Multi-centered higher spin solutions from $W_N$ conformal blocks

Motivated by the question of bulk localization in holography, we study the problem of constructing multi-centered solutions in higher spin gravity which describe point particles in the interior of AdS$_3$. In the Chern-Simons formulation these take into account the backreaction after adding Wilson line sources. We focus on chiral solutions where only the left-moving sector is excited. In that case it is possible to choose a gauge where the dynamical variables are a set of Toda fields living in the bulk. The problem then reduces to solving the $\mathcal{A}_{N-1}$ Toda equations with delta function sources, which in turn requires solving an associated monodromy problem. We show that this monodromy problem is equivalent to the monodromy problem for a particular $\mathcal{W}_N$ vacuum conformal block at large central charge. Therefore, knowledge of the $\mathcal{W}_N$ vacuum block determines the multi-centered solution. Our calculations go beyond the heavy-light approximation by including the backreaction of all higher spin particles.


Introduction
The question of how local bulk physics in anti-de-Sitter space emerges holographically from the boundary CFT is an important one and has received considerable attention in recent years following the work of HKKL [1]. Ultimately it has ramifications for our understanding of the black hole interior and the information puzzle [2] [3].
One of the basic questions one can ask in this context is how a bulk state containing a collection of classical particles, tracing out worldlines in the interior of Anti-de Sitter (AdS) space, is described in the dual CFT. In the context of AdS 3 /CFT 2 , it was established [4][5] [6] [7][8] that these are intimately linked to 'classical' Virasoro conformal blocks at large central charge, in that the classical block computes the action of the classical particles. This connection has been further explored and generalized in various ways. In this work we will focus on the generalization to particles in higher spin AdS 3 gravity and their relation to classical blocks of the W N algebra [9][10] [11].
In most of the investigations into the relation between bulk particles and conformal blocks, a 'heavy-light' approximation is made in which at most one of the particles is assumed heavy enough to backreact on the geometry. To go beyond this approximation, one must consider fully backreacted multi-centered solutions in the bulk, which is a daunting task in general. In [12], it was shown that the backreaction problem in Lorentzian AdS 3 simplifies if one considers spinning particle trajectories which, on the boundary, only excite the left-moving sector. In this case, as we shall review in section 2 below, the spacetime contains a submanifold of constant negative curvature except at the particle locations where there are conical singularities. The problem then reduces to solving a Euclidean Liouville equation on the unit disk with ZZ boundary conditions [13] and delta-function sources. This requires solving a monodromy problem which is identical to the monodromy problem which determines the classical vacuum block in a specific channel. Therefore the multicentered solution can in principle be constructed once the classical vacuum block is known, and vice versa.
The goal of the current work is to generalize the above results to the construction of multi-centered solutions in higher spin gravity theories, coming from backreacting particles with higher spin charges. This problem was analyzed in [10] [11] in the heavy-light limit. To go beyond this approximation, we first recast the pure gravity results in Chern-Simons variables. We show that there exists a gauge which is convenient for the problem at hand, in which the left-moving gauge field is a Lax connection for the Liouville theory. Similarly, for the higher spin case, we show in section 3 that we can go to a gauge where one of the gauge fields is a Lax connection for the A N −1 Toda field theory. One somewhat suprising feature to come out of this analysis is that the standard higher spin theory with sl(N, R) gauge symmetry leads to a non-standard reality condition on the Euclidean Toda fields, while the standard reality condition instead describes higher spin theory with su( N 2 , N 2 ) gauge symmetry.
The Toda fields that we construct in this manner live in the bulk, and we argue that by including suitable singular sources in the Toda equations we can take into account the backreaction of localized bulk particles. Point particle sources can be coupled consistently to higher spin gravity in the form of Wilson lines [14] [9], and we consider here a class of 'chiral' particles which only couple to the left-moving gauge field. The problem then reduces to solving the Toda system on the unit disk with delta function sources under suitable boundary conditions. Upon taking into account the boundary conditions through a doubling trick, the problem can be reduced to a certain monodromy problem on the complex plane. We show in section 4 that the same monodromy problem arises in the determination of a classical W N vacuum block in a specific channel. Therefore, from the knowledge of the classical W N vacuum block we can in principle construct the multicentered solution and vice versa. While we treat the case of spin-3 gravity in the most detail, we discuss the generalization to the spin-N case in section 5.
To summarize our results in the context of bulk locality, we provide in this work a prescription to construct a state in the Lorentzian bulk theory containing localized particles from a classical block in a Euclidean Toda CFT. This seems close in spirit to H. Verlinde's 'CFT-AdS' idea, where states in AdS with localized particles or black holes are constructed directly from the Euclidean CFT [15] (see also [16], [17]). The present work can presumably be viewed as a concrete realization of this idea, albeit in a highly symmetric setting.

Chiral Solutions in Pure Gravity
In this section we review and expand on an earlier observation [12] that a class of 'chiral' solutions in 2+1 dimensional AdS gravity can be conveniently described in terms of a Liouville field living in the bulk. Here, the term chiral means that, from the point of view of the boundary CFT, only the left-moving sector is excited. We will first derive this Liouville description in the metric formulation, where it has a clear geometric origin, and subsequently in Chern-Simons variables, which will facilitate the extension to higher spin gravity in the following sections.

Chiral sector and bulk Liouville field
The general solution to pure AdS 3 gravity which has a cylindrical boundary and satisfies Brown-Henneaux boundary conditions [18], is parametrized by two arbitrary periodic functions T (x + ),T (x − ). The metric 1 reads, in Fefferman-Graham coordinates, Here, the boundary is at y = 0, and x ± = t ± φ are light-cone coordinates on the boundary cylinder, with (x + , x − ) ∼ (x + + 2π, x − − 2π). For example, the global AdS 3 solution is given by In this work, we will be interested in the class of 'chiral' solutions, whereT = −1 takes the same 2 value as for global AdS , while T (x + ) can be arbitrary. The metric (2.1) can then be rewritten in the fibered form The main observation is that the metric on the 2D base manifold, given by the last two terms in the above expression, has constant negative curvature as one can easily verify. Therefore, we can make a coordinate transformation which brings this base metric in conformal gauge, such that The field φ(z,z) then satisfies the Liouville equation ∂∂φ + e −2φ = 0, (2.5) and we will see below that the function T (x + ) is essentially the boundary value of the Liouville stress tensor (see (2.27)). Note that the Liouville field φ is a bulk field depending on the original coordinates y and x + . For later convenience it will useful to work out this coordinate transformation in more detail. It follows from (2.4) that there exists a real function α(y, x + ) such that Furthermore, one can show that the 3D Einstein equations imply the relation In summary, for chiral solutions to pure 3D gravity the metric can be brought in the form where φ is a Liouville field satisfying (2.5) and we have defined 1 Note that we work in units where the AdS radius is set to one. 2 One can easily generalize this discussion to the case whereT is an arbitrary negative constant, as was done in [12], though we will focus onT = −1 in what follows.

