Conformal Bootstrap, Universality and Gravitational Scattering

Conformal Bootstrap, Universality and Gravitational Scattering

Abstract

We use the conformal bootstrap equations to study the non-perturbative gravitational scattering between infalling and outgoing particles in the vicinity of a black hole horizon in AdS. We focus on irrational 2D CFTs with large and only Virasoro symmetry. The scattering process is described by the matrix element of two light operators (particles) between two heavy states (BTZ black holes). We find that the operator algebra in this regime is (i) universal and identical to that of Liouville CFT, and (ii) takes the form of an exchange algebra, specified by an R-matrix that exactly matches with the scattering amplitude of 2+1 gravity. The R-matrix is given by a quantum 6j-symbol and the scattering phase by the volume of a hyperbolic tetrahedron. We comment on the relevance of our results to scrambling and the holographic reconstruction of the bulk physics near black hole horizons.

\institution

PU

Department of Physics, Princeton University, Princeton, NJ 08544, USA

\authors

Steven Jackson, Lauren McGough and Herman Verlinde1

1 Introduction and Summary

In this paper we study the role and implications of the conformal bootstrap equations for the AdS/CFT correspondence. We are particularly interested in how the non-perturbative gravity regime emerges from the conformal field theory, and vice versa. 2D CFT has special properties, thanks to the infinite conformal symmetry and left-right factorization of its correlation functions [1, 2, 3, 4, 5]. Similarly, 2+1-D gravity is special due to the absence of a local metric excitations [6, 7]. Both sides possess a precise topological and analytic structure, in the form of braiding and fusion matrices, that satisfy non-trivial consistency conditions. Our plan is to display this structure on both sides, and use it to establish a precise non-perturbative holographic dictionary.

The existence of this geometric dictionary is not a new insight, and key elements were anticipated since long before the discovery of AdS/CFT [8, 9]. We will not add any major technical advance to the subject: most of the formulas that we will use have been obtained over the past two decades in work by others [10, 11, 12, 14, 13, 15, 16, 17, 18, 19, 20]. Our main new observations are that

  • there exists a general kinematical regime in which the operator algebra of irrational CFTs with takes a universal form identical to that of Liouville CFT

  • this operator algebra can be cast in the form of an exchange algebra, specified by an R-matrix that exactly matches with the scattering amplitude of 2+1 gravity.

Because we have tried to make the exposition somewhat self-contained, this paper is rather long. We will therefore start with a statement of the problem and a summary of our results. Other recent papers that exploit the properties of conformal blocks to extract new insights for holography include [21, 22, 23, 24, 25].

1.1 Statement of the problem

Consider a 2D CFT with large central charge and a weakly coupled holographic dual. It has a dense spectrum of primary states with asymptotic level density governed by the Cardy formula. In the dual gravity description, the heavy states , with energy are expected to describe BTZ black holes with metric [26, 27]

(1)

with the AdS radius and the Schwarzschild radius.

Consider the Heisenberg state obtained by acting with a local CFT operator on a primary state2 . The corresponding Schrdinger state

(2)

for describes a bulk geometry with a matter perturbation that is predestined to reach the boundary at time . It seems reasonable to interpret this perturbation as a particle A that escapes from near the horizon of the black hole of mass . One of the goals in this paper is to put this identification through some non-trivial tests and make it more explicit.

The specific test that we would like to perform is to exhibit the gravitational backreaction of particle A through its interaction with some other particle B, and vice versa. To this end, it is useful to make a spectral decomposition of the local operator

(3)

into components that increase the energy of the primary state by a specified amount

(4)

Here denotes the relevant OPE coefficient. The descendants are terms given by products of left and right-moving Virasoro generators acting on the primary state .3 We interpret this component as describing the partial wave in which the bulk particle A has specified energy . We wish to confirm this interpretation, by exhibiting how the CFT encodes the full semi-classical backreaction of particle A on the black hole geometry.

Figure 1: Penrose diagram describing an eternal BTZ black hole with an infalling and outgoing matter perturbation [29]. The two particles collide near the horizon and affect each others’ trajectory via a gravitational shock wave interaction. Alternatively (as shown in the figure) one may represent the effect of the shock wave as a shift in the location of the event horizon.

Let us now introduce [29] a second particle B, created by acting with another local operator at some earlier time

(5)

Here we fixed the total energy of the bulk particles and to be .

