Worldline approach to semi-classical conformal blocks

We extend recent results on semi-classical conformal blocks in 2d CFT and their relation to 3D gravity via the AdS/CFT correspondence. We consider four-point functions with two heavy and two light external operators, along with the exchange of a light operator. By explicit computation, we establish precise agreement between these CFT objects and a simple picture of particle worldlines joined by cubic vertices propagating in asymptotically AdS$_3$ geometries (conical defects or BTZ black holes). We provide a simple argument that explains this agreement.


Introduction
Conformal field theories (CFTs) in two dimensions are specified by a central charge, a list of primary operators, and their OPE coefficients, this data being subject to the consistency requirements of modular invariance and crossing symmetry [1,2]. The CFT data appears directly in the decomposition of correlation functions in terms of conformal blocks, where the coefficient of each conformal block is given in terms of the OPE coefficients among primary operators. The conformal blocks correspond to the virtual exchange of a given primary operator and all of its Virasoro descendants. The conformal blocks are completely fixed by conformal invariance, although no closed form expression for them is known, except in special cases. At the same time, efficient recursion relations exist allowing one to compute the conformal block to any desired order in the conformally invariant cross ratio [3].
While all of this is ancient history, conformal blocks have received renewed attention recently, both due to their central role in the revival of the conformal bootstrap program [4], and also as a useful way to think about how local physics can emerge in the bulk in examples of AdS/CFT duality [5,6,7,8,9,10]. 2 It is the latter perspective which is closest to our considerations here. Recent work in this general direction includes the study of time evolution of entanglement entropy in excited states created by local operators [13,14,15]; the issue of universality in 2d conformal field theories [16,17]; entanglement entropy in higher spin theories [18]; and the emergence of chaos in thermal systems [19].
We are interested in conformal blocks in the semi-classical limit, corresponding to taking the central charge and operator dimensions to infinity while keeping their ratios fixed. There is excellent evidence that in this limit the conformal blocks exponentiate [20,21] as e − c 6 f ( h i c ;x) , although there exists no proof of this directly from the fundamental definition as a sum over Virasoro descendants. It is then natural to expect that the function f ( h i c ; x) can be computed from a saddle point analysis, and indeed several realizations of this are known, as we discuss.
Since conformal blocks are fixed by conformal symmetry, they can be computed in any theory with this symmetry, provided of course that the desired values of the central charge and operator dimensions are available. If we have a family of theories with a variable central charge, where 1/c plays the role of , then we can think of taking c → ∞ and computing f ( h i c ; x) by solving some "classical equations". An example is provided by Liouville theory, and this motivates the monodromy approach to the computation of semi-classical conformal blocks [20,21]. This approach boils down to solving an ordinary differential equation, ψ ′′ (z) + T (z)ψ(z) = 0, with prescribed monodromy. The monodromy condition fixes certain "accessory parameters" in T (z), and these can be used to reconstruct the conformal block by integration. 2 Most of this work is in the context of d > 2 dimensional CFT, where the conformal group is finite dimensional, and explicit formulas for the conformal blocks are known [11,12].
Another useful realization is in terms of gravity in AdS 3 . As was emphasized in the context of holographic entanglement entropy [22,23], the monodromy approach can be recast in terms of the problem of finding a solution of Einstein's equations with specified boundary behavior. In this context, T (z) is identified as the boundary stress tensor of AdS 3 gravity. The authors of [22,23] were thereby able to derive from first principles the Ryu-Takayanagi formula in this context, equating the entanglement entropy with the (regulated) length of a bulk geodesic.
Especially relevant for our purposes is the illuminating paper [10] which, among other things, gave an AdS 3 bulk interpretation of vacuum conformal blocks for the case in which two of the external operators are heavy and two are light, in a sense made precise below. The picture is that the heavy operators set up a classical asymptotically AdS 3 geometry corresponding to a conical defect or BTZ black hole, and the light operators are described by a geodesic probing this background solution.
Here we extend the results of [10] in several directions. Firstly, we consider the case of nonvacuum conformal blocks corresponding to an exchanged primary O p , and also allow for the light operators to have distinct conformal dimensions. The bulk picture is that we now have three geodesic segments in the background geometry, one segment for each of the light operators, with the segments meeting at a cubic vertex; see figure 1. The segment corresponding to the operator O p has one endpoint on the vertex, with the other ending at the conical defect or BTZ horizon. The resulting formula for the conformal block is more intricate than the case considered in [10], but is still quite compact. By expanding this result for small cross ratio, we are able to check against the corresponding result obtained directly from the CFT recursion relation, and we indeed find agreement. We also verify agreement with the monodromy approach.  Fig. 1: Geodesic configuration. The disk represents a slice of a conical defect or BTZ geometry, as supported by the heavy operator. Light operators are represented by geodesic segments as shown.
We also provide a simple argument that explains the agreement between the bulk geodesic approach and the monodromy approach. This is done by thinking about the backreaction on the metric produced by the configuration of particle geodesics. The relation to the CFT monodromy approach is especially transparent in the Chern-Simons formulation of AdS 3 gravity [24,25], and this lets us establish that the solution sourced by the geodesics is in direct correspondence with a solution of the monodromy problem.
The geodesic approximation is only valid to first order in the light operator dimensions, while the full conformal block of course receives contributions at all orders. In the bulk we can think about solving Einstein's equations order by order to compute these corrections. Equivalently, this can be phrased in terms of solving the monodromy problem at higher orders. We verify that the second order solution indeed yields a result in agreement with that obtained from the recursion relation.
Before proceeding, we would like to emphasize that a motivation for carrying out the work presented here is to eventually apply these results to computations that are not entirely dictated by symmetry. For instance, semi-classical correlation functions computed in the BTZ geometry display a specific form of information loss [26]. In the present context such correlation functions arise in a manner in which it is clear what effects have been thrown out, namely non-vacuum blocks and 1/c corrections, and this might be a useful way to think about what is needed to restore purity. For a calculation of one loop corrections of holographic entanglement entropy, see [27].

