripts used in this section to distinguish the various sectors of solutions to (4.30)-(4.31).
Note that, as elements like f λ L ,λ R are in general non-analytic in Y for λ L ∈ C, expanding the master-fields over such a basis seems incompatible with a physical interpretation of the expansion coefficients in terms of fields of various spins. However, we shall show in Section 5.1 that this effect is peculiar to having started the construction with the master fields restricted at the unfolding point, and that reinstating the spacetime dependence via the gauge function L in fact removes this problem, provided that the star products with L are performed prior to taking the limit back to the unfolding point.
Finally, it is useful to note that in the limit λ L − λ R → 0, the integral presentation (4.55) of where now ε = sign(λ R ), in the sense that the divergence of each Gamma function at the denominator cancels exactly the one of the corresponding τ -integral in the limit. This is of course in agreement with the result of the limit taken on the non-integral presentation (4.45).
The above results apply to each of the (commuting) (a + 1 , a − 1 ) and (a + 2 , a − 2 ) systems, so in the following sections we can directly use the above results by adding the labels "1" and "2".

Reality properties of the eigenfunctions
Later in this paper, we will discuss the reality condition imposed on fields. To prepare for that discussion, we first investigate the reality properties of the eigenfunctions.
Using the convention (4.18) and (4.19), we have for both the "1" and the "2" system i.e. the creation and annihilation operators behave respectively like real and imaginary numbers under hermitian conjugation.
The complex conjugate of (4.30) and (4.31) are: which shows that, due to the fact that w † = −w, the right (left) eigenvalue of f † is the opposite of the complex conjugate of the left (right) eigenvalue of f .
Thus, the Hermitian conjugate of f λ L ,λ R , with λ R + 1 2 ∈ Z + , is an element with left eigenvalue λ ′ L = −λ R ∈ Z − + 1 2 and complex right eigenvalue λ ′ R = −λ * L , and as such admits a regular presentation as (see Eqs. (4.55) with (4.51)-(4.52)) (4.60) Indeed, we can compare with the hermitian conjugate of f λ L ,λ R from Eq. (4.55) with λ R + 1 2 ∈ Z + , which reads where a minus sign coming from the complex conjugation of the i in the integration measure is compensated by an overall minus sign due to reversing the orientation of the contour. Changing integration variable as ς * = −ς ′ and dropping the prime, where the phase factor was extracted taking into account the phase conventions on (4.55) when λ L ∈ C. Indeed the expression obtained above coincides with (4.60) provided that 27 4.5 Sp(4; R)-covariant notation for the eigenfunctions Introducing the notation we are now ready to expand the master field over the functions 27 Note that the condition (4.63) is indeed compatible with ((f λ L ,λ R ) † ) † = f λ L ,λ R , as it can be shown by repeating the reasoning that leads to (4.63) for the case when λR ∈ C, which results in N −λ * R ,−λ L = (N λ L ,λ R ) * e iπ(λ * R + 1 2 ) (−1) 1 2 +λ L , and nesting the two formulas to get with regular presentation We recall that the second equality in (4.65) is due to the fact that the creation and annihilation operators (4.18) and (4.19), commute under star-product. For λ L = λ R ∈ Z − 1 2 one retrieves the projectors studied in [8,10,11] and recalled as a special case in Section 4.1.
We can now rewrite the complete eigenfunctions f λ in an Sp(4; R)-covariant notation, and break the latter into SL(2; C)-covariant blocks when convenient. This will be useful to highlight the physical meaning of the various structures involved, and will facilitate the evaluation of the star products with the gauge function. We shall do it in general for an arbitrary family of solutions (4.13), and later specify to the case studied in the present paper.
In order to shorten the expressions, let us also introduce the notation Then, ignoring for now the normalization constants, we can write where, using matrix notation A α B α =: AB = ab +āb := a α b α +āαbα, withκ αβ := idemκ, and ΘY = θy +θȳ , As explained in Section 4.1, each one of the matrices is an Sp(4, C) Gamma matrix, so it is either block-diagonal (v (q)αβ = 0 =v (q)αβ ), for the π-even generators, or it is block-off-diagonal (with κ (q)αβ = 0 =κ (q)αβ ) for π-odd generators. As we shall see later (see [10] for more details), the star-products with the gauge function (2.24) will result in a conjugation of the K (q)αβ matrices by an x-dependent Sp(4; R) matrix, giving rise to K L (q)αβ matrices with all blocks non-vanishing: in particular, the off-diagonal blocks v L (q) are the Killing vectors corresponding to the rigid isometry generator K (q) and the diagonal blocks κ (q) ,κ (q) the selfdual and anti-selfdual part of the corresponding Killing two-form. In the case that we are studying in this paper, K (+) = iB and K (−) = iP , and in particular, as is clear from (A.21), Note that choosing the integration contour to encircle the points ς 1 = ±1, ς 2 = ±1, with signs correlated in such a way that ς 1 ς 2 = 1, corresponds to choosing iB as principal Cartan generator: that is, to fixing the lowest-weight state of the Fock space to be 4e −4iB (and the corresponding anti-Fock space highest-weight state to be 4e 4iB ), with the commuting generator iP only appearing in the excited states (see [10] for details). This also implies that the lowest-weight state, as well as the corresponding anti-Fock highest-weight state, have enhanced symmetry under an so(2) B ⊕ so(2, 1) {M 12 ,P 1 ,P 2 } residual isometry algebra, any other diagonal state f λ iL =λ iR in the module is biaxially symmetric, with isometry algebra so(2) B ⊕ so (2)  Let us look at the generic element in the twisted sector in Φ ′ in terms of these building blocks.