The Penrose diagram of the semi-classical geometry associated to the state (5) is shown in figure 1. It shows an eternal BTZ black hole of mass with two matter pulses, one outgoing and one infalling. The stress energy of each particle induces a corresponding shift of the BTZ mass parameter across its trajectory. The two particles collide close to the horizon, creating a future black hole region of mass . The value of is determined by the collision energy of the two particles, which in turn is set by , the energies and , and the time difference between when B departs and A arrives at the AdS boundary.

The total geometry of figure 1 is governed by 6 mass parameters

(6)

We will assume that particle and are both massless and set . Our formulas in sections 2 and 3, however, remain valid for general and .

Particles A and B affect each others trajectory via a gravitational shock wave interaction. Initially, the outgoing particle A hovers very close to the event horizon, until just before it escapes to infinity. When the infalling particle B passes through, it slightly shifts the location of the event horizon. As a result, particle A ends up moving even closer to the event horizon, or it may even end up inside the black hole. Either way, its future trajectory is strongly affected by the shift. Thanks to the exponential redshift associated with black hole horizons, this effect grows exponentially with the time difference between when B falls in and when A was supposed to come out. In a suitable kinematical regime, this gravitational effect dominates all other interactions between the two particles.

Gravitational shock wave interactions near black holes were studied by Dray and ’t Hooft [28] and in subsequent work [30].In the context of AdS/CFT, they were recently used by Stanford and Shenker to provide new evidence that CFTs with AdS duals are fast scramblers [29]. It was shown in [30] that the geometric effect of the shock wave can be captured by means of a so-called exchange algebra between infalling modes and outgoing modes of the schematic form

(7)

where are the matrix elements of a unitary scattering operator that shifts the location of one operator by an amount proportional to the Kruskal energy of the other.

This exchange algebra is a characteristic property of infalling and outgoing modes near black horizons in any dimension [30]. However, gravitational interactions in 2+1 dimensions have special characteristics that make this description especially natural. In 2+1 gravity, backreaction due to localized matter sources is described by means of simple global geometric identifications [6, 26, 27]. As a result, point particles and black holes enjoy long range interactions similar to the topological braiding relations in systems with non-abelian statistics.4 The basic operation in theories with braid statistics is the R-operation, that interchanges the location of particles, or equivalently, the ordering of two operators. We therefore expect that the braid statistics of 2+1-D theories with gravitationally interacting quasi-particles can be captured by a similar type of R-operator.

How would this gravitational exchange interaction show up in the dual CFT? It has long been known that the operator algebra of (chiral) vertex operators in 2D CFT can be cast in the form of an exchange algebra [5]. This property underlies the celebrated relationship between rational CFTs and 2+1-D topological field theories of particles with braid statistics [31]. In the following we will argue, with considerable supporting evidence, that the same algebraic structure extends to irrational CFTs with large central charge. Specifically, we will show that the components of local operators and in 2D CFT must satisfy a universal exchange algebra of form

(8)

This exchange relation is an exact property of the CFT, that follows by analytic continuation from the braid relations between chiral vertex operators in the euclidean domain. It is a causal relation that holds whenever the local operators and are time-like separated. The matrix depends on all mass parameters and is known as the R-matrix or braid matrix, which prescribes the monodromy properties of conformal blocks of the CFT. It encodes the value of OPE coefficients, and satisfies a number of non-trivial consistency relations analogous to Yang-Baxter equations. Finding the explicit form of the R-matrix is equivalent to finding a solution to the conformal bootstrap (and thus in general a difficult task). Hence in a concrete sense, the exchange algebra provides a complete characterization of the CFT.

Our goal in this paper is to derive the exact form of the R-operation and its associated exchange algebra, both from 2+1 gravity and in 2D CFT. We will find that the answers precisely match, for the simple reason that the two calculations are essentially identical.

1.2 Semi-classical scattering

How would one derive the explicit form of the scattering matrix from the point of view of the gravity theory? As a useful warm-up exercise, we first look at the classical scattering process. Let us pick values of the four masses , in eqn (6), i.e. we fix the initial mass of the black hole, the masses and of the two particles, and the total energy .

From fig. 1 it is clear that once we know the initial energy of the outgoing mode A and the initial time difference , the subsequent time evolution is uniquely determined: the final energy of particle B and the new arrival time are given functions of and . We could also reverse time and pick the final energy of the infalling mode, and the time difference between the final arrival time of particle A and the time when particle B would have originated from if particle A had not been there. The initial energy and the actual departure time are determined as given functions of and .