Conformal blocks
In this section we briefly review the definition of conformal blocks in 2D CFT, as well as some of the methods available to compute them, either in a series expansion or in the semi-classical limit.

Definitions
We follow the conventions of [2]. The correlation function of four primary operators is expanded as 3) Higher order terms are readily computed using a convenient recursion relation [3].

Heavy-light correlators, and the semi-classical limit
It will now be convenient to define a rescaled stress tensor and rescaled conformal weights. If T CF T and h denote the usual stress tensor and conformal weight, we now define T and ǫ as

4)
In terms of which the OPE is The semi-classical limit of the conformal blocks is defined by taking c → ∞ at fixed ǫ, where ǫ refers to the external operators O 1,2,3,4 as well as the internal primary O p . In this limit there is good evidence, though no direct proof, that the conformal blocks exponentiatẽ This can be verified directly to the first few orders in the x expansion using the recursion relation obtained in [3] and reviewed in appendix A. It has been verified to high order in [28]. We will be interested in the case in which the four-point function involves two operators of equal dimension ǫ h , with the other two operators having dimensions ǫ l and ǫ ′ l . Here the subscripts stand for "heavy" and "light", in the sense that we will be consider ǫ l , ǫ ′ l ≪ 1 so that we can expand perturbatively in these quantities. Note though that our usage of "light" is somewhat nonstandard, since this label is often applied to operators whose dimension h is held fixed as c → ∞, which is not the case here.
In particular, the correlator of interest is (suppressing henceforth the dependence on anti-holomorphic quantities) expanded in the x → 1 OPE channel. As indicated, we can think of this correlator as the two-point function of light operators in the excited state created by the heavy operators.
To expand in conformal blocks, first use invariance under so that the expansion is Suppressing the labels, we then write, in the semi-classical limit, As noted above, the recursion relation can be used to computef (x) in a power series, although the results rapidly get complicated. To simplify we further expandf (x) in the light operator dimensions. More precisely, we replace and then expand in δ. At linear order in δ we find while at quadratic order in δ we havẽ 13) The expansions continue to higher orders in x and δ.

