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date: 05 July 2022

AdS3 Gravity and Holographyfree

AdS3 Gravity and Holographyfree

  • Per KrausPer KrausMani L. Bhaumik Institute for Theoretical Physics, Department of Physics & Astronomy, University of California, Los Angeles

Summary

General relativity in three spacetime dimensions is a simplified model of gravity, possessing no local degrees of freedom, yet rich enough to admit black-hole solutions and other phenomena of interest. In the presence of a negative cosmological constant, the asymptotically anti–de Sitter (AdS) solutions admit a symmetry algebra consisting of two copies of the Virasoro algebra, with central charge inversely proportional to Newton’s constant. The study of this theory is greatly enriched by the AdS/CFT correspondence, which in this case implies a relationship to two-dimensional conformal field theory. General aspects of this theory can be understood by focusing on universal properties such as symmetries. The best understood examples of the AdS3/CFT2 correspondence arise from string theory constructions, in which case the gravity sector is accompanied by other propagating degrees of freedom. A question of recent interest is whether pure gravity can be made sense of as a quantum theory of gravity with a holographic dual. Attempting to answer this question requires making sense of the path integral over asymptotically AdS3 geometries.

Subjects

  • Gravitation and Relativity
  • Particles and Fields

1. Introduction

Few facts about our universe are more apparent than that it is composed of four dimensions: three of space and one of time. Yet many have speculated that this is perhaps an illusion reflecting our limited perception: There may be a real sense in which the universe possesses more, or even fewer, than four dimensions. Regardless, it has proven fruitful to ask the question “What would the laws of physics look like in a world of nonstandard spacetime dimensionality?” Such a question can shed light on which aspects of nature’s laws are universal and which are contingent, and can reveal new avenues for exploration. Most ambitiously, perhaps it will be possible to provide an answer to the question of “why” the observed dimensionality of spacetime is what it is.

A broad range of observations strongly indicates that spacetime is governed by Einstein’s general theory of relativity, and so one is led to consider the formulation of this theory in varying spacetime dimensionality. The Einstein field equations in the presence of matter and cosmological constant Λ,

Rμν12Rgμν+Λgμν=8πGTμν, (1.1)

where the speed of light is set to unity (c=1), certainly make mathematical sense for spacetime dimensionality D4. However, many basic implications of these equations depend dramatically on the precise value of D. The observed value D=4 is special in at least one regard: It is the lowest dimensionality in which the gravitational field has local dynamics of its own, as manifested, for example, in the existence of gravitational waves. Mathematically, this is due to the fact that the Riemann tensor is not an independent object for D<4. In D=3, it obeys the identity

Rμναβ=2gμ[αRβ]νgβ[αRν]αRgα[μgβ]ν, (1.2)

where […] denotes anti-symmetrization; while in D=2, the corresponding identity reads Rμναβ=Rgα[μgν]β. The Einstein field equations then imply that for D<4, the Riemann tensor, which is what characterizes the local geometry of spacetime, is determined locally by the stress tensor Tμν, which is to say that the gravitational field carries no independent degrees of freedom. For this reason, one might conclude that gravity in dimension D<4 is too trivial to be interesting. However, this conclusion is premature: While local gravitational degrees of freedom are indeed absent, in appropriate circumstances there exists a rich class of what one might call “global degrees of freedom” whose existence depends in some way on the topology of the spacetime under consideration. These global degrees of freedom can support many recognizable features of gravity in higher dimensions, notably black-hole solutions. For this reason, lower dimensional gravity is of interest as a theoretical arena in which many problems, often of a quantum nature, can be explored in a simplified setting. The focus here is on D=3 with a negative cosmological constant, where one powerful feature that emerges is a close connection to two-dimensional (2D) conformal field theory, providing an example of the AdS/CFT (anti–de Sitter/conformal field theory) correspondence (Aharony et al., 2000; Gubser et al., 1998; Maldacena, 1999; Witten, 1998).1 The powerful features of 2D CFTs that allow for their solvability in certain circumstances—the operator product expansion, Virasoro algebra, and modular invariance—are all realized in gravitational language, aspects of which were already evident in Brown and Henneaux (1986).

2. Solutions of Three-Dimensional (3D) Gravity

Einstein’s equations for pure gravity and negative cosmological constant may be written as

Rμν=22gμν. (2.1)

The cosmological constant has been traded for the length scale . It will be convenient to set =1. This suffices, since if gμν obeys Rμν=2gμν, then dimensional analysis implies that gμν=2gμν will satisfy (2.1). In any dimension and for any value of the cosmological constant, the maximally symmetric solution of the Einstein equations may be conveniently described as a hypersurface embedded in an ambient flat spacetime of one higher dimension. Among other advantages, this description makes the symmetry of the spacetime manifest. In the present case, consider the surface

X12+X22X32X02=1 (2.2)

embedded in the signature 22 spacetime with line element ds2=dX12+dX22dX32dX02. The symmetry group is SO22. Global coordinates are obtained by solving the hypersurface constraint as

X1=rcosϕX2=rsinϕX3=1+r2costX0=1+r2sint, (2.3)

yielding the induced line element

ds2=1+r2dt2+dr21+r2+r2dϕ2. (2.4)

It may be verified that this satisfies (2.1) with =1. Although t appears in (2.3) as a periodic coordinate, implying closed timelike curves, one is free to drop this identification in (2.4) and work on the covering space. This corresponds to taking the symmetry group to be not SO22 but rather its universal cover. However, ϕ is taken to be 2π periodic. The resulting spacetime is referred to as AdS3. The induced metric on a surface of fixed r is that of a cylinder, and so in the AdS/CFT correspondence, gravitational observables in these coordinates will be related to observables of a CFT on the cylinder.