The scattering process of fig 1 for spherical matter shells was recently studied in [29]. We can read off the relations between the four parameters ( and from their paper. As explained in [29], following [28], one can obtain the colliding shock wave geometry by means of a simple gluing procedure. The radial location of the collision point is determined by a matching relation that equates the values of the factors (see eqn (1)) of all four BTZ black hole geometries,

(9)

We are interested in the universal behavior when the collision takes place very close to the horizon. In this regime, the distance between collision radius and the two Schwarschild radii and scales exponentially with the time differences and

(10)

where is the surface gravity and is a time delay, given by

(11)

Here we recognize the characteristic exponential redshift effect near black hole horizons. Combining eqns (9) and (10), one derives the following relations [29]

(12)
(13)

Eqn (12) determines as a function of and the time difference . We see that, due to the exponential growth in , quickly becomes bigger than . Once this happens, eqn (13) no longer yields a real solution for . This is not surprising: as seen from fig 1, when the energy of mode becomes negative, indicating that its trajectory has been shifted to behind the horizon.

Now let us replace particle A and particle B by quantum mechanical wave packets. As explained in detail in [30], we should anticipate that the second quantized mode operators and that create both asymptotic wave packets do not commute but satisfy an exchange relation. For spherical wave-packets – which can be simultaneously localized in time and energy – and in the leading order semi-classical limit, we expect that this exchange relation takes the form

(14)

Since acts in the future of , this relation is in perfect accord with causality. It expresses the causal effect that the trajectory of A, after its encounter with B, is shifted by the specified amount, relative to its original trajectory. The time shifts can be computed by a similar calculation as the one that gave us the relations (12)-(13). One finds that

(15)

Note that the time delay indeed becomes infinite when approaches .

The dependence of the scattering phase on the initial energy and final energy follows from the usual rules of geometric optics, via the Hamilton-Jacobi equations

(16)

These equations ensure (or can be derived from the fact) that both sides of the exchange relation (14) have the same dependence on and . To integrate the H-J equations, we are instructed to compute the time differences and as functions of and .

(17)

Standard H-J theory tells us that the result for is equal to the total action evaluated on the unique classical space-time trajectory that interpolates between the state where A has initial energy and B has final energy . In the regime that there exists a real trajectory, , , the above equations integrate to

(18)

Equivalently, defines the generating function of the canonical transformation between the initial and final canonical variables and .

In section 2 we will present a more refined and general description of this semi-classical scattering process, in terms of holonomy variables. We will obtain a more general exact expression for the scattering phase in terms of dilogarithms that reduces to the above answer in the regime of interest. As we will see, the more general answer has a natural geometrical interpretation as the volume of a hyperbolic tetrahedron with dihedral angles specified by the holonomies of the BTZ geometry and conical defects created by the particles involved in the scattering. Our main goal in the following will be to reproduce this semi-classical scattering matrix from the CFT side.

1.3 Scattering phase from the modular bootstrap

The R-matrix that specifies the exhange algebra of chiral vertex operators in 2D CFT satisfies several consistency relations that are equivalent to the conformal bootstrap equations. The relations express the physical principles of locality and associativity of the operator algebra, and analyticity of the correlation functions. The conformal bootstrap usually refers to a set of equations for conformal blocks and OPE coefficients. The R-matrices are part of the general set of modular matrices that specify the monodromy properties of chiral correlation functions. The program of finding the monodromy matrices via their consistency relations is called the modular bootstrap.

For integrable theories like rational CFTs, the R-matrices that solve all the above consistency relations are explicitly known. Our interest, however, is in irrational CFTs with large central charge and with holographic duals. These theories are not integrable and the R-matrices are not known and seemingly impossible to compute. So how can one hope to get mileage from considering the modular bootstrap for this case?

The main distinction between irrational and rational CFT is that the sum over intermediate states on the right-hand side of (8) runs over a dense spectrum with an infinite set of labels, rather than over a small finite set. In our specific setting, the sum is dominated by intermediate states in the densely populated neighborhood of the heavy state . It is reasonable to approximate the spectral density in this region by a continuum, and replace the sum over states by an integral, weighted with an appropriate measure that reflects the Cardy growth of states. This simplification, which is similar to the hydrodynamic approximation of a gas or liquid, comes with an important benefit: just like the hydrodynamic limit, it ignores irrelevant microscopic details that distinguish individual CFTs and produces a universal and more tractable set of equations that capture the essential macroscopic properties. The appearence of such a universal regime in the CFT is of course expected from the point of view of dual bulk theory, whenever one enters a kinematical regime in which gravitational effects dominate over all other interactions.