Monodromy method
A convenient method for computing the semi-classical conformal block is the monodromy method. This is well reviewed in [21,10], and so we will be brief. We consider the differential equation T (z) can be thought of as the stress tensor in the presence of the operators appearing in the four-point function (2.7). Up to the free parameter c x , its form is fixed by demanding that it have the correct double poles and asymptotic behavior. Noting that the simple pole term at z = x is T (z) ∼ c x z−x , along with the OPE (2.5), we see that c x is related to the x-derivative of the conformal block. We can then integrate as is determined by demanding that the two independent solutions of (2.14) undergo a specific monodromy as we go around a contour that encloses the singularities at z = 1, x, the monodromy being fixed by the dimension of the primary ǫ p . Specifically, the monodromy matrix M is required to have eigenvalues The problem is tractable in perturbation theory, where we can expand in the dimensions of the light operators, or in 1 − x. For the former we write The equations are then The zeroth order solutions are We then obtain solutions at the next two orders as In this form it is easy to read off the monodromy matrix as we encircle z = 1, and similarly for M (2) . The integrals are easily computed using residues. Now let us give a few examples.
• First order in ǫ ′ l = ǫ l , ǫ p This case was consired in [10]. Using M yielding This gives (choosing the integration constant so thatf (0) = 0) and expanding in x we verify agreement with (2.12).
• First order in ǫ ′ l = ǫ l , ǫ p We now allow for two independent light operators. The result turns out to agree precisely with (3.30), the latter being obtained from a bulk computation.
• Second order in ǫ ′ l = ǫ l with ǫ p = 0 In this case we solve M (2) = 0. In general this is quite complicated due to the complexity of the second order solution ψ (2) . To give a very simple illustration, we choose the light operators as above, and further expand to lowest nontrivial order in 1 − x, which gives This matches the result from the recursion relation (the term one order beyond those given in (2.13)). It is simple to extend to higher orders in x, if desired.

Semi-classical conformal blocks from bulk geodesics
In [10] it was observed that the conformal block with ǫ ′ l = ǫ l and ǫ p = 0 can be reproduced by computing the length of a geodesic in an asymptotically AdS 3 background. In this section we show how to extend this to reproduce the conformal block for general values of ǫ ′ l , ǫ l , ǫ p , assuming all are small.

Setup
It is conceptually easiest work in global coordinates corresponding to CFT on the cylinder. The geometry related to the operator O h by the state-operator correspondence is which is obtained from global AdS 3 by t → αt, φ → αφ, although in (3.1) we take φ ∼ = φ + 2π. Here α is the same quantity as in (2.21). For α 2 > 0 (3.1) represents a conical defect with a singularity at ρ = 0. We define For α 2 < 0 we instead have a BTZ black hole with event horizon at ρ = 0, provided that we change the identifications to t ∼ = t + 2π. Since we take φ ∼ = φ + 2π, we will mainly restrict to the α 2 > 0 case. Now consider computing AdS correlation functions of the operators O l and O l ′ . The light operators are placed at: We work on a fixed time slice of the conical defect, so that w = φ. An operator of dimension (h, h) is dual to a bulk field of mass m = 2 h(h − 1). Since we are always assuming h ∼ c ≫ 1, this is m = 2h ≫ 1. In this regime, the scalar field is well approximated by a point particle, and correlation functions can be computed in terms of regulated geodesic lengths. To compute the conformal block involving the exchanged primary O p we propose the following simple prescription, which we will subsequently verify in particular cases, and then justify on general grounds. Working on a fixed t slice, we take the external light operators to be inserted as in (3.2). Attached to the boundary point w = 0 is a segment of geodesic corresponding to a particle of mass m l ′ = 2h l ′ . Similarly, a m l = 2h l geodesic segment is attached at w. The primary operator is represented by a geodesic segment attached at ρ = 0 in the background (3.1). The three geodesic segments meet at a cubic vertex, located at some point in the interior of (3.1). The worldline action is, after stripping off a factor of c 6 , where the Ls denote the regulated lengths of the geodesics. 4 The location of the cubic vertex is obtained by minimizing S. This yields a contribution to the correlation function on the cylinder G(w) = e − c 6 S(w) .

