Wilson lines and Ishibashi states in AdS3/CFT2

We provide a refined interpretation of a gravitational Wilson line in AdS3 in terms of Ishibashi states in the dual CFT2. Our strategy is to give a method to evaluate the Wilson line that accounts for all the information contained in the representation, and clarify the role of boundary conditions at the endpoints of the line operator. This gives a novel way to explore and reconstruct the local bulk dynamics which we discuss. We also compare our findings with other interpretations of Ishibashi states in AdS3/CFT2.

the approach initiated by [10], where it was argued that a Wilson line in an infinite-dimensional highest-weight representation R under the bulk SL(2, R) × SL(2, R) gauge group could be used to compute boundary theory correlators, i.e.: where we have picked coordinates x µ = (r, y i ) with r an AdS holographic coordinate and y i a CFT coordinate. Here the Wilson line W R ends on the boundary at r → ∞, and Ψ denotes the CFT 2 state dual to a particular configuration of Chern-Simons gauge fields that constitute the gravitational background in the interior.
The representation space R was generated from the Hilbert space of an auxiliary SL(2, R)valued quantum mechanical degree of freedom U (s) that lives on the Wilson line. The quadratic Casimirs of the representation R mapped in the usual manner to the conformal dimensions (h,h) of the dual CFT operator. While this represented progress towards extracting geometric observables from the Chern-Simons formulation of 3d gravity, several issues remained obscure: 1. The relation (1.1) was understood to hold only if a particular boundary condition was used for the auxiliary field U , demanding that it approached the identity element of SL(2, R) at the two endpoints of the Wilson line. While this is perhaps a somewhat natural choice, its precise interpretation in the CFT was not made clear.
2. All previous treatment of the U (s) path integral was performed in a semi-classical limit, i.e. one in which h ≫ 1. At a calculational level this allowed the path integral to be evaluated using its saddle-point; nevertheless this restriction seems somewhat artificial from the point of view of the dual CFT. Is it possible to go away from this limit?
3. How can one obtain other bulk observables from the Chern-Simons formulation, e.g. bulk-tobulk propagators or one-loop determinants for scalar fields on the gravitational background?
In this work we answer these questions by providing a careful and fully quantum mechanical treatment of the Wilson line described above. In particular, we will show that the U (s) worldline degree of freedom originally introduced in [10] can be understood as a particular SL(2, R) rotation of the global part of an Ishibashi state (familiar from boundary CFT). We use this technology to develop a purely algebraic method for computing open-ended Wilson lines, and demonstrate equivalence (in the semi-classical limit) with the path-integral techniques used in [10].
The outline of this paper is as follows. In Sec. 2 we review the path integral representation of W R (x i , x f ) proposed in [10], which will serve as a comparison to our quantum mechanical analysis.
In Sec. 3 we turn to a detailed analysis of the quantum mechanics responsible of the geometrical features in W R (x i , x f ). This motivates the introduction of coherent states which we denote as rotated Ishibashi states. Using these states, we relate W R (x i , x f ) to their inner product; we rederive the path integral formulation by discretizing this inner product; and we show that W R (x i , x f ) is a Green's function on the group manifold SO (2,2). In Sec. 4 we tie the quantum mechanical aspects of W R (x i , x f ) to its geometrical features. We show that W R (x i , x f ) is a Green's function on spacetime created by the Chern-Simons connections (which is a distinct statement from the properties on the group manifold). For global AdS 3 and the BTZ black hole, we show how to build local bulk fields by a suitable decomposition of W R (x i , x f ). This provides a new local probe of AdS 3 in the Chern-Simons formulation of 3d gravity. In Sec. 5 we discuss the CFT interpretation of our results. And in Sec. 6 we discuss future directions and related results in AdS/CFT that make use of Ishibashi states.

Path integral representation
In this section we will consider the path integral representation of a Wilson line operator in the Chern-Simons theory. As we review below, this object should be thought of as the Chern-Simons description of the worldline of a massive particle moving in the bulk. This section is a brief summary of the results in [10].
The gauge group of the Chern-Simons theory is SO(2, 2) ≃ SL(2, R) × SL(2, R), and the bulk sl(2, R) gauge connections are A,Ā. 1 The natural observables in Chern-Simons theory are Wilson loops in a certain representation R of the bulk gauge group; in this work we will always take R to be a product of two infinite-dimensional highest-weight representations in sl(2, R) ⊕ sl(2, R).
We may now consider the following Wilson loop operator: and C is a closed loop in the bulk of AdS 3 . This observable is fully gauge-invariant, and will typically be an interesting observable if the bulk loop wraps some non-trivial object in the bulk (e.g. the horizon of a BTZ black hole). Note that the trace involves a sum over the infinitely many states of the highest-weight representation.
We may also consider an open-ended Wilson line operator. To define this object we specify the locations of its endpoints (x i , x f ). We must also specify boundary data in the form of two specific 1 In appendix B we present our conventions on the Chern-Simons description of AdS3 gravity.
states |U i , |U f ∈ R at these endpoints. We may then define the following operator: where now γ(s) is a curve with bulk endpoints (x i , x f ) parametrized by s. W R (x i , x f ) is no longer fully gauge-invariant; clearly it depends in a gauge-covariant manner on the choice of boundary data |U i , |U f . Nevertheless, for flat connections, W R (x i , x f ) only depends on the topology of γ, but not on the shape of the curve.
From a geometric point of view, the Wilson line described above describes the physics of a massive point particle propagating from x i to x f on AdS 3 . A point particle in the classical limit is characterized by at least one continuous parameter: the mass m. This data is stored in the choice of highest-weight representation R that defines the Wilson line. Further details of this representation are given in full detail in Sec. 3. For now we require only that the representation is specified by two constants (h,h) which determine the Casimirs of the sl(2) algebra. Their identification with the mass m and orbital spinŝ of the particle is given by where c 2 = 2h(h − 1) andc 2 = 2h(h − 1) are the quadratic Casimirs; note that the AdS radius is set to unity.
From the point of view of AdS/CFT, the developments in [10][11][12][13][14] show that if the endpoints x i , x f are taken to infinity, the Wilson line operator defined in (2.2) is a bulk observable that computes correlation functions of light operators Ψ|O(y i )O(y f )|Ψ in the dual CFT. Here |Ψ is a "heavy" state whose gravitational dual is given by the bulk connections (A,Ā) and O(y) is a "light" operator whose scaling dimensions (h,h) are encoded in the choice of representation 2 R.
In what follows we limit the discussion to h =h; see [15,16] for a discussion whenŝ = 0.