Figure 2: Overview of our results. We study irrational 2D CFTs with a string theory dual on AdS. If the CFT has large , a sparse light spectrum [32] and only Virasoro symmetry, the AdS theory is expected to have two regimes in which gravity dominates: a high energy regime close to a black hole horizon, and a low energy regime at long distances. In the CFT, the dynamics in these regimes is dominated by the high entropy part of the spectrum, and the modular properties of conformal blocks are captured by Liouville theory. There exists a precise mathematical identification between 2+1 quantum gravity, defined as the quantization of its classical phase space, and Liouville modular geometry.

To formulate a self-consistent set of conformal bootstrap equations in this approximation, in essence all one needs to do is to replace sums over a dicrete spectrum by integrals over a continuum. The resulting bootstrap program is equally restrictive as the discrete variant. Moreover, through contributions by many authors, it has essentially been completed! By now the modular geometry of Virasoro conformal blocks is fully worked out [11, 12, 13, 14, 34]. Unique and explicit expressions for the universal R-matrix, spectral density, and OPE coefficients are available and ready to be applied to the study of AdS/CFT.

Results about Virasoro modular geometry are usually reported in the context of studies of Liouville CFT, or equivalently, in the context of the quantum theory of Teichmller space5 – which as we will see, beautifully connects with the quantum geometry of 2+1-D gravity. We have collected the basic elements of this story, and some of the explicit formulas, in sections 2, 3 and 4 and in the Appendix. The main statements are summarized as follows.

Virasoro conformal blocks span a linear vector space, which can be endowed with a Hilbert space structure and an inner product [13, 14]. The spectral density of the Hilbert space is uniquely fixed by the requirement that modular and braid operations are implemented as unitary transformations. Crucially, as we will review in section 4, one finds that the level density for matches with the Cardy formula [33].

The fusion and braiding properties of the Virasoro conformal blocks are captured by the category of representations of the quantum group , the -deformed universal enveloping algebra associated with the non-compact group [13]. Here each factor relates to a chiral sector of the CFT. In particular, the R-matrix which (as we will explain in section 5) governs the exchange algebra between local operators in the Lorentzian CFT [5], is given by the square (product of left times right) of the corresponding -deformed -symbol

(19)

with:

(20)

Here labels the representation of , and , , and are all related and determined by the central charge via

(21)

Explicit formulas for the quantum -symbol are known [13] but complicated (see the Appendix) and therefore perhaps not very illuminating to non-experts. However, their geometrical meaning becomes more apparent in the semi-classical limit , i.e. the limit of large central charge . In the dual theory, this is the large radius limit of AdS.

To exhibit the semi-classical behavior of the R-matrix, we write

(22)

has two natural geometrical interpretations. First, from the known expression of the quantum -symbols (19), one can show by direct calculation that, for small , becomes (up to trivial phase) equal to the volume of a generic hyperbolic tetrahedron [17, 18, 20]

(23)

specified by six dihedral angles , determined by the mass parameters (6) via

(24)

This geometric interpretation of the quantum 6j-symbols was used by Murakami and Yano to derive their exact mathematical formula for the volume of the hyperbolic tetrahedra [35]. In the present context, this relation encourages a physical interpretation of the R-matrix in terms of gravitational scattering amplitude. In the following sections, we will confirm this intuition and make it precise.

Figure 3: The hyperbolic tetrahedron . The dihedral angles at the six edges are determined by the six mass paramaters in (6) via .

The relations (8) and (22)-(23) form a cornerstone of the non-perturbative holographic dictionary between 2D CFT and 2+1 quantum gravity. On the CFT side, they express an exact property of Liouville theory which, according to our proposal, captures the universal behavior of generic irrational CFTs in the regime of interest. This proposal finds support from the gravity side, where a matching calculation of represents the geometric phase associated with a high energy scattering process dominated by the gravitational shock wave interaction.

1.4 Scrambling behind the horizon

Besides being a useful testing ground for studying and clarifying the appearence of gravitational interactions from CFT, another fascinating aspect of the shockwave dynamics near black holes is that they provide a realization of the butterfly effect [29]. On the gravity side, it is clear how this comes about: the perturbance created by the mode B, however small in energy, can still have a dramatic effect on the future time evolution of the mode A. If the moment of infall of B precedes the would-be-arrival of mode A by more than a critical value , the mode A will never emerge at the asymptotic boundary of AdS. This is a dramatic effect. Moreover, the critical time difference grows only logarithmically in the black hole mass , which is remarkably short.