From the line element (3.4), it is apparent that null geodesics can reach r= in finite t. This indicates the need to impose boundary conditions in order to obtain well-defined dynamics in AdS (Avis et al., 1978; Balasubramanian et al., 1999; Breitenlohner & Freedman, 1982; Klebanov & Witten, 1999; Witten, n.d.-a). For instance, in order to arrive at a well-defined initial value problem for the Klein–Gordon equation 2m2ϕ=0, one needs to impose boundary conditions in the form of consistent large r falloff conditions on initial data. In some cases, these boundary conditions are fixed by demanding compatability with the SO22 symmetry of AdS3. More precisely, the demand is that there exist Noether charges whose Poisson brackets yield the SO22 Lie algebra. At the quantum level, one further requires that the Hilbert space furnishes a unitary representation with energy bounded from below. In the case of the Klein–Gordon equation, this turns out to be possible if and only if m2mBF2, where mBF2=1 is the so-called Breitenlohner–Freedman bound (Breitenlohner & Freedman, 1982). For m20, there is a unique choice of allowed boundary conditions, while for 1m2<0, there are two possibilities, often referred to as standard and alternate quantization.

Another standard choice of coordinates is suitable for matching to CFT on the plane. These so-called Poincaré coordinates are obtained by writing:

X1=xzX2=1z2x2+t22zX3=tzX0=1+z2+x2t22z, (2.5)

yielding:

ds2=dz2dt2+dx2z2. (2.6)

The restriction z>0 is imposed. Unlike global coordinates, Poincaré coordinates do not cover the full spacetime, as seen from the fact that X0+X2=1z>0. Geodesics reach z= in finite proper or affine time, necessitating attaching additional Poincaré coordinate patches in order to describe the full spacetime.

A third choice of coordinates is appropriate for the description of black holes. Define BTZ (Banados et al., 1992, 1993) coordinates as

X3±X1=r2r2r+2r2e±r+ϕrtX2±X0=r2r+2r+2r+2erϕr+t, (2.7)

with r+r0. The line element takes the form

ds2=r2r+2r2r2r2dt2+r2r2r+2r2r2dr2+r2r+rr2dt2. (2.8)

As for Poincaré coordinates, multiple coordinate patches are required to cover the full spacetime. As defined in (2.7), the coordinate ϕ takes values on the real line. To obtain a black hole one imposes the identification ϕϕ+2π. Equation (2.8) then describes a rotating black hole with event horizon at r=r+ (Banados et al., 1992, 1993; see Carlip, 1995, for a review; see Hemming et al., 2002, for pictures that help with visualizing the regions of AdS covered by the various coordinate systems).

In the absence of matter, every solution of the Einstein equations is locally equivalent to AdS3. Global distinctions can have a dramatic impact; for example, it was noted that black holes can be be made out of global AdS3 by imposing a particular periodic identification. A more subtle point, to be developed in more detail, is that two solutions that differ only by an infinitesimal coordinate transformation should be identified as physically distinct configurations provided that the coordinate transformation acts sufficiently nontrivially at the asymptotic boundary of the spacetime. In this sense, pure AdS3 gravity does not support local degrees of freedom in the conventional sense, but there do exist “boundary degrees of freedom,” often referred to as “boundary gravitons” (Brown & Henneaux, 1986).

3. Asymptotic Structure

These statements rely on making precise the notion of an asymptotically AdS3 spacetime and its associated asymptotic symmetries (Brown & Henneaux, 1986). The signature of spacetime is taken to be Euclidean; Lorentzian signature solutions are recovered by the usual Wick rotation tit. It is useful to put the metric in “Fefferman–Graham” gauge (Fefferman & Graham, 1985) by writing

ds2=dρ24ρ2+hijxρdxidxj, (3.1)

where i,j=1,2. In particular, the choice of radial gauge with gρi=0 has been made. This choice is not necessarily possible (i.e., while demanding that the metric remain invertible and differentiable) globally, but this fact will not affect the present discussion regarding the asymptotic structure. The definition of the radial coordinate ρ has been chosen so that the asymptotic boundary lies at ρ=0, and such that hij admits a simple series expansion in ρ. In these coordinates, the Einstein equations Rμν=2gμν take the form

K2KijKij=Rh+2, (3.2)
iKijKhij=0, (3.3)
2ρρKijhijK+2KikKjk3KKij+12hijKmnKmn+K2hij=0, (3.4)

where i is the covariant derivative with respect to hij, and the extrinsic curvature tensor Kij is given by

Kij=ρρhij. (3.5)

Note that i type indices are raised by hij, defined as the inverse of hij via hikhkj=δji. Finally, K=Kii is the trace of the extrinsic curvature.

Asymptotically locally AdS3 metrics are those for which hij has a simple pole in ρ at ρ=0. One writes hijxρ1ρgij0x+. The metric gij0 is referred to as the “conformal boundary metric”; in the AdS3/CFT2 context it is the metric on which the CFT “lives.”2 To proceed, consider the series expansion of hij in ρ. This can, in fact, be done more generally for AdSd+1, but the d=2 case considered here offers dramatic simplifications, as the series contains only three terms. For d>2 the series contains more terms and lnρ can also appear. Restricting to d=2 the equation becomes

hijxρ=1ρgij0x+ρgij2x+ρ2gij4x. (3.6)

The Einstein equations (3.2)–(3.4) reduce to

gij4=14g2g01g2ij0iTij=0Trg01T=18GRg0. (3.7)

Here i0 is the covariant derivative built out of the metric gij0, and 0i is defined by raising its index using the inverse metric g01ij; note that one can write g0 or g0 interchangeably based on typographical convenience. The object Tij is defined as

Tij=14Ggij2Trg01g2gij0. (3.8)

Tij is a symmetric tensor that is conserved with respect to the metric gij0 and has a trace proportional to the Ricci scalar of this metric. These are also the fundamental properties of the stress tensor in a 2D conformal field theory. Indeed, in the AdS3/CFT2 correspondence, the “boundary stress tensor” Tij is identified with the CFT stress tensor (Balasubramanian & Kraus, 1999; Brown & York, 1993; de Haro et al., 2001; Henningson & Skenderis, 1998; Kraus, 2008; Skenderis, 2002).