Path integral representation of the Wilson line
This particular Wilson line is somewhat more complex than those normally studied in compact gauge theories, simply due to the fact that R has infinitely many states in it. We now review the work of [10], who constructed R as the Hilbert space of an auxiliary quantum mechanical system that lives on the Wilson line, replacing the trace over R by a path integral over a worldline field U . We pick the dynamics of U so that upon quantization the Hilbert space of the system is the 2 Here light denotes an operator that, as the central charge c goes to infinity, its conformal weight is fixed, while a heavy operator has a scaling dimension that is linear with c. Equivalently, in gravity we would say that it is a particle with a small mass in Planck units. desired representation R. More concretely, we rewrite (2.1) as where the the auxiliary system can be described by the following action: The variable s parametrizes the curve γ, and we pick s ∈ [s i , s f ]. Here the trace Tr(...) is a short-cut notation for the contraction using the Killing forms, i.e. if P ∈ sl(2, R) where P = P a L a and L a is a generator of sl(2, R). There is also a (classically) equivalent first-order formulation of this action that is more convenient for certain applications (such as the generalization to higher spin gravity). In the first order formulation it is manifest that c 2 is the Casimir of the representation, and satisfies c 2 = 2h(h − 1). This action requires that h =h. As the entire action is multiplied by a factor of √ c 2 , h → ∞ defines a semi-classical limit of the path integral, and for the remainder of this section we will follow [10] and work only in this limit. In subsequent sections we relax this restriction.
This action is invariant under a local SL(2, R) × SL(2, R) symmetry: in particular the covariant derivative is defined as where A(x) andĀ(x) are the connections that determine the background, and in the action (2.5) they are pulled back to the worldline x µ (s). Under an SL(2, R) × SL(2, R) gauge transformation by finite group elements L(x), R(x), the gauge fields transform as The worldline action is then invariant under the following transformation of the worldline field: Now for an open ended Wilson line as in (2.2), we must still specify boundary data on U (s) at the endpoints of the curve. 3 We thus pick two SL(2, R) elements U i , U f and require that For a semi-classical level this is sufficient, and in later sections we will explain in detail the relationship between this choice of boundary data and the quantum states |U i and |U f defined in (2.2).
We now consider the evaluation of this Wilson line on a fixed classical background defined by A andĀ. In the h → ∞ limit, this can be done by evaluating the on-shell action (2.5) for the field U (s) subject to the boundary conditions described above. This computation was explained in detail in [10]. Here we write the answer in a way that will generalize simply to our results in the next section.
In particular the answer only depends on the SL(2, R) evolution of the state from the starting point to the endpoint. If we thus consider flat connections 10) and the following group elements then the on-shell action S can be written as where α labels the conjugacy class of the group element g L ( (3.43) in this state is then given by Note that the role of the boundary data U i , U f in (2.12) is to tie together the two sectors, left and right; we will return to this point in what follows.

Geometric interpretation: proper distances
So far, our review has been very abstract, with no physical interpretation given to A andĀ. However we know that for appropriate choices of these gauge connections, this system should represent the physics of a particle moving on AdS 3 ; we now explain how the result above is related to geometry.
In particular, α defined in (2.12) turns out to be related to the proper distance from x i to x f .
To understand this, note that the action (2.5) can be suggestively written as where the dependence on U (s) in (2.18) has been hidden in the definition ofÃ ν : Note that if we now define a generalized vielbein 4 along the trajectory as then we may write the action very simply in terms of the metric associated to this vielbein as which is manifestly the proper distance associated to the metric g µν . Thus the Wilson line is probing a geometry that is assembled in a particular manner from the connections A,Ā, where the dynamics of the auxiliary field U is playing a role in tying together the two connections into a vielbein. Note that the prefactor √ c 2 indicates that the value of the Casimir controls the bulk mass of the probe, as we alluded to previously.
We also consider the equations of motion obtained from varying (2.5) with respect to U : Normally one considers these as equations for U (s): nevertheless, if one fixes U (s) and thinks of the variable as being the choice of path x µ (s), then this is precisely the geodesic equation for the metric g µν . From here it is clear that the value of the Wilson line between any two bulk points is where D(x i , x f ) = 2α is the length of the bulk geodesic connecting these two points. Here '∼' denotes the limit of large c 2 , where c 2 = 2h(h − 1) ∼ 2h 2 , and hence the classical saddle point approximation is valid.
In what follows we will provide a proper quantum-mechanical treatment of this Wilson line.

Hilbert space representation
The path integral approach to evaluate (2.2) provides insight into the transformation properties for the field U : this choice is in great part responsible of the geometric interpretation of W R (x i , x f ) in AdS 3 gravity. Based on this, in this section we will carefully explain the relationship between the field U (x) and quantum mechanical states in the highest-weight representation. This will allow us to evaluate W R (x i , x f ) without the need of taking a classical limit -in contrast to (2.19)-and, in later sections, have a refined geometric and holographic interpretation of our Wilson line.

Highest weight representations
We first review some facts associated with highest-weight representations. Some words on notation are appropriate: when we are discussing an abstract realization of the sl(2, R) algebra with no particular representation in mind, we will denote the generators with capital L a . We denote the generators of sl(2, R) acting on the highest weight state by ℓ a . A highest-weight representation is defined with respect to a reference state |h that is an eigenstate of ℓ 0 and is annihilated by ℓ 1 : We may now define excited states by acting on |h with ℓ −1 , and the correctly normalized states are defined by where the state |h, k has L 0 eigenvalue (k + h): i.e. k counts the energy above the ground state, and |h, 0 = |h . The Casimir of this representation is 2h(h − 1): where η ab is the Killing form.
We will be interested in states that transform in a highest-weight representation under a tensor product of two independent copies of sl(2, R) × sl(2, R) with h =h, and so we will label them as |h, k ⊗ |h,k ≡ |h; k,k , where the ground state is |h, 0, 0 . We denote the sl(2, R) generators acting on the first k index (the "left") by ℓ a and those acting on thek index (the "right") as ℓ a .
These form two independent sl(2, R) algebras, and we have The group action of each copy of SL(2, R) on these states is the usual one: in particular, we have with the D's the representation matrices for the adjoint representation of sl(2, R). Note that we