How can one understand this effect from the CFT side? At some early time, the signal A is encoded in subtle phase correlations of a strongly interacting many body state. These correlations are prearranged to produce a localized coherent effect at the prescribed time , which is the arrival of particle A at the AdS boundary. The butterly effect highlighted in [29] exploits the apparent delicacy of this type of prearrangement in a strongly interacting system: it only takes a very small perturbation of the many body state to destroy the phase correlations and completely thermalize the signal. The main surprise, however, is how fast and efficient the scrambling process appears to be.

Thermalization is a complicated dynamical phenomenon and in general hard to quantify. The dual gravity description gives us new ways of analyzing the strongly coupled dynamics of CFTs. But rather than using gravity, it would be more satisfying to explain the effect directly within the CFT and in the process gain insight into the holographic reconstruction of the bulk physics and into the nature of AdS black hole horizons. The methods described in this paper are well suited for this particular problem. Once a CFT question can be translated in terms of Virasoro modular geometry and Liouville theory, the transition to the gravity side becomes relatively easy since both are weakly coupled in the regime of interest, . Hence it should be possible to gain more insight into the scrambling process with pure CFT techniques.

Figure 4: The scrambling of a signal (operator ) due to the a perturbation (operator ) at some earlier time . An observer that measures the state can detect signal only if acts on the state from the left. Passing A through B produces a new intermediate channel with energy , which for exceeds . Signal A becomes scrambled: its coherent phase information get washed out by the large entropy region of the spectrum near .

Let us imagine the following scenario. Person prepares a message A for person , destined to arrive at time . Person then perturbs the system by sending in some noise B at time . Both the signal and the noise takes the form of a wave packet, localized in time and with some approximate frequency

(25)

We imagine that wave packet A carries some information encoded in the relative phases in this sum. Observer has access to the state

(26)

Let us suppose that knows how to decode , but has no knowledge of what looks like. In other words, since can only detect , she can act with on the left of the state and decode whatever phase coherent signal she can extract from it. But since is located on the right of , the signal is hidden behind the noise. No problem, so far: all has to do is to interchange the order of and , so that acts from the left. So the exchange algebra becomes an intrinsic part of the story.

The interchange proceeds via the R-operation . Here is a microscopic property of the CFT that, according to our proposal, is well approximated by the quantum 6j-symbol (19). After applying R to the wave packets, we can use geometric optics to localize the sums over frequencies and deduce that observer sees the state

(27)

with and determined by the stationary phase condition. Given our result that matches the gravitational scattering phase, we recover the relation (12). We again find (but now from the CFT side!) that quickly grows and soon becomes bigger than . This is when, on the gravity side, signal A has disappeared from view.

The shifted time variables and are defined via the saddle point equations (15). We see that the equation for has no real solution for . Instead it gives that

(28)

The time variable has moved off to a different Riemann sheet. Picking the leading branch, we find that the semi-classical state accessible to observer takes the form (up to a phase)

(29)

This supercritical situation is depicted in fig 4. It shows that represents the intermediate channel in the transition from the initial state to the final state at energy . In the space-time diagram in fig 1, the intermediate state represents the future black hole interior. The mode , that encodes the signal, is now a lowering operator that maps the excited state to the lower state . Crucially, the entropy of the state is larger than that of the final state , in the sense that its neighborhood has the larger spectral density.

Observer tries to measure the phase information encoded in the lowering mode . Can she do it? If had been a raising operator, the answer would have been: Yes. In that case, it would map states from a lower energy band with lower level density , to a higher energy band with higher level density . So states produced by are relatively sparse within the final energy band. The excess entropy is the information that observer has to her disposal to decode the signal. On the other hand, if is a lowering operator, it maps a high energy band with entropy to a lower energy band with entropy . In this case, there is no reliable way for observer to distinguish the final state from a random state without any phase coherence. In this sense, signal has indeed disappeared behind a black hole horizon.

Now suppose that observer does in fact know what mode looks like. With this extra bit of knowledge, the entropy of the state at energy is reduced to . This is smaller than the entropy of the final state. So we’re back in the first situation in which is able to read the signal. From the dual perspective, giving observer the knowledge of mode amounts to giving her access to the black hole interior. From the interior, the lowering mode is a real particle and the state (29) is obtained by acting with a creation operator

(30)

This formula for the interior creation operator, including the inverse of the square root of the Boltzman factor, is reminiscent of the recovery operator of a quantum error correcting code. By giving observer access to the information of mode , she can perform the projection on a suitable code subspace of the Hilbert space, and recover the signal [36].