The analysis just presented shows that the asymptotic data associated with an asymptotically locally AdS3 solution are the conformal boundary metric gij0 and the boundary stress tensor Tij, the latter having a fixed divergence and trace. However, it is important to emphasize that not every such choice of gij0 and Tij leads via (3.6) to a globally acceptable solution of the Einstein equations. Rather, a singularity will typically be encountered at some ρ. This may represent either a coordinate singularity, indicating a breakdown of radial gauge, or a true curvature singularity. Avoidance of the latter imposes additional constraints on the boundary stress tensor that are not visible from a small ρ analysis.

Further insight into the boundary stress tensor is achieved by obtaining it from the variation of the Einstein–Hilbert action, which in Euclidean signature is

S=116πGMd3xgR+2+Sbndy. (3.9)

The boundary terms are fixed by demanding that the variation of the action yields the equations of motion and that the action is finite when evaluated on an asymptotically locally AdS3 solution (Balasubramanian & Kraus, 1999; de Haro et al., 2001; Henningson & Skenderis, 1998; Kraus, 2008; Skenderis, 2002). For the purposes of asymptotic analysis it is convenient to write the action evaluated on metrics restricted to radial gauge (3.1). Including boundary terms, the result is

S=116πGρcρ+2ρd2xhK2KijKij+2+Rh+18πGd2xhρc+Sanom. (3.10)

Several comments are in order. First, the asymptotic boundary is at ρ=0 but a cutoff at ρc has been imposed, which will eventually be taken to 0, since individual terms diverge as ρ0. One also includes an upper limit ρ+, which could indicate either a value for which the geometry smoothly ends or where the coordinates break down, in which case one should introduce a separate coordinate patch to extend the geometry. The specific value of ρ+ plays no role in the present asymptotic analysis. Next,

Sanom=lnρc32πGMd2xhRh. (3.11)

The boundary term Sanom is needed to cancel a divergence proportional to lnρc that arises from the rest of the action when evaluated on a general solution. Its appearance is linked to the Weyl anomaly, as will become clear momentarily. Due to the restriction to radial gauge, the Euler–Lagrange equations derived from (3.10) yield only the third line of the Einstein equations (3.4). The other equations (3.2 and 3.3), which in terms of radial evolution represent initial data constraints, must be included by hand.

When the Einstein equations are obeyed, the action is stationary under variations with δgij0. Allowing for nonzero δgij0 lets one define the boundary stress tensor in terms of the on-shell variation of the action,

δS=14πMd2xg0Tijδg01ij. (3.12)

Direct computation establishes agreement with the previous expression (3.8). It is useful to derive the stress tensor conservation and trace equations from the action perspective. These follow from considering the behavior of the action under coordinate transformations. In this regard note that, although the choice of radial gauge obscures this somewhat, the action is invariant under coordinate transformations, except for the anomaly term (3.11), which depends explicitly on a choice of radial coordinate ρ via the appearance of lnρc. First consider a coordinate transformation that does not change the value of ρc as ρc0. Writing xixi+ξi this acts on the boundary metric as δgij0=iξj+jξi, where ξi=gij0ξj. Using invariance of the action δξS=0 and integrating by parts, (3.8) gives the conservation equation in (3.7). To obtain the trace equation one considers a coordinate transformation that does act on ρc. The rescaling ρλρ acts on the conformal boundary metric as gij0λ1gij0. According to (3.8), δS=4πMg0Trg01Tλ. But using that (3.11) is the only non-coordinate invariant contribution to the action one can also read off δS=λ32πGMd2xg0Rg0, where the small ρ form of hij was used. Equating these expressions gives the trace relation in (3.7). In 2D CFT, the nonzero trace of the stress tensor is often referred to as the “Weyl anomaly,” and it typically arises by regulating UV divergences appearing in loop diagrams. In the present context it arises from the need to add a non-coordinate invariant boundary counterterm. Note, however, that this boundary counterterm is proportional to a topological invariant—the Euler number—of the boundary metric, and so has no local variation and therefore does not affect the derivation of the Euler–Lagrange equations.

Now consider the case in which the boundary metric is the Euclidean cylinder,

gij0dxidxj=dϕ2+dt2=dwdw¯,w=ϕ+it, (3.13)

with ϕϕ+2π. In this case, (3.7) yields the stress tensor components

Tww=Tw,Tw¯w¯=T¯w¯,Tww¯=0, (3.14)

and from (3.8), one reads off

gij2dxidxj=4GTij. (3.15)

The full metric is found to be

ds2=dρ24ρ2+1ρdw4T¯w¯dw¯dw¯4GρTwdw. (3.16)

Comparing with the global AdS3 metric in (2.4) using r=1ρρ4 one finds agreement upon setting T=T¯=116G.

The space of solutions (3.16) are often referred to as “Banados geometries” (Banados, 1999); they represent the general solution with cylinder boundary. They are typically not smooth everywhere; relatedly, the range of ρ has so far not been specified. The metric tensor is non-invertible at ρ1=4GTT¯, indicating either a breakdown of the coordinates or a true singularity. The different possibilities will be discussed.

Now turn to the important concept of asymptotic symmetries. A generic solution of the form (3.1, 3.6) will not possess any Killing vectors and hence no symmetries in that sense of the term. However, one of the main applications of symmetries is their implications for the existence of conserved quantities. In gravitational theories conserved quantities exist not by virtue of symmetries of any one particular metric but rather by virtue of asymptotic symmetries shared by a class of metrics. In the present context, an asymptotic symmetry is a coordinate transformation that leaves invariant the conformal boundary metric gij0. Turn now to the form of such asymptotic symmetries.