Chern-Simons variables
With a view towards generalizing this observation to higher spin gravity, it will be useful to describe the above change of variables in the Chern-Simons formulation [19], [20], where the gravitational field is described by a flat connection taking values in sl(2, R) ⊕ sl(2, R).
In this formulation the gauge connections A,Ā are related to the dreibein e and the spin connection ω as For solutions obeying the Chern-Simons equivalent of the Brown-Henneaux boundary conditions, the gauge connections can be taken to have the Fefferman-Graham form [21] A HW = −L 0 dy y 11) where in the second formula we have already restricted to chiral solutions withT = −1.
Our claim is that we can perform a gauge transformation to a gauge where the connections take the form: The connectionÃ is a Lax connection for the Liouville equation: one verifies that the flatness ofÃ is equivalent to the Liouville equation (2.5). Indeed, the connectionÃ can be brought into the standard form in the literature for the Lax connection of Liouville theory. For this purpose we perform a by a complex change of basis in the Lie algebra generated by V ≡ e iπ 4 (L 1 −L −1 ) , (2.14) resulting in the the standard Lax connection for the Liouville equation (see e.g. [22]) The right-moving connection becomes in this basis The associated field strengthF = dÃ +Ã ∧Ã satisfies and its vanishing is indeed equivalent to the Liouville equation. Although the connections A T andĀ T don't live in the original Lie algebra, they are useful to define since they give a field strength in the Cartan subalgebra and because they can be easily generalized for the case of arbitrary spin that we examine in section 5.
It remains to describe the gauge transformation relating (2.11) and (2.13) in more detail. A generic gauge transformation can be decomposed into a translation part involving the AdS translation generators L m −L m , and a local Lorentz part involving the Lorentz generators L m +L m . As shown in [20], the translation part can be traded for a coordinate transformation (y, x + , x − ) → (z,z,t), which we take to be precisely the transformation derived before: One can verify that this transformation takes (2.11) into (2.12,2.13), i.e.
To find the precise relation between the bulk Liouville field φ(z,z) and the boundary stress tensor T (x + ), we will work out this gauge transformation near the boundary. First of all, by making a conformal transformation we may assume z to take values in the unit disk, with the AdS boundary located at |z| = 1. For global AdS, T = −1, this leads to and the following relation between the coordinates: More generally, we allow Liouville solutions on the unit disk with the same blow-up behaviour near the boundary, i.e.
This boundary condition (2.21) on the Liouville field was considered in [13] and is often referred to as the 'ZZ boundary condition'. In general, we can construct a holomorphic quantity from the Liouville field, the 'Liouville stress tensor' where the overall normalization is chosen for later convenience. Like φ, this is a bulk quantity, though its value at the boundary turns out to be closely related to the boundary stress tensor T (x + ). Using the Liouville equation one shows that solutions which behave like (2.23) near |z| = 1 can be expanded as Plugging this into eq. (2.6), one finds that the desired gauge transformation near the boundary is of the form (2.18), (2.19) with and that the boundary stress tensor is related to the Liouville stress tensor as This relation was derived previously in [12] using the metric formulation. It reflects the fact that, on the boundary, the coordinates z and x + are related through z = e ix + : the relation (2.27) is the standard conformal transformation of the stress tensor, with the constant term arising from the Schwarzian derivative.

Point-particle sources
As motivated in the Introduction, in what follows we will not restrict ourselves to pure gravity, but we will introduce point particle sources in the bulk which backreact on the metric. We want to choose the quantum numbers of the particles and their trajectories in such a way that the backreacted metric falls in the chiral class (2.8). The quantum numbers specifying the particle can be taken to be taken to be the sl(2, R) ⊕ sl(2, R) weights (h,h), or equivalently the particle mass m = h +h and spin s = |h −h|. In order not to turn on the right-moving stress tensorT , we takeh = 0, in other words we consider 'extremal' spinning particles with m = s. We should keep in mind that the particle mass m refers to the coefficient in front of the proper length part of the action m ds. The extremal particles we are considering are in fact excitations 3 of massless fields with spin, since we recall that the field theory mass in AdS 3 is given by Note that both concepts of mass coincide in the limit m s. Spinning particles are described in metric variables by the Matthisson-Papapetrou-Dixon equations [23], [24], [25]. As was shown in [26], in locally AdS 3 spaces spinning particles still move along geodesics. One can easily check that, in backgrounds of the chiral form (2.8), the curves of constant z are geodesics, and these curves will be our spinning particle trajectories. In global AdS 3 these are helical trajectories spinning around the center of AdS 3 at constant radius.
These considerations have a nice and elegant counterpart in Chern-Simons variables, which will easily generalize to the higher spin case. In the Chern-Simons description, point particles can be introduced by inserting Wilson lines W R (C)WR(C) into the path integral [14][9], with W R (C) = tr R P exp where C is a curve and R andR are (unitary, infinite-dimensional) sl(2, R) representations with lowest weight h and highest weight 4 −h respectively. For the case of interest wherē h = 0, theR representation is trivial and our spinning particles only couple to A. As advocated in [14][9], it's useful to rewrite the trace over Hilbert space states in (2.29) as a quantum mechanical path integral as we review in Appendix D. For large h, it's justified to make a saddle point approximation which amounts to adding a source term (D.5) to the classical Chern-Simons action. As was shown in [10], the resulting description is indeed equivalent to the Matthisson-Papapetrou-Dixon equations. As we show in Appendix D, the effect of adding the spinning particles is to introduce delta-function sources on the right hand side of the Liouville equation (2.5): ∂∂φ where the m i are the point-particle masses and z i their locations in the z coordinate of (2.8). We see that the particles must be 'heavy', in the sense that m i ∼ G −1 ∼ c in order to produce backreaction. It was derived in [12] that solving the sourced Liouville equation with the boundary condition (2.23) amounts to solving a certain monodromy problem. It was also shown there that this monodromy problem is equivalent to the one which determines a certain CFT vacuum conformal block; knowledge of the conformal block therefore allows for the construction of the multi-centered solution and vice versa. In the following sections we will generalize these results to the construction of multi-centered solutions in higher-spin gravity.
We end this section with some comments.
• We would like to emphasize that the multiparticle configurations we are considering here are specific to Lorentzian AdS 3 . Indeed, our particle geodesics, which have constant AdS radial coordinate and constant value of x + , do not analytically continue to geodesics in Euclidean AdS 3 , except for the geodesic at the center of AdS 3 . Unlike their Euclidean counterparts, our Lorentzian geodesics do not reach the boundary and therefore do not correspond to localized sources in the boundary CFT; this is why their description as localized sources of the bulk Liouville field is particularly convenient.
• The setup described above might be viewed as a special case of Verlinde's 'CFT-AdS' idea [15] (see also [16], [17]), where a state in the Lorentzian bulk theory is directly corresponds to a state in the Euclidean CFT. In our case, the bulk state contains a number of spinning particles and corresponds to a state created by the insertion of heavy operators in Euclidean Liouville theory on the unit disk with ZZ boundary conditions. Our bulk Liouville field φ should also not be confused with the boundary Liouville field constructed in [27], though the two are related: the left-moving components of their stress tensors obey the relation (2.27).