The star-product of (4.68) with κ y can be written as where we have defined the modified oscillators y := y − ivȳ, and we recall that, κ (q)αβ being a symmetric 2 × 2 matrix, κ 2 (q) := det κ (q) = 1 2 κ αβ (q) κ (q)αβ and κ −1 . It is clear from this expression that the elements of the twisted sector is, in fact, regular if κ (q)αβ of the principal Cartan generator is non-singular, and otherwise non-real-analytic -realizing, in fact, a delta-function in Y space, as we shall show in Appendix D (see also [10] for the diagonal case). Distributional masterfields in the twisted sector are then smoothened into regular functions almost everywhere once the x-dependence is reinstated via the gauge function L, but maintain the delta-function behaviour on the spacetime surface where (κ L (q) ) 2 (x) = 0. This is the case for the twisted sector of the family M(E; J), investigated in [8,10], where the singular behaviour occurs in r = 0 and results in a spherically-symmetric black-hole-like behaviour in the tower of Weyl tensors encoded in Φ (L) . For the π-even principal Cartan generators J and iB, instead, κ 2 (J) = κ 2 (iB) = 1, so regular and twisted sector are, in fact, degenerate, as anticipated. This, however, does not imply that the L-rotated elements of the twisted sector are regular, as this depends on whether (κ L (q) ) 2 (x) has a region where it vanishes or not (which ultimately depends on whether the corresponding Killing vector has positive defined norm or not). While solutions based on J as principal Cartan generator will always be regular, as we shall see in this paper the solutions based on iB possess a singular surface, and we shall study the smoothening of the corresponding singularity in Appendix D.
For the family M(iB; iP ) regular and twisted sector are equivalent (the lowest/highest weight elements e ∓4iB are in fact eigenstates of κ y ). However, the integral presentation that we use selects the twisted sector in Φ ′ , in the sense that, as we shall explain in Appendix E, the elements of the regular sector, once transformed by means of the gauge function L, will give rise to integrands that are incompatible with the small-contour integral representation employed in this paper, at least in some spacetime region 28 . For this reason we shall discard them in the following. Thus, we shall expand Φ ′ on the regular sector, i.e.
and we shall now turn our attention to imposing constraints on the states allowed in such expansion.

Identification conditions
Fluctuation fields over the four-dimensional BTZ-like background need to be left invariant by a full spatial transvection along the S 1 cycle. We shall now impose this condition on the Weyl zero-form master field.
The Weyl zero-form (see Section 1) transforms as (4.78) where in L gauge (4.79) and γ ′ induces the corresponding transformation on the rigid Y -space element Φ ′ , A finite transvection generated by P := P 0 ′ 1 is therefore implemented via (4.81) as explained in Appendix B, the K αβ of which is here specified to iP αβ , and φ is rescaled for convenience of later discussion. Thus the BTZ-like periodicity conditions are imposed by where √ M φ 0 represents the circumference of the S 1 cycle of the BGM background. We can choose Imposing the identification condition on (4.77) amounts to imposing it on each factor f λ . The transformation of f λ can be written as (4.83) and requiring that the transformation is periodic in φ amounts to imposing the condition Since we assume that λ 1,2 R + 1 2 ∈ Z + , this condition reduces to Furthermore, if we require that the transformation for φ = 2π be an identity, we need to further We can therefore expand Φ ′ over states f λ compatible with the BTZ-like identification by restricting the eigenvalues to those satisfying (4.85)-(4.86), and we write All valid values of λ ν λ f λ 1 a ± 1 f λ 2 a ± 2 ⋆ κ y + conj , (4.87) with the assumption that λ 1R + 1 2 ∈ Z + and λ 2R + 1 2 ∈ Z + (4. 88) in the first term of the sum. The notation "conj" for the second term in the sum stands for the conjugate terms required by the reality conditions, which we now turn to determining. We recall that this expansion only captures one subsector of the full spectrum, the one distinguished by discrete eigenvalues. Had we started from the more complicated setup envisaged in Section 3.4, even after imposing the identification condition we would still be left with one unconstrained complex eigenvalue, and one of the sums in (4.87) would then be substituted by an integral over the latter.
The discrete spectrum contains simpler states, and the linearized analysis that we undertake in this paper does not require the introduction of the full spectrum of states. For this reason we shall limit our expansion to the discrete spectrum, and this is the reason for the subscript on Φ ′ used in (4.87).

The conjugate terms
There are more kinematic conditions to impose on the Weyl zero-form that further constrain the eigenvalues: the bosonic projection and the reality conditions (2.19) .
Satisfying the bosonic projection condition amounts to imposing that which means that f λ is invariant when the sign of Y α is flipped. To this end, it is convenient to represent the elements f λ with λ iR ∈ Z + − 1 2 as where we have written explicitly the lowest weight state f 1/2,1/2 = 4e − 1 Then Therefore, (4.90) implies that 29 e iπ λ 1L +λ 2L +λ 1R +λ 2R = 1 , (4.93) i.e., under the assumption that λ 1,2 R + 1 2 ∈ Z + , (4.94) 29 As usual, since λ1,2 L are complex, we obtain the overall phase (4.93) by extracting the −1 from the first two factors in (4.92) within standard branch cut conventions, assigning phases in such a way that Arg(−a + i ) ∈ (−π, π], i = 1, 2 and taking into account the reality conditions (4.57) and the assumption Re(a + i ) > 0 used in defining (4.53) and (4.54). Now we consider the reality condition. Using π 2 = 1, it is clear that any Φ ′ written as the (4.95) Therefore, the "conj" terms in (4.87) are obtained by applying the π-automorphism to the Hermitian conjugate of the first term, i.e., Using that and that λ 1,2 R + 1 2 ∈ Z + , we derive that (4.96) is equal to where in the last equality we have used the results of Section 4.4. Note that the first condition for the bosonic projection in Eq. (4.94) also ensures that which is a necessary condition for the element κ yκȳ ⋆ f λ 1 f λ 2 , that appears in the reality condition (4.96), to be an admissible element of an associative algebra, since κ y andκȳ are unimodular 30 (see Eqs. (2.6)-(2.7)).