(3.4)
This is related to the conformal block on the plane by the conformal transformation z = e iw , where the factor of z −h l takes into account the transformation of the operator O l . In terms of the functions f andf , Each geodesic segment is obtained by extremizing the standard worldline action I = ǫ dλ g µν dx µ dλ dx ν dλ , where we choose λ to be proper length. Geodesics thus obey where the conserved momentum conjugate to φ is This gives cos ρ = 1 The regulated length is defined by imposing a cutoff near the boundary, at cos ρ = Λ −1 . As a simple example, consider the case ǫ ′ l = ǫ l with ǫ p ≪ ǫ l . In this regime we can first work out the geodesic connecting the two boundary points. Attached to this will be a geodesic connecting the midpoint with ρ = 0. To first order in ǫ p /ǫ l we can neglect the fact that the latter geodesic will "pull" on the former. A simple computation yields for the ǫ l geodesic cos ρ = sin αw 2 cosh λ . (3.10) Its regulated length is given by 2L l = 2λ cos ρ=Λ −1 = 2 ln sin αw 2 + 2 ln Λ 2 . (3.11) The length of the ǫ p geodesic is The contribution to the correlator is then where we dropped an uninteresting w-independent prefactor. After a little bit of algebra, we then find in agreement with (2.26). Note that although we assumed ǫ p ≪ ǫ l , since the result is linear it turns out to agree with (2.26), where no such assumption was made.
In the above, the worldines were taken to extremize the action, which implies that the stress tensor of the particles is covariantly conserved. This is needed in order that the particles can consistently couple to gravity, a fact that we will need later when we explain the relation between the geodesic approach and the monodromy approach. In this regard, we also note that we only need to require that the worldlines extremize the action, and they need not furnish a global minimum.

Result for general light operators
Working in a fixed-t slice, our goal is to find the minimum value of the worldline action (3.3) as a function of the angular separation ∆φ of the light operators O l , O l ′ . We are interested in the regime where the action is minimized by a configuration in which all three worldlines' lengths are nonzero. This requires each of the three weights ǫ l , ǫ ′ l , ǫ p to be less than the sum of the other two. In the complementary case, where one of the three weights is greater than or equal to the sum of the other two, it is simple to minimize the action but the relation to the CFT picture breaks down. By varying the action with respect to the location of the cubic vertex, one finds that at the vertex ǫ lẋ where the dots denote derivatives with respect to a proper length parameter that increases away from the vertex. The problem of computing the minimum value of S is identical to that of computing the energy in equilibrium of three elastic "rubber bands" whose tensions are ǫ l , ǫ ′ l , ǫ p , independent of their lengths. Equation (3.15) is the equilibrium condition of vanishing net force on the cubic vertex.
The angular and radial components of equation (3.15) are where ρ is the radial coordinate of the cubic vertex and is the conserved momentum along worldline l that comes from the metric's φ-translation isometry, with p ′ φ defined similarly. The sign of each square root in equation (3.16b) is positive if the corresponding worldline approaches the cubic vertex from radially outward and negative if it approaches from inward. When |ǫ 2 l − ǫ ′2 l | < ǫ 2 p equations (3.16a, b) can only be true if both square roots are positive, meaning both worldlines must approach the cubic vertex from the outward direction. On the other hand when |ǫ 2 l − ǫ ′2 l | > ǫ 2 p equations (3.16a, b) require the worldline corresponding to the smaller of ǫ l , ǫ ′ l to approach the vertex from the inward direction. Equations (3.16a, b) also imply where µ > 0 is defined by µ 2 is positive as a consequence of the assumption that each of the three weights ǫ l , ǫ ′ l , ǫ p is less than the sum of the other two.
In light of equation (3.16a) it is useful to characterize the shapes of geodesics l, l ′ with a single parameter rather than the redundant set (p φ , p ′ φ ). We define and assume without loss of generality that P ≥ 0. Equation (3.18) implies that P is in fact the tangent of the ρ coordinate of the cubic vertex. When both worldlines approach the vertex from outward, the angular separation ∆φ between their endpoints is The integration variable u is related to the radial coordinate ρ by u = (tan ρ)/P . When one of the worldlines approaches the vertex from inward instead, one must add to the right hand side of equation (3.21) the angle that worldline sweeps out while its radial coordinate is smaller than that of the cubic vertex. An expression for that angle is where ǫ s is the smaller of ǫ l , ǫ ′ l . We will need to solve equation (3.21) for P as a function of ∆φ. One can put the equation in the form e iα∆φ = cos γ cos ψ + i sin ψ cos γ + i sin γ sin ψ cos γ ′ cos ψ + i sin ψ cos γ ′ + i sin γ ′ sin ψ , (3.23) where the angles ψ, γ, γ ′ are all between 0 and π/2 and are defined by cot ψ = P , cos γ = µ/ǫ l , cos γ ′ = µ/ǫ ′ l . We may trade out ψ for a new variable z defined by cos ψ = (z + z −1 )/2, sin ψ = (z − z −1 )/2i. Equation (3.23) is equivalent to This is a quadratic equation for z. It can be rewritten as a quadratic equation for P and solved to yield We have introduced θ := α∆φ/2.
The expression for P in equation (3.25) gives, via equation (3.20), the conserved momenta p φ , p ′ φ of worldlines l, l ′ in terms of the angular separation ∆φ between their endpoints. In the case where one worldline approaches the cubic vertex from inward equation (3.25) continues to hold. Now, the variation of the action (3.3) with respect to the locations x µ l , x µ l ′ of the worldlines' boundary endpoints is Here the dot denotes a derivative with respect to a proper length parameter that decreases away from the boundary. (In the picture where the worldlines are rubber bands and S is their energy, the right hand side of equation (3.26) is the work required to move the ends of the rubber bands.) It follows that the change in the action from a small increase in the angular separation ∆φ of the wordlines' endpoints is The endpoints are at the locations of the light operators, (3.2). Their angular separation is w, and so ∂S ∂w = αµP (3.28) where P is given by equation (3.25), and the variable θ is related to w by θ = αw/2. Equation (3.28) can be integrated to give S(w) = (ǫ l + ǫ ′ l ) ln sin θ + ǫ p arctanh cos θ 1 − β 2 sin 2 θ − |β|ǫ p ln |β| cos θ + 1 − β 2 sin 2 θ (3.29) where β := (ǫ ′ l − ǫ l )/ǫ p . Plugging this into (3.6), it yields This result can be verified by checking that it agrees with the result from the monodromy approach, and also by expanding in z and verifying agreement with the result of the recursion relation.