1.5 Organization

The organization of this paper is as follows. In section 2 we describe in some detail the calculation of the semi-classical scattering matrix from 2+1-D gravity. In sections 3 and 4 we summarize the main results of the modular bootstrap and of Virasoro modular geometry and its connection with quantum Teichmller space. In section 5 we discuss the exchange algebra of Lorentzian 2D CFT and establish the match between CFT and gravity. We present our concluding remarks in section 6. Some technical formulas are collected in the Appendix. Most of the rest of this paper is review of known results. The main new observation is the application of these exact results to gravitational scattering in AdS/CFT.

The motivating question of our paper, how the effect of the shockwave interactions can be derived from the structure of CFT conformal blocks, has also been studied by Dan Roberts and Douglas Stanford [37]. We are coordinating the submission of both our papers.

2 Gravitational Scattering Matrix

In this section we present the computation of from 2+1-D gravity. On the bulk side of the holographic duality, the R-matrix defines the scattering amplitude between an outgoing mode and the ingoing mode in the background geometry of a BTZ black hole. Our definition of the scattering amplitude, and strategy for computing it, are as follows.

First we identify the classical phase space of two particles inside the eternal BTZ black hole geometry. If we fix the mass of the black hole and the particles, the configuration space is parametrized by picking two points on the 2D hyperbolic cylinder. Dividing out by the global isometries leaves a two-dimensional configuration space, that can be identified with the Teichmller space of the sphere with four punctures – or more accurately, with two holes (the two asymptotic regions of the black hole) and two punctures (the two particles). If we would ignore backreaction, the phase space would simply be the tangent space to the configuration space. As we will see, after including the backreaction, the phase space takes the form , the product of two copies of Teichmller space. This space has a standard symplectic form, which follows from the 2+1-D Einstein action.

We then parametrize the initial and final kinematical state of the two particles, in terms of a convenient set of canonical variables on phase space. The initial and final quantum state are then defined as eigenstates of a commuting subset of these variables. For the initial state, this subset includes the mass of the initial black hole, while for the final state it includes the mass of the final black hole (see figs 1 and 5). The initial and final coordinate variables do not commute with each other. So in particular . The quantum scattering matrix is then defined as the unitary operator that implements the associated canonical transformation.

Figure 5: The embedding of the Penrose diagram of fig 1 in the AdS cylinder. The left figure shows the two Poincaré wedges, connected at the corner points. The black hole horizon is indicated by the purple dashed line connecting the two corner points. The right figure shows the two trajectories of the infalling and out going particle and .

2.1 Classical geometry

The classical geometry of the BTZ black hole and of point particles in 2+1-d gravity is obtained by making global identifcations in AdS. AdS is isomorphic to the group manifold; space-time points in AdS can thus be specified by group elements . The AdS metric is given by the group invariant metric

(31)

with the AdS curvature radius. The AdS isometry group acts on the group element via left- and right multiplication

To describe a non-rotating BTZ black hole of mass , one useful coordinate choice is

(32)

with , and . Here denotes the black hole Schwarzschild radius, . In this parametrization, the BTZ space-time takes the static form

(33)

The constant time slice is a 2D hyperbolic cylinder with two asymptotic regions connected by an ER bridge. The horizon is the geodesic at with geodesic length .

The BTZ metric is invariant under shifts in , but the group element (32) is not. In particular, a shift under a full period amounts via an isometry with . Note that can take any positive real value. The BTZ black space-time is thus obtained by taking a quotient of the group manifold by the subgroup generated by the holonomy . The space-time geometry of spinning black holes of mass and spin is obtained by taking the quotient via a hyperbolic element

(34)

Here and denote the location of the outer and inner horizon, given by

(35)

The Bekenstein-Hawking entropy of the BTZ black hole is equal to 1/4 times the geodesic length of the black hole horizon

The simplicity of the BTZ black hole space-time rests on two basic geometric facts: (i) the Einstein equations in 2+1 dimensions enforce that the space-time outside any matter source is locally AdS, (ii) in 2+1 dimensions, the space-time manifold outside localized matter excitations is non-simply connected. Every matter excitation creates a non-contractible loop in , and thereby adds one extra element to the fundamental group . For every element of one can associate a non-trivial space-time isometry , called the holonomy around the loop . The conjugacy class of the holonomy is determined by the mass and spin of the matter particles.