Chiral Solutions in Spin-Gravity
Our main objective is to generalize the above results for pure gravity to the Chern-Simons formulation of higher spin gravity. In this and the next section we will perform the calculations explicitly for the spin-3 theory, and we will extrapolate to arbitrary spin in section 5.
It is natural to guess that the role of Liouville theory in the pure gravity problem will be replaced by Toda field theory in the higher spin case, and this will turn out to be so. One subtlety we will pay particular attention to is the following. One can consider two different higher spin theories based on the two noncompact real forms of the algebra A 2 , namely sl(3, R) and su (1,2). We will see that chiral solutions in these theories are described by different real sections of complex Toda field theory. Somewhat surprisingly, we will see that the standard Euclidean Toda field theory describes the non-standard higher spin theory with gauge algebra su(1, 2) while the standard sl(3, R) higher spin theory requires a nonstandard reality condition on the Toda fields.

Spin-3 Gravity
We start by recalling some standard facts about gravity coupled to a massless spin-3 field in Lorentzian AdS 3 , referring to [28] for more details. The action is given by where the gauge fields A andĀ take values in a real form of the algebra A 2 . There are two real forms of this algebra which contain sl(2, R) as a subalgebra and give rise to a higher spin extension to gravity: these are the noncompact real forms sl(3, R) and su (1,2). The theory based on sl(3, R) is the most prevalent in the literature as it leads [28] to an asymptotic symmetry which is the standard Zamolodchikov W 3 algebra [29], while the su(1, 2) leads to a different real form of the complex W 3 algebra which leads to problems with unitarity at the quantum level [30]. The two real forms can be conveniently treated simultaneously by introducing a parameter σ = ±1, such that The parameter σ enters the commutation relations which take the form: and similarly for the barred generators. Note that the two real forms are related by the 'Weyl unitary trick' of multiplying the spin-3 generators W a with a factor i. An explicit matrix realization is given in Appendix A. The connections A andĀ are linear combinations of the generators L m , W a (and their barred counterparts) with real coefficients. The generators L m form an sl(2, R) subalgebra which is said to be 'principally' embedded in the full algebra. The restriction of the gauge field to this subalgebra defines the gravitational subsector of the theory, while the coefficients of W a andW a describe a massless spin-3 field. As shown in [28], field configurations obeying the higher spin equivalent of the Brown-Henneaux boundary conditions [18] can be brought in a standard form which generalizes (2.11) and is called the 'highest weight gauge'. Restricting once more to chiral solutions whereĀ takes the same form as in global AdS 3 , solutions in this gauge take the form Here, T (x + ) and W (x + ) are arbitrary periodic functions.

The Toda field gauge
We will now show that we can make a gauge transformation which brings the connection A into the form of a Lax connection for A 2 Toda field theory, which will facilitate the construction of backreacted multi-particle solutions in the next subsection. More precisely, we will go to a gauge where

9)
As before, the only role of the complex change of basis generated by V defined in (2.14) is to simplify the right hand side. The flatness conditionF = dÃ +Ã ∧Ã = 0 is equivalent to the A 2 Toda field equations If we initially view φ 1 and φ 2 as complex-valued fields, the requirement thatÃ has real coefficients imposes a certain reality condition on the fields, which depends on the chosen real form, i.e. on the value of σ. On the components of A T in the expansion A T = m l m L m + a w a W a , the condition thatÃ has real coefficients imposes that Applying this to (3.10) one therefore finds that the appropriate reality condition is We observe that, somewhat surprisingly, the standard form of the Toda field equations, where φ 1 and φ 2 are real fields, is relevant for the nonstandard higher spin theory with algebra su(1, 2) ⊕ su(1, 2) and vice versa. As shown in Appendix C 5 the Toda equations imply that the following combinations are purely holomorphic: (3.14) One can show that T transforms as a stress tensor under conformal transformations, while W transforms as a spin-3 primary. We note from comparing (3.10) to the pure gravity expression (2.13) that the gravity subsector is obtained by setting Note that we have W = 0 in the gravity subsector as was to be expected. From the pure gravity expression (2.21) and the Toda equations it follows that the global AdS 3 solution is given by and has T = W = 0. For the σ = −1 case the fact that we have real Toda fields imposes that n = 0. More generally, we allow Toda solutions on the unit disc with the same leading blow-up behaviour near the boundary circle. These boundary conditions are the Toda generalization of the ZZ boundary conditions [13]. Using the Toda equations one shows that such solutions behave near the boundary as Combining (3.18) with the reality conditions (3.13) on the fields, we see that the currents T and W should satisfy the following reality conditions on the boundary circle: To simplify notation for this section we have relabeled the currents such that compared to Appendix C we have T ≡ W (2) , W ≡ W (3) .
It remains to spell out the gauge transformation between the highest weight gauge (3.8) and the Toda field gauge (3.10). We will only work out the near-boundary behaviour of this transformation, from which we will derive the precise relation between the bulk Toda currents T (z), W(z) and the boundary currents T (x + ), W (x + ). A generic gauge parameter contains a spin-2 and a spin-3 part, each of which consists of a 'translation' and 'local Lorentz' part. As in the gravity case, we will trade the spin-2 translation for a coordinate transformation (y, x + , x − ) → (z,z,t). It turns out that, to the order required, the latter coordinate transformation is unchanged from the pure gravity case, i.e. it still takes the form (2.26). The required remaining gauge parameter turns out to be where Λ,Λ are given in (2.19). Performing the gauge transformation and making use of the near-boundary expansion (3.18) we find that (3.8) transforms into (3.10) with the currents related as As a check we note that T (x + ) and W (x + ) are indeed real upon imposing the reality conditions (3.13). These relations are consistent with the conformal transformation properties of the currents and the fact that the boundary coordinates are related as z = e ix + .
Point-particle sources Having found convenient variables to describe chiral solutions in pure higher spin-3 gravity, we will now study solutions in the presence of point-particle sources which are compatible with this chiral ansatz. Such higher-spin point particles are once again described by Wilson lines [9], and in order to preserve the chiral structure we study sources which couple only to A and not toĀ: Such sources correspond to the spin-3 generalization of extremal spinning particles. The representation R is an infinite-dimensional representation of sl(2, R) resp. su(1, 2) built on a primary state |h, w satisfying As shown in Appendix D, in the presence of such backreacting point particles the Toda equations acquire nontrivial delta function sources on the right hand side where the parameters α 2 are related to the conformal dimension and spin-3 charge of the i-th particle as (3.25) It's a useful consistency check that these sources are compatible with the reality conditions (3.13) which for σ = 1 require that α

Properties of the solutions
The associate ODE problem: Toda field theory is an integrable system and its solutions can be constructed from solutions to a set of auxiliary ordinary differential equations (ODEs). For more details on these auxiliary ODEs we refer the reader to Appendix C.
Here we give just the essentials needed to make the discussion coherent.
The A 2 -Toda field theory has two systems of auxiliary ODEs associated with it. One shows that the fields e φ i satisfy the following sets of equations with T (z) the stress tensor and W(z) the primary spin-3 current given in (3.14). We will denote by ψ i (z), i = 1, 2, 3 a set of linearly independent solutions of the equation and by χ i (z), i = 1, 2, 3 a set of independent solutions of We will now show that we can build Toda solutions e φ i from the ψ i , χ i in a holomorphically factorized form.
Solutions for complex conjugate Toda fieldssl(3, R) case: By substituting (3.35) into the first Toda equation (3.11) and solving for e φ 2 we find that Thus for the Toda fields we have