Consider infinitesimal coordinate transformations xμxμ+ξμx that admit an expansion around ρ=0, starting from the leading behavior ξiρxi=εixj+ and ξρ=0+. This acts on the conformal boundary metric as δξgij0=i0εj+j0εi. One option for εi is to use Killing vectors of gij0, if any exist. More generally, if gij0 admits conformal Killing vectors such that δgij0gij0 then one can try to combine these with an xi dependent rescaling of ρ to achieve δξgij0=0. This is achieved by taking ξρ=i0εiρ+. This coordinate transformation takes one out of radial gauge (3.1), since it induces a term in the line element 12ρji0εidxj. However, this can be removed by correcting ξ to ξiρxi=εi14g0ijjk0εkρ+. This is as far in the ρ expansion as one needs to go; further terms in the expansion of ξμ can be chosen arbitrarily, which means that all such choices will be seen to lead to the same conserved charge.

Specializing to the case of the cylinder (3.13), conformal Killing vectors are given by taking εw=εw and εw¯=ε¯w¯ and the asymptotic Killing vectors are

ξw=εw12w¯2ε¯ρ+ξw¯=ε¯w¯12w2ερ+ξρ=wε+w¯ε¯ρ+. (3.17)

Since (3.16) is the general radial gauge metric with cylinder boundary, it must be that the coordinate transformation only acts on the functions TwT¯w¯, and indeed one finds

δξTw=2Twε+w+18Gw3εδξT¯w¯=2T¯w¯ε¯+w¯T¯ε¯+18Gw¯3ε¯. (3.18)

Continuing to restrict attention to the cylinder, there is a conserved charge associated to each asymptotic Killing vector,3

Qξ=i2π02πTtiξi, (3.19)

where the integrand is evaluated at ρ=0. Conservation follows from the fact that Jj=Tjiξi is a conserved current, which in turn follows from the fact that ξi a conformal Killing vector of the boundary metric combined with conservation and tracelessness of the boundary stress tensor. Separating the two independent components of ξi, the equation becomes

Qε=12πTwεwdw,Q¯ε¯=12πT¯w¯ε¯w¯dw¯, (3.20)

where the integration contours go once around the cylinder as in (3.19).

A famous result of Brown and Henneaux (Brown & Henneaux, 1986) is that the conserved charges Q and Q¯ each furnish a Virasoro algebra with central charge c=32G, where is the AdS3 radius that is here set to 1. This result is essentially contained in the expressions (3.18), which can be recognized as the standard expression for the conformal transformation of the stress tensor in a 2D CFT. To make this more precise one first needs to discuss the canonical formulation of AdS3 gravity, although it will be schematic. Given the choice of a boundary cylinder, the phase space can be taken as the space of Banados metrics (4.16).4 Next, a symplectic form Ω can be defined on this phase space. Recall that a symplectic form is by definition a nondegenerate closed two-form. One then defines a vector field Vξ on phase space that generates the phase space flow corresponding to the flow of metrics defined by the spacetime vector field ξ. An important fact is that the phase space variation of a conserved charge, δQξ, is related to the corresponding phase space vector field contracted against the symplectic form as

δQξ=iVξΩ, (3.21)

where we are using the standard notation i to denote the contraction operation. Equation (4.21) is described by saying that Vξ is the Hamiltonian vector field corresponding to the phase space function Qξ. This equation provides the link to Poisson brackets, since the latter is defined in terms of the symplectic form. Omitting the details (see Compère & Fiorucci, n.d. for a pedagogical treatment), the result is

Qξ1Qξ2=δξ1Qξ2. (3.22)

Although it is not obvious from what has been written, the Poisson bracket defined in this way is antisymmetric and obeys the Jacobi identity. Indeed, it is not hard to show that

Qξ1Qξ2=Qξ1ξ2+central, (3.23)

where ξ1ξ2 denotes the standard Lie bracket of the vector fields, and “central” denotes central elements that have vanishing Poisson bracket with all functions on phase space.

It is conventional to work with modes in a Fourier expansion, so

Tw=nQneinw,T¯w¯=nQ¯neinw¯, (3.24)

corresponding to Qn=Qeinw and Q¯n=Qeinw¯. Putting things together gives the Poisson brackets

iQmQn=mnQm+n+18Gm3δm+n,0iQ¯mQ¯n=mnQ¯m+n+18Gm3δm+n,0iQmQ¯n=0, (3.25)

which is the Virasoro algebra with central charge c=32G.5 To quantize, AB replaces iAB and unitary highest weight representations are considered. The structure of such representations is, of course, well understood from the study of 2D CFT.

Energy (mass), M, and angular momentum, J, are the charges associated with translation in t and ϕ,

M=Qξ=it=12π02πT+T¯=Q0+Q¯0J=Qξ=ϕ=12π02πTT¯=Q0Q¯0. (3.26)

4. Spectrum of Solutions

In interpretation of solutions with constant TT¯, as previously noted, global AdS3 corresponds to Q0=Q¯0=116G=c24. It is assumed that this is the ground state of the theory, as can be justified by the nonexistence of any smooth solutions with smaller mass, and (anticipating the quantum theory) by the fact that this is the smallest energy for which there exist unitary representations of the Virasoro algebra. In particular, even if matter, such as a scalar field, were to be added to the theory, it is reasonable to suppose that energy is bounded from below by global AdS3.

In pure gravity, there are no smooth solutions in the range c12<M<0. The closest substitute are conical deficit solutions, obtained by removing a wedge from global AdS3 and identifying edges, leaving a conical singularity at the tip. In more detail, returning to (3.16) with T=T¯=1/16G and performing the coordinate transformation: ρtϕ=ρ/n2t/nϕ/n, now identifying ϕϕ+2π, corresponding to ϕϕ+2πn, the metric is

ds2=dρ24ρ2+1ρdw14n2ρdw¯dw¯14n2ρdw. (4.1)

These solutions carry charges Q0=Q¯0=c24n2. While these solutions are singular, they serve as effective descriptions of the geometry produced by point particles of mass m1/G (although the mass should not be so large as to produce a black hole). Such a description is accurate further than a Compton wavelength, λ1/mG, from the particle, which is useful, provided this distance is much smaller than the AdS radius, that is, for G=1.