Rotated Ishibashi states
We will now define a family of quantum states that have the same transformation as the classical field U (x) in (2.9). To do so it is convenient to consider the following triplet of sl(2, R) operators, labeled by an element U ∈ SL(2, R): This is a linear combination of the generators on the two sides, with one side rotated by U . We will denote a state that is annihilated by Q a (U ) for all a as |U , i.e.
Q a (U )|U = 0 . This defines a rotated state, each labeled by an element U of SL(2, R). We now explore some of the properties of these states. First consider commuting G(L)G(R) through Q m (U ). We find Acting with this relation on the state |U , we find that the state G(L)G(R −1 )|U is annihilated by Q a (LU R). But by the definition of the U states, this means that Thus we see that acting on a U state with an element of SL(2, R) × SL(2, R) causes it to transform inhomogenously precisely as the classical U field did in (2.9). We also note that every U state is It will be useful to have some explicit examples of |U in terms of the highest weight repre-sentation discussed above. As a start let us consider the state |U = |Σ Ish whose action on the generators is 11) and as a group element is Using (3.11), (3.8) becomes This equation has the following unique solution, |h; k, k , (3.14) which is (in its Virasoso incarnation [17]) called the "Ishibashi state." Another choice for our states is setting |U = |Σ cross whose action is and as a group element it reads For this choice (3.8) becomes 17) and the unique solution to this equation is which is usually referred to as the "crosscap (or twisted) Ishibashi state" [17]. The state |Σ cross (rather than |Σ Ish ) will play an important role in section 4, for reasons that we will elaborate on there.
If we can construct any reference state in this family, then we can find any other state by acting on it with an appropriately chosen G(L) and/orḠ(R −1 ). 5 And for this reason we will call the states |U (in a slight abuse of notation) rotated Ishibashi states. Our rotated Ishibashi states are coherent states that live in the product of two highest weight representations and only involve the global part of the conformal group, unlike the states used for boundary CFT [17,18].

Inner product
An important object in our analysis is the inner product of a rotated Ishibashi state. These states are not orthogonal-they form an overcomplete basis-which leads to a non-trivial expression. The relevant matrix element to evaluate any such inner product is where |Σ is a reference state from our family of rotated Ishibashi states. For concreteness we will take |Σ to be either |Σ Ish or |Σ cross , (3.20) as defined in (3.14) and (3.18).
Evaluating (3.19) leads to (3.21) In the first equality we used (3.10). In the second line we used (3.14) and (3.18); the coefficient a k is equal to 1 and (−1) k , respectively. In the third line we used that |a k | 2 = 1, which reduces the computation to a trace of the group element inside the bracket. In the last line we decomposed the group element as where α controls the conjugacy class of the group element in question. The last equality is our final result, which is just a sl(2, R) character of G(L Σ R Σ −1 ). From here the role of |Σ is becoming more evident: it controls how the right element R would act as left element relative to L and vice versa.
The result (3.21) immediately generalizes to the inner product between any two of the U -states as defined in (3.8): any rotated state continuously connected to Σ will satisfy In other words, the inner product between any two U -states U 1 and U 2 is a function only the "magnitude" α of the conjugacy class of the group element that relates U 1 to U 2 . α can be thought of as an invariant distance between the two elements on the group manifold (and indeed we will develop its geometric interpretation in the next subsection). Note that as U 1 approaches U 2 , α → 0 and thus the norm of any U state itself is infinite: this divergence can be seen immediately from noting that the norm of |Σ diverges.
Finally, the U states satisfy a completeness relation. It is shown in Appendix A.2 through explicit computation that for 2h > 1 we have where dU is the Haar measure on SL(2, R). In pedestrian terms, this simply means that we treat SL(2, R) as being locally AdS 3 and integrate over it using the usual volume measure, taking care to integrate over SL(2, R) and not over its universal cover.

The Green's function on the group manifold
Here we discuss a few further properties of the inner product U 1 |U 2 computed above. In particular, the inner product (3.23) is actually a Green's function with respect to the invariant Laplacian on the SL(2, R) group manifold.
We begin by placing coordinates σ α on the group manifold SL(2, R). Let us denote the usual generators of sl(2, R) in the fundamental representation by L a . As SL(2, R) is a group manifold, there exist vector fields ξ α a andξ α a that generate the group action on a point in the manifold from the left and from the right, i.e. (3.25) As the U -states (3.10) transform in the same way, they satisfy: as well as a similar relation for the barred sector. Now we act with this relation twice on the σ 2 coordinates parametrizing the inner product U (σ 1 )|U (σ 2 ) with U (σ 1 ) = U (σ 2 ). In particular, denote the Killing form on sl(2, R) by η ab and compute where in the last equality we have used the Casimir relation (4.31). It is straightforward to verify that the second-order differential operator on the left-hand side of (3.27) is (up to a factor of 1 2 ) the invariant Laplacian on SL(2, R), which we denote by U . As our analysis holds only for non-coincident U 1 , U 2 , we conclude that Here δ(U 1 , U 2 ) is a delta function on the group manifold that is nonzero only if U 1 = U 2 , and which is normalized to satisfy dU δ(U 0 , U ) = 1 with dU the Haar measure on SL(2, R) and U 0 a reference group element. This can of course also be checked by explicitly verifying that (3.23) satisfies the appropriate Laplacian; this is also the fastest way to verify the existence of the delta function on the right-hand side.

Relationship to path integral
In this section we will demonstrate that the in the large-h limit, the inner product defined above can be computed from a path integral over a classical field U (s), as used in [10] and reviewed in Sec. 2. Essentially we will perform the analogue of the usual construction of the path integral for quantum mechanical systems, where the non-compact nature of the representation, and therefore of the U states, provide some extra wrinkles.
Consider computing an inner product of the form To give this a quantum-mechanical interpretation, we will represent the group elements L and R as path-ordered exponentials of gauge fields A(s) andĀ(s), where s should be thought of as "time", i.e.
To make contact with conventional quantum mechanics, one can imagine that A andĀ define a Hamiltonian for the system defining time-evolution along s. We will now derive a path integral expression for the inner product (3.29). We follow the normal algorithm of dividing the path from s i to s f into many small intervals of size ǫ, discretizing the path as We may then break up each path-ordered exponential: and similarly for the right sector. The inner product takes the form We now use (3.24) to insert a complete set of U states at each time step. We find where we have introduced an overall prefactor N to absorb factors of the form (2h − 1) ∞ into the usual ambiguities in the measure of the path integral. We see that we must evaluate many inner products of the form To evaluate this inner product, we make the usual assumption that most contributions to the path integral come from reasonably smoothly varying U (s), so that we may assume that U (s j+1 ) = . Thus to lowest order in ǫ we are evaluating We use the transformation property of the U states (3.10) to move all of the group elements to the ket on the right to obtain Next, we use the general form for the inner product (3.23) to conclude that where α(s j ) is given by the conjugacy class of the SL(2, R) element where to obtain this expression we expanded all terms up to order ǫ, and then re-exponentiated the resulting expression. It should be understood that this expression is correct only up to order ǫ. We have encountered a version of (3.38) in (3.22) and (3.23), and we will encounter again in subsequent sections. The a simple way to read off α(s j ) is by noticing that (3.38) -and its counsins (3.22) and (3.23)-are independent of the sl(2) representation. With these freedom, we choose to solve this equation in the fundamental representation of sl(2), described by the 2 × 2 traceless matrices, where α is given by a trace: Here the gauge-covariant derivative D s U is that defined in (2.7), and our conventions for the fundamental representation are given in appendix A.
We have thus computed the contribution of one infinitesimal piece of the path. Assembling all of these pieces by taking the product, we see that the full inner product (3.29) is given by We now consider taking the continuum limit ǫ → 0; the product of integrals dU (s j ) over each group element at each point on the path becomes a path integral [DU ] over a continuous worldline field U (s). We first consider the numerator of the above expression: this naturally becomes an integral over a smooth action: i.e. precisely the exponential of the action S[U ] postulated on physics grounds in [10].
We now turn to the denominator 1 − e −α(s j ) . In the limit ǫ → 0, each α(s j ) is infinitesimal, and thus we may write: where we have introduced a new auxiliary field σ(s j ) at each point on the worldline; integrating out this field generates the denominator (up to an overall ill-defined prefactor that depends on the discretization). The full path integral is thus where the full continuum action is (3.44) In the h → ∞ limit, we may ignore the second term in the action: this is then precisely the path integral (3.43)-(2.5) which was proposed on symmetry grounds in [10].
We can now see that at finite h, the path integral proposed in [10] must be corrected by additional "quantum" terms arising from the measure of the path integral when integrating over U states. This additional term -the wrinkle we alluded to at the start of this subsection-arises from the fact that the inner product of two nearby U states is divergent, which is itself a direct consequence of the non-compactness of SL(2, R) and the resulting infinite tower of highest weight states. It would be interesting to understand better the physical significance of this term; however in this paper we will not attempt to treat the path integral (3.43) at finite h, and will instead simply directly compute matrix elements from the algebraic approach developed above.

