Unravelling Holographic Entanglement Entropy in Higher Spin Theories

There are two proposals that compute holographic entanglement entropy in AdS$_3$ higher spin theories based on $SL(N,\mathbb{R})$ Chern-Simons theory. We show explicitly that these two proposals are equivalent. We also designed two methods that solve systematically the equations for arbitrary $N$. For finite charge backgrounds in AdS$_3$, we find exact agreement between our expressions and the short interval correction of the entanglement entropy for an excited state in a CFT$_2$.


Introduction
Entanglement entropy is an interesting probe in a holographic setup: it is a boundary observable that directly probes the local geometrical data of the bulk gravitational theory.Holographic entanglement entropy might describe how geometry emerges in a classical theory of gravity from a quantum theory.
There are, however, many gravitational theories where the notion of local geometrical quantities is rather unnatural.Higher spin theories, as originally formulated by Vasiliev, are such a class of gravitational theories with non-local interactions among a generically infinite tower of fields.The enlarged gauge redundancies of the fields act nontrivially and unfamiliarly on the metric; the usual Riemannian definitions fall short for these theories.Nevertheless, these theories do have a seemingly healthy dual description in terms of a CFT.In this case, entropy (either thermal or from entanglement) in the dual theory will provide guidance: the object that captures holographic entanglement entropy in higher spin theories can give a generalized definition of geometry.One of the goals of this program is to quantify this new definition and its possible repercussions.
Progress has been made towards this direction.For a simple class of higher spin theories based on three dimensional SL(N, R) Chern-Simons theory, there are two proposals for what is the appropriate object that captures entanglement entropy [1,2].Both of these proposals consider a Wilson line as the correct object that replaces the notion of geodesic length.In [2] one of the key ingredients was to search for a composite line operator that was invariant under local Lorentz transformations.In [1] the goal was to design a Wilson line that captured the dynamics of a massive particle.The details of each proposals are rather different, but there was some evidence that both proposal were equivalent since the reported results for SL(3) higher spin gravity are the same.Here we will prove that they are the same.
In order to construct our proof, the first step will be to generalize the proposal of [1] to SL(N, R) Chern-Simons. 1 This is done in Section 3. We will give an explicit construction of the Wilson line, and in this process we will discuss how to evaluate the saddle point value of the operator.These methods depend on the representation used for the background connections.It turns out, that the composite operator constructed in [2] is a clever way to get the final answer.The authors in [2] were only guided by symmetries and consistency conditions, which shows that, in this particular case, simple physical requirements on the observable are enough to capture the dynamics.
In Section 4, we develop as well two methods to explicitly evaluate the Wilson line as a function of the background SL(N, R) fields.The first method uses the fundamental representation of the algebra.It has the advantage that it gives an exact answer for any range of the parameters, but it is somewhat tedious to extract certain features from the answer.The second method relies on a small interval expansion of the composite Wilson line defined in [2].This small interval expansion captures first correction to the relative entropy of a pure state with respect to the vacuum.The result is well known to be universal in a CFT 2 [3], and our results match up perfectly with this universality for any value of N .We end our discussion with some open questions in Section 5.

The shortest introduction to AdS 3 higher spin gravity
The simplest way to craft a higher spin theory follows from the famous observation that three dimensional Einstein gravity with a negative cosmological constant can be reformulated as a SL(2, R) × SL(2, R) Chern-Simons theory [4,5].By simply taking instead the gauge group to be SL(N, R) × SL(N, R), we will produce a non-trivial theory for symmetric tensors of spin s = 2, 3, . . ., N [6].
The action of the SL(N, R) × SL(N, R) Chern-Simons theory is given by Here M is the 3-manifold that supports the sl(N, R) algebra valued connections A and Ā, and the trace 'tr' denotes the invariant quadratic form of the Lie algebra as defined in appendix A. The equations of motion following from (2.1) are 2) The metric and higher spin fields are obtained from the Chern-Simons connection as symmetric, traceless tensors that transform in the spin s representation of SL(2, R).For example, the metric and the spin three field can be expressed as follows g µν ∼ tr e µ e ν , φ µνρ ∼ tr e (µ e ν e ρ) , (2.3) where, in line with the pure gravity case, one defines and we introduced the AdS radius ℓ.The metric and higher spin fields can then be expressed in terms of trace invariants of the vielbein [7,8], with the total number of inequivalent invariants being N − 1 for sl(N, R).This definition for metric-like fields is appropriate for the principal embedding of sl(2, R) in sl(N, R). 2The relation between the Chern-Simons level and the gravitational couplings is in accordance with the pure gravity limit.The notation tr f denotes a trace in the fundamental representation of sl(N, R), and L 0 is given in (A.8).The central charge of the asymptotic symmetry group is [9,7] For the immediate purpose of this work this is all we need to know about AdS 3 higher spin gravity.For a more in complete discussion see for example [10,11,12].