We are finally ready to give the Weyl zero-form that we are going to focus on in the remainder of the paper, ⋆ κ y , (4.102) 30 The analogous condition involving the star multiplication with κyκȳ from the right is trivially satisfied with our choice of λiR + 1 2 ∈ Z + . The condition (4.101) also constrains the real parts of the left eigenvalues as Re(λ1L +λ2L) ∈ Z, which, again due to λiR + 1 2 ∈ Z + , is anyway implied by the second condition in (4.94). Note however that imposing (4.101) together with its right analogue would restrict Re(λ1L + λ2L) ∈ Z and Re(λ1R + λ2R) ∈ Z even without fixing where the elements f λa,λ b ,λc,λ d entering the expansion of the Weyl zero-form admit the integral presentation and where we recall that: • for simplicity of the integral presentation we set and from which it follows that λ 1L + λ 2L = (λ 1R + λ 2R )mod 2 ; (4.107) • the contour encircles ±1 according to the pole of the integrand, i.e., it encircles +1 for the elements encoded in the first term in the sum (4.102) and −1 for those in the second one; • the previously used normalization constants N λ i have been absorbed into the deformation parameters ν λ .

Fluctuation fields in spacetime
Having determined the Weyl zero-form encoding fluctuation fields of all integer spins over a 4D BTZlike background at the unfolding point, we shall now spread this initial data over a spacetime chart and reinstate the x-dependence via the star products with the gauge function L (as in (2.24)). We shall examine the resulting behaviour of the individual spacetime fields, and resolve their apparent singularities at the level of their embedding into the master fields living on the full (x, Y )-space.

The Weyl zero-form in L-gauge
Gauge function. We choose L to be [86] L (x; y,ȳ) = 2h where σ a are the Van der Waerden symbol (see Appendix A for explicit realizations), and 1, 1, 1) . (5.2) With this choice the AdS 4 background one-form connection is given by are the vierbeins and spin-connection and the x a are stereographic coordinates, which are related to the embedding coordinates (3.33) by We refer to Appendix A for more details on the relation of stereographic coordinates with the other coordinate systems that we use in this paper.
Spacetime-dependent Weyl zero-form. The evaluation of (2.24) is facilitated by noting that the adjoint action of L on a Z-independent symbol f amounts to a rotation of the Y oscillators, viz.
In order to compute Φ (L) , it is therefore useful to write the eigenfunctions f λ in the Sp(4, R)covariant notation of Section 4.5. The L-rotation of the Y oscillators induces a spacetime-dependent transformation of all the structures contracted with them, the polarization spinor Θ α and the K αβ matrix (with its κ αβ ,καβ and v αβ blocks) entering Eqs. (4.69)-(4.72). Ultimately, their transformations all descend from the L-rotation induced on the spin-frame basis spinors u ± α andū ± α . We shall henceforth denote with a label (L) the corresponding L-transformed quantities, and refer to the so-transformed master fields as being in L-gauge. Thus, where we have defineď where the matrix L is given in (5.7), and Θ Lα (x; τ 1 , τ 2 ) = (Θ(τ 1 , τ 2 )L(x)) α . (5.12) In our specific case K (+) = iB and K (−) = iP , sǒ It will be useful in the following to write all van der Waerden symbols in terms of a spin-frame (u +α , u −α ), u +α u − α = 1, idem their complex conjugates (see Appendix A for details). As a conse-quence, (5.15)-(5.18) can be rewritten as It is easy to show that Moreover, We also note that, once all constraints (4.105)-(4.107) on the eigenvalue that we allow in the expansion of the Weyl zero-form have been taken into account, Eq. (5.9), in particular the τintegrals, can be rewritten in the more suggestive form where we have defined p := Im(λ 1L ) = −Im(λ 2L ), ∆λ := λ 1L − λ 1R = λ 2L − λ 2R (the last equality following from the identification condition (4.105)).
In terms of the L-rotated quantities above introduced, the star product of (5.9) with κ y reads with the modified oscillators y L := y −iv Lȳ and where we recall that (κ L (q) ) −1 This is the expression of the generic term in the expansion of our Weyl zero-form Φ (L) in (5.8). In particular, the generating function of the scalar and the (self-dual part of the) generalized Weyl tensor fields is Real-analyticity at the horizon. While the expression for Φ (L) (5.29) obtained above may seem, at a first glance, essentially identical to its x-independent counterpart at the unfolding point on the horizon (4.76), a closer look reveals an important difference: the Y -independent term at the exponent -bilinear in θ L α , that is, blinear in the τ i -is here actually non-vanishing, whereas its counterpart in (4.76) is in fact trivial. Indeed, asκ αβ reduces to κ (iB)αβ (see Eqs. and (A.15)), clearly the quadratic term in θ at the exponent of (4.76) vanishes. However, the situation changes once the star products with the gauge function displace the Weyl zero-form from the unfolding point at the horizon:κ L αβ (x) and θ L α (x) acquire extra, spacetime-dependent terms that complicate their spinorial structure, giving rise to non-trivial, Y -independent terms bilinear in τ i . This is crucial in order for the linearized Weyl zero-form to be considered a proper generating function of fluctuation fields. Indeed, the scalar field (s = 0) and the (self-dual part of the) spin-s linearized Weyl tensor C α(2s) (s = 1, 2, 3, . . . ) are extracted from Φ (L) via (analogously, with the roles of y andȳ interchanged for the anti-self-dual part of the Weyl tensors). Therefore, in order for Φ (L) to contain the propagating degrees of freedom in the coefficients of its Y expansion it is crucial that it be real-analytic in Y = 0. In this respect, the expansion (4.102)-(4.103) that we start from at the unfolding point is problematic, since, as we have seen, whenever complex eigenvalues are involved, as it is in this case necessary in order to have nontrivial momentum on S 1 . This leads to complex powers of the oscillators, which reflect themselves, in the integral presentation, into ill-defined τ -integrals in the limit Y → 0 (4.103). However, as we have commented on above, displacing the Weyl zero-form away from the unfolding point by means of the star products with the gauge function leads to an integrand of schematic form e O(τ 2 )+O(τy)+O(y 2 ) , (5.33) which, after taking the derivatives w.r.t. y-coordinates, gives e O(τ 2 )+O(τy)+O(y 2 ) Polynomial(τ, y) .