Wilson lines: Local Fields and Geometry
Our goal in this section is to give a geometric interpretation to the algebraic construction in Sec.
3. We will start in Sec.

Gravitational Wilson line as an overlap of two states
The results in Sec. 3 gives a prescription to evaluate overlap of states in the highest weight representation. In this section we would like to implement those results to a gravitational Wilson line. More concretely, we would like to analyse as an overlap of a suitable initial and final |U state. We keep the reference state |Σ generic so far, and we will discuss the different choices Σ Ish , and Σ cross in Sec. 4.3. As in Sec To recast (4.1) as an inner product, it is useful to rewrite the flat connections as Using the transformation of the path ordered exponential under (4.2): and therefore To write this expression as an overlap between to states, we define and with this, we can rewrite the previous amplitude as It is important to note that in this expression we have implicitly assumed that the group element and similarly for g R . All of our manipulations will use group elements that are unitary. And we should stress that |U (x) is not gauge invariant. In its definition in (4.5) we implicitly made a choice: we are splitting the path from x i to x f to a mid point where g L = g R = 1, and without any further specification of the connections, we have not motivated nor justified this choice. This bug does not affect (4.6), and we will ignore it for now. We will return to this point in Sec. 4.3 when we directly analyse |U (x) .
Having casted the gravitational Wilson line as an inner product in (4.6), we can now use the same logic that leads to (3.21) and (3.22). In particular we find that where, following (3.22) for this case, α(x i , x f ) is given by the solution to and we defineg Note that while, by definition, A andĀ act on different spaces, the role of Σ is to tie together these two sectors;Ã can be thought of as the 'left' version of the 'right' connection.
To solve for α(x i , x f ) in (4.8), it is useful to note that this equation is independent of the sl(2, R) representation, and hence we can simply use a finite dimensional representation. 6 Using the fundamental representation of sl(2, R) (see appendix A), and after taking the trace both sides of (4.8), gives where Tr f is the trace in the fundamental representation. Using (4.2) together with (B.7) and is the geodesic length of an effective metric given by The relevant metric for global AdS and BTZ is given in (B.9). Therefore, the inner product is This is the familiar bulk-to-bulk propagator of a minimally coupled scalar field in a locally AdS 3 background [19,20]. In the semi-classical limit, where the numerator is negligible and h is large, the saddle point approximation of the path integral in (4.12) precisely agrees with (2.19). The background metric (4.11) is in agreement with (2.17), and (4.8) is equivalent to (2.12).
At the level of evaluating (4.12), the detailed nature of |Σ can be overlooked: provided the endpoint states satisfies we will obtain (4.12), and interpret it as the bulk-to-bulk propagator of a scalar field with background metric (4.11). With this perspective, if the input is g µν , we could just infer the values of (A,Ã) and use them in (4.12), without making explicit reference to the difference betweenĀ and Ã , and hence neglect the role of |Σ . However, |U (x) is an object sensitive to |Σ , and as we will discuss in section 4.3, this will disentangle the different features that |Σ captures as we build local probes in AdS 3 .

Algebra meets geometry
An expression such as (4.12) makes rather evident that the Wilson line is a propagator, and hence its ties to geometry. The drawback however is the brut aspect of the observation: it relied on evaluating explicitly the observable on AdS 3 and the BTZ background. In this section we will do better. We will show that the object can be understood as a bulk-to-bulk propagator with respect to the bulk spacetime metric associated with the flat connections A,Ā. The important improvement here relative to our prior observations is that here we treat the Wilson line quantum mechanically, and as such it will capture the geometry as perceived by a bulk field of an arbitrary mass.
We begin by assuming that the bulk spacetime is simply connected (e.g. for pure AdS 3 ). In this case all paths from x i to x f are topologically equivalent, and (4.14) is a well-defined function of the two endpoints.
We first recall that in (3.28) it was shown that the object U 1 |U 2 was a Green's function on the group manifold SL(2, R). This is logically distinct from showing that the matrix element (4.14) is a Green's function on the bulk metric defined by A,Ā.
To make a connection between these two objects, we first need to establish how the matrix elements in (4.14) change if we move, for instance, the point x f . The dependence on endpoints x i and x f enters in (4.14) as follows: using (4.2)-(4.3), the matrix element reads In the second line we made use of the transformation properties of our reference states (3.10), and used the definitiong R ≡ Σ −1 g R Σ . We note that this is where the choice of |U to be rotated states is crucial: the state combines both sectors, which will lead to a geometric interpretation of W R (x f , x i ) in the subsequent steps. From (4.15), the full dependence on x i and x f enters through the following group element Taking an x f derivative of this group element, we have Recall now that (3.28) was shown by exploiting the fact that the left and right action of the group generated a set of vector fields on the group manifold (3.25). We would now like to extend this idea to the geometric bulk, i.e. we seek a set of vector fields ζ µ a ,ζ µ a defined on AdS 3 such that Multiplying both sides of these equations by L b and taking a trace, we see that the defining relations become If the generalized vielbeins shown above are invertible, then the ζ,ζ exist, and we have shown that movement in bulk spacetime is equivalent to movement on the group manifold. Furthermore the condition (4.18) guarantees that they satisfy the sl(2, R) × sl(2, R) algebra as Killing vectors on the bulk spacetime. Thus following through the same steps as in (3.28), we conclude that where now x f is the Laplacian on the bulk AdS 3 spacetime. The construction of ζ,ζ will be carried out explicitly in Sec. 4.3.
We now consider the case where the bulk spacetime is not simply connected, e.g. the BTZ In this case, if we would like to obtain an unambiguous answer that depends only on the endpoints, one prescription is to sum over all inequivalent paths, i.e., we define a path-summed Wilson line as where the sum is over all topologically inequivalent paths C(x f , x i ) that connect x f to x i . An example of such situation is nicely capture by the BTZ black hole. In this case the inequivalent paths correspond to geodesics winding around the horizon multiple times, and the resulting propagator is a sum over these windings. For the static black hole, the resulting propagator is Here we are using the geodesic length in (B.14), and n controls the number of times the path encloses the horizon. In the metric formulation this sum can be understood as the sum over images that gives the propagator the correct periodicity condition (see e.g. [21]), which in complete agreement with our expression.