Relation between geodesic and monodromy approaches
In this section we explain why computing the action for bulk geodesics gives answers for the semi-classical conformal blocks that agree with those of the monodromy approach. The argument is very simple. Given a geodesic configuration, we can work out the linearized metric perturbation sourced by the particles. This perturbation gives rise to a boundary stress tensor that can be identified with T (z) appearing in the monodromy method. The monodromy conditions arise by requiring that Einstein's equations are obeyed at the geodesics. Each geodesic segment carries a conserved momentum p φ , which can be identified with the accessory parameter c x , since it appears as the residue of a simple pole in T (z). Finally, p φ = dS dφ , just as c x is related to the derivative of f . This establishes the relation between the geodesic action S and the function f appearing in the monodromy method.
We first show how to relate the simple pole in T (z) to p φ . We consider a metric with the usual Fefferman-Graham expansion near the boundary, and take g (0) µν dx µ dx ν = dwdw. We also include a particle with worldline action with λ equal to proper length. We consider the case that the worldline pierces the boundary at some location w 0 , and we wish to consider the Einstein equations near this point. Expanding the Einstein equations for large ρ to first order, the only non vanishing equations read g where the (rescaled) mass and canonical momentum are c 6 ǫ = h and c 6 p µ = 2h dx µ dλ , c being the usual Brown-Henneaux central charge, c = 3ℓ/2G. Now, the components of g (2) µν are just the boundary stress tensor, g (2) µν = T µν (rescaled as in (2.4)). Thus, Using ∂ w 1 w = 2πδ (2) (w), and p w = p φ 2 we find where . . . denote non-singular terms. By the usual relation between canonical momentum and the variation of the action under a change of boundary conditions we have c 6 p φ = dS h dφ . The relation to the formulas appearing in monodromy approach is now clear: just as c x appeared as the residue of the simple pole in the stress tensor and was related to derivative of f , the same is true of p φ , now related to the derivative of the geodesic action.
The final step to demonstrate the equivalence of the two approaches is to to show how the monodromy conditions arise in the bulk. We first note that the full bulk metric will take the form This is a solution of the source free Einstein's equations if ∂ w T = ∂ w T = 0. At the location of the particle worldlines these equations are corrected, as in (4.4).
We now pass to the Chern-Simons formulation of 2+1 gravity with negative cosmological constant. This was employed in a closely related context in [18]. The metric is replaced by an SL(2) × SL(2) connection, Now consider the holonomy of this connection around a closed contour C, which we take to lie at fixed ρ. Focussing just on A, In the absence of matter the holonomy would be trivial, since Einstein's equations are equivalent to flatness of the connections. But for a contour that encircles a particle worldline, the holonomy will pick up a contribution fixed by the mass of the particle. To relate this to the monodromy approach, we note that computing Hol[C] is equivalent to computing the monodromy of the system of differential equations where ψ is a two component vector. The bottom component obeys which we recognize as the ODE appearing in (2.14). We focus on a contour that encircles the two operator insertion points at 0 and w. This contour encircles the worldline corresponding to the exchanged primary O p . It is then clear that if T is such that the monodromy of the differential equation (4.10) is related to ǫ p in the correct way, then the holonomy of the Chern-Simons connection will be such that we solve the Einstein equations in the presence of the particle worldline.
Summarizing, we see that given a linearized solution of Einstein's equations in the presence of particle worldlines we can find a solution of the monodromy problem. Further, the action of the bulk solution agrees with the function f appearing in the exponent of the semi-classical conformal block.