It is the appearance of the non-trivial Y -independent terms bilinear in τ i at the exponent that helps the convergence of the Mellin transforms and restores analyticity in Y (at least for generic spacetime points), as we shall show with examples in the next Subsection.
Resolution of membrane-like singularities. On specific surfaces the Weyl zero-form (as well as each spin-s component field) may have an analytic curvature singularity. As clear from (5.29), this happens where the Killing two-form becomes degenerate, that is, where (κ L ) 2 = 0, which, with our restriction on the eigenfunctions as in (4.105)-(4.107), reduces to the surface ∆ 2 ≡ 1 + X 2 3 − X 2 0 = 0. Singularities of this type were already studied in the context of spherically-symmetric higher-spin black holes in [8,10,11] and, as we shall see, it is possible to generalize the conclusions of those papers to our present case of fluctuations of type (5.29) over a BTZ-like background: what happens is that, as anticipated in Section 2.6, the embedding of such curvature singularities in a higher-spin covariant theory -where higher-spin symmetries force the appearance of one such singular Weyl tensor field for every component of an infinite-dimensional multiplet, all packed as coefficients of the expansion of the Weyl zero-form onto an infinite-dimensional, non-commutative fibre algebra Y -effectively trades the space-time singularities of the component fields for a delta-function-like behaviour in Y of the corresponding master field. In practice, the quantity √ ∆ 2 enters the formula (5.29) as the vanishing parameter of a delta sequence: away from the surface ∆ 2 = 0 the Weyl zeroform is a smooth Gaussian function of the oscillators, while it approaches a Dirac delta function on Y -space in the ∆ 2 → 0 limit. However, unlike the delta function on a commutative space, the delta function in the non-commutative Y -space, thought of as a symbol for an element of a star product algebra, is smooth. In other words, the mapping of the spacetime curvature singularities to a distribution in the fibre has the advantage that the latter type of singularity can be handled better, as the resulting distributions have good star-product properties and can be considered elements of an associative algebra 31 , see [8,10,14]. 31 Furthermore, it is to some extent possible to consider a delta function of the oscillators as a bounded function (which would give an even stronger meaning to the notion of resolution of curvature singularities) in the sense that, on a non-commutative space, a change in the ordering prescription can turn a delta function into a smooth symbol (e.g., an exponential [10]). Changes of ordering are formally part of the gauge transformation that leave the classical observables of the Vasiliev system invariant (with important subtleties that are currently being studied [51,87]), so in this sense the above resolution of curvature singularities would amount to saying that the latter are an artifact of the ordering choice for the infinite-dimensional symmetry algebra governing the Vasiliev system.
We shall spell out the details of the limit in Appendix D for the simplest non-trivial choice of eigenvalues (all real parts of left and right eigenvalues take the lowest-weight value 1/2) . Qualitatively, the result will be that lim where f (X) is a function of the spacetime coordinates andŷ := lim ∆ 2 →0 y L . We defer to Appendix D the precise result and more comments.
Thus, from the considerations above we expect that the Weyl zero-form can be analytically continued through the horizon (to which the unfolding point belongs), and that the membrane-like curvature singularities in ∆ = 0 are resolved at the master-field level in the sense specified above.
Limit to the singularity of the BGM background. Furthermore, one can observe that ∆ 2 | X 2 =0 = ξ 2 /M . The analysis of the membrane-like singularity therefore suggests that also ξ = 0 is a regular point, in the sense that the master field is given here by a well-defined regular prescription. Therefore, we expect that the master field configuration can be continued through the singularity, thus exploring the full background manifold AdS 3 × ξ S 1 .
In what remains, we shall turn our attention to extracting and studying the behaviour of the Lorentz scalar fluctuation field.

The scalar field
Choice of quantum numbers. For simplicity, we shall begin by studying the scalar field from the simplest non-trivial choice of eigenvalues that our kinematical conditions allow: that is, The complex conjugate, denoted with c.c. is extracted by means of the identical projection on the conjugate term conj (L) of (5.8) (or, equivalently, (5.30)).
Recalling the realization of iP in terms of number operators (4.16), it is evident that, operating on an element f λ with eigenvalues (5.37), the twisted adjoint action of iP (according to (4.9)) on it extracts the eigenvalue λ 2L −λ 1L symmetry. This is also why it makes sense to use the Schwarzschild-like coordinates, not adapted to the action of the Killing vector corresponding to iB, to write the fluctuation fields for this choice of eigenvalues. Note that a left-right asymmetric choice of real parts of the eigenvalues would give rise to exponentially growing/decreasing quasi-normal modes in the coordinate dual to B.