To understand the appearance of black-hole solutions, note that for T=T¯= constant the metric (4.16) takes the form

ds2=dρ24ρ2+1ρ14GQ0ρ2dt2+1+4GQ0ρ2dϕ2. (4.2)

For Q0<0, as one heads radially into the geometry (by increasing ρ), one first encounters the vanishing of gϕϕ at ρ=1/4GQ0, which is the location of the conical defect singularity. For Q0>0 one instead encounter the vanishing of gtt at ρ=1/4GQ0, which signals the presence of an event horizon in the Wick-rotated Lorentzian solution.

More generally, for the black-hole solutions (3.8), tit converts to Euclidean signature, while also continuing r to pure imaginary values in order to keep the metric real, and then

r2=1+4GQ0+Q¯0ρ+16G2Q0Q¯0ρ2ρr±=2GQ0±GQ¯0, (4.3)

to convert to (3.16). Then the mass and angular momentum are

M=r+2+r28G,J=r+r4G. (4.4)

The Euclidean black-hole solution is singular unless time is identified periodically (Gibbons & Hawking, 1977). To determine the periodicity, the local geometry can be studied near the horizon at r=r+, or equivalently ρ=ρ+14GQ0Q¯0. In particular, the norm of the vector field v=t+irr+ϕ behaves near the horizon as vμvμκL, where L is the proper distance to the horizon, and

κ=r+2r2r+ (4.5)

is the surface gravity. Writing v=t where tϕ=tϕ+irr+t, demanding the absence of a conical singularity requires tt+β, where β=2π/κ. In terms of the original coordinates therefore tϕtϕ+2πt+βϕ+irr+β. In terms of w=ϕ+it,

ww+2πw+2πτ,τ=irr+β+2π. (4.6)

It follows from this that the conformal boundary geometry corresponding to a Euclidean BTZ black hole is a torus of modular parameter τ. Also note that TH=1/β is the Hawking temperature of the black hole, while irr+β serves as a chemical potential for angular momentum.

Of course, there are other solutions whose boundary is also a torus of modular parameter τ. For instance, starting from global AdS (2.4) the same identification as in (4.6) can be imposed. This is usually referred to as “thermal AdS3”. In fact, although the explicit transformation is not written out here, it is not hard to see that a BTZ black hole of modular parameter τ can be related by a change of coordinates to thermal AdS at some other value of the modular parameter τ=1τ. The relation between the thermal AdS and BTZ corresponds to a choice of which cycle on the boundary is contractible when extended into the bulk; see Figure 1. So at the level of Euclidean solutions, the black-hole metrics can be avoided and instead the space of thermal AdS geometries can be referred to.

Figure 1. Relation between BTZ and thermal AdS. The bulk topology is that of a solid cylinder. For BTZ, the t cycle on the boundary is contractible when extended into the bulk; for thermal AdS it is the ϕ cycle that plays this role.

The Lorentzian BTZ black hole has many similarities with the physically relevant Schwarzschild and Kerr black holes, although its asymptotic structure is, of course, very different. The BTZ black hole can be formed from the collapse of matter, assuming it is present in the theory, and the resulting black hole will emit particles according to the Hawking process at the temperature TH=1/β defined previously. Also, by the same logic as for Schwarzschild/Kerr, the black hole may be assigned an entropy S=A4G=πr+2G. Thus, all the major puzzles involving quantum-mechanical black holes, such as the (non)unitarity of black-hole evaporation, the microscopic interpretation of black-hole entropy, and so on, may be posed in the simplified context of BTZ black holes, which is the main reason for their appeal.

5. AdS3/CFT2 Correspondence

Holographic duality (Susskind, 1995; ‘t Hooft, 1993) can be motivated both from general considerations (Heemskerk et al., 2009) and from detailed constructions in string theory or other microscopic models (Maldacena, 1999). The two perspectives are mutually reinforcing. Starting with the general perspective, it was observed that classical gravity in AdS3 has conserved charges whose Poisson bracket furnishes two copies of the Virasoro algebra at central charge c=3/2G. Purely at the level of symmetry, a 2D CFT with this value of the central charge is a candidate for a dual description. Under some mild assumptions, this is already enough to guarantee a match between the black and CFT entropies in the high-energy regime (Strominger, 1998), since this is fixed in terms of the central charge by Cardy’s formula (Cardy, 1986), which in turn relies on modular invariance and the existence of an SL(2,) ×SL(2,) invariant ground state. Of course, to establish a fully fledged duality, more than just thermodynamics must be checked, in particular the spectrum of operators and their correlation functions. On the gravity side, some collection of weakly coupled fields can be introduced into the theory. Boundary correlators may then be computed via Witten diagrams (Witten, 1998). Interpreted in a 2D sense, such correlators are seen to obey standard axioms of 2D CFT. Each bulk field corresponds to a primary operator and a convergent operator–product expansion is available. So, order by order in the bulk couplings, a 2D CFT can be defined to which it is holographically dual by construction (Heemskerk et al., 2009). Of course, on its own this bulk-based procedure can only provide answers to questions that are already calculable in the bulk, whereas preferably the duality should shed light on such questions as the structure of spacetime at short distances and the microscopic interpretation of black-hole entropy. To answer such questions, a CFT that has an independent definition and that matches the perturbative physics in the bulk can be sought. However, there is also an intermediate approach, in which basic consistency conditions common to all conventional CFT can be used to constrain bulk physics in ways not manifest in bulk perturbation theory.