Properties of the currents
Pole structure: If near the sources z i we can neglect the potential term compared to the kinetic term, then the Toda fields φ i behave near z = z i as Thus the solution of the Toda equations on the disk specifies meromorphic (quasi-)primary currents T and W. The sources α (i) j fix the most singular terms in T , W which take the form where the ellipses denote lower order poles and a regular part. The constants T i and W i are specified in terms of α (i) 1 and α (i) 2 by examining the form of the Toda fields φ 1 and φ 2 near a specific source.
Doubling trick: Having established the pole structure of the currents T (z) and W(z) as functions on the unit disk, we should also make sure that they obey the reality conditions (3.19) on the unit circle. As usual, this is imposed by using a 'doubling trick' and extending T (z) and W(z) to meromorphic functions on the complex plane in a suitable manner. Using the Schwarz reflection principle we find that the appropriate reflection properties are In particular, this means that the currents have poles both at the z i and at their image pointsz −1 i . Thus they are of the form 43) where in the above equations we assumed for simplicity that there is no source located at the origin and as a result there is also no image source at infinity.
The parameters c are called accessory parameters and are not directly determined by the α (i) j . Instead we will see below that they are determined by solving a monodromy problem which arises from demanding that the Toda fields are single-valued. Not all the parameters in (3.42) and (3.43) are independent however, since we still have to impose the reflection properties (3.41). We will now determine the number of independent accessory parameters after imposing the reflection properties. Substituting (3.42) and (3.43) into (3.41) and requiring equality of the poles at each order near z i we get from T One can check that these conditions guarantee that (3.41) holds everywhere. Regularity at infinity requires that T falls of as z −4 and W as z −6 , which is implied by the reflection conditions (3.41) and the fact that T , W are regular in the origin. So this does not lead to any further conditions. After taking into account these constraints we are left with 2K − 3 independent real accessory parameters coming from T and 4K − 5 parameters coming from W, giving in total 6K − 8 real accessory parameters to be specified by solving the monodromy problem. Note that the above conditions were derived in the assumption that none of the poles are in the origin. It is straightforward to generalize to the case of a pole in the origin and to check that the number of independent accessory parameters remains unchanged.

Doubling trick for ψ i
The reflection property of the currents (3.41), together with the associate ODEs, implies there should be a reflection property for the ψ i . For the su(1, 2) case we find while for the sl(3, R) case we have To prove these relations we start from the associated ODE (3.28) satisfied by ψ i (z): Under a conformal transformation z →z = f (z) the fields transform as The last transformation can be derived from the fact that ψ i should transform like e φ 1 , i.e. as a primary of weight −1, under holomorphic reparametrizations in order for (3.50) to be conformally invariant. Using these transformations, we perform a conformal transformation with f (z) = 1/z on (3.50), and use the reflection conditions (3.41) to obtain, after a relabeling of the coordinate: For the su(1, 2) case, upon choosing σ = −1 and comparing with the ODE (3.29) satisfied by χ i , we see that there should exist a constant matrix S such that Here, S is a proportionality matrix whose exact form is independent of the particular solution, and can be fixed by examining the simple solution displayed in appendix C. 4 where M i ∈ SL(3, C) is a monodromy matrix. However further restrictions are imposed on this monodromy matrix by requiring that the Toda fields are single valued. We will see that in each case the monodromy matrix must be an element of one of the real sections of SL(3, C).
Real Toda fields -su(1, 2) case: As we derived in (3.30) we have Thus for e φ 1 to be single-valued we should demand that . This means that we must have M i ∈ SU (1,2). In other words in order for the solution of the Toda system to be single-valued we must adjust the accessory parameters such that all the monodromy matrices M i are elements of SU(1, 2).
Complex conjugate Toda fieldssl(3, R) case: For the first field in the Toda system we have (3.37) e φ 1 = − abc ψ aψb∂ψc . (3.60) In order for it to be single-valued we should have that which can be rewritten as At this point we make use of the definition of the adjugate matrix Which means that M i ∈ SL(3, R)..

Monodromies of image points
Here we will prove that the monodromy matrix of a contour that encircles a singular point z i and its image 1/z i (and none of the other singularities) is the identity matrix. This property will be important in making the connection to classical W 3 blocks in section 4. To do so lets us first consider the monodromy matrix M γ i of a point z i which is encircled by a counterclockwise contour γ i which has a base point p at the boundary. The mirror contourγ i encircles the image point 1/z i in an opposite, clockwise, orientation and results in a monodromy matrix Mγ i . We denote the contour encircling the image point counterclockwise asγ −1 i . Then We will use the reflection property of ψ i to show that this is indeed the case. For simplicity we are going to drop from M the subscript γ i and adopt a new notation such that Then after encircling the point z i we have (3.71) At this point we make use of the definition of the adjugate matrix (3.63) which we can rewrite as In the present case F = ΛN Λ. Thus (3.71) becomes Complex conjugate Toda fieldssl(3, R) case: Again we start by rewriting (3.49) as ψ a (z) = i e −i π 6 z 2 ψ a (1/z) . (3.78) Then after encircling the point z i we have Since N is real, by combining the last equation with (3.78), we immediately see that (3.68) holds.

Monodromy around all points
In this section we will show that the monodromy of a contour encircling all points within the unit disk takes values in one of the real forms SL(3, R) resp. SU (1,2), without making the assumption that the monodromy around each of the points belongs to a real form of SL (3, C). This fact will be of use in the counting of accessory parameters that follows this section. Let us denote by Γ a counterclockwise contour encircling all the points within the unit disk, which has a base point p at the boundary. The mirror contour,Γ encircles all the image points in an opposite, clockwise, orientation. We denote the contour encircling all the image points counterclockwise asΓ −1 . Then For a counterclockwise contour encircling all of the points and their images we have The latter is similar to (3.68). Then the rest of the proof follows the same steps in the case of monodromies of image points by inverting the logic of that section. Specifically, instead of starting with the group the monodromy matrix belongs to and trying to prove (3.68), we start with (3.82) and we find the conditions that specify the group the monodromy matrix belongs to.

Parameter counting and monodromy reduction
We will now show that the condition that the monodromy matrices take values in SL (3, R) resp. SU (1,2), in the generic situation, imposes precisely as many constraints as the number of available undetermined accessory parameters in the ODE, namely 6K − 8. Therefore, for generic values of the particle positions and quantum numbers, we expect our equations to have a unique solution. To count the number of required constraints, we follow the following logic (see [12]): we first compute the dimension of the space in which the monodromy matrices around the K singular points in the unit disk take values if the accessory parameters are arbitrary, and subtract from this the dimension of the space in which they take values when the SL (3, R) (resp. SU(1, 2)) condition is imposed.
To compute the dimension of the former space, we note that the monodromy matrix M i takes values in SL (3, C) which has real dimension 16, leading to 16K parameters. Not all of these are independent however, since the coefficients (s) i of the most singular pole terms in T and W are fixed. The latter fix the trace invariants trM i and trM 2 i and therefore subtract 4K parameters. Furthermore, as we proved, the reflection property for ψ i implies that the monodromy, when encircling all singular points in the unit disk, must take values in one of the real forms SL (3, R) resp. SU(1, 2) , thus subtracting 8 parameters. Furthermore, by making a change of basis in the space of solutions of the ODE, all monodromies are conjugated by a constant matrix in SL (3, R) resp. SU (1,2). Since this conjugation generically acts effectively (i.e. it sweeps out an 8-dimensional subspace), we should subtract another 8 parameters, leading to the desired dimension Now we compute the dimension of the space of monodromy matrices after imposing the SL (3, R) resp. SU (1,2) conditions. The dimension of these real groups is 8, leading to 8K parameters a priori. Again the invariants trM i and trM 2 i are fixed in terms of the (s) i , but now these invariants are automatically real, therefore subtracting 2K parameters. The reality constraint on the monodromy when encircling all points in the unit disk is now automatically satisfied, and the overall conjugation by a constant matrix in SL (3, R) resp. SU(1, 2) subtract 8 parameters. This leads to a dimension of 8K − 2K − 8 = 6K − 8.
Computing the difference of these two dimensions leads to the number of 6K − 8 constraints we need to impose to reduce the monodromy to SL (3, R) resp. SU (1,2), and matches precisely the number of undetermined accessory parameters at our disposal. We conclude that, generically, imposing the single-valuedness of the Toda fields will precisely fix all the accessory parameters.

