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 $\mathrm{\Lambda }$,

where the speed of light is set to unity ($c=1$), certainly make mathematical sense for spacetime dimensionality $D\ne 4$. 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

where […] denotes anti-symmetrization; while in $D=2$, the corresponding identity reads $R_{\mathit{\text{μναβ}}}={\mathit{Rg}}_{\alpha [\mu }{g}_{\nu ]\beta }$. 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 $T_{\mathit{\mu \nu }}$, 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

The cosmological constant has been traded for the length scale $\mathrm{\ell }$. It will be convenient to set $\mathrm{\ell }=1$. This suffices, since if $g_{\mathit{\mu \nu }}$ obeys $R_{\mathit{\mu \nu }}=-2g_{\mathit{\mu \nu }}$, then dimensional analysis implies that $g_{\mathit{\mu \nu }}^{\prime }={\mathrm{\ell }}^2{g}_{\mathit{\mu \nu }}$ 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

embedded in the signature $\left(2,2\right)$ spacetime with line element ${\mathit{ds}}^2={\left({\mathit{dX}}^1\right)}^2+{\left({\mathit{dX}}^2\right)}^2-{\left({\mathit{dX}}^3\right)}^2-{\left({\mathit{dX}}^0\right)}^2$. The symmetry group is $\mathit{SO}\left(2,2\right)$. Global coordinates are obtained by solving the hypersurface constraint as

yielding the induced line element

It may be verified that this satisfies (2.1) with $\mathrm{\ell }=1$. 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 $\mathit{SO}\left(2,2\right)$ but rather its universal cover. However, $\varphi$ is taken to be $2\pi$ 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 $r=\infty$ 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 $\left({\nabla }^2-m^2\right)\varphi =0$, 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 $\mathit{SO}\left(2,2\right)$ symmetry of AdS₃. More precisely, the demand is that there exist Noether charges whose Poisson brackets yield the $\mathit{SO}\left(2,2\right)$ 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 $m^2\ge m_{\text{BF}}^2$, where $m_{\text{BF}}^2=-1$ is the so-called Breitenlohner–Freedman bound (Breitenlohner & Freedman, 1982). For $m^2\ge 0$, there is a unique choice of allowed boundary conditions, while for $-1\le m^2<0$, 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:

yielding:

The restriction $z>0$ is imposed. Unlike global coordinates, Poincaré coordinates do not cover the full spacetime, as seen from the fact that $X^0+X^2=\frac{1}{z}>0$. Geodesics reach $z=\infty$ 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

with $r_{+}\ge r_{-}\ge 0$. The line element takes the form

As for Poincaré coordinates, multiple coordinate patches are required to cover the full spacetime. As defined in (2.7), the coordinate $\varphi$ takes values on the real line. To obtain a black hole one imposes the identification $\varphi \cong \varphi +2\pi$. 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 $t\to \mathit{it}$. It is useful to put the metric in “Fefferman–Graham” gauge (Fefferman & Graham, 1985) by writing

where $i,j=1,2$. In particular, the choice of radial gauge with $g_{\mathit{\rho i}}=0$ 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 $\rho$ has been chosen so that the asymptotic boundary lies at $\rho =0$, and such that $h_{\mathit{ij}}$ admits a simple series expansion in $\rho$. In these coordinates, the Einstein equations $R_{\mathit{\mu \nu }}=-2g_{\mathit{\mu \nu }}$ take the form

where ${\nabla }_i$ is the covariant derivative with respect to $h_{\mathit{ij}}$, and the extrinsic curvature tensor $K_{\mathit{ij}}$ is given by

Note that $i$ type indices are raised by $h^{\mathit{ij}}$, defined as the inverse of $h_{\mathit{ij}}$ via $h^{\mathit{ik}}{h}_{\mathit{kj}}={\delta }_j^i$. Finally, $K=K_i^i$ is the trace of the extrinsic curvature.