Performing parametric integrals in global coordinates. The contour integrals in (5.29) can be evaluated immediately, and simply set ς 1 = ς 2 = 1. Projecting onto Y = 0, (5.36) is reduced to where we have defined p := n √ M . The exponent is a quadratic form in the τ i , which can be computed by substituting (5.19) and (5.27) (alternatively, see Appendix D for an "adapted" spin-frame, in which the basis spinors are chosen as the eigenspinors of κ L (iB) ), where, in embedding coordinates, and where we recall that ∆ 2 ≡ (κ L (iB) ) 2 = 1 + X 2 3 − X 2 0 = X 2 0 ′ − X 2 1 − X 2 2 . Finally, the remaining integrals over τ 1 and τ 2 can be computed using the following formula (see Appendix F for the details of the derivation, and Appendix G for a succinct analysis of the extraction of component fields on the seemingly problematic surface X 0 + X 3 = 0): Thus, the scalar field profile can be written as where we recall . The first factor in (5.42) guarantees periodicity along the direction of identification, as will become manifest in Schwarzschild-like coordinates. In fact, as we showed in Section 3, writing the background solution by means of a globally defined gauge function facilitates its extension to the full topologically extended eternal spinless BGM black hole, consisting of two eternal BGM black holes glued together across their singularities. The natural question, that we addressed next, is whether fluctuation fields can be thought of as smooth at the BGM singularity (as well as on other submanifold, such as the horizon), which seems impossible in the usual gravitational analysis.
We have showed that higher-spin gravity provides interesting mechanisms for resolving classical singularities in gravity. These mechanisms rely on an interplay between differential and operator algebras. The former can be used to treat fluctuations on manifolds with degenerate metrics.
Moreover, the unfolded machinery requires the introduction of infinitely many form fields, which can be packaged into master fields taking their values in operator algebras. In the presence of higher spin symmetry, these operators become density matrices on non-commutative symplectic manifolds. As we have seen in this paper, the linearized master fields can be continued across horizons and singularities (and other surfaces), where individual Lorentz tensorial fields have fatal singularities: typically, the singular behaviour of individual fields on such surfaces manifests itself in the fact that the master fields become delta function in the fiber coordinates. As the latter are non-commutative variables, however, the master fields remain well-defined as symbols of operator algebra elements, and in that sense the limit to the horizon or the singularity is uneventful. In that sense, the fluctuations do explore the full topologically extended eternal spinless BGM black hole.
In order for such mechanisms to survive at the fully non-linear level, one has to show that the aforementioned operator algebras admit a well-defined quantum star product. We shall address this important issue, which involves the composition of operators in the image of the Wigner-Ville map applied to wave-functions that are not L 2 , in a future publication [84]. Preliminary results show that Holder duality [82] as well as the particle/black hole duality (2.29) may play an important role in constructing these algebras.
The formalism also leads to a natural twistor space regularization of the self-energy for Coulomb-like solutions on AdS 4 , which clearly deserves further scrutiny. Another physically interesting feature that emerges naturally from our construction is the appearance of quasi-normal modes on the BGM black-hole background. They arise essentially as a result of the fact that if the adjoint action of ad ⋆ K of the oscillator realization K of the identification Killing vector − → K has an integer spectrum, then the spectrum of ad ⋆ K , where − → K is a dual Killing vector, contains imaginary parts. The latter are responsible for the appearance of exponentially growing/decaying modes, which we interpret as quasi-normal modes. Our construction based on the unfolded formulation provides a systematic way of obtaining them analytically, and may therefore prove useful to study quasi-normal modes and the properties of their dual thermal states in greater detail.
using Dirac matrices obeying (Γ A ) α β (Γ B C) βγ = η AB C αγ +(Γ AB C) αγ , and van der Waerden symbols and raising and lowering spinor indices according to the conventions In order to avoid cluttering the expression with many spinor indices, in the paper we also use the matrix notations The van der Waerden symbols can be realized in a given spin-frame In particular, the SO(3, 2) generators that define the families of solutions (4.13) are realized as The so(3, 2)-valued connection In these conventions, it follows that where R ab := 1 2 e c e d R cd,ab and T a := e b e c T a bc are the Riemann and torsion two-forms. The metric g µν := e a µ e b ν η ab . The AdS 4 vacuum solution Ω (0) = e (0) + ω (0) obeying dΩ (0) + Ω (0) ⋆ Ω (0) = 0, with Riemann tensor R (0)µν,ρσ = −λ 2 g (0)µρ g (0)νσ − g (0)νρ g (0)µσ and vanishing torsion, can be expressed as Ω (0) = L −1 ⋆ dL where the gauge function L ∈ SO(3, 2)/SO (3,1). The stereographic coordinates x a are related to the coordinates X A of the five-dimensional embedding space with metric ds 2 = dX A dX B η AB , in which AdS 4 is embedded as the hyperboloid X A X B η AB = − 1 λ 2 = −l 2 , as The familiar global spherical coordinates (t, r, θ, ϕ) in which the metric reads are related locally to the embedding coordinates by providing a one-to-one map if t ∈ [0, 2π), r ∈ [0, ∞), θ ∈ [0, π] and ϕ ∈ [0, 2π) defining the single cover of AdS 4 . This manifold can be covered by two sets of stereographic coordinates, x µ (i) , i = N, S, related by the inversion x µ N = −x µ S /(λx S ) 2 in the overlap region λ 2 (x N ) 2 , λ 2 (x S ) 2 < 0, and the transition function T S N = (L N ) −1 ⋆ L S ∈ SO(3, 1). The map x µ → −x µ /(λx) 2 leaves the metric invariant, maps the future and past time-like cones into themselves and exchanges the two space-like regions 0 < λ 2 x 2 < 1 and λ 2 x 2 > 1 while leaving the boundary λ 2 x 2 = 1 fixed. It follows that the single cover of AdS 4 is formally covered by taking x µ ∈ R 3,1 .
For simplicity, we set l = λ −1 = 1 in the body of the paper.