Monodromy problem for classical W 3 blocks
In the previous section we have shown that constructing backreacted bulk solutions containing certain spinning particles in the bulk higher spin theory reduces to solving a certain monodromy problem, where the accesssory parameters in an ODE are fixed by requiring monodromies to be in SL(3, R) resp. SU (1,2). Another context where a similar monodromy problem appears, is in the construction of W 3 blocks at large central charge. In this case, the monodromy properties of the ODE are determined by the quantum numbers of the exchanged W 3 families. In this section we will show that the bulk monodromy problem is in fact identical to the one which determines a W 3 vacuum block in a specific channel. We start with a brief discussion of the properties of W N blocks. W 3 blocks As for the Virasoro algebra, the W N blocks are defined to be the building blocks of correlators which capture the purely kinematical information which is fixed by the W N Ward identities. The W N -blocks are much less studied objects than their Virasoro counterparts, see however [31][32][33] [34].
One peculiarity of W N blocks as opposed to Virasoro blocks goes back to the wellknown fact that the W N Ward identities don't allow one to express arbitrary correlation functions in terms of correlators involving only W N primaries [31]. This leads to the property that generic W N blocks depend on an infinite number of extra parameters besides the familiar dependence on the chosen channel, the cross-ratios and the quantum number of the external and exchanged primaries. This arbitrariness is however absent for the vacuum blocks which will turn out to be the ones relevant for our purposes.
Let us illustrate these features for the case of the four-point block (which will be relevant for constructing a two-centered bulk solution), and comment on the the generalization to higher point blocks at the end. We start by considering a correlation function of four W 3 primary operators 6 Here, O η is a W 3 -primary operator characterized by it's weights η = (h, w) with respect to L 0 and W 0 . The four-point function can be decomposed into a sum over exchanged where Π η is a projector onto a particular W 3 descendant family of a primary state |η = O η |0 , Π η = I 1,2 ,J 1,2 Here capital I is a multi-index so that L I is an abbreviation for L i 1 L i 2 . . . L i k . The G J 1 I 1 ,I 2 J 2 are elements of the inverse of the inner product matrix G J 1 I 1 , Mimicking the definition of Virasoro conformal blocks, we consider the ratio However, unlike in the Virasoro case, the quantity B η is not completely fixed by W 3 kinematics and still depends on dynamical information. As shown in [31], [32], B η generically still depends on the ratios of three-point functions One therefore defines the W 3 block to be an object depending on an extra set of free parameters C n ,C n , which in a specific W 3 CFT should be specialized to take the appropriate values (4.4). It will be important to note that this infinite arbitrariness is absent 7 for the vacuum block, since the C n ,C n vanish when |η = |0 is the vacuum. Summarizing, we have represented the correlation function A 4 as Note that in writing this expansion, we have also chosen a 'channel', i.e. we have chosen to fuse O η 1 with O η 2 and O η 3 with O η 4 . Similarly, one can perform a decomposition of an n-point correlation function by inserting n − 3 projectors for the exchanged primaries and dividing by n − 2 three point functions, analogous to (4.3). 6 In order to simplify equations, we are displaying only the holomorphic coordinate dependence in our formulas. 7 Another case where the arbitrariness is absent is when the external operators are semi-degenerate [32].
Inserting a degenerate primary As is the case for Virasoro blocks (see e.g. [4]) the W N blocks in a certain large-c limit can be determined by solving a monodromy problem [34] [35]. To derive this, one considers the amplitude with an insertion of an additional degenerate primary and makes use of the constraints imposed by the decoupling of its null descendants.
To this end we consider the auxiliary object B η [Ψ], which differs from the block B η defined in (4.3) by an insertion of an extra degenerate primary Ψ in the numerator: Here, Ψ is a specific degenerate operator associated to a state which satisfies shortening conditions of the form: (4.7) The explicit coefficients 8 can be found in [34]. Upon inserting the last equation into the correlation function (4.6), we obtain a shortening relation of the form The operatorsT ,Ŵ which act on the B η [Ψ] are defined aŝ where h i and w i are the conformal dimension and spin-3 charge of the i th operator. By W (i) −k we denote the negative k th mode acting on i th operator inside the B η [Ψ], for example W As it stands, equation (4.8) is a differential equation which couples different amplitudes.
We will now argue that, in a suitable large-c limit, it reduces to an ODE for a single function.
Classical, large c, limit We now consider the limit of large central charge c, in which the primary operators O η i are assumed to be "heavy" in the sense that η i /c remains finite in the limit. It has been argued, see e.g. [36], although not proven, that in this limit the general conformal block exponentiates lim c→∞ B η = e − c 6 bν . (4.12) Here, the parameters ν i = ( i , δ i ) on the right hand side contain the rescaled "classical" weights The behaviour of (4.12) is reminiscent of a saddle-point approximation, where b ν is the action of the saddle-point. Now let's consider the quantity B η [Ψ] in (4.6) with the insertion of the degenerate primary Ψ. Since Ψ is "light", in the sense that its charges are of order 1 at large c, it is natural to assume that its presence does not change the saddle point and that B η [Ψ] factorizes as lim (4.14) We expect a similar factorization to occur in the case of W An important assumption here is that this factorization involves the same function ψ ν as in (4.15), see [36] for a justification. Under these assumptions, the decoupling equation (4.8) reduces in the large-c limit to at holomorphic ODE for the "wave function" ψ ν Here T , W are the functions with coefficients i , δ i defined in (4.13), and the accessory parameters a i , d i , c i given by the following limits The exponentiation of the classical block (4.12) implies that the c i are finite and given by It is natural to expect that the a i and d i similarly remain finite in the limit.
To summarize, we have found associated to a classical W 3 block an ODE (4.16) which is of the same form as the auxiliary ODE (3.28) determining the solutions of the A 2 Toda system. Similarly to what happens for classical Virasoro blocks [4], the choice of the exchanged W 3 primary η determines the monodromy properties of the ODE (4.16). To see this, note that the quantity (4.6) contains the four-point function Using the OPE between the light operator Ψ(z) and the heavy operator O η (0) one can show that the trace invariants trM, trM 2 of the monodromy matrix as Ψ(z) encircles the origin are fixed in terms of η, and this is of course also the same monodromy when z encircles both z 1 and z 2 . This discussion generalizes straightforwardly to classical n-point blocks: they are similarly determined by an ODE whose monodromy properties are determined by the choice of exchanged primaries in the chosen channel.
Relating the monodromy problems Now we are ready to relate the monodromy problem for the classical W 3 four-point block to the monodromy problem which determines a 2-centered solution in spin-3 gravity. In studying the latter problem we encountered an ODE on the complex plane of the form (4.16) with identical singularities in two pairs of image points. Therefore we consider a CFT four-point function with a primary O η 1 inserted in the points z 1 and 1/z 1 , and a second primary O η 2 inserted in z 2 and 1/z 2 . Next we should choose a channel in which to perform the conformal block expansion. It turns out that the relation between the two monodromy problems is the simplest if we choose the "mirror channel" in which we fuse the operators located in image points (i.e. z 1 and 1/z 1 resp. z 2 and 1/z 2 ) together in pairs. The reason is that we derived in paragraph (3.6.2) above that, in the bulk problem, the monodromy when encircling a pair of image points is the identity. From our discussion in the previous paragraph, this means that the exchanged primary in the corresponding W 3 block in the mirror channel is the identity operator. In other words, our bulk monodromy problem determines a W 3 vacuum block.
In summary, we have argued that the monodromy problem determining a 2-centered solution in spin-3 gravity is equivalent to that which determines the classical vacuum 4point block .