In general, holonomies come in three types, depending on whether the conjugacy class of the group element is hyperbolic, parabolic, or elliptic. For a black hole, both holonomies are in a hyperbolic conjugacy class; for localized point particles with mass in the subcritical regime , both elements are in the elliptic conjugacy class. For spinless particles . The local geometry then has a simple conical singularity with angle related to the mass via

Classical 2+1-D space-times with localized matter souces thus have a completely topological characterization in terms of a homomorphism from the fundamental group to the discrete subgroup of generated by the holonomies around all non-contractible loops . The holonomy is defined with respect to some arbitrary base-point, together with a choice of local frame. Changing the base point and local frame is a gauge symmetry that acts by overall conjugation of the holonomy elements.

This description can also be derived by considering the first order formulation of classical 2+1-D gravity in terms of a triad and spin connection . The linear combinations constitute two independent connections, in terms of which the Einstein action splits into a difference of two Chern-Simons actions

(36)

The Einstein equation and torsion constraint take the form of flatness constraints: are flat everywhere except at the localized matter sources. The group elements then coincide with the holonomies of the flat gauge fields around the closed loops surrounding each matter source.

The classical phase space of a dynamical system is identical the space of classical solutions. We thus find that the phase space of 2+1-D gravity defined on factorizes6 into the product where each factor is given by the moduli space of flat connections over , with specified holonomies around each local matter source. The phase space includes all positions and momenta of the matter particles. Unless the background space-time has non-trivial topology, we can identify the geometric phase space with the phase space of the matter particles. This identification is supported by simple counting: each additional point particle adds one extra holonomy element . Of the six extra phase space variables per particle, four are the position and momentum, the other two are associated with the spin.

Given that all particle trajectories must pass through any given time slice, we can measure all holonomies of the 2+1-D space time within a constant time slice. Via the uniformization theorem the space of flat bundles on a 2D surface is isomorphic to Teichmller space, the space of constant negative curvature metrics on . We will further clarify this identification in section 4, where we will use it to establish a precise connection between 2+1-D gravity and the modular geometry of conformal blocks in 2D CFT.

Figure 6: The Teichmller space of the hyperbolic cylinder (spatial section of a BTZ black hole) with two asymptotic regions and two punctures is parameterized by four holonomies that satisfy the relation , modulo overall conjugation.

Let us finally specialize to the case of interest. From the above general description, we can now confirm that the phase space of two matter particles moving in the background of an eternal BTZ space-time is given by the product of two copies of the Teichmller space of the four-punctured sphere. For our purpose, the most convenient description of this space is via its identification with the space of flat bundles

(37)

Here the symmetry acts via simultaneous conjugation . The relation follows from the corresponding relation in . For our application, we must pick two of the holonomies to be hyperbolic, and the other two elliptic

(38)

To simplify our formulas somewhat, we will often restrict to the case of a non-rotating black hole and spinless particles with zero total angular momentum . In this case, the conjugacy classes of the four holonomies are given in terms of the mass parameters (6) via

(39)

Here and specify the horizon ‘area’ (length) of BTZ black holes of mass and , and are the conical angle associated with particle and .

In the following we will keep all four mass parameters fixed. The Teichmller space (37) is therefore dimensional.7 The total phase space

(40)

of the two particles is therefore 4-dimensional. This is four less than the usual , since we have chosen to fix the total energy and angular momentum of the two particles. These are both first order constraints, that each reduce the phase space dimension by two.

2.2 Classical phase space

As our next task, we need to determine the commutation relations between the physical phase space variables. Luckily, this work has already been done for us in [19]. We will now summarize their elegant calculation. Since the phase space takes the factorized form (40), and because the Einstein action splits into a sum of two independent terms as in (36), we can focus on each of the two factors separately, and combine them afterwards.

The reader needs to be on guard, however, for a minor point of possible confusion. A point on Teichmller space is commonly specified by a 2D constant curvature metric on the hyperbolic cylinder with two punctures. It is tempting to identify this 2D constant curvature metric with the metric on a spatial slice of the 2+1-D geometry. However, this is in general not the correct identification: the 2D metric on a spatial slice is obtained from the space components of the triad, which together with one of the spin connections , also combines into a flat gauge field.8 In contrast, the space components of the two gauge fields define two flat connections, which are linear sums of the tetrad and spin connection . We will try to steer clear of this possible confusion, by restricting ourselves to the case where the black holes and particles in the bulk all have zero spin and angular momentum. In this simplified situation, the quantum numbers are such that the spin connection effectively decouples, and all three Teichmller spaces can essentially be identified. It is completely straightforward, however, to extend the following analysis to the general case.