Discussion
We have achieved a clean AdS 3 bulk interpretation of the semi-classical conformal blocks, extending the observations in [10]. To linear order in the light operator dimensions, we simply have to find the equilibrium configuration of three geodesic segments joined at cubic vertex, propagating in a geometry dual to the heavy operators. We close with a couple of observations and questions for the future.
It is straightforward in principle to go to higher orders in the light operator dimensions. From the bulk point of view, this just corresponds to solving Einstein's equations order by order in Newton's constant. Phrased in the language of the Chern-Simons formulation, the problem consists of finding locally flat connections with specified holonomies around contours representing the locations of the worldlines. As at linear order, this is the same problem as in the monodromy approach.
It should also be straightforward to consider conformal blocks for higher point functions. In this case we would have more worldlines attached with additional cubic vertices. Conformal blocks on higher genus Riemann surfaces could also be considered.
We have focused here on the semi-classical conformal block, which is the leading term in the large c expansion. More generally, we can think of writing In terms of the loop expansion in the bulk, we expect thatf (1) is given by the effects of 1-loop fluctuations around the classical background. We can use the recursion relation to compute terms inf (1) (x). For example, setting ǫ ′ l = ǫ l and ǫ p = 0, we find, at linear order in δ, (5296ǫ 2 h + 37392ǫ h − 9675)ǫ l x 4 + . . . (5.2) where the . . . denote terms higher order in ǫ l and in x. It is important to note that to obtain this we first extract the large c asymptotics for general values of ǫ l , ǫ ′ l and ǫ p , and only afterwards set ǫ ′ l = ǫ l and ǫ p = 0. The terms in the first line are puzzling, as they are nonzero even upon setting ǫ l = 0. It will be interesting to understand their physical origin.
Once the bulk interpretation of the full conformal block is established, including the subleading 1/c effects, we can think of using this information to interpret specific CFT correlators. It will be interesting to apply this to the black hole context, where the problem of information loss can be phrased in terms of such correlators.
Another interesting direction to consider is the extension to higher spin theories, where the Virasoro algebra is enhanced to a W-algebra. In [18] it was established that the semi-classical vacuum block admits a bulk realization in terms of a Wilson line [29,30,31] embedded in an asymptotically AdS 3 background with higher spin fields excited. A natural question is how to extend this story beyond the vacuum block.