Wilson lines in SL(N, R) Chern-Simons
There are currently two seemingly different proposals to compute holographic entanglement entropy in AdS 3 higher spin theories.The proposal of the authors in [1] states that entanglement of the dual theory is captured by a massive Wilson line where W R (C) is a bulk Wilson line defined as: Here A and Ā are the connections representing a higher spin background in SL(N, R) Chern-Simons theory.The representation R is the infinite-dimensional highest-weight representation of sl(N, R), and C is a curve in the bulk that connects the end points of the interval of width ∆x in the boundary.
The other proposal in the market to compute entanglement entropy is given by the following object [2]: where the quantity inside the logarithm is a composite Wilson line defined as where C is the same curve as in (3.2).The trace is taken in a finite-dimensional representation, denoted R N , which is different for every N .It was noted in [1,2] that for explicit backgrounds in SL(3, R) both (3.1) and (3.3) reported the same answer.However, the proposals look very different!In this section we will show that they are equivalent for an open boundary interval.To do so, we will first generalise the proposal of [1] to SL(N, R).In the process of finding an efficient and systematic way to evaluate (3.2), we will show how the composite Wilson line (3.4) makes its appearance.
In this section we will only focus on higher spin theories based on the principal embedding of SL(2, R) in SL(N, R).See appendix D for the generalization to non-principal embeddings.

Wilson line and massive particles
As anticipated, we would like the Wilson line (3.2) to give information about the entanglement entropy of an open interval ∆x in the CFT.In 3d Einstein gravity, the calculation of the entanglement entropy is equivalent to computing the length of a geodesic connecting the endpoints of ∆x [13,14,15].A geodesic can be understood as the trajectory followed by a massive point particle.Our Wilson line should mimic the dynamics of this massive particle, and hence as a minimal requirement it should be able to carry the data of this particle.
A point particle in the classical limit is characterized by at least one continuous parameter: the mass m.This data is stored in the representation R that defines the Wilson line.An infinitedimensional representation of sl(N, R) ⊕ sl(N, R) will do the trick: it allows for continuous parameters which we can identify with a mass. 3In particular, we will work with the so-called highest-weight representation.Consider the sl(N, R) algebra in (A.1), and we define the highest-weight state of the representation as |hw ≡ |h, w 3 , ..., w N with the following properties: j |hw = 0 , j = 1, . . ., s − 1 . (3.5) The constants h and w s with s = 3, . . ., N are the parameters defining the representation.|hw is annihilated by the lowering operators; a descendant state is created by acting with the raising operators: W −j and L −1 .With this, the Wilson line in the infinite-dimensional highest-weight representation of sl(N, R) × sl(N, R) is labelled by two towers of quantum numbers: (h, w s ) and ( h, ws ).In particular the mass m and orbital spin ŝ are given by For the purpose of computing entanglement entropy we would like for the representation to only carry mass and no other quantum numbers.Hence we will set h = h , w s = ws = 0 , ∀s .
We have to fix as well the value of m in order to make contact with entanglement; this will be done in section 3.3.Of course this choice of representation can be modified, but this will change the interpretation of the Wilson line in terms of the dual theory.For instance one could designed probes that carry higher spin charge or orbital spin; the interpretation of this object in the dual CFT interpretation will be different, but still rather interesting.See [16] for the case when w 3 = w3 = 0 in SL(3, R) higher spin theory, and see [17] for a discussion when ŝ = 0.

Path integral representation of the Wilson line
The more complex step is to actually evaluate the trace in (3.1).Following [1], we will interpret R as the Hilbert space of an auxiliary quantum mechanical system that lives on the Wilson line, and replace the trace over R by a path integral.This auxiliary system is described by some field U , and we will pick the dynamics of U so that upon quantization the Hilbert space of the system will be precisely the desired representation R.More concretely, where the action S(U, A, Ā) C will have SL(N, R) × SL(N, R) a local symmetry.The auxiliary system is appropriately described for the following action: (3.9)Here P is the canonical momentum conjugate to U that lives in the Lie algebra sl(N, R).The variable y parametrizes the curve C, and we pick y ∈ [y i , y f ].The trace Tr(...) is a short-cut notation for the contraction using the Killing forms in (A.12): where P = P a T a and T a is a generator of sl(N, R).The functions λ m (y) represent Lagrange multipliers which enforce constraints on P .The elements c m are the Casimir invariants C m (A.13) applied to the highest weight state, and contain the information of the highest-weight quantum numbers h and w s .Note that in this action we already implemented that h = h and w s = ws , since there is only one momenta variable P .This will suffice for the discussion here, but the generalization is worthwhile studying [17].
The covariant derivative is defined as where A and Ā are the connections that determine the background.With these definitions we have achieved our first goal: the system is invariant under the local symmetries along the curve.The transformation properties of the fields are and U (s) → L(x µ (y))U (s)R(x µ (y)) , P (y) → R −1 (x µ (y))P (y)R(x µ (y)) , with L and R being element of the group SL(N, R).
The equations of motion are: For an open curve C we need to choose boundary conditions for U (y) at the endpoints of the curve.In the pure gravity case, it is natural to ask that the answer is invariant under Lorentz transformations (since the geodesic length shares this property).In SL(2, R) × SL(2, R), the group elements R and L that parametrize the local Lorentz subgroup is: (3.16) A natural condition is to impose that U (y i ) and U (y f ) are invariant under a gauge transformation of the form (3.16); this will assure that S EE is insensitive to Lorentz transformations.From (3.13), we see that the only boundary conditions that satisfy this condition are: (3.17) For higher spin gravity, the symmetry group is SL(N, R) × SL(N, R) and we cannot say that the Lorentz subgroup is described by (3.16); the condition (3.16) is much bigger in this case!Still, we will impose (3.17) in the higher spin case since it is the natural generalization of the gravitational case.

On-shell action
In this subsection we will find a practical way to compute the action (3.9) for any background connection.The derivations will be applicable for both open and closed curves, and we will keep w s = 0 in this subsection.
To evaluate (3.9), we start by eliminating the dependence of U using equation (3.14): where in the last line we used the Casimirs constraints to eliminate P .Recall that the curve C is running from y ∈ [y i , y f ].It will be useful for us to define: and with this simplified notation, the action becomes: We need to determine ∆α m as a function of the connections A and Ā; we will follow the method used in [1].We start by building a solution when A = Ā = 0: this defines for us U 0 (y) and P 0 (y) which from (3.14) read U 0 (y) = u 0 e −2 α 2 (y)P 0 −3 α 3 (y)P 0 ×P 0 +...−N α N (y)P 0 ×...×P 0 , P 0 (y) = P 0 , where u 0 is a constant matrix, and α m (y) is defined in (3.19).From here, building a solution with A = 0 and Ā = 0 is rather simple.As consequence of the flatness condition (2.2), every connection can be expressed locally as a gauge transformation where the group elelemnts L and R will reproduce different background connections.This means that we can build any solution to (3.14) for connections (3.22) by simply acting with L and R on (3.21).This gives Next, we impose the boundary condition (3.17); enforcing this condition on (3.23) gives If we combine both previous equations to eliminate u 0 we obtain where we define and for a general representation with (h, w s ) = ( h, ws ) we would have At this stage it useful to note that M , as defined in (3.25), can also be written as and hence W comp R N (C) = Tr R N M .However, (3.27) is still not casted in the appropriate way to show that (3.1) is equivalent to (3.3).

Lines: Entanglement Entropy
As advertised, we are interested in using W R (C) to evaluate entanglement entropy.With this application in mind, we will focus our attention to open intervals that are anchored at the boundary.Furthermore, as argued in [1], we have to choose the massive representation (3.7).This implies that P 0 ∈ sl(2, R).Without loss of generality, it is convenient to set With this choice several of simplifications occur.In particular, the Casimirs c m = 0 for m ≥ 3 due to our choice of Killing forms in (A.14), and hence he on-shell action (3.20) reduces to To solve for ∆α 2 we just need to decode P in (3.26).Since (3.25) and (3.32) are independent of the representation, for simplicity, we will first focus in the fundamental representation.Using (3.31) and the identities listed in appendix A, we can rewrite P as where all the sums of the series belong to the Cartan subalgebra, and we defined Since the Cartan elements are diagonal in the fundamental representation, P is a diagonal matrix.
In order to solve (3.25), we should put both sides in the same basis.So, we will diagonalize M : where λ P and λ M , are the eigenvalue matrices for P and M .In principle one could evaluate the eigenvalues and match both sides, and solve for ∆α 2 .Instead, noting that tr and hence, in the saddle point approximation, we have This result assumes an ordering of the eigenvalues of M ; we will discuss about the implications of the ordering in the following subsection.
Our goal is to compute entanglement entropy and, for that to be the case, the massive particle described by (3.31) needs to implement the correct type of singularity in the background solution [18].This requirement determines uniquely c 2 , and this can be done by analyzing the backreaction of W R on (A, Ā).We will skip the details here since it follows in a straight forward manner from either the arguments in [1] or [16] applied for SL(N, R) theory.We find with n being the number of replicas that define Renyi entropies, and k is the Chern-Simons level.Holographic entanglement entropy is then given by Operationally, may then simply write the entanglement entropy as and substitute in the final answer.This automatically takes care of the n-dependence, but it should be kept in mind that the motivation is actually the reasoning in (3.39).

Primary ordering
In evaluating (3.37) there is an implicit choice of the ordering for the eigenvalues of M .In this subsection we want to make this choice explicit.P 0 was fixed according to (3.31), but this information is washed away in (3.37).The problem arises because potentially there are many choices of P 0 that give (3.36) as a valid solution of (3.25).Actually, the different orders of λ M correspond to different momenta configurations with c 2 = 0 and c m = 0, but not necessarily compatible with (3.31).Since we need to assure that the Wilson line does no carry higher spin charges, we will fix the order of λ M which is compatible with P 0 ∈ sl(2, R).
To fix the ordering, lets study first (3.35)when the background is in the gravitational sector: A, Ā ∈ sl(2, R).In this case we have that M ∈ SL(2, R).Any diagonal matrix belonging to SL(2, R) must be conjugated to e L 0 , therefore the eigenvalues of M have the form: , z in the fundamental representation of SL(N, R)(see appendix A), and z is a function of the parameters of the background connections.Since M ∈ SL(2, R), from (3.25) we have that P ∈ sl(2, R).But this is not enough: we need as well that P 0 ∈ sl(2, R), and from (3.33) and (3.25) it sets ∆ αm = 0 for m > 2. The eigenvalues of exp(P) are then Now lets compare (3.43) with (3.42).Since these equations are invariant under ∆ α2 → −∆ α2 , and z → z −1 , there are two possible orders to match the eigenvalues: e ∆ α2 = z ±1 .By construction, one of the orders gives ∆ α2 positive, and the other negative.Since S EE ∼ ∆ α2 and the entropy must be positive, we will pick the order in which ∆ α2 > 0. This determines the ordering uniquely in the SL(2, R) limit.When we turn on higher spin vevs in the background connections, the eigenvalues of M will change giving raise to non zero ∆ αm in P.However, the matching of the eigenvalues will be determined by continuity with the SL(2, R) limit.Provided a solution to ∆ αm was found from (3.35), this solution must satisfy lim This determines the ordering of eigenvalues in λ M which is compatible with P 0 ∈ sl(2, R), and it will be referred to as primary order.This prescription will assure that the Wilson line indeed carries the quantum numbers (3.7).

The proof
Finding the eigenvalues λ M is a tedious task.However, we just need the leading divergent pieces as the endpoints of C asymptote to the boundary.The goal of this section is to find a different way to solve for ∆α 2 when C ends on an open interval at the boundary. 5In this process we will be able to prove that (3.1) is equivalent to (3.3).Our task will be divided in two steps: 1. To first understand the divergent properties of M as C approaches the boundary.
2. Find a representation that easily projects out ∆α 2 from M .
To achieve our first task, we need to spell out more what it is assumed about the background connections A and Ā.We have that Since we are only interested in traces of M , i.e. its eigenvalues, we will conjugate M by R(y i ) to get We are interested in connections (3.22) of the form where b(ρ) ≡ exp(ρL 0 ), a = a t dt + a x dx and ā = āt dt + āx dx.The limit ρ → ∞ defines the boundary of the space.All the connections will as well satisfy which means that the backgrounds are asymptotically AdS 3 in accordance with e.g.[19,20].This guarantees that all backgrounds have a well defined SL(2, R) × SL(2, R) limit.Our curve satisfies boundary conditions and we take ρ 0 → ∞.From (3.45) and (3.47) we have where ∆a x is the integral of a x with boundary conditions (3.49), and an analogous definition for ∆ā x .It is rather clear here that divergent piece as ρ 0 → ∞ is governed by L 0 .If we solve for (3.35) while enforcing the primary ordering of eigenvalues, we find that the leading order solution in e ρ 0 is e ∆ α2 ∼ e 4ρ 0 , and e ∆ αm ∼ 1, m > 2 . (3.51) In this expression we are highlighting the divergent piece: the symbol "∼" denotes equality up to a non-zero function of (∆a x , ∆ā x ).The derivation of (3.51) is presented in appendix C. Now we move on to our second task.We consider again equation (3.35) but this time we will take its trace: Tr R (e P ) = Tr R (M ) , where R is a representation that we have not fixed yet.Actually, the goal is to find a representation for which it is easy to read off ∆ α2 from equation (3.52).In a general representation, P is not diagonal anymore.However, since all its elements are Cartans, we can write: where the index j = 1, . . ., dim(R), and the powers (n R ) are the eigenvalues of the Cartan elements in P, which depend on the representation R.
We want to take ρ 0 → ∞ in (3.53); from (3.51) the dominant term is the one with the biggest power of e ∆ α2 .In the following we will find a representation R for which the dominating term has as well null powers of e ∆ αm for m > 2. For this representation, the limit ρ 0 → ∞ of Tr R (M ) will depend only on ∆ α2 .
In order to find the suitable representation we need a way to generally characterize the powers in (3.53).Since P is an element of the Cartan subalgebra h, we define p as its dual element in the root space h * .From equation (3.33), the explicit form of p: where l 0 , h s are the dual elements of the Cartans L 0 , H s .We can write each diagonal element of P for a general representation R using its defining weights where ..., ... is the inner product on the root space h * , defined in Appendix B. Using this notation, (3.53) reads Tr R (e P ) = where the sum runs for all the weights − → Λ (j) R of the representation R. As ρ 0 → ∞, the leading term is the one with the weight that maximizes l 0 , − → Λ (j) R .
In the principal embedding l 0 is a dominant weight, which means that l 0 has positive and integer Dynkyn labels.Moreover, where α i are the simple roots of the algebra.There exists as well a unique highest weight, − → Λ hw R , which has the biggest coefficients in the basis of the simple roots, i.e., all other weights are calculated subtracting simple roots to the highest weight: where n (j) i are positive integers that can be found for every weight R .With (3.57) and (3.58), we see that the maximum value of the inner product l 0 , − → Λ Now, we are ready to pick a representation that extracts just the information of the parameter ∆ α2 .It is obvious that we have to choose a representation R N whose highest weight is where we have used l 0 , l 0 = tr f (L 0 L 0 ).Using equations (3.32) and (3.34) we finally find The dimension of R N is given by the Weyl formula: With formula (3.61), we find the leading term in e ρ 0 (the UV cutoff) of the entanglement entropy for higher spin theory with less computational effort than using (3.37).The difficult portion is to write M in the representation R N .Furthermore, formula (3.61) allows us to do identify proposals in [1] and [2].As noted in (3.30) we have Moreover, the representation R N is exactly the same that in (3.3).As a consequence, we have proven that formula (3.3) captures the most divergent piece of (3.1), and hence both proposals capture holographic entanglement for AdS 3 higher spin gravity.We consider (3.61) our most important result.

Loops: Thermal entropy
In this subsection, we will show how to find the thermal entropy for a higher spin black hole using a Wilson loop.In this case, we consider periodic boundary conditions where x is the spatial coordinate with periodicity x ∼ x + 2πℓ.In SL(2, R) × SL(2, R), the Wilson loop in the infinite dimensional representation computes the length around the horizon, which is the thermal entropy of the black hole [1].Analogously as we did for the entanglement entropy, we will show that for the representation (3.7), the Wilson loop in SL(N, R) × SL(N, R) will recover the thermal entropy for higher spin black holes in agreement with [21].From Section 3.2.1 we found a general expression for the on-shell value; however this expression simplifies greatly for a closed path.We start by noticing that the auxiliary variables of the Wilson line require U (y f ) = U (y i ) , P (y f ) = P (y i ) .
(3.66) Imposing these periodic conditions for U in (3.23), and we get Using (3.47), we rewrite the previous equation as: Here we are assuming that (a x , āx ) are constant connections.Demanding periodicity in P (y) in equation (3.23) we obtain the following condition: which says that P 0 and āx simultaneously diagonalize, and, therefore, the same do P and āx .If we denote V as the matrix of eigenvectors, and λ x and λ P represent the eigenvalues, equation (3.68) reduces to exp(λ Since the left-hand-side is diagonal, consistency of the previous equation requires to choose u 0 such that u 0 V is the matrix which diagonalizes a x , and the right-hand-side of (3.70) is diagonal as well.With this choice: exp(λ P ) = e 2πℓ(λx− λx) . (3.71) Analogously to Section 3.2.1,we use Tr(PP 0 ) = S on-shell to find: To compute thermal entropy we choose again P 0 as (3.31), and use (3.41).In this case, the Wilson line computes gives This is the generalization to SL(N, R) for the thermal entropy found in reference [1] for SL(3, R).
With this result we have reproduced by means of our formalism the thermal entropy for higher spin black hole, proposed originally in [21].If we choose P 0 ∼ W (3) 0 we would reproduce the thermal results for spin-3 entropy defined in [16].

Entanglement entropy on finite charge backgrounds
In this section we will evaluate our Wilson line and obtain S EE for any background connection that satisfies Drinfeld-Sokolov boundary conditions.The backgrounds represent finite charge solutions, and the results in this section are valid for either higher spin black holes [22,19,20] or a conical defect [23].
More explicitly, we will consider connections of the form (3.47)-(3.48),and we will implement these boundary conditions by writing where b(ρ) ≡ exp(ρL 0 ), a = a t dt + a x dx and ā = āt dt + āx dx, and In this decomposition, (a x , āx ) contain the information about the higher spin charges of the solutions, which up to some normalization are (q (s) , q(s) ).In particular the conformal weights are given by h The components (a t , āt ), which are constrained by the equation of motions, contain the conjugate potentials to the charges. 6This decomposition of the connection follows from the discussion in [19,20,24]; note that this is different from the holomorphic decomposition used in [22].
In the following we will develop two different methods explicitly evaluate S EE as function of q (s) and the length of the interval ∆x.The first method will elaborate on solving for the eigenvalues of M and evaluating (3.39).This method gives an exact answer for any range of ∆x, but it is somewhat tedious to extract certain features from the answer.The second method relies on a small interval expansion of (3.61).The main appeal of this limit is that the first correction relative to the vacuum is universal [3] which gives a direct check of the proposals in [1,2] for any N .

Method I
Our goal is to characterize the leading divergent behavior of the eigenvalues of the matrix M in (3.25) so we can evaluate (3.39).Working in the fundamental representation of sl(N, R), the characteristic polynomial is given by The coefficients c i can be written in terms of traces of powers of the matrix M , see e.g.[25].The coefficient c i is the result of the following determinant: where Since we are interested in the behavior of the eigenvalues when ρ 0 → ∞, it will be useful to find the leading order term of the coefficients c i .To do so, we will first determine c i as a function of the eigenvalues λ (j) M .The first coefficient is explicitly: Using (C.6), we see that only the first eigenvalue contributes to the leading order of c 1 : where ε = e −ρ 0 is the UV cutoff.The second coefficient can be written as: M λ (2) whose leading term is: M λ (2) 3) .(4.9) If we keep analyzing (4.5) for different values of i, we will see that the coefficient c i will always be a sum of terms of the type: where each term in the series is a multiplication of i eigenvalues, and the sum runs over all possible combinations of the j, k, l..., which do not repeat an eigenvalue more than once.The symbol ∝ means that both sides are equivalent up to a numerical factor.Using (C.6), the leading divergence in ρ 0 of c i is the term that contains the first i-th eigenvalues: M λ (2) This determines the leading power in ε.Defining m i as the factor which multiplies the leading ε-power in c i , and using (C.2), we have where the dots stand for subleading order terms when ρ 0 → ∞.It is useful to notice that c N = (−1) N det(M ) = (−1) N , and hence m N = (−1) N .In this regime, we can rewrite (4.4) as: We can find the eigenvalues as a function of m i solving the previous equation.We just need to substitute in (4.13) the leading term of one eigenvalue found in (C.6), and we will see that only two terms are dominant in the equation.Not considering the rest of the terms, we can solve for each eigenvalue, and find its expression as a function of m i 's.However, we have actually already found the solution through the reasoning above.From (4.11), we see that we can write an eigenvalue in terms of the coefficients c i , and with (4.12), we write it in terms of m i : where i = 1, ...N , and we have defined c 0 ≡ 1 and m 0 ≡ 1.Now, we are ready to write the entanglement entropy as a function of the factors m i , which are easier to compute than the exact eigenvalues.First, we write (3.39) in the following form: M ) 2(N −1) ... (λ Substituting (4.14) in the previous formula, we obtain the leading term of the entanglement entropy in the UV cutoff: It is important to notice that if we want to compute the entanglement entropy of an specific background using the previous formula when N is even, we will only need to find N/2 coefficients m i .When N is odd we will need (N + 1)/2.This is due to (C.7), from which we infer that c i = c N −(i−1) using (4.11).Consequently, from (4.12), m i = m N −(i−1) .

Example: SL(3) excited states
As an example, we evaluate (4.16) for excited states in SL(3) Chern-Simons theory.The connections are given by and the Casimirs are given by where λ i are the eigenvalues of a x , t (s) j are the traces in (A.10), and the definitions for āx are analogous.In this notation, a higher spin black hole corresponds to a solution with λ i ∈ R (real eigenvalues), while a conical defect is a solution with λ i ∈ iZ 3 (imaginary eigenvalues that exponentiate to the center of SL(3, C)).In this notation the conformal dimensions and spin-3 charge of the background solution (for general N ) are which follow the conventions in [23].
The relevant traces to evaluate the Wilson line are From (4.16), and setting N = 3, we have The values of m 1,2 for a connection of the form (4.17), as a function of its eigenvalues, are where i λ i = 0, and We would like to emphasize that (4.23) is a different function of the background charges for the spin-3 black hole relative to those reported in [1,2].The reason is simple: here we used (4.2),where the spatial (a x , āx ) contain the information about the spin charges [19,20,24], while in [1,2] the holomorphic version of the connection was used [22].What is interesting to note is that our results for entanglement entropy will reproduce the same answer as the holomorphic proposal in section 5 of [2].The holomorphic proposal was designed such that the Wilson line was only influenced by the portion of the connection that contained the charges explicitly (with the weakness that the composite line was not gauge covariant); we achieved the same result using (4.2) with the advantage that gauge covairance is restored.
It is interesting to evaluate the small interval expansion of (4.22).We get This correction to the vacuum entanglement is universal for a CFT 2 and it was computed [3]: the reported result their was which agrees perfectly with (4.25), since (4.20) implies There is another comparison that one could make.There has been progress in evaluating entanglement entropy in CFT 2 with W 3 symmetry at finite spin-3 chemical potential [26,27].Our results for EE are casted as function of the charges of the background, i.e. the eigenvalues of (a x , āx ), and in order to make the comparison, we need to cast the charges as function of the potentials.This was done in [26,27,2] using holomorphic variables, but those results apply here as well.The agreement between the bulk and boundary computation was already noted in [26].

Method II
Our second method starts from (3.61) which reads and from (3.45) we have where we used (3.47) and (4.1).The simplicity in this formula is that we only need to evaluate one trace; the difficulty is that the representation R N can be rather horrible.In any case, our objective is to extract the most divergent piece in ρ 0 and simultaneously make a small interval expansion.
To understand the divergent structure we first consider the vacuum configuration, i.e.
which is simply AdS 3 in Poincare coordinates.Then it is rather simple to show that However this series terminates at some finite power of L ±1 , and the reason being that the matrices e ±L 0 ρ 0 will not give an arbitrarily divergent power of e ρ 0 as we showed in appendix C. The largest power of e ρ 0 is determined by the largest eigenvalue of L 0 which in this case is tr f (L 0 L 0 ); this follows from the definition of R N which sets − → Λ hw R N = l 0 .Furthermore, this implies that L ±1 are nilpotent matrices of degree n + 17 : Therefore, for the vacuum we find lim where we used (4.31) and (4.32).Hence which is the well known universal result for the vacuum entanglement entropy in a CFT 2 .Recall that c is given by (2.6) and ε = e −ρ 0 .8For the general connections of the form (4.2), the logic is rather similar.Since adding background charges does not affect the most leading power of e ρ 0 , but it will affect the coefficient in front of e nρ 0 .If we Taylor expand as in (4.31), schematically we will have e −L 0 ρ 0 e ∆ax e L 0 ρ 0 ∼ 1 + ∆x(e ρ 0 L 1 + e −(s−1)ρ 0 q (s) W (s) We are still interested solely on the terms which grow like e nρ 0 in (4.36).The complication now is that this can be achieved, for example, by having additional n powers of L 1 interlaced with n ′ powers of W (s) −s+1 such that n − (s − 1)n ′ = n.But say we are only interested in the first correction in ∆x away from the vacuum.Then, by inspection of (4.36), we get that the relevant term comes from terms involving L 1 and L −1 solely: where Basically the first correction in ∆x comes from a term in (4.36) that has n + 1 powers of L 1 and one power of L −1 , with the condition that L −1 cannot sit at the edge of the string.It is useful to notice that (4.38) can be rewritten as: (See Appendix A.1 for details).Analogously, for the barred sector we have where which can be as well rewritten as Tn+2 = −(n/6)(n + 2)(n + 1)(L −1 ) n.Using (4.37) and (4.40) in (4.29) we find that9 S EE = k log lim We can compare as well with the universal correction of the entanglement entropy for the vacuum state due to the insertion of a single primary field of weight (h, h) [3].The results perfectly agree.This provides a non-trivial check that our method to compute holographic entanglement entropy.

Discussion
We have explicitly constructed and evaluated a Wilson line in SL(N, R) Chern-Simons theory with the purpose of computing holographic entanglement entropy in higher spin theories.We showed that the two proposals [1,2] are consistent with each other.Furthermore, we checked that our results are in perfect agreement with the universal corrections computed in [3] using CFT 2 techniques.This is a non-trivial test that W R (C) is an observable that can generalize the notion of geometry in this class of theories.Our results, applied to SL(3) higher spin gravity, are as well in agreement with the perturbative results reported in [26,27] for CFT 2 with W 3 symmetry.We would like to end this work with some open questions and future directions: 1. Despite our very general results, our derivations fall short in describing entanglement when infinitely many higher spin fields are present.The simplest example of such a theory would be hs[λ] × hs[λ] Chern-Simons theory.W R (C) should still capture both thermal and entanglement entropy in this case.The obstruction is that both methods developed in Section 4 use heavily finite dimensional representations of the algebra in order to analyze (3.25).Evaluating a Wilson line with gauge group hs[λ] is not impossible, but some tricks might be needed to apply our results to the more general case.
2. It was noticed both in [1,2] that the entanglement entropy on a higher spin black hole violated strong sub-additivity.In both papers, the holomorphic formulation of the black hole was used.
Here we used canonical description of the higher spin black hole, along the lines of [19,20].
What is rather interesting is that for N = 3 our results in (4.23)-(4.22)behave accordingly to the strong subaddivity bounds. 10It is not clear under which conditions holographic entangle-ment entropy should obey strong sub-additivity: higher derivative corrections or deviations from the null energy conditions could violate these inequalities [28,29].It is not obvious how non-local interactions tamper our expectations.Still it would be interesting to study if the decomposition (4.2) would give the desired behavior for S EE .
3. In [16] a new "spin" to the Wilson line was given by adding higher spin charges to the representation R.This is not only a novel definition in the bulk, but a new and rather mysterious observable in the CFT.The discussion presented here easily accommodates for this new observable, with one caveat: what is the generalization of the composite Wilson line (3.4)?W comp R N (C) is designed to only capture entanglement entropy.Perhaps the proof in Section 3.3.2can be adjusted to instead find a composite Wilson line that gives spin-3 entanglement [16].
4. One aspect that has been not studied properly in this context is entanglement entropy for multiple intervals.Homology conditions, and analogous properties of the HRT formula [15] should be tested in this context as well.Understanding the effect of junctions when several Wilson lines are present in the bulk might provide better insight to global properties of these operators and their interpretation in the CFT.
5. It will be rather useful to have further independent derivations of entanglement in a CFT 2 that could corroborate our results.This could be made either by considering the large central charge limit of theories with W N symmetry (along the lines of [30,31,32]), by using modular properties of the CFT 2 , or by exploiting conformal perturbation theory.Some progress has been made in conformal perturbation by [26,27,33].We hope to report on related topics soon [34].
6. Our discussion here is strictly classical.Quantum corrections to entanglement entropy in AdS 3 /CFT 2 have been discussed in [35,36,37,38].It would be interesting to see if the expectation value of the Wilson line has anything to add to this topic.
A Conventions for sl(N, R) algebra We follow the same conventions as in [23].A convenient basis for the sl(N, R) algebra is represented by {L 0 , L ±1 }, the generators in the sl(2, R) subalgebra, and W (s) j , the higher spin generators with j = −(s − 1), ...(s − 1).Their commutation relations are: In this notation, L 0 and W (s) 0 are elements of the Cartan subalgebra, and the rest of generators are raising and lowering operators.These commutation relations represent the principal embedding of sl(N, R).We will often use the notation where H 2 = L 0 .An explicit representation for the other sl(N, R) generators, which is independent of the representation, is as follows: With this definition we have −s+1 = (L −1 ) s−1 . (A.5) We write the fundamental representation of sl(N, R) as follows.The {L 0 , L ±1 } generators for the principal embedding of sl(2, R) are In T n+2 there are n/2 pair of terms of this type (n is always an even number as defined in (4.32)).We can repeat the previous trick for every term T r , where r = 1, . . ., n/2.We first swap L −1 with L 1 and identify T r−1 .In the rest of the terms we exchange L 0 with L 1 , to obtain only elements proportional to (L 1 ) n.We will easily notice that T r will be of the form: To find T n+2 , we need to sum over all T r .Using little bit of algebra, we arrive to: An analogous procedure can be repeated for (4.41) to find:

B Representation theory of simple Lie algebras
All definitions and useful properties of representation theory utilized in this work can be found in general text books (we particularly used [39] and [40]).However, we would like to make a special comment about the normalization chosen for the scalar product in the Lie algebra.We consider a general simple Lie algebra g with dimension g, prepared in the Cartan-Weyl basis: where the indexes run as i, j = 1, ...h, and α = 1, ...g − h.The generators H i are elements of the Cartan subalgebra h, and E α are ladder operators.We can associate a h-dimensional vector α = (α (1) ...α (h) ) to every element E α .These vectors α are called roots, and they belong to the dual space of the Cartan subalgebra, denoted by h * .We can define a scalar product in g through the Killing form.In the basis (B.1), the Killing forms will always follow: where δ ij is a Kronecker delta.Moreover, we can define a bilinear form in h * , denoted by ..., ... , which is directly related to the Killing form in g: We would like to remark that in this work we have used the following convention for the Killing forms in g: C Details on section 3.3.2 In this appendix we provide the details of deriving (3.51) in Section 3.3.2.
We will find what is the growth of ∆ αm with ρ 0 given in (3.51).Following the logic used in Section 3.3.1,we first look at the limit case M ∈ SL(2, R) and make use of the fundamental representation.From (3.45) and (3.47) we have M = e −2L 0 ρ 0 e ∆ax e 2L 0 ρ 0 e −∆āx , (C.1) where for now ∆a x and ∆ā x live in sl(2, R).In the limit ρ 0 → ∞, the eigenvalues M can be approximated by its leading order term in ε ≡ e −ρ 0 : The index j runs from 1 to N .The numbers n j are integers whose value depends only on the b(ρ), and f j are continuous functions of ∆x whose behavior depends on ∆a x and ∆ā x .Considering L 0 in the fundamental representation (A.8), we see from (C.1) that the maximum power of e ρ 0 in M will be 2(N − 1), and then {n j } max = (N − 1)/2.
We know that the eigenvalues of M ∈ SL(2, R) follow (3.42).Consequently, the same relation will hold for their leading terms.Using the freedom to z → z −1 , we pick λ M to retain the most negative power of ε.Then, n 1 will be the highest eigenvalue of L 0 .For simplicity, we redefine f 1 (∆x) ≡ κ −2(N −1) .Then, we find that the leading term of the eigenvalues of M ∈ SL(2, R) follows {λ (j) M } = {(κ ε) −2(N −1) , (κ ε) −2(N −3) , ... , (κ ε) −2(N −(2j−1)) , ... , (κ ε) 2(N −3) , (κ ε) 2(N −1) } .This is the leading divergent behavior of ∆α 2 when the background connections approach the SL(2, R) limit.We now turn on the vevs for the higher spin charges in the connection.M is still of the form (C.1), with the only difference that the elements to ∆a x and ∆ā x belong to sl(N, R), while preserving the boundary condition (3.48).Because we assume continuity with the SL(2, R) limit, when the background has higher spin particles, we can characterize (λ M ) i as follows: 11(λ M ) j ∼ g j (∆x)ε −4n j . (C.5) Since b(ρ) does not change in the presence of higher spin charges, each eigenvalue remains with the same leading power of ε: n j is the same that in (C.2).However, adding new elements to a x and āx does change f j into a different function g j .Consequently, connecting continuously the higher spin charges to (C.3), we can characterize the leading term of the eigenvalues of M as: 1) , g 2 ε −2(N −3) , ... , g j ε −2(N −(2j−1)) , ... , where we have used the following property of the eigenvalues of M : to redefine g j = 1/g N −(j−1) .The previous formula is due to the fact that λ M is conjugated to e H , where H is an element of the Cartan algebra in SL(N, R).By construction of the Cartans, the diagonal elements of e H are inverse of each others.We assume continuity in ∆ αi , and we know that the leading ρ 0 -dependence in the eigenvalues is always the same.Therefore, we infer that when we connect the higher spin elements, ∆ α2 has the same ρ 0 power that in (C.4) while ∆ αi , i > 2 remains ρ 0 -independent: e ∆ α2 ∼ e 4ρ 0 , and e ∆ αi ∼ 1, i > 2 . (C.8) Although the analysis has been done for the fundamental representation, a solution to (3.35) must be independent of the representation.Consequently, for any representation in the principal embedding, the leading term of ∆ αm , m > 2 does not depend on ρ 0 .

D Non-Principal Embedding
The discussion in the main sections focused on the principal embedding of SL(2, R) in SL(N, R).In this appendix we will extend the results of Section 3 to other embeddings.Actually the discussion in subsections 3.1-3.2 is basically embedding independent (any modification is trivial); only portions of subsection 3.3 need to be revisited.The first modification is L 0 : the Cartan element in the sl(2, R) subalgebra is not (A.8) for non-principal embeddings.This fact affects our reasoning in subsection 3.3.1.Equations (3.42)

. 26 )
For a given P 0 , (3.25) determines ∆α m as a function of the background A and Ā. Solving (3.25) is the most difficult task we have ahead of us.To determine the on-shell action we note that Tr(PP 0 ) = S on−shell .Hence using (3.25) we find − log W R (C) = S on−shell = Tr(log(M )P 0 ) .(3.27)This gives a very general expression for the on-shell value of the effective action for both open and closed curves C. The specific choice of P 0 will determine the representation R. For instance if we wanted to evaluate the Wilson line for a representation like (3.7) we would use

2 N
) with ρ the Weyl vector and α > 0 are positive roots.For the principal embedding l 0 = ρ and that the number of positive roots of SL(N, R) is N (N − 1)/2.Consequently the dimension of the representation that calculates the entanglement entropy in (3.61) is dim(R N ) =