A fruitful example of the latter intermediate approach comes from imposing modular invariance. It is conventional to define the partition function of a 2D CFT as6

Zττ¯=TrqL0c24q¯L¯0c24,q=e2πiτ,q¯=e2πiτ¯. (5.1)

For a theory with a local diffeomorphism invarant action, Zττ¯ has an equivalent definition as a path integral over fields defined on a Euclidean torus of modular paramter τ, as follows from standard facts about path integrals. In passing from the path integral to the Hilbert space of states, a choice of which direction on the torus is “space” and which is “time” is made. Assuming diffeomorphism invariance, this choice is simply an arbitrary choice of labeling and should not affect the result. This logic leads to the statement of modular invariance of the partition function,

Zττ¯=Zττ¯,τ=+b+d, (5.2)

where abcd are integers that obey adbc=1. As noted, modular invariance of a standard Lagrangian theory is automatic, but it is highly constraining in building up the theory abstractly without use of a Lagrangian. The modular S-transformation τ=1/τ inverts the temperature and so modular invariance implies constraints between the spectrum at high and low energies. Depending on context, a violation of modular invariance may or may not signal an inconsistency, but at the very least it does mean that Zττ¯ does not admit a geometrical interpretation in terms of a theory on a torus. In the AdS3 context, such an interpretation is desired since such a torus is evident from the asymptotic geometry, hence modular invariance is usually demanded. Modular invariance becomes even more constraining if unitarity and compactness of the CFT are demanded. This implies that the partition sum takes the form

Zττ¯=iNiqhic24q¯h¯ic24, (5.3)

where Ni are non-negative integers. Coming back to the AdS3 description, introducing a collection of weakly interacting fields fixes the spectrum at low energies; however, the assumed low energy spectrum may turn out to be inconsistent in the sense that it admits no completion that defines a modular invariant partition function of a compact unitary CFT. Similar statements hold for correlation functions.

The program of using modular invariance to constrain or rule out candidate theories is known as the modular bootstrap; see Hellerman (2011) for its application in the current context. The simplest application of this general idea is provided by Cardy’s derivation (Cardy, 1986) of the asymptotic density of states of a CFT on a circle, assuming only modular invariance and the existence of a normalizable ground state with h0=h¯0=0. The latter fixes the low temperature behavior Zττ¯eπcτ26 as τ2. Invariance under τ1/τ then gives Zττ¯eπc6τ2 as τ20+. Applying standard formulas from statistical mechanics to extract the entropy from the partition function yields the asymptotic behavior

S2πc6Q0+2πc6Q¯0,Q0,Q¯0. (5.4)

The energy and momentum are E=Q0+Q¯0, J=Q0Q¯0. Equation (5.4) holds for a circle of circumference 2π, but the result for any other size circle is obtained from extensivity of the entropy in this regime.

The argument leading to (5.4) can be run equally well on the CFT or AdS sides. On the AdS side, global AdS3 provides the vacuum state, and its contribution to the partition function is given by periodically identifying Euclidean time, as described. Since the geometry is static, the Euclidean action is given by I=E0β, and then ZeI=eπcτ26 as shown previously. A coordinate transformsation of this geometry gives BTZ with τ=1/τ, and the coordinate invariance of the action then gives ZeI=eiπc6τ2 for a high temperature BTZ black hole. Extracting the entropy gives (5.4) again. Assuming the bulk is governed by the Einstein–Hilbert action, the entropy formula (5.4) is equivalent to the area law S=AH4G, as is readily verified. The present derivation makes clear (Kraus & Larsen, 2005) that (5.4) holds for more general actions including higher-derivative corrections where the entropy is not governed by the area law but instead by the more general Wald formula (Wald, 1993). As far as (5.4) is concerned, the only thing that changes is the relation between the central charge c and the parameters appearing in the action, generalizing the Brown–Henneaux relation c=3/2G.

A great success of string theory is in providing a microscopic counting of states that make up the entropy of certain black holes (Strominger & Vafa, 1996). The original derivation appeared miraculous, since the nature of the computations on the string-theory side versus gravity looked completely different, and the concordance of the entropy formulas only arose at the very end after all charge normalizations and so forth are carefully accounted for. The discussion of (5.4) helps demystify this. For a string-theory construction whose dynamics are described by a 2D CFT and that models a black hole with a near horizon BTZ factor, agreement between the entropies in the thermodynamic regime is assured, provided that the central charges agree. So for this class of models, the success boils down to computing the central charge of the CFT and comparing to the Brown–Henneaux formula (or its generalization to include higher-derivative terms). This comparison can be made without direct reference to black holes. In the time since the original work, the subject of black-hole state counting has progressed enormously, with successes that go well beyond the thermodynamic regime just discussed (Dabholkar et al., 2015; Dijkgraaf et al., n.d.). Currently, there does not exist a general-principles argument that explains the agreement between the entropy formulas in the general context. This represents one facet of the statement that the understanding of quantum black holes is incomplete.

It is interesting to ask whether pure 3D gravity makes sense as a quantum theory of gravity (Maloney & Witten, 2010; Witten, n.d.-b). To put this in context, note that the black-hole solutions of string theory that are described by the AdS3/CFT2 correspondence contain additional features. First, these are solutions originating in 10 or 11 dimensions, with a near-horizon geometry taking the form of a product space, for example AdS3×S2×M or AdS3×S3×M where M is some compact space such as T4 in the case of IIB string theory. The sphere factor typically has a curvature scale of the same magnitude as the AdS factor, so even if M is of microscopic size these solutions are macroscopically five- or six-dimensional. So the perturbative spectrum includes gravitons of this higher-dimensional spacetime, as well as the other massless and massive modes coming from quantizing strings in this background. The whole system is dual to a CFT2 which by necessity must be quite complicated, at least in the regime of parameters for which the macroscopic gravity description is reliable.