Local fields
In the last portion of this section we will evaluate and interpret |U (x) as defined in (4.5). As mentioned there, this definition is gauge dependent. A definition of |U (x) that reinstates this dependence is We will decompose the state (4.24) as a sum over local functions in the infinite-dimensional representation and evaluate Φ k,k (x). Alternatively, the function Φ k,k (x) is This function is the object that will provide local bulk information in the Chern-Simons formulation of 3D gravity.
The explicit calculation of this Φ k,k (x) can be a complicated task. A way to proceed is by using the technique in Appendix A of [22]. The aim there is to find a differential operators L a (x) whose action in the inner product (4.26) is where ℓ a is the infinite-dimensional generator that acts as in (3.2), and L a (x) is a differential operator acting on the x variables, whose explicit form depends on the state |U (x) . Analogous formulas can be found for the barred sector. These operators are precisely the vector fields introduced in Φ 0,0 (x) can be fully determined by solving following differential equations together with its barred version. Therefore, we will be able to infer the form of Φ k,k (x), by successively applying L −1 (x), andL −1 (x) to the seed Φ 0,0 (x). From here it follows that Φ k,k (x) obeys the Casimir equation where In other words, Φ k,k (x) is a local bulk field of mass m 2 = 4h(h − 1) and whose boundary conditions are given by the highest weight conditions (4.29)-(4.30).
Finally, once we have the explicit expression of the functions Φ k,k (x), we will compute the inner products of two states (4.25) as Note that when we evaluate (4.32) we will not make use of (3.21), and hence the derivations in this portion give an alternative and more direct derivation of (4.12). In the following, we will carry out this procedure for two explicit backgrounds. Sec. 4.3.1 is devoted to global AdS 3 , which agrees completely with the results in [22], and Sec. 4.3.2 focuses on the static BTZ black hole.

Global AdS 3
Let us consider the state |U for global AdS 3 and build explicitly Φ k,k (x) for this background. To start we will first infer the group elements from the standard metric for AdS 3 , i.e. ds 2 = − cosh 2 ρ dt 2 + dρ 2 + sinh 2 ρ dφ 2 .
where x ± = t ± φ. We will use the definition (4.24) with g L (x 0 ) = 1 = g R (x 0 ); this places |Σ at the origin of AdS in accordance with the results in [22][23][24]. This gives where we used (3.10) and (4.34). For most of the following derivations we will drop the subscript "AdS" and restore it when needed.
The next step is to find the differential operators L a (x) in (4.27) for global AdS 3 . For that we use the inner product as Taking derivatives with respect to the global coordinates gives and using commutation relations, we can move the generators that are not in the exponents to the right, to get Now, it is straight forward to obtain the differential operators that follow (4.27) for global AdS 3 ; these read It is important to remark that these differential operators were built without making direct reference to |Σ .
To find the barred differential operators we follow a procedure analogous to what we did in (4.37)-(4.38), but using the following inner product: where we are rewriting the action of the left group elements as an action via the right, i.e.
While in (4.36) we could ignore Σ, we are now forced to understand how Σ acts on the states to infer the differential operatorsL a . A sensible choice is to require thatL a are related to L a by This is the familiar assignment of Killing vectors in AdS 3 ; the interesting twist here is that not any choice of probe Σ will achieve this assignment. A choice of |Σ that delivers Our starting point in this subsection was the metric for AdS 3 in (4.33). Another starting point is to use the fact that global AdS 3 is maximally symmetric, and the group elements that label rotations and translations in this space are as it was done in [22,25]. For the crosscap state, (4.44) is in complete agreement with (4.34). The choice |Σ Ish would lead to different group elements, which is tied to the fact that in this case we have a non-stantard relation between algebra elements and Killing vectors of the geometry.
The differential operators (4.39), and (4.42) are Killing vectors of global AdS 3 , as advocated in Sec. 4.2. Moreover, L 2 (x) +L 2 (x) in (4.31) is the usual d'Alembertian for AdS 3 . Therefore, Φ k,k (s) is a scalar field with mass m 2 = 4h(h − 1) in a global AdS background. Now, we can solve (4.30) using the previous differential operators, as done in [26]; the highest weight state is (4.45) To find Φ k,k (x) we simply need to identify the solutions to (4.31) and organize them as L −1 (x), andL −1 (x) acting on (4.45). This leads to where P (a, b) n are Jacobi polynomials, and C k,k = (−1) k k!(2h+k−1)! k!(2h+k−1)! is a constant that has been chosen to match the normalizations in (4.29). Therefore, we found the state (4.25) in a global AdS background. This is in complete agreement with the known results of normalizable wavefunction in AdS 3 as in, e.g., [27].
We are ready to compute the overlap of two states at different positions in the bulk. Using (4.32) with (4.46) gives The previous sum is performed in the Appendix C. If we choose x = tanh 2 ρ i , y = tanh 2 ρ f , r = e −i∆x − , and s = e −i∆x + , the left hand side of (C.3) is equal to (4.47). Applying (C.3), we find