B Finite transformations of the Cartan generators
In this appendix, we use a simple calculation to illustrate in the star-product language the finite transformations corresponding to the Cartan generators. We investigate the simple example: and its star-inverse φ is any real or imaginary number. Using the property K αβ K βγ = δ α γ , we can derive If we replace φ with iφ, we obtain From the above formulas we can see that, for φ ∈ R, γ (iφE) and γ (iφJ) are periodic transformations, and γ (iφB) and γ (iφP ) are non-periodic transformations, which well-correspond to their (non-)compact nature that we expect from AdS 4 isometries.

C Further comments on the eigenfunctions
In this Appendix we shall show explicitly how the eigenfunctions (4.55) arise starting from the integral presentation of the projectors (4.11). For the latter, i.e., for the case λ L = λ R and λ L,R + 1 2 ∈ Z + , it was established in [10] that different eigenfunctions are related by creation and annihilation operators, with f 1 2 , 1 2 (a + , a − ) being the ground state, i.e. a + ⋆ f 1 Moreover, diagonal elements with different half-integer eigenvalues are orthogonal with respect to the star product, and form an associative algebra (which can be extended by the corresponding twisted sector, see [8]).
Things are much more complicated, however, for general complex eigenvalues. In this paper we have not yet constructed a well-defined quantum system, and in particular we shall defer to a forthcoming paper the study of their algebraic properties under star product [84]. Below we will only qualitatively show that different eigenfunctions can be brought from one to another by using creation and annihilation operators with complex powers, which can be in their turn realized by means of the integral transform (4.53).
We will first show that, starting from an eigenfunction with equal eigenvalues λ L + 1 2 = λ R + 1 2 ∈ Z + , and by acting the creation operator on the left, we obtain an eigenfunction with a different left eigenvalue λ L ∈ C, i.e.
To show that the r.h.s. of (C.1) produces the eigenfunction with λ L , we use the results from Section 4.2. We substitute (4.53) and into the r.h.s. and evaluate the star-product between the Y -dependent factors of the integrands: Using this result, we have On the other hand, following the discussion around (4.38), if we assume that λ L + 1 2 ∈ Z + always holds, we can similarly derive the relation between different complex right eigenvalues: The above discussion is still far from a systematic study to build up a quantum system. In particular, as stressed in Section 4.3, allowing both left and right eigenvalues to take complex values involves alternative choices of contour other than the small circle around ±1. Moreover, such different contour-integral presentation has also the advantage of giving well-defined star-product properties between eigenfunctions with arbitrary complex eigenvalues. This goes beyond the scope of this paper, and we will continue our report on this issue in a forthcoming work [84].

D Analysis of membrane-like curvature singularities
In this Appendix we shall study the limit ∆ 2 → 0, corresponding to an analytic singularity for every individual fluctuation fields extracted from the generating function (5.8), at the level of Weyl zero-form master field -that is, in terms of the behaviour of the latter in the full (x, Y )-space.
It is instructive to first study the limit in the diagonal case, i.e., for λ iL = λ iR = λ ∈ Z − 1/2, i = 1, 2, that is (see Eq. (4.56)) for elements (5.29) of the form with ϵ = sign(λ), and where the quantities in the integrand were defined in Section 5.1. We shall also restrict our discussion to the simplest non-trivial choice of eigenvalues (5.37), that in the diagonal limit n = 0 reduces to studying the Weyl zero-form resulting from the lowest-weight element λ iL = λ iR = λ = 1 2 only. In this case the contours encircle the points ς 1 = 1 = ς 2 , and as a consequenceκ L αβ andv L αβ (5.13)-(5.14) reduce to κ L (iB)αβ and v L (iB)αβ . As the dependence on iP disappears, modulo a redefinition of the normalization factor (which we shall ignore here) we can simplify the notation by substituting the two contour integrals with a single one [10], where now y L := y − iςv Lȳ , and for notational simplicity we are now omitting the label (iB), which is henceforth understood everywhere unless specified otherwise. We recall that, as discussed in Section 4.2, keeping the contour integral is part of the regular presentation of our eigenfunctions: we evaluated them in Section 5.2 purely for the purpose of looking at the spacetime dependence of the component fields, but whenever the behaviour on the non-commutative Y space is of relevance, as will be in the interpretation of the ∆ 2 → 0 limit, we should keep them as they are an integral part of the definition of the eigenfunctions from the point of view of their algebraic behaviour.
In the ∆ 2 → 0 limit κ L αβ andκ L αβ become degenerate, and, as we shall now show, the integrand takes the form of a delta-sequence whereŷ α := (y+Sȳ) α , with S αβ a van der Waerden symbol coming from the ∆ 2 → 0 limit of −iςv L αβ , and D αβ : in terms of an x-dependent spin-frame that we shall now introduce. To study the limit precisely, it is convenient to perform a local SL(2, C) transformation to rewrite κ L αβ and v L αβ on a common "adapted" spin-frame (b +α , b −α ), b +α b − α = 1 on which κ L αβ takes the canonical form This condition determines the matrix of the Lorentz transformation only up to a free complex where Q := 1 + Performing this local Lorentz transformation corresponds to choosing a basis of the tangent space in which all spin-s Weyl tensors extracted from (D.3) are manifestly of Petrov-type D [10].