Quantum corrections to semiclassical gravity in AdS3 are controlled by G1/c, so a sequence of CFTs that can be said to be dual to pure AdS3 gravity should reproduce this semiclassical limit as c. In the classical limit, the spectrum of the bulk theory corresponds to the space of classical solutions. Restricting attention to smooth solutions with AdS3 asymptotics, the space of solutions is very restricted. First, there is global AdS3 representing the vacuum, with M=c12 and J=0. Nontrivial asymptotic symmetry transformations can be applied to global AdS3 to obtain new solutions; although these are simply coordinate transformations of global AdS3, they are “new” in the sense that they carry different charge QnQ¯n. In terms of Virasoro representation theory, these solutions are all Virasoro descendants of the vacuum. In order to focus on aspects not dictated purely by representation theory, attention can be restricted to solutions representing primary states. At the level of classical solutions, primary states correspond to the lowest-energy configurations after using the freedom to perform coordinate transformations. There are no primary solutions in the range c12<M<0. There are singular conical defect solutions in this range, which, as discussed, represent the metric produced by massive particles that can be excluded in the minimal conception of pure AdS3 gravity.

For M>0 there are BTZ black-hole solutions, with a mass spectrum that is continuous and unbounded from above. Viewed as solutions in Lorentzian signature, these geometries have event horizons past which the solutions can be extended. Extending a spacelike slice past the horizon either the black-hole singularity or a second AdS3 region are eventually encountered, as happens for the more familiar Schwarzschild black hole. So smooth solutions whose asymptotics are given by a single AdS3 region can be found by choosing to discard these black-hole solutions. In pure 3D gravity, this is not immediately inconsistent, given that there are no propagating degrees of freedom with which to form a black hole from collapse.

If the black holes are indeed discarded and the spectrum declared to consist of the vacuum together with all of its Virasoro descendants, a sensible quantum theory, with a Hilbert space that is a single representation of (two copies of) the Virasoro algebra, is obtained. However, the partition function is not modular invariant. In terms of Euclidean gravity solutions, the origin of the problem is clear: it was noted previously that the modular S-transform of thermal AdS3 at modular parameter τ is a BTZ solution at modular parameter 1/τ, so discarding the BTZ solutions is not a modular-invariant procedure.

As was discussed, including Lorentzian black-hole geometries as states is problematic since they are not smooth solutions with a single asymptotic AdS3 region. The flip side of this is the lack of a pure gravity understanding of the high-energy microstates whose existence is dictated by modular invariance. Putting these issues aside, the more modest goal of asking whether Euclidean 3D gravity can produce a sensible partition function as a sum over geometries can be persued. The precise rules for doing so are not obvious: in standard nongravitational quantum field theory, path-integration rules can be fixed by the requirement that they reproduce results from canonical quantization, but it was already said that the desired black-hole microstates are not visible in the canonical quantization of pure 3D gravity.

Keeping this in mind, the simplest option is to sum over classical solutions whose conformal boundary is a torus of specified modular parameter, along with quantum fluctuations around these solutions (Dijkgraaf et al., n.d.; Maldacena & Strominger, 1998; Maloney & Witten, 2010). Formally, this is a simple procedure. There exists a so-called SL(2,) family of Euclidean BTZ black-hole solutions, all of which can be coordinate transformed back to thermal AdS. This can be thought of in terms of the solid torus that fills in the boundary torus; see Figure 1. There is a unique noncontractible cycle on the boundary that is, however, contractible when extended into the bulk. However, there is freedom in how this cycle is related to the cycles implied by the periodic identifications of ϕt. For example, for global AdS3, the ϕ cycle is contractible, while for usual BTZ it is the t cycle that is contractible. This can be generalized by taking arbitrary integer combinations of these with coefficients cd. However, a restriction to cd coprime should be made, since the ncnd cycle is already contractible in the solution labeled as cd. Such a solution with parameter τ can be coordinate transformed to thermal AdS with parameter τ=+b/+d with adbc=1. The values of ab are not uniquely fixed; however, the Euclidean action only depends on τ through q=e2πiτ and the ambiguity in ab does not affect q.

This line of reasoning therefore leads to the candidate partition function

Zqq¯=cd=1Zthermalqγq¯γ,qγ=e2πi+b/+d,adbc=1. (5.5)

Zthermal gets contributions from the states that are directly visible in gravity—the vacuum and its Virasoro descendants. Namely, any string of operators of the form a LknkL3n3L2n2 can be applied to global AdS, (along with their L¯ partners which have been suppressed) where the nm are nonnegative integers.7 Including also the classical action I=βM=iπc12ττ¯ gives

Zthermalqq¯=m=2nm=0qmnmeiπcτ/122=1q2ητ2qq¯c1/24, (5.6)

where the Dedekind eta function is

ητ=q1/24n=11qn. (5.7)

Plugging into (5.5) gives a candidate partition function that is modular invariant by construction. Unfortunately, the result is unsatisfactory because the sum over cd turns out to be divergent. However, under some plausible assumptions there is a unique regularization of the sum that preserves modular invariance (Maloney & Witten, 2010). From its construction, it is far from evident that the resulting regularized partition function is that of a compact, unitary CFT, and indeed this turns out not to be the case. After writing the partition function in the form of (5.3) the spectrum can be seen to be continuous rather than discrete, and the spectral density is negative in some regions.

There are a few possible lessons to be learned from this result. One possibility is that pure AdS3 quantum gravity simply does not exist in the sense that it has been defined. However, perhaps preferable is to admit additional states below the black-hole threshold, such as the conical-defect geometries (Benjamin et al., 2019). Of course, even if inclusion of these in the partition function could be shown to yield a sensible result, since they are singular geometries the rules for including them in correlation functions are not clear. Another possibility is that the full path integral is not given just by a sum over classical solutions and additional 1-loop fluctuations (Maxfield & Turiaci, 2021).8 Perhaps it is possible to make sense of the path integral over all metrics with specified conformal boundary. Even if this were achieved, the question of how to describe the resulting partition function as a sum over states with a gravitational description would remain. This may be too much to ask for; perhaps pure gravity is capable of producing sensible results for suitable “coarse grained” quantities like the density of states but cannot describe the individual states; see Mathur (2005) for discussion of issues related to this.