Asymptotically locally AdS₃ metrics are those for which $h_{\mathit{ij}}$ has a simple pole in $\rho$ at $\rho =0$. One writes $h_{\mathit{ij}}\left(x,\rho \right)\sim \frac{1}{\rho }{g}_{\mathit{ij}}^{\left(0\right)}\left(x\right)+\dots$. The metric $g_{\mathit{ij}}^{\left(0\right)}$ 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 $h_{\mathit{ij}}$ in $\rho$. This can, in fact, be done more generally for AdS_d+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 $ln\rho$ can also appear. Restricting to $d=2$ the equation becomes

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

Here ${\nabla }_i^{\left(0\right)}$ is the covariant derivative built out of the metric $g_{\mathit{ij}}^{\left(0\right)}$, and ${\left({\nabla }^{\left(0\right)}\right)}^i$ is defined by raising its index using the inverse metric ${\left(g_{\left(0\right)}^{-1}\right)}^{\mathit{ij}}$; note that one can write $g^{\left(0\right)}$ or $g_{\left(0\right)}$ interchangeably based on typographical convenience. The object $T_{\mathit{ij}}$ is defined as

$T_{\mathit{ij}}$ is a symmetric tensor that is conserved with respect to the metric $g_{\mathit{ij}}^{\left(0\right)}$ 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” $T_{\mathit{ij}}$ 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 $g_{\mathit{ij}}^{\left(0\right)}$ and the boundary stress tensor $T_{\mathit{ij}}$, the latter having a fixed divergence and trace. However, it is important to emphasize that not every such choice of $g_{\mathit{ij}}^{\left(0\right)}$ and $T_{\mathit{ij}}$ leads via (3.6) to a globally acceptable solution of the Einstein equations. Rather, a singularity will typically be encountered at some $\rho$. 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 $\rho$ 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

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

Several comments are in order. First, the asymptotic boundary is at $\rho =0$ but a cutoff at ${\rho }_c$ has been imposed, which will eventually be taken to $0$, since individual terms diverge as $\rho \to 0$. One also includes an upper limit ${\rho }_{+}$, 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 ${\rho }_{+}$ plays no role in the present asymptotic analysis. Next,

The boundary term $S_{\text{anom}}$ is needed to cancel a divergence proportional to $ln{\rho }_c$ 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 $\delta g_{\mathit{ij}}^{\left(0\right)}$. Allowing for nonzero $\delta g_{\mathit{ij}}^{\left(0\right)}$ lets one define the boundary stress tensor in terms of the on-shell variation of the action,

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 $\rho$ via the appearance of $ln{\rho }_c$. First consider a coordinate transformation that does not change the value of ${\rho }_c$ as ${\rho }_c\to 0$. Writing $x^i\to x^i+{\xi }^i$ this acts on the boundary metric as $\delta g_{\mathit{ij}}^{\left(0\right)}={\nabla }_i{\xi }_j+{\nabla }_j{\xi }_i$, where ${\xi }_i=g_{\mathit{ij}}^{\left(0\right)}{\xi }^j$. Using invariance of the action ${\delta }_{\xi }S=0$ 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 ${\rho }_c$. The rescaling $\rho \to \mathit{\lambda \rho }$ acts on the conformal boundary metric as $g_{\mathit{ij}}^{\left(0\right)}\to {\lambda }^{-1}{g}_{\mathit{ij}}^{\left(0\right)}$. According to (3.8), $\mathit{\delta S}=4\pi {\int }_{\partial M}\sqrt{g^{\left(0\right)}}\mathrm{T}r\left(g_{\left(0\right)}^{-1}T\right)\lambda$. But using that (3.11) is the only non-coordinate invariant contribution to the action one can also read off $\mathit{\delta S}=-\frac{\lambda }{32\mathit{\pi G}}{\int }_{\partial M}{d}^{2}x\sqrt{g^{\left(0\right)}}R\left(g^{\left(0\right)}\right)$, where the small $\rho$ form of $h_{\mathit{ij}}$ 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,

with $\varphi \cong \varphi +2\pi$. In this case, (3.7) yields the stress tensor components

and from (3.8), one reads off

The full metric is found to be

Comparing with the global AdS₃ metric in (2.4) using $r=\frac{1}{\sqrt{\rho }}-\frac{\sqrt{\rho }}{4}$ one finds agreement upon setting $T=\overline{T}=\frac{1}{16G}$.

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 $\rho$ has so far not been specified. The metric tensor is non-invertible at ${\rho }^{-1}=4G\sqrt{T\overline{T}}$, 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 $g_{\mathit{ij}}^{\left(0\right)}$. Turn now to the form of such asymptotic symmetries.

Consider infinitesimal coordinate transformations $x^{\mu }\to x^{\mu }+{\xi }^{\mu }\left(x\right)$ that admit an expansion around $\rho =0$, starting from the leading behavior ${\xi }^i\left(\rho ,x^i\right)={\epsilon }^i\left(x^j\right)+\dots$ and ${\xi }^{\rho }=0+\dots$. This acts on the conformal boundary metric as ${\delta }_{\xi }{g}_{\mathit{ij}}^{\left(0\right)}={\nabla }_i^{\left(0\right)}{\epsilon }_j+{\nabla }_j^{\left(0\right)}{\epsilon }_i$. One option for ${\epsilon }^i$ is to use Killing vectors of $g_{\mathit{ij}}^{\left(0\right)}$, if any exist. More generally, if $g_{\mathit{ij}}^{\left(0\right)}$ admits conformal Killing vectors such that $\delta g_{\mathit{ij}}^{\left(0\right)}\propto g_{\mathit{ij}}^{\left(0\right)}$ then one can try to combine these with an $x^i$ dependent rescaling of $\rho$ to achieve ${\delta }_{\xi }{g}_{\mathit{ij}}^{\left(0\right)}=0$. This is achieved by taking ${\xi }^{\rho }={\nabla }_i^{\left(0\right)}{\epsilon }^i\rho +\dots$. This coordinate transformation takes one out of radial gauge (3.1), since it induces a term in the line element $\frac{1}{2\rho }{\partial }_j{\nabla }_i^{\left(0\right)}{\epsilon }^i{\mathit{dx}}^j\mathit{d\rho }$. However, this can be removed by correcting $\xi$ to ${\xi }^i\left(\rho ,x^i\right)={\epsilon }^i-\frac{1}{4}{g}_{\left(0\right)}^{\mathit{ij}}{\partial }_j{\nabla }_k^{\left(0\right)}{\epsilon }^k\rho +\dots$. This is as far in the $\rho$ expansion as one needs to go; further terms in the expansion of ${\xi }^{\mu }$ 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 ${\epsilon }^w=\epsilon \left(w\right)$ and ${\epsilon }^{\overline{w}}=\overline{\epsilon }\left(\overline{w}\right)$ and the asymptotic Killing vectors are

Since (3.16) is the general radial gauge metric with cylinder boundary, it must be that the coordinate transformation only acts on the functions $\left(T\left(w\right),\overline{T}\left(\overline{w}\right)\right)$, and indeed one finds

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

where the integrand is evaluated at $\rho =0$. Conservation follows from the fact that $J_j=T_{\mathit{ji}}{\xi }^i$ is a conserved current, which in turn follows from the fact that ${\xi }^i$ a conformal Killing vector of the boundary metric combined with conservation and tracelessness of the boundary stress tensor. Separating the two independent components of ${\xi }^i$, the equation becomes

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 $\overline{Q}$ each furnish a Virasoro algebra with central charge $c=\frac{3\mathrm{\ell }}{2G}$, where $\mathrm{\ell }$ 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 $\mathrm{\Omega }$ 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 $V_{\xi }$ on phase space that generates the phase space flow corresponding to the flow of metrics defined by the spacetime vector field $\xi$. An important fact is that the phase space variation of a conserved charge, $\mathit{\delta Q}\left[\xi \right]$, is related to the corresponding phase space vector field contracted against the symplectic form as

where we are using the standard notation $i$ to denote the contraction operation. Equation (4.21) is described by saying that $V_{\xi }$ is the Hamiltonian vector field corresponding to the phase space function $Q\left[\xi \right]$. 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

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

where $\left[{\xi }_1,{\xi }_2\right]$ 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

corresponding to $Q_n=-Q\left[e^{-\mathit{inw}}\right]$ and ${\overline{Q}}_n=Q\left[e^{\mathit{\text{in}}\overline{w}}\right]$. Putting things together gives the Poisson brackets

which is the Virasoro algebra with central charge $c=\frac{3}{2G}$.⁵ To quantize, $\left[A,B\right]$ replaces $i\left\{A,B\right\}$ 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 $\varphi$,

4. Spectrum of Solutions

In interpretation of solutions with constant $\left(T,\overline{T}\right)$, as previously noted, global AdS₃ corresponds to $Q_0={\overline{Q}}_0=-\frac{1}{16G}=-\frac{c}{24}$. 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 $-\frac{c}{12}<M<0$. 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 $T=\overline{T}=1/16G$ and performing the coordinate transformation: $\left(\rho ,t,\varphi \right)=\left({\rho }^{\prime }/n^2,t^{\prime }/n,{\varphi }^{\prime }/n\right)$, now identifying ${\varphi }^{\prime }\cong {\varphi }^{\prime }+2\pi$, corresponding to $\varphi \cong \varphi +\frac{2\pi }{n}$, the metric is

These solutions carry charges $Q_0={\overline{Q}}_0=-\frac{c}{24n^2}$. While these solutions are singular, they serve as effective descriptions of the geometry produced by point particles of mass $m\sim 1/G$ (although the mass should not be so large as to produce a black hole). Such a description is accurate further than a Compton wavelength, $\lambda \sim 1/m\sim G$, from the particle, which is useful, provided this distance is much smaller than the AdS radius, that is, for $G\ll \mathrm{\ell }=1$.

To understand the appearance of black-hole solutions, note that for $T=\overline{T}=$ constant the metric (4.16) takes the form

For $Q_0<0$, as one heads radially into the geometry (by increasing $\rho$), one first encounters the vanishing of $g_{\mathit{\varphi \varphi }}$ at $\rho =-1/4{\mathit{GQ}}_0$, which is the location of the conical defect singularity. For $Q_0>0$ one instead encounter the vanishing of $g_{\mathit{tt}}$ at $\rho =1/4{\mathit{GQ}}_0$, which signals the presence of an event horizon in the Wick-rotated Lorentzian solution.

More generally, for the black-hole solutions (3.8), $t\to \mathit{it}$ converts to Euclidean signature, while also continuing $r_{-}$ to pure imaginary values in order to keep the metric real, and then

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

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 $\rho ={\rho }_{+}\equiv \frac{1}{4G\sqrt{Q_0{\overline{Q}}_0}}$. In particular, the norm of the vector field $v={\partial }_t+\frac{{\mathit{ir}}_{-}}{r_{+}}{\partial }_{\varphi }$ behaves near the horizon as $\sqrt{v^{\mu }{v}_{\mu }}\sim \mathit{\kappa L}$, where $L$ is the proper distance to the horizon, and

is the surface gravity. Writing $v={\partial }_{t^{\prime }}$ where $\left(t,\varphi \right)=\left(t^{\prime },{\varphi }^{\prime }+\frac{{\mathit{ir}}_{-}}{r_{+}}{t}^{\prime }\right)$, demanding the absence of a conical singularity requires $t^{\prime }\cong t^{\prime }+\beta$, where $\beta =2\pi /\kappa$. In terms of the original coordinates therefore $\left(t,\varphi \right)\cong \left(t,\varphi +2\pi \right)\cong \left(t+\beta ,\varphi +{\mathit{ir}}_{-}{r}_{+}\beta \right)$. In terms of $w=\varphi +\mathit{it}$,

It follows from this that the conformal boundary geometry corresponding to a Euclidean BTZ black hole is a torus of modular parameter $\tau$. Also note that $T_H=1/\beta$ is the Hawking temperature of the black hole, while $\frac{{\mathit{ir}}_{-}}{r_{+}}\beta$ serves as a chemical potential for angular momentum.

Of course, there are other solutions whose boundary is also a torus of modular parameter $\tau$. 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 $\tau$ can be related by a change of coordinates to thermal AdS at some other value of the modular parameter ${\tau }^{\prime }=-\frac{1}{\tau }$. 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.

Open in new tab

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 $\varphi$ 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 $T_H=1/\beta$ defined previously. Also, by the same logic as for Schwarzschild/Kerr, the black hole may be assigned an entropy $S=\frac{A}{4G}=\frac{\pi r_{+}}{2G}$. 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 ${\text{SL}}_{(2,ℝ)}\times {\text{SL}}_{(2,ℝ)}$ 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

For a theory with a local diffeomorphism invarant action, $Z\left(\tau ,\overline{\tau }\right)$ has an equivalent definition as a path integral over fields defined on a Euclidean torus of modular paramter $\tau$, 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,

where $\left(a,b,c,d\right)$ are integers that obey $\mathit{ad}-\mathit{bc}=1$. 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 ${\tau }^{\prime }=-1/\tau$ 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 $Z\left(\tau ,\overline{\tau }\right)$ 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

where $N_i$ 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 $h_0={\overline{h}}_0=0$. The latter fixes the low temperature behavior $Z\left(\tau ,\overline{\tau }\right)\sim e^{\frac{\mathit{\pi c}{\tau }_2}{6}}$ as ${\tau }_2\to \infty$. Invariance under $\tau \to -1/\tau$ then gives $Z\left(\tau ,\overline{\tau }\right)\sim e^{\frac{\mathit{\pi c}}{6{\tau }_2}}$ as ${\tau }_2\to 0^{+}$. Applying standard formulas from statistical mechanics to extract the entropy from the partition function yields the asymptotic behavior

The energy and momentum are $E=Q_0+{\overline{Q}}_0$, $J=Q_0-{\overline{Q}}_0$. Equation (5.4) holds for a circle of circumference $2\pi$, 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 $I=E_0\beta$, and then $Z\sim e^{-I}=e^{\frac{\mathit{\pi c}{\tau }_2}{6}}$ as shown previously. A coordinate transformsation of this geometry gives BTZ with ${\tau }^{\prime }=-1/\tau$, and the coordinate invariance of the action then gives $Z\sim e^{-I}=e^{\mathit{i\pi c}6{\tau }_2}$ 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 $S=\frac{A_H}{4G}$, 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 ${\mathrm{AdS}}_3\times S^2\times M$ or ${\mathrm{AdS}}_3\times S^3\times M$ where $M$ is some compact space such as $T^4$ 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 $G\sim 1/c$, so a sequence of CFTs that can be said to be dual to pure AdS₃ gravity should reproduce this semiclassical limit as $c\to \infty$. 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 $M=-\frac{c}{12}$ 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 $\left(Q_n,{\overline{Q}}_n\right)$. 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 $-\frac{c}{12}<M<0$. 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 $\tau$ is a BTZ solution at modular parameter $-1/\tau$, 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 ${\text{SL}}_{(2,\mathbb{Z})}$ 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 $\left(\varphi ,t\right)$. For example, for global AdS₃, the $\varphi$ 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 $\left(c,d\right)$. However, a restriction to $\left(c,d\right)$ coprime should be made, since the $\left(\mathit{nc},\mathit{nd}\right)$ cycle is already contractible in the solution labeled as $\left(c,d\right)$. Such a solution with parameter $\tau$ can be coordinate transformed to thermal AdS with parameter ${\tau }^{\prime }=\left(\mathit{a\tau }+b\right)/\left(\mathit{c\tau }+d\right)$ with $\mathit{ad}-\mathit{bc}=1$. The values of $\left(a,b\right)$ are not uniquely fixed; however, the Euclidean action only depends on