(4.22)
The argument can be generalized in a straightforward manner to show that the monodromy problem for a K-centered solution determines a classical vacuum 2K-point block in a "mirror" channel where the operators in image points are fused together in pairs (see fig.1). We should stress that, due to the fact that W 3 vacuum blocks are unique and don't depend on extra parameters like (4.4), the derived correspondence is really one-toone: solving the monodromy problem for the K-centered solution determines the 2K-point vacuum block through (4.20) and, conversely, from the knowledge of the vacuum block we can in principle derive the accessory parameters in the ODE and construct the bulk multi-centered solution. 9

Generalization to Arbitrary Spin
Here we present generalization of the above results to higher spin theories with spins 2, . . . , N . Our results in this section are either based on generic arguments or are extrapolated from explicit calculations we carried out for N = 3 and N = 4. Many of the details of Toda theory related to this section are discussed in appendix C. We focus on gauge connections that belong either to the maximal, sl(N, R), or to the next to maximal, su( N 2 , N 2 ), non-compact real form of sl(N, C).

The Toda gauge connection
First we observe that for N = 3, the left connection (3.9) in Toda gauge can be rewritten for both values of σ as where the matrices H i and E ± i are defined in appendix B. From (5.1) we can generalize to arbitrary spin by writing the connection

Then (5.2) becomes
is the Cartan matrix and is obtained from In matrix form we have The flatness of the A T is equivalent to the Toda system being satisfied.
The argument to be made is that (5.3) can be brought into a form similar to (3.9) with the presence of a parameter σ such that Similarly we choose the right connectionĀ T to be proportional to the principally embedded Cartan element of sl(2, R) inside sl (N, R) respectively su N 2 , N 2 which is trivially flat. As discussed in appendix D in the presence of backreacting point particles the equations of motion become where for the i-th particle we assumed the decomposition of the momentum (5.10) The constants α (i) j can be expressed in terms of the quantum numbers of the particles from the constraints coming from the Lagrange multipliers, as explained in appendix D.1.
As explained in appendix C there are two systems of associated ODEs (C.11) and (C.14) expressed in terms of holomorphic and antiholomorphic currents U

Properties of the currents
Pole structure: If near the sources z i we neglect the potential term compared to the kinetic term, then the Toda fields φ i behave near z = z i as where the ellipses denote lower order poles and a regular part. The constants The reality condition is imposed by using a doubling trick and extending W (s) (z) to meromorphic functions on the complex plane in a suitable manner. Using the Schwarz reflection principle we find that the appropriate reflection properties are Constraints on the accessory parameters: As a result of (5.14) W (s) now has poles both at the z i and their image pointsz −1 i . Thus W (s) is of the form where in the above equation we assumed for simplicity there is no source located at the origin and as a result there is also no image source at infinity. As in the sl (3)  j . Instead they are determined by solving a monodromy problem, which arises from demanding that the Toda fields are single-valued. Once again not all of the parameters of (5.15) are independent since we have to impose the reflection condition (5.14). For each of the currents W (s) we have in total 4sK real parameters from which 4(s − 1)K are accessory parameters, where K denotes the number of insertions within the unit disk. Substituting (5.15) in (5.14) and requiring the equality of poles at each order near the z i we get 2sK real conditions. 2K out of them are e (s) Thus the parametersẽ j . So we are left with 4(s − 1)K accessory parameters and 2(s − 1)K real conditions on them which gives 2(s − 1)K parameters left. Substituting these real conditions in (5.14) and requiring regularity at the origin we get another 2s − 1 additional constraints. Thus from each W (s) (z) we get parameters. Summing up s from 2 to N we get that from all the currents W (s) we have in total parameters left that should be fixed by solving the monodromy problem.

Doubling trick for ψ i
To prove the doubling trick for ψ i one needs the associated ODE expressed in terms of the (quasi-)primary currents W (s) . That however is not known for general N and has to be worked out case by case. Thus here we will only sketch the general steps of the proof for arbitrary N , which justify our choice when it comes to the doubling trick of ψ i . Under a conformal transformation z →z = f (z) the fields in the associated ODE for ψ i (z) transform as The transformation of ψ i above was derived from the transformation of φ 1 (C.15) and the fact that ψ i is the holomorphic part of e φ 1 . These transformations should leave the associated ODE conformally invariant. Upon performing a conformal transformation with f (z) = 1/z and using the reflection property (5.14) we expect to get an ODE with the presence of the parameter σ from which one can deduce the following. For the su N 2 , N 2 case we have σ = −1 and thus with X(z) given by (C.39). Then For the sl (N, R) case we have σ = 1 and thus The matrices S andS are proportional to the identity and are respectively given by (C.54) and (C.62).