6. Other Developments

The focus here has been on pure 3D gravity with asymptotically AdS3 boundary conditions. Various extensions and modifications of this setup have been considered. A few of them will now be mentioned, mostly to provide an entryway into the extensive literature on these topics.

6.1 Other Boundary Conditions

Asymptotically, AdS3 solutions give rise to left- and right-moving boundary Virasoro algebras with central charges cL=cR=3/2G. Conformal field theories can have cLcR, and it is natural to look for their gravitational realization. The simplest way to achieve this is to add to the action the parity-odd gravitational Chern–Simons term SCS=γΓdΓ+23Γ3, where Γ is the Christoffel connection 1-form. This leads to (Kraus & Larsen, 2006) cLcR=96πγ. The resulting theory is known as “topologically massive gravity,” or TMG (Deser et al., 1982a, 1982b). Any solution of the ordinary Einstein equations is also a solution of TMG, but TMG admits additional solutions. These include “warped AdS3” solutions (Anninos et al., 2009), whose asymptotic behavior is not that of AdS. The CFT interpretation of such solutions is subtle but is of interest due to the similarity with the near-horizon geometry of Kerr black holes (Guica et al., 2009). Warped AdS solutions also exist in other theories besides TMG.

6.2 Chern–Simons Description

The Einstein equations for 3D gravity with a negative cosmological constant can be recast as flatness conditions for a gauge field valued in the SL(2,) ×SL(2,) Lie algebra (Achucarro & Townsend, 1986; Witten, 1988). The connections A and A¯ are formed as linear combinations of the 3D vielbein e and spin connection ω as

A=ω+e,A¯=ωe. (6.1)

The Einstein equations are then equivalent to the flatness conditions dA+AA=dA¯+A¯A¯=0, which are the Euler–Lagrange equations for the Chern–Simons action

S=k4πTrAdA+23AAAk4πTrA¯dA¯+23A¯A¯A¯, (6.2)

with k related to the Newton constant as k1/G. This repackaging has the nice feature of factorizing into separate A and A¯ contributions. This feature makes the appearance of a factorized asymptotic symmetry algebra manifest from the outset. Cotler and Jensen (2019) contains a discussion of extracting the effective boundary CFT dynamics from the Chern–Simons action. One difference between the metric and Chern–Simons formulations is that a configuration such as A=A¯=0 seems acceptable in Chern–Simons theory yet is equivalent to a vanishing metric tensor gμν, which is regarded as singular. Therefore, the path integral over all “smooth configurations” would seem to define different quantities in the two descriptions. The Chern–Simons description is also considerably more rigid; for example, there is no simple way to extend the action (6.2) to include, say, a scalar field while maintaining the SL(2,) ×SL(2,) gauge symmetry.

6.3 Higher-Spin Gravity

The simplifying feature of pure 3D gravity of having no local degrees of freedom can be generalized to pure higher-spin gravity (Campoleoni et al., 2010; Prokushkin & Vasiliev, 1999). In this theory, the metric tensor is accompanied by a tower—either finite or infinite—of symmetric traceless tensor fields and an associated higher-spin gauge symmetry. These degrees of freedom are conveniently realized in the Chern–Simons formulation simply by replacing SL(2,) ×SL(2,) by some higher rank gauge algebra such as SL(N,)×SL(N,), which gives rise to fields of spin 2,3,N. The asymptotic symmetry can be shown to be (two copies of) a nonlinear W algebra. The BTZ black hole in these theories has generalizations that carry higher spin charge (Ammon et al., 2013). The theories with an infinite tower of higher-spin fields contain some elements of string theory, and may be viewed as models of string theory in a phase with unbroken gauge symmetry (Gaberdiel & Gopakumar, 2013). Indeed, there is a certain tensionless limit of string theory that gives rise to a higher-spin theory and furthermore has a holographically dual CFT that can be identified explicitly (Eberhardt et al., 2019).

6.4 Ensemble Interpretation

The fact that natural proposals for the partition function of pure gravity lead to results that are incompatible with the existence of a dual unitary compact CFT points to a new interpretation: perhaps pure gravity is dual to an ensemble of dual CFTs, and the gravity partition function is really computing the partition function averaged over the ensemble. Compelling evidence for such an interpretation arises in 2D gravity (Jackiw, 1985; Teitelboim, 1983), where a precise connection to random matrix theory has been established (Saad et al., n.d.). Some hints in three dimensions are also available (Cotler & Jensen, n.d.). In the AdS3/CFT2 context, an outstanding challenge is to identify the relevant ensemble of CFTs.

Acknowledgments

Work supported in part by NSF grant PHY-19-14412.

References

Notes

  • 1. Early papers on 3D gravity include Deser and Jackiw (1984), Deser et al. (1984), Gott and Alpert (1984), Giddings et al. (1984), Martinec (1984), and Witten (1988).

  • 2. More accurately, gij0 is a representative of a conformal class of metrics related by Weyl transformations. This follows from the fact that the freedom to rescale the ρ coordinate implies that there is no unique gij0 associated with a given bulk solution.

  • 3. The factor of i is inserted because t is Euclidean time.

  • 4. This statement is a bit imprecise due to the breakdown of the metric at ρ=4GTT¯, but this can be ignored here since asymptotic symmetries only depend on the asymptotic form of the metric.

  • 5. The more familiar form iLmLn=mnLm+n+c12m3mδm+n,0 is obtained by the identification Ln=Qn+c24δn,0.

  • 6. Recall that Q0Q¯0=L0c24L¯0c24.

  • 7. Recall that the vacuum is annihilated by Ln, n1.

  • 8. The c dependence of (5.6) can be interpreted as the statement that there are no loop corrections beyond 1-loop.