Retracing the analysis of the spherically-symmetric higher-spin black holes of [8,10], it is convenient to use the free parameter w in order to realize the Killing vector v L αβ on the adapted spin-frame in a canonical form, i.e., in terms of a single van der Waerden symbol. However, a novel feature arises here from the fact that both the norm of the Killing vector field and the determinant of its Killing two-form are not positive-definite: v L is spacelike for |X 3 | > |X 0 | and there det κ L is also positive; while v L is timelike for |X 0 | > |X 3 | and in this region det κ L is positive as long as X 2 3 < X 2 0 < X 2 3 +1 and negative when X 2 0 > X 2 3 + 1. As a consequence, the specific canonical form for v L changes in these three different spacetime regions: in particular, one can take (D.14) Of the above three regions, the relevant ones for the study of the singularity on (κ L ) 2 = 0 are clearly X 2 3 < X 2 0 < X 2 3 +1 and X 2 0 > X 2 3 +1. It is then easy to show that in the limit 1+X 2 3 −X 2 0 → 0 the integrand in (D.3) becomes (D. 16) where the realization of the van der Waerden symbols in terms of a spin-frame 32 has been given in Appendix A. As a consequence, 32 We note that while the entries of the SL(2, C) matrix (D.8) with w given by (D.12)-(D.14) separately scale like ∆ −1/2 , its determinant remains finite and equal to 1 everywhere, including in the limit ∆ → 0. This implies that b +α b − α = 1 also for ∆ = 0, i.e., that b ± α give a good spin-frame everywhere, thus in particular enabling to split y L α into y L± = b ±α y L α components which remain non-commuting, in such a way that, in particular, [ŷ − ,ŷ + ]⋆ = 2i(1 − s 2 ), which is in turn crucial to defining a proper non-commutative two-dimensional delta function.
A number of observations are now in order. First, it is interesting to note that, differently from the spherically-symmetric black-hole-like solutions where this singular behaviour was first observed, in this case the singular, delta-sequence limit is not obtained at the unfolding point x a = 0 (a point on the horizon of the gravitational background, in this paper) where the master-field (D.3) is, instead, regular. This is because, as evident from the discussion above, such distributional behaviour that characterizes the curvature singularities at the master-field level is strictly connected to the points at which the Killing two-form is degenerate. The latter was strictly vanishing at the unfolding point for the black-hole solutions (and for all solutions based on a π-odd Cartan principal generator -that is, on E and iP up to SO(3, 2) transformations) whereas the Killing two-form of the solutions studied in the present paper is clearly non-degenerate for x a = 0, and the only reason that it can have zeroes outside the horizon is due to the fact that the corresponding Killing vector has indefinite norm.
As already concluded in [8][9][10] for the spherically-symmetric solutions, this delta-function-like limit indicates that, even though at (κ L ) 2 → 0 every Weyl tensor diverges, the Weyl zero-form remains well-defined at X 2 0 − X 2 3 = 1 as an operator. Indeed, a delta function of noncommutative variables has well-defined star product composition properties (and, in fact, is part of the associative algebra to which the exact solutions studied in [8,14] belong). In this sense, thought of as a symbol for an element of a star product algebra, such a master field remains smooth in the (κ L ) 2 → 0 limit.
We stress that the integral presentation (D.17) is crucial to the interpretation of the resulting distribution as an associative algebra element, and, therefore, to the above interpretation of the Weyl zero-form in the ∆ 2 → 0 limit. In fact, using contour integrals to represent Fock-space endomorphisms by means of so(2, 3) enveloping-algebra elements (with the obvious prescription to take all star products before performing the contour integrals) is the core of the regular presentation scheme that was essential to the solution-building method presented in [8,10,14]). In the case at hand, performing the auxiliary contour integrals first would lead to the delta function δ 2 (ŷ | ς=1 ) which has divergent star product with itself asŷ | ς=1 are abelian oscillators. On the other hand, using the regular presentation ensures that the element (D.17) has good star product properties [8,14].
In this sense, the regular presentation can be thought of as way of regulating the star products of non-polynomial elements by introducing auxiliary, complex integration variables to achieve a sort of point-splitting procedure in Y space, that gets rid of divergent terms and keeps the finite part of the star products above in a way which is compatible with associativity (at least within the Fock-space projectors with identical left and right eigenvalues and its dual space, obtained via star-multiplication with κ y ) [8].
However, note that, differently from the the cases treated in detail in [10], in this case the Weyl zero-form admits a delta-function limit of modified oscillatorsŷ α that are specific to each side of the surface of apparent singularity (κ L ) 2 = 0, which would correspond to a discontinuity in the component fields at (κ L ) 2 = 0 if the component field description would make any sense there. The Weyl zero-form stays anyway regular in the sense above as a master field, which is the only suitable description in the strong coupling region.
This concludes the discussion of the diagonal limit.
Such a resolution of the curvature singularity that arises in the limit (κ L ) 2 → 0 can be shown, in fact, to still take place when λ iL − λ iR is non-vanishing. For definiteness, let us first focus in greater detail on the choice λ 1L = 1 2 + ip, λ 2L = 1 2 − ip, λ 1R = 1 2 = λ 2R , which is the case treated in greater detail at the end of Section 5.2. The master field in (5.29) reads in this case with θ L given in (5.27), ς := ς 1 +ς 2 2 , and where we denote with O(ς 2 − ς 1 ) all the terms weighted by the combination ς 2 − ς 1 (i.e., carrying the dependence on iP ), that vanish once one evaluates the contour integrals. Indeed, such terms will have no effect on the result, since with the choice the basic effect of the two contour integrations, featuring a simple pole, is just to set ς 1 = ς 2 = 1. All relevant quantities are therefore projected onto the iB sector and therefore, for the sake of brevity, we shall henceforth omit the evanescent terms altogether. Now, away from the surface ∆ 2 = 0 we can use the SL(2, C) transformation (D.7)-(D.8) to write all quantities on the adapted spin-frame b ± α . Let us first approach the limit from the region X 2 3 < X 2 0 < X 2 3 + 1. Inverting the transformation (D.7)-(D.8) with (D.12), we get that is, θ ′L α = −(l * (X)τ 1 + k * (X)τ 2 )b + α + (l(X)τ 1 + k(X)τ 2 )b − α .