Monodromies
The monodromies work out similarly to the SL (3, C) case. Thus for this section we mostly state the results as the proof is a direct generalization of what was described in section 3.6.
In general after encircling a singular point z i the solution transforms as where M i ∈ SL(N, C) is a monodromy matrix. However requiring the Toda fields to be single valued constrains the monodromy matrix. Specifically for real Toda fields, since we have Requiring the field to be single valued and by using the definition of the adjugate matrix we get which implies that M ∈ SL (N, R).
Using the reflection property of Ψ, (5.22) resp. (5.21) one can show that, provided the monodromy matrices belong to SL (N, R) resp. SU( N 2 , N 2 ), the monodromy matrix of a contour encircling both a point and its image is the identity matrix. In a similar manner one can show that the monodromy matrix of a contour encircling all points within the unit disk is an element of SL (N, R) resp. SU( N 2 , N 2 ).
Parameter counting and monodromy reduction Similarly to section 3.6.4 we will show that the condition the monodromy matrices take values in SL(N, R), respectively SU( N 2 , N 2 ) imposes in general precisely as many constraints as the number of available undetermined accessory parameters in the ODE. Once again we first compute the dimension of the space in which the monodromy matrices around the K singular points in the unit disk take values and subtract from this the dimension of the space in which they take values when the SL (N, R), resp. SU( N 2 , N 2 ) condition is imposed. In general the monodromy matrix M i of a single point is an element of SL (N, C). Thus for the monodromy matrices of all K points we have in total 2K(N 2 − 1) real parameters. However not all of these parameters are independent. The conjugacy class of each matrix M i is fixed due to the relation of the class functions tr M k i (for k = 1, . . . , N − 1) to the higher spin charges of the particle. The latter subtracts 2K(N − 1) parameters. Furthermore the monodromy matrix, when encircling all singular points in the unit disk, must take values in one of the real forms SL(N, R), resp. SU( N 2 , N 2 ), thus subtracting N 2 − 1 parameters. The last constraint comes from the conjugation of all monodromies by a constant matrix, in SL(N, R), resp. SU( N 2 , N 2 ), after making a change of basis in the space of solutions of the ODE. Thus we should subtract another N 2 − 1 parameters. This leads to 2KN (N − 1) − 2(N 2 − 1) independent parameters.
Next we compute the dimension of the space of monodromy matrices after imposing the SL(N, R), resp. SU( N 2 , N 2 ) conditions. These real forms have dimentions N 2 − 1, leading to K(N 2 − 1) parameters. Again the conjugacy class is fixed, but now the invariants tr M k i are automatically real, therefore subtracting K(N − 1) parameters. The reality constraint on the monodromy when encircling all points in the unit disk is now automatically satisfied, and the overall conjugation by a constant matrix in SL(N, R), resp. SU( N 2 , N 2 ) subtracts N 2 − 1 parameters. This leads to a dimension of KN (N − 1) − (N 2 − 1).
Computing the difference of these two dimensions leads to the number of KN (N − 1) − (N 2 − 1) constraints we need to impose to reduce the monodromy to SL(N, R), resp. SU( N 2 , N 2 ), and matches precisely the number of undetermined accessory parameters at our disposal. Thus, generically, imposing the single-valuedness of the Toda fields will precisely fix all the accessory parameters and guarantees that for generic values of the particle positions and quantum numbers, our equations have a unique solution.
Relation to classical W N blocks Our arguments in section 4 for the W 3 case can be generalized in a straightforward manner to show that the monodromy problem determining a K-centered solution in the spin-N theory is equivalent to the one determining a classical vacuum 2K-point W N block in a "mirror" channel where the operators in image points are fused together in pairs.

Outlook
To conclude, we list some open problems and possible generalizations: • In this work we argued for a connection between backreacted multi-centered solutions and W N blocks from studying the bulk equations of motion. It would also be of interest to evaluate the (regularized) bulk action on these solutions, which one would expect, based on results in the heavy-light approximation [7], to compute the W N vacuum block B 0 itself. Such an analysis would extend the holographic computation of Virasoro and W N blocks beyond the heavy-light approximation.
• We restricted our attention to the description of a rather special class of particles localized in the bulk, which source only the left-moving Chern-Simons connection. It would be of obvious interest to generalize this to the case where both left-and right-moving connections are excited and make contact with other approaches to bulk localization [17], [37].
• As we remarked before in section 2, the configurations we studied in this work are particular to the Lorentzian bulk theory, because the geodesics on which our particles move do not analytically continue to geodesics in Euclidean AdS. In contrast, geodesics in the Euclidean AdS begin and end on the boundary, and are described by localized excitations on the boundary. It seems likely that this Euclidean setup has a simple description in terms of the free field variables on the boundary introduced in [38].
• In our multi-centered solutions, we restricted the individual centers to be particles rather than black holes. It would be desirable to generalize our approach to include black hole centers. It would also be worthwhile to construct solutions where the individual centers are particles but the monodromy when encircling all centers is that of a black hole. These would be toy models for black hole microstates in which questions about information loss could be addressed [39].
• The real forms appearing in our discussion are SL(N, R) and SU( N 2 , N 2 ). However, one might wonder whether more general SU(p, q)'s are physically relevant. There is seemingly only one restriction given by the existence of SL(2, R) embeddeding in the SU(p, q) which corresponds to the gravitational subsector. A full answer would requre a thorough study of reality conditions of Toda fields and appropriate SL(2, R) embeddings. Secondly, an investigation of embeddings other than the principal one might also be of interest. also like to thank the Yukawa Institute for Theoretical Physics at Kyoto University for hospitality during the workshop YITP-T-18-04 "New Frontiers in String Theory 2018" while this work was in progress.

A Explicit representation matrices
For general N , we can take the explicit representation of [40] with all odd spin generators multiplied by √ σ. Concretely, for N = 2: For N = 3:

B Chevalley basis for A N −1
Here we write the matrix basis we used to write down the form of the Toda connection for general N , (5.3). The N × N matrices can be written down in terms of Kronecker's delta where i = 1, . . . , N − 1. Specifically for N = 3 we have

C Aspects of Toda theory
In this appendix we review general aspects of the A N −1 Toda theory that we use in the main text. For the case of real Toda fields, this is material known in the literature and can be found in various sources including [22,41,42]. Here we will start by assuming the Toda fields to be complex and we will later specify two different reality conditions on them. The system of Toda equations consists of N − 1 fields φ i satisfying ∂∂φ 1 + e −2φ 1 +φ 2 = 0 , ∂∂φ 2 + e −2φ 2 +φ 1 +φ 3 = 0 , . . .
We observe that the above system has a Z 2 symmetry under exchanging φ i ←→ φ N −i . Also if we know one of the Toda fields, let's say e φ 1 , we can find the rest by substituting in the Toda equations and solving them iteratively. The field e φ 1 can be found by solving an N -th order associated ODE and the related monodromy problem.

C.1 The associated ODE
Two expressions for the N -th order ODE are where we have assumed U (1) = N i=1 J i = 0. The relation between J i and U (i) can be seen by expanding the above equation. Specifically for the A 1 Toda (Liouville) we have where J 1 = −J 2 ≡ J and T is the Liouville stress tensor. For the case of the A 2 Toda we have (C.10) The currents V (s) can be taken from the currents U (s) simply by replacing ∂ with∂. Then we have two ODEs, one for the holomorphic and one for antiholomorphic currents, which are both satisfied by e φ 1 Because of the Z 2 symmetry of the Toda system we can also make the identification Then we get a second set of holomorphic and anti-holomorphic currents (C.14) In other words we can get (C.14) from (C.11) be exchanging φ i ↔ φ N −i .

C.2 Primary currents
One can check that under a conformal mapping z → f (z) the Toda system (C.1) is invariant provided the fields φ i transform as The constants q i are the proportionality constants φ i = q i φ grav that appear after restricting to the gravity subsector of the theory as From this we immediately deduce q 1 = q N −1 = N − 1. For example for N = 2 we have q = 1 and for N = 3 we have q 1 = q 2 = 2. Out of the currents U For s = 2 the current transforms as a quasi-primary where β N is a constant and is the Schwarzian derivative. Specifically for the sl(2) case we have β 2 = 1 2 while for sl (3) Thus for the case of sl(3) we observe that 2 .
(C. 23) In general for the primary currents we have the relation that transform as primaries under transformation (C.15). Now we have already assumed (C.25) and the sub-index b simply denotes the primary currents constructed out of V (s) instead of U (s).