BTZ
As we did for global AdS 3 , we will now find the local functions Φ k,k (x) for the static BTZ background. Our starting point is to build the group elements (g L ,g R ) from the metric, which for the black hole reads (4.49) In  where we casted all the elements as acting on the left, and we introduced Following the same procedure as in Sec. 4.3.1, we can find differential operators defined as (4.27) for the BTZ state. Using 8 we find the non-barred differential operators In order to obtain the barred generators, we proceed as done for global AdS 3 in (4.40)-(4.41), i.e.
we rewrite the state |U (x) BTZ as having an action only via right group elements. This gives As before, we will fix Σ such that the barred differential operators,L a , are equal to the non-barred operators with x + ↔ x − , as it is natural in the metric formulation. A quick inspection singles out |Σ cross as the appropriate choice rather than |Σ Ish . Using |Σ = |Σ cross in (4.55) we find Having evidence that the state |Σ cross is a natural probe (with usual geometric properties we associate to BTZ), we can infer from (4.50) that One can obtain |U (x) BTZ from the gauge transformation that relates global AdS 3 and BTZ, and using (4.44). We found, however, instructive to take a perspective where the metric is the first input and from there build (4.57).
The Ishibashi state |Σ Ish also leads to barred differential operators. Acting on (4.55) with (3.11), we get barred differential operators similar to those in (4.56), but with an overall minus sign inL ± . These differential operators are still Killing vectors and they follow the sl(2, R) L × sl(2, R) R algebra by definition.
We now return to building Φ k,k (x). To start consider (4.30): given (4.53), it is clear that Φ 0,0 (x) is non-separable in any of its variables, which makes (4.30) very difficult to solve. In order to simplify (4.30), we will make a change of variables; using (B.15) we now have (4.58) The barred operators are defined analogously with X + ↔ X − . The advantage of (4.58), relative to (4.53), is that the differential operators just involve powers on the coordinates, and hence we can find a suitable polynomial solution to (4.30). The unique solution to (4.30) reads where in the second line we have changed to BTZ coordinates in (B.11). And as expected the solution (4.59) not separable in this coordinate system. Acting with L −1 (x), andL −1 (x) in (4.59), and inspired by the the Jacobi polynomial form of the global case (4.46), the general expression for a descendant of (4.59) reads where a ≡ i+r + −i+r + , and C k,k is same factor as in (4.46). It is straight forward to verify that Φ k,k (x) in (4.60) satisfies the d'Alembertian equation on the static BTZ background.
Having an explicit expression for Φ k,k (x), we can compute the overlap of two states (4.25) for the BTZ black hole. Using (4.60), we see that he sum we need to perform in (4.32) is exactly equal and Using the result for the sum (C.3), with the previous definition for X, Y, r, and s, we find the the overlap of the two states in the BTZ black hole: where σ(x i , x f ) is the geodesic distance for Poincare (B.17), which can be rewritten as the geodesic length in BTZ (B.14) using (B.18). With no surprises, this is in complete agreement with (4.12).
It is interesting to analyse the behaviour of the field (4.60) in the BTZ coordinates. Looking at (B.18), we see that the BTZ boundary r → ∞ is located at Z → 0, and in this limit we have Φ k,k → 0. The horizon r = r + is at the Poincare boundary (X + , X − , Z) → ∞, where Φ k,k as well vanishes. This behaviour, together with the fact that solves the BTZ wave equation, shows that (4.60) behaves as a quasi-normal mode for the black hole. However, it is not a traditional BTZ quasi-normal mode as those built in, e.g., [28][29][30][31]. There are a few discrepancies, and a few similarities, with this literature that are worth highlighting.
1. Highest weight condition. As it was observed in [32,33], imposing the highest weight conditions (4.29)-(4.30) leads to eigenfunctions that obey the quasinormal modes conditions. This is a first indication that Φ k,k (x) should have been regular throughout, as they certainly are.
2. Separability of eigenfunctions. The most canonical way to find solutions to the Casimir equation (4.31) is by casting the basis of solutions in a Fourier decomposition in (t, φ), which are the natural directions for the Killing symmetries of the black hole. This leads a eigenfunctions that are separable functions in the coordinate system (r, t, φ), in strike contrast to (4.60). The construction of the operators L a in [32], which is used to build a basis for quasinormal modes, is as well compatible with the separability ansatz. From a technical point of view, our lack of separability could be attributed to the unitary condition we enforce in (4.57): this leads to a group elements that are simply different to those used in prior work. 9 3. Periodicity conditions. By design, the connections (A,Ā) that characterize BTZ in the Chern-Simons formulation have the following feature [34,35]: they are single valued along the thermal cycle in Euclidean signature (smoothness of the Euclidean cigar geometry), and carry a nontrivial holonomy around the spatial cycle (an indication that the connection has a finite size horizon). This is reflected in (4.60) by the fact that our eigenfunctions are not periodic as we take φ ∼ φ + 2π, but are periodic under t ∼ t + i2π/r + . This is clearly not a feature of the modes built in [28][29][30][31], which are decomposed in periodic Fourier modes along the φ direction.
4. Inner Product. Despite the two differences above, it is interesting to note that if we evaluated the overlap (4.32) using the quasinormal modes in [29], it would lead to (4.63). The derivations are shown in appendix D. This indicates that the bulk-to-bulk correlation functions are not sensitive to how we represent Φ k,k (x).

CFT interpretation
Here we discuss the CFT interpretation of the results above. In particular, consider computing a Wilson line in AdS 3 , ending at the AdS boundary at the two boundary points z 1 , z 2 at radial coordinate ρ 1 , ρ 2 with generic boundary conditions U 1 , U 2 at each endpoint. What, precisely, is this object in the CFT?
The considerations of the previous section should make it clear that the resulting object is a suitably smeared two-point function, and here we simply provide a purely boundary interpretation of this smearing procedure. The kinematics of these procedure are very familiar from the language of the the HKLL construction [36,37] and this section may be understood as a translation of some of those results into the language of Chern-Simons gravity. Since |Σ cross leads to the standard conventions in the metric formulation, relative to |Σ Ish , we will focus on the role of the crosscap Mathematically this is essentially the same construction as [22][23][24]. There are two main differences: in all of these works the specification of the SL(2, R) element was interpreted to specify a point in AdS 3 rather than a boundary condition on a Wilson line. Furthermore in [23,38,39] the full Virasoro group was considered rather than just its global subgroup. The former is just a matter of interpretation, and we will touch briefly on the latter in the conclusion.