The most general element in the twisted sector that satisfies the conditions (4.105)-(4.107) is (see with q := Re(∆λ) ∈ Z, and it may have a membrane-like curvature singularity at ∆ 2 = 0 for our choice of λ iR + 1 2 ∈ Z + . Higher λ iR will in general increase the order of the pole in ∆ 2 = 0, and we shall defer a full analysis of the general case to future work, focussing here on elements with the lowest right eigenvalue λ 1R = λ 2R = 1 2 . In such case, again omitting evanescent terms O(ς 2 − ς 1 ) and studying the ∆ 2 → 0 limit from above, one is reduced to the expression Clearly, in the case that q < 0 we end up with a master fields than is more regular that the q = 0 case studied above. On the other hand, the higher powers of ∆ that appear at the denominator for q > 0 can be interpreted as giving rise to derivatives of a delta function. The latter can however still be considered part of an associative algebra, in the sense that they admit a star-factorization in terms of delta functions, as Y -derivatives of δ 2 (ŷ) can be rewritten as (linear combinations of) star products of the type y α ⋆ δ 2 (ŷ) andȳα ⋆ δ 2 (ŷ).

E Removing the ambiguity between regular and twisted sector
In where this time y L := y − iv Lȳ . The different treatment of the holomorphic and anti-holomorphic dependence induced by the star-multiplication with κ y give rise to all the difference with respect to the twisted sector, resulting in particular in a more complicated form of the Killing two-form and Killing vector: Equivalently, in terms of a rigid spin-frame, In particular, note that, while for ς = 1 the above expressions reduce to (5.15)-(5.16) and (5.19)- (5.20), respectively, in this case the ς-dependence cannot be factored out of each of the 2 × 2 blocks ofK L as it happened for the twisted sector. This means that, as we shall see, the integrand of the contour integrals will differ from those so far examined, and will in fact be incompatible with a small-contour integral presentation of type (4.103) that we consider here. In particular, the study of the limit ∆ 2 → 0 elucidates the problem.
In fact, recalling that (κ L ) −1 αβ = −κ L αβ (κ L ) 2 it is immediate to see that the ς dependence is now nested with the spacetime dependence in the integrand, via It is clear that, evaluating the contour integrals first, the Weyl zero-form (E.5) reduces to the corresponding one in the twisted sector. This is expected, since the element 4e −4iB , which corresponds to f L after the contour integral is evaluated, is an eigenstate of κ y , so there is no distinction between the x-independent elements f L based. As a consequence, as long as the non-integral presentation of such elements is concerned, the regular and the twisted sector are equivalent. However, as mentioned above, this is not the case at the level of the integral presentation. Indeed, the integrand in (E.5), coming from the regular sector, develops a branch cut due to (E.10), and for ∆ 2 = 0 the latter inevitably crosses over the integration contour, making the small-contour integral presentation ill-defined for the the expansion of the Weyl zero-form over the regular sector. This conclusion still holds when one considers non-diagonal element and gives an imaginary part to the left eigenvalues, as the extra dependence on τ i (contained in θ L ) coming from the Mellin transform does not modify the poles in ς 1 and ς 2 of the contour integrals. This is the reason that we discarded the regular sector in the expansion of the Weyl zero-form in this paper. We defer the analysis of alternative, more general integral presentations to a future publication [84].

F An integral formula using parabolic cylinder functions
In this Appendix we shall prove the formulae (5.41) and (5.45), which are crucial to extract the scalar field fluctuation (5.42). One way to do it is the following. First, one can compute one of the two τ i -integral, say the one in τ 2 , by regularizing it via multiplication by a factor lim ϵ→0 + τ ϵ 2 e −ϵτ 2 2 . We will find in the end that the result can be analytically continued to ϵ → 0 + , so it will be possible to remove the regulator from the final expression. We can then recognize that the τ 2 -integral corresponds to the integral realization of a parabolic cylinder function We can now change variable to t = τ 2 1 and use the formula [88] ∞ 0 dt t G Apparent singularity at X 0 + X 3 = 0 In this appendix, we discuss a subtlety arising in the generalized Weyl tensor computation for the particular choice of eigenvalues (5.37).
We first look at (5.45) and (5.40) for the scalar field computation. One can see that the integral may not be well-defined at X 0 + X 3 = 0, in which case the exponent vanishes. In this situation, we should first compute the integrals for X 0 + X 3 ̸ = 0 and then analytically continue the result to X 0 + X 3 = 0.
For the scalar field it is easy to see that X 0 + X 3 = 0 is not a real problem, because the factor X 0 + X 3 does not appear at all in the result (5.42). Another way to see this is that in (5.45) the factor X 0 + X 3 can be simply absorbed by redefining the integration variables without creating any extra factors in the integrand.
However, for spin s > 0 we need a more careful discussion. In this case, due to the derivatives of the Weyl zero-form master field with respect to the y α -coordinates, the integrand has an extra factor in comparison to the spin-0 case, a polynomial in the τ i : and thus the redefinition (G.1) may or may not give rise to vanishing denominators in the limit X 0 + X 3 → 0, depending on the coefficients of the polynomial.
We have checked the integrands of the spin-1 Faraday tensor C αβ and of the spin-2 Weyl tensor C αβγδ , the "Polynomial(τ 1 , τ 2 )" factors of which respectively correspond to where W α := (κ L ) −1 θ L α . (G.5) After the redefinition (G.1) these polynomial factors do contain components that blow up at X 0 + X 3 = 0, but the Lorentz-invariants C αβ C αβ , C αβγδ C αβγδ , C αβγδ C γδεζ C εζ αβ and C αβγδ C αβγδ approach finite constant values in the limit X 0 + X 3 → 0.
This suggests that the generalized Weyl tensors can be made finite by a frame rotatioň y α = y α ′ Λ −1 α ′ α , (G. 6) i.e. the tensors are finite after the transformation Λ α For example, we checked that by the rotation