First we need to find some suitable coordinates . Since the gauge invariant traces of single group elements are all held fixed, let us next look at the traces of products of two group elements. There are four combinations

(41)

These holonomy variables correspond to four topologically distinct loops. However, is two-dimensional, so there must be two relations among the four variables. These take the following form [19]

(42)

We will use these relations to solve for and . So our independent coordinates are and . They are related to the mass parameters and via

(44)

Here , , and are the mass and horizon length of the initial and final black hole regions in figs 1 and 5.

The variables and do not commute. Their commutator follows from the Einstein action. The first order form (36) is most convenient for this purpose.9 The CS action can be rewritten in 2+1 notation as with . Here and denote 2D one-forms and wedge product. Applying the usual rules, one deduces [31] that the symplectic form is given by (here as before)

(45)

restricted to the space of flat connections, , modulo gauge transformations. is known as the Weil-Petersson symplectic form. It is the basic starting point for the quantum theory of Teichmller space. We will denote the associated Poisson bracket by . In terms of the gauge potential, it takes the local form

(46)

We can now compute the Poisson bracket between the holonomy variables and . The two variables do not commute because (as seen in fig 5), the corresponding paths and have an intersection point. From the commutator (46) and the identity for general matrices and , one deduces the so-called skein relation

(47)

where and are the two paths obtained by replacing the intersection via the anti-symmetric smoothing operation

With a little bit of mental gymnastics, one recognizes that the two holonomy variables on the right-hand side of (47) are in fact equal to and . We thus arrive at the following remarkably simple-looking result [38]

(48)

This formula was first obtained by Goldman, without the use of the gauge theory formalism. Indeed, the variables represent the geodesic lengths of the corresponding curves, rather than traces of holonomies of a gauge field.

Figure 7: The variables , , and are associated with four topologically distinct loops each of which surrounds a hole and puncture (left) or two punctures (right).

In spite of its apparent simplicity and geometrical beauty, eqn (48) is not yet a suitable starting point for making a transition to the quantum theory. The reason is clear: once we eliminate and via eqns (42) and (2.2), the right-side becomes a complicated function of the phase space coordinates and . To make the quantization task more manageable, one typically looks for a set of Darboux coordinates, for which the symplectic form and Poisson bracket take a canonical form. A useful set of Darboux coordinates on Teichmller space [39] are the so-called Fenchel-Nielsen coordinates, also known as ‘length and twists’.

In our case, there are two possible choices for the length coordinate: or . Each have their own canonical conjugate ‘twist’ variable and defined such that

(49)

In terms of the complex geometry, a shift in the twist variable acts by cutting the 2D surface open along the corresponding cycle , rotating one side by an angle , and gluing the two parts back again. The result that the length and twists are Darboux variables was first shown by Wolpert [39]. As we will see shortly, in terms of our scattering problem, is a direct measure of the time-difference between the would-be arrival time of particle and the moment when particle is sent in.

2.3 Scattering matrix

We are finally ready to define and compute the gravitational scattering amplitude between the outgoing particle and the infalling particle in the BTZ black hole background. First we specifiy the initial and final states.

The initial state decribes three objects: a BTZ black hole of mass , particle A that travels just outside of its horizon, and particle B that is falling in from asymptotic infinity. Particle A adds a finite amount of energy to the black hole mass: from a distance, the geometry look like a single black hole of mass . We take as a basis of initial states the eigenstates of the total mass operator that measures . So in particular

(50)

Particle adds an additional amount of energy equal to . Again, we pick our basis states to be eigenstates of the operator that measures the total energy .

The final state also describes three objects: a BTZ black hole of mass , particle B that has fallen in, and particle A that has escaped to infinity. Particle B has added to the black hole mass, so from the outside, it looks like a black hole of mass . We choose as our basis of final states the eigen states of , satisfying

(51)

Particle adds an additional amount of energy equal to . As before, we pick our basis states to be eigen states of the total energy operator .

The gravitational scattering matrix is now simply defined as the overlap between an initial and a final basis state

(52)

From the previous discussion, we have learned that the horizon lengths and of the initial and final black holes do not commute with each other. The scattering matrix should thus be thought of as the unitary operator that implements the canonical transformation between the Darboux coordinates associated with the initial state and the variables associated with the final state. To construct this operator, we need to find the explicit relation between the two sets of Darboux variables. Luckily, also this calculation has already done for us at the semi-classical level in [19], and at the full quantum level in [14, 34]. For now, we proceed with the semi-classical analysis. So we set

(53)

Here is the generating function of the canonical transformation between the initial and final Darboux variables and