C.3 Properties of the solutions
As we alluded in the beginning of this section to fully solve the Toda system we only need to know one of the Toda fields which can be found by satisfying an associated ODE. For the field φ 1 we showed that the holomorphic and antiholomorphic ODEs are given by (C.11). Thus we can write where ψ i andψ i are independent holomorphic and anti-holomorphic solutions of (C.11) Similarly for the field φ N −1 we have where χ i andχ i are independent holomorphic and anti-holomorphic solutions of (C.14) To further analyze the solutions we will need to impose a reality condition on the Toda fields. Here we will consider two different reality conditions. In the first case we will take the Toda fields to be real. In the second case we will take the Toda fields to be complex conjugate of each other such that φ i =φ N −i .

C.3.1 Real Toda fields
For convenience we arrange the solutions into a column vector Ψ = (ψ 1 , . . . ψ N ) T . When the Toda fields are real we observe that ThusΨ(z) andΨ solve the same ODE. Then in general,ψ i will be linear combinations of where Λ ∈ GL (N, C) is a constant matrix such that Taking the conjugate transpose of the above and demanding e φ 1 to be real we get the condition Λ † = Λ . (C. 35) As we mentioned before by knowing the solution for e φ 1 and by substituting in the Toda equations we can iteratively obtain all the e φ i with i = 1, . . . , N − 1. To perform this procedure we actually need only N −2 from the total of N −1 Toda equations. Substituting all the e φ i in the last Toda equation we obtain the condition where W ψ is the Wronskian. Also because U (1) =Ū (1) = 0, since i J i = 0, we have that the Wronskian is constant and we can set it equal to one. Therefore (C.36) becomes To fully specify Λ we will consider a specific solution of (C.27) which we will describe in the next subsection. Similarly from the ODEs (C.14) we have Starting from e φ 1 and by iteratively solving the Toda equations we deduce where N 1,2 ∈ GL (N, C) are constant matrices such that Since we have φ 1 =φ N −1 we deduce that N † 1 = N 2 . By substituting in the Toda equations we find that Thus we can write

C.4 A simple solution
A simple solution to the associated ODE can be found if we set all the currents to zero, U we find the proportionality matrix S to be is chosen again such that the Wronskian is equal to one. By using (C.56) together with (C.46) we find for N being odd while for N being even The solution specifies again the zeroth order term in the expansion of φ 1 which is now complex for N being odd where n is an integer. For the reflection property of ψ i we find the proportionality matrix to bẽ

D Coupling higher spin particles via Wilson lines
The dynamics of massive point particles coupled to gravity in three dimensions is captured, in the Chern-Simons formulation, by Wilson lines. This connection was initially established for the case of asymptotically flat gravity [14,43]. In the context of entanglement entropy it has also been extended to AdS 3 both for higher spin theories [9,10] and for spinning particles [26]. Here we review this construction in for the situation relevant to us, namely that we consider the Lorenzian higher spin theories with independent gauge fields A,Ā taking values in sl(N, R) or su( N 2 , N 2 ), and that we restrict attention to 'chiral' pointparticle sources which couple only to A.

D.1 Wilson Lines
We start from the higher spin theory described by two Chern-Simons fields A,Ā taking values in sl(N, R) or su( N 2 , N 2 ). The most general Wilson line we can add to this theory is of the form W R (C)WR(C), where W R (C) = tr R P exp (D.1) It depends both on the choice of the curve C and on two representations R,R of the gauge group. It is natural to expect that, if we take R andR to be unitary irreducible representations, the Wilson line describes a coupling of a point particle to the higher spin theory, with R andR carrying the information about the physical properties point particle such as mass, spin and higher spin charges. This interpretation was made precise in [9], see also [26], [44]. Restricting attention to representations where the energy P 0 = L 0 −L 0 is bounded below, we are led to the infinite-dimensional, lowest-weight representations. These are built on a lowest weight or primary state |h, w = |h, w 3 , ..., w N satisfying −j . The physical properties of the particle are encoded in the 2 × (N − 1) independent Casimir invariants of the representation R andR. These can be expressed in terms of the primary weights (h, w) (and their right-moving cousins) as we will work out below for N = 2, 3. In this work we are be interested in the subclass of particles wherē is the singlet representation; so from now on we will deal only with the left-moving Wilson line W R (C). A useful representation of the Wilson line is obtained by interpreting R as the Hilbert space of an auxiliary quantum mechanical system that lives on the Wilson line. The auxiliary quantum system is described by a field U taking values in the gauge group (SL(N, R) or SU( N 2 , N 2 )) and it's conjugate momentum P taking values in the Lie algebra. The dynamics of U, P is picked so that upon quantisation the Hilbert space of the system will be the representation R. Then the trace over R is replaced by a path integral Here A s denotes the pullback of A to the world-line C and P is a canonically conjugate momentum to U and takes values in the Lie algebra sl(N, R) resp. su( N 2 , N 2 ). We note that the λ (i) are Lagrange multipliers which fix the trace invariants of P in terms of the Casimir invariants c (i) R of the representation R. We refer to [44] for more details on the equivalence between (D.1) and (D.5).
The action (D.5) is invariant under the gauge symmetry A → Λ(A + d)Λ −1 under which the worldline fields transform as where in this formula Λ = Λ(x µ (s)) is pulled back to the worldline. In the above action we take the trace "tr" to be normalized as in the N -dimensional representation. This defines the Killing forms where P = P a T a . We will be interested in the regime of the parameters (h, w) where the path integral (D.4) is well approximated by its saddle point value; in this regime we have to find solutions the equations following from the total action (D.14) The equation from varying U is ∂ s P + P, ∂ s U U −1 = 0 , (D. 15) and from varying the momentum we obtain 1 2 D s U U −1 + 2λ (2) P + 3λ ( Let us now simplify the system equations (D.12,D.15,D. 16,D.18) for the situation at hand. As we argued and showed explicitly for N = 2 and N = 3, we can choose a coordinate system (t, z,z) and a suitable gauge such that A is a Lax connection for the A N −1 Toda system. In particular, A is of the form A z dz + Azdz and the worldline C the particle moves on has constant z = z 0 . In this case we can choose the worldline coordinate s such that t(s) = s and we observe that The remaining equation (D.12) is then 10 where the constant Lie algebra element P is constrained to satisfy (D.18). Since A is a Lax connection for the A N −1 Toda system, the above equations reduce to the Toda equations with delta-function sources. In the following two paragraphs we will work the precise coefficients in front of the delta-functions for N = 2 and N = 3.

D.3 Spin 2 case
In this case we have (see (2.17)) where V is constant element defined in (2.14). Plugging the latter into (D.20) shows that V −1P V should be proportional to L 0 : The constant α is determined by the Casimir constraint (D.18). From (D.9) the quadratic Casimir takes the value c Using (D.13) and generalizing to several point-particle sources leads to ∂∂φ + e −2φ = 4πG which is the equation found in [12] using the metric formulation upon identifying m i = h i .

D.4 Spin 3 case
In this case we have from (3.9)