Example: CFT on the plane
We now present some elementary computations to explain how this works in the basic case of Poincaré AdS 3 in coordinates: dual to the CFT on the plane with complex coordinates z,z. Rather than working with boundary data on the edge of an excised disc at each endpoint, it is more convenient to perform the stateoperator correspondence to map each descendant on the edge to a local operator at the center of the disc. As there are an infinite number of states in the sum, this is a very non-local operator which we denote by O U i (z i ,z i ). We will use a variant of the HKLL construction to compute the two-point function and then reproduce this answer from a Wilson line computation.
Focus on the first endpoint at (z 1 ,z 1 ). We first consider the case where the boundary state U 1 is the crosscap Ishibashi state |Σ cross itself. Consider the disc centered at z 1 in the CFT, with radius e −ρ ; we would like to place the boundary data corresponding to the crosscap Ishibashi state: where the normalization constant in each sector is c m = Γ(2h) Γ(m+1)Γ(m+2h) . We now use the stateoperator correspondence to replace each state on the disc ℓ m −1l m −1 |0 with the operator ∂ m∂m O(z 1 ) at the center. However, we should note that the evolution from the center of the disc to the edge will cause each state's amplitude to be multiplied by a factor of e +ρ(2h+2m) . Compensating for this, the operator that creates the crosscap state on a disc of radius e −ρ 1 is The sum over derivatives of the local operator O can be written as an integral over the following kernel [25,40] O (ρ 1 ) where in the last line we have introduced some new notation. We note that this is nothing but the usual HKLL smearing kernel in Euclidean signature, though our interpretation is somewhat different.
We now consider deforming away from the crosscap state to a more general U -state. The SL(2, R) generators have a simple geometric action on the plane, and this geometric action results in a transformation of the parameters in kernel K. In particular, if we parametrize the SL(2, R) element U 1 in a convenient way as then it is shown in Appendix E that the appropriate smeared operator is We now pause to interpret this result from the point of view of HKLL. Recall that the smearing function (5.7) corresponds to the HKLL representation of a bulk field in Poincaré coordinates (5.1), where the precise coordinate values of the bulk reconstructed field are (ρ a , z a ,z a ) = (ρ 1 + µ 1 , z 1 + e ρ 1 w 1 ,z 1 + e −ρ 1w 1 ) , (5.8) In particular, the proper distance D within AdS 3 between two points with coordinate values Let us now consider computing the two-point function (5.2) of two U states inserted at distinct points on the boundary z 1 , z 2 . This is a well-posed CFT computation involving integrals over two K kernels. Rather than repeat it here, we simply note that it is a standard HKLL computation, and by construction the result is the usual bulk-to-bulk AdS 3 propagator between points with Poincare coordinate values given by (5.8) (and a corresponding relation relating (ρ a , z a ,z a ) to (ρ 2 , z 2 ,z 2 )), i.e.
We will now reproduce this CFT result from a Chern-Simons computation. In particular, we consider the following gauge connections for Poincaré AdS 3 in Euclidean signature: As usual the full connections are related to the objects recorded here by Full conventions are in Appendix B; in particular these connections are equivalent to those in (B.8) with C → 0, together with the usual Euclidean continuation x + → z, x − → −z and a rescaling of the field-theory directions by a factor of 2; the last step is convenient so that the resulting coordinate system is precisely equivalent to (5.1).
The prescription above states that the two point-function (5.2) is calculated in the Chern-Simosn representation by the following matrix element: We may easily verify this relation. (3.23) now tells us that the right-hand side of this expression is equal to this matrix element is equal to where as usual α is defined as the L 0 conjugacy class of the following group element: Computing α from here and the explicit representation of U 1,2 as in (5.6), we find that it is equal to 2D as defined above (5.9); thus we find that the Chern-Simons computation agrees with the CFT result, confirming (5.12).
Note that everything in this computation is fixed by kinematics, and we have simply shown how the SL(2, R) parameters characterizing the boundary conditions combines with the geometric data to give the familiar HKLL result.

Discussion
We provided a full quantum mechanical description treatment of worldline degree of freedom of a Wilson line in SO(2, 2) Chern-Simons theory. This degree of freedom allowed us to build a local probe in the Chern-Simons description of AdS 3 gravity. There are a few striking features of this probe which we highlight.
1. We designed states in the worldline quantum mechanics such that they would couple to both (A,Ā). This condition naturally introduced the notion of rotated Ishibashi states, which we denote as |U , and their coupling to the connections creates a background spacetime metric 2. Using purely the Chern-Simons formulation, we can build local bulk fields that probe the background geometry (6.1). These local probes are defined in (4.26) and we investigated some their properties for global AdS 3 and the static BTZ black hole.
It is very satisfactory that our choice |U = |Σ cross is compatible with the proposals in [22][23][24], and we also reproduce the smearing functions of the HKKL [36,37] proposal for vacuum solutions.
This is expected since the symmetries of AdS 3 constrain heavily the resulting bulk field, leaving little room for disagreement at this level of discussion. Perhaps the interesting difference of our approach is that our construction leaves room to consider other probes |U , and highlights some of the gauge dependence in the construction of the local field Φ k,k (x), which we emphasised around (4.24). For black holes the situation is more delicate: for instance, it would be interesting to compare and complement the proposals in [25,[41][42][43][44] with our derivations in Sec. 4.3.2. Along these lines it would be interesting to carry out our derivations for the rotating BTZ black holes, and other backgrounds in 3D gravity we have not explored.
We comment very briefly on one other aspect; as we have been able to reproduce bulk-tobulk propagators from the Chern-Simons description of 3d gravity, it is worth wondering whether all of the aspects of the quantum field theory of a scalar field on a gravitational background can be obtained from the Chern-Simons computation, e.g. can we obtain a one-loop scalar field determinant on a BTZ black hole background? As this is essentially the same information as the bulk-to-bulk propagator, we might think so. Indeed we expect the logarithm of the one-loop determinant W to be the sum over connected Feynman diagrams, which in our context is the sum of Wilson lines that each wrap the horizon n times on topologically distinct paths C n . We find: Here we have assumed that the topologically trivial path does not contribute; the factor of 2 arises from positive and negative n. The combinatoric factor 1 n is a symmetry factor 10 and as usual α is the conjugacy class of the holonomy of A (orĀ) around the black hole; on the BTZ background it evaluates to α = 2πr + . The result above is then precisely the logarithm of the usual one-loop scalar determinant on a black-hole background; see e.g. [45] for details and a repackaging of this result in CFT language.
An important issue that we have not addressed is quantum corrections due to fluctuations of the background connections. This would capture 1/c corrections, i.e. corrections controlled by the AdS radius in Planck units, or equivalently subleading terms controlled by the level of the Chern-Simons theory. Work in this direction has been done for SL(2) Chern-Simons theory, where Virasoro conformal blocks are known to be tied to appropriate Wilson line in Chern-Simons [46][47][48]. Recent developments for this holomorphic theory include [49][50][51][52][53][54]. It would be interesting to evaluate 1/c corrections of our worldline quantum mechanics; in this case we expect that the intertwining of the two copies of sl(2) will produce interesting features. For example, we should be able to probe if the global conditions in (3.17) are enhanced to the Virasoro conditions on the Ishibashi state [38,39], or something completely different, such as the conditions proposed in [40]. We leave these questions for future work.
Another natural direction forward is to use our construction to build probes in SL(N ) × SL(N ) Chern-Simons theory. This would provide a unique way to build local probes in higher spin gravity.
A discussion of Ishibashi states for W 3 algebra was done in [55], which is a natural starting point for future investigations.
A Properties of so (2,2) representations In this appendix we collect a set of definitions, conventions and identities that are relevant for our manipulations in the highest weight representation, and the rotated Ishibashi states.

A.1 sl(2, R) conventions
The Lie algebra for sl(2, R) is given by Our conventions for the fundamental representation of sl(2, R) is In our conventions, the Lie algebra metric reads For the highest weight representation of sl(2, R) × sl(2, R), we denote the generators as (ℓ a ,l a ).

A.2 Completeness of rotated Ishibashi states
Here we establish a completeness relation for the |U states: We note that if the right-hand side exists, it must be equal to a multiple of the identity by SL(2, R) invariance; thus the only question is whether or not the integral converges, and what the normal-ization factor is if it does. As the group is non-compact the convergence is not (to our knowledge) actually guaranteed. Thus we perform an explicit computation in coordinates. In particular we view the SL(2, R) group manifold as global AdS 3 and place on it the usual global coordinates (ρ, t, φ). It is important to note that we work here not SL(2, R) and not with its universal cover, and thus both coordinates t and φ are periodic with period 2π.
The explicit matrix elements between the |U (ρ, t, φ) states and the discrete highest-weight states |h, k,k can be constructed via the usual methods of finding the highest-weight state and systematically acting with the raising operators. The result is precisely that given in a (slightly) different context in the bulk of the paper (4.46): where P (a, b) n are Jacobi polynomials, and C k,k ≡ (−1) k k!(2h+k−1)! k!(2h+k−1)! . Note also that in these coordinates on the group manifold the Haar measure is just the usual volume element on AdS 3 , i.e. dU = dρdtdφ sinh ρ cosh ρ . (A.7) With this in hand, we simply directly compute the following matrix elements: The normalization factor is given by This is difficult to evaluate for generic k. However by SL(2, R) invariance it must be independent of k,k (a fact we have also checked directly by numerical evaluation of the integral), allowing the integral to be performed for k =k = 0, resulting in .

B Chern-Simons formulation of AdS 3 gravity
With the purpose of setting up conventions, in this appendix we give a very short review of the Chern-Simons formulation of AdS 3 gravity. We refer the reader to the original articles [1,2] and more recently in, e.g., [56,57] for further details.
The relevant Chern-Simons gauge group for AdS 3 gravity is G = SO(2, 2). The Einstein-Hilbert action can be written as with A ∈ so(2, 2). Here k is the level of the Chern-Simons theory. The relation to the conventional gravitational vielbein and spin connection is where M a are Lorentz generators and P a are translations in so(2, 2).
It will be convenient to write the gauge group SO(2, 2) as SL(2, R) × SL(2, R). The flat connection A can then be decomposed as two pairs of connections with L a = 1 2 (M a + ℓP a ), andL a = 1 2 (M a − ℓP a ). Here ℓ is the AdS radius, which for most of our work we will set ℓ = 1, and Newton's constant is related to the Chern-Simons level via We will denote the generators of sl(2, R) simply as L a . After performing this decomposition the action can be written where the trace operation used in defining the Chern-Simons form is now the usual bilinear form on the sl(2, R) Lie algebra.

B.1 Metrics, connections, and geodesic distances
In this appendix we gather various properties used for global AdS and the BTZ background. We present the relevant information in Chern-Simons formulation, and the metric formulation. For the later, we gather the different coordinate systems used and the relevant geodesic distances.
In Chern-Simons formulation, we write the pair of sl(2, R) × sl(2, R) as In this section we add the tilde in the right sector, for consistency with the conventions used in the main text. When the connections are constant in boundary coordinates, we can cast the group elements as where b(ρ) parametrizes the choice of radial variable, and a µ , andā µ are constant elements of the sl(2) algebra. We can use the previous reparametrization to express the BTZ and global AdS metric, with 11 Here ρ is the radial direction and x ± = t ± φ with φ ∼ φ + 2π. Via (4.11), these connections (B.8) correspond to the metric: (B.9) 11 We chose these explicit form of the connections because they result into unitary group elements (B.7) when we consider the highest weight representation with (ℓn) † = ℓ−n. This is required by the purposes of the main text. For readers familiar with the previous literature in 3d gravity in CS formalism, it will be comforting to know that (B.8) is related to the more familiar form of the BTZ connections: The automorphism labelled by R is performed in the right sector, and analogously for the left sector.

C Generating function of Jacobi polynomials
In this Appendix we will perform a double sum of multiplication of two Jacobi polynomials which is used in the main text. For that, we use the review on generating functions in [58]; formula (62) in Sec. 2.3 of [58] reads ∞ n n!(−α − β)! (−α − β + n)! (x − 1) n (y − 1) n t n P (α−n,β−n) n x + 1 x − 1 P (β−n,α−n) n y + 1 We need also the identity ∞ n P (α,β) Combining the previous formulas, with y → 1/y, and other basic identities of hypergeometric functions, we can derive the following sum: k!(2h +k − 1)! k!(2h + k − 1)! r k sk(xy)k where σ is defined as For the examples worked out in section 4.3, σ is directly related to the geodesic distance between two endpoints.

D Inner product with quasi-normal modes eigenfunctions
In this appendix we explore what will happen if in (4.32), given by we replaced (without justification) Φ k,k the more familiar quasi-normal modes for the BTZ black hole.
The quasi-normal modes are defined as the fields in black hole geometries that are purely ingoing at the horizon, and that vanish at infinity. For the BTZ black hole, solutions to 2 Φ = m 2 Φ with these conditions are found in [29], imposing separability in its variables: where we have considered the non-rotating case (r − = 0), and that the mass of the scalar field is related to the conformal dimension as h = 1 2 (1 + √ 1 + m 2 ). The vanishing boundary condition gives the left and right quasi-normal modes: ω ± = ±l − 2ir + (n + h) . We have named the previous field Φ k,k by analogy with the global case, but it does not follow (4.29) for the BTZ differential operators in (4.53).
Inspired by the global case, we will compute the overlap of two states (4.25) in the bulk. Evaluating connection chosen to be the AdS 3 connection; for the choice (5.11) the assignment is: Thus the operation we want to realize is where we use the notation from (5.5). The more interesting one is the dilatation, which acts by rescaling z 1 (note: actually z 1 , not the second argument of K) by a factor of e 2σ : exp (2µz 1 ∂ 1 ) K(ρ, z 1 ,z 1 ) = K(ρ, e 2µ z 1 ,z 1 ) [O] . (E.7) However, due to the form of the integral kernel, we have the following relation: K(ρ, e 2µ z 1 ,z 1 ) = K(ρ + µ, z 1 ,z 1 ) . (E.8) (where this is now a relation that works for the arguments of K). To see this, note that dzdz e −2ρ − (z − e 2µ z 1 )(z −z 1 ) which is (5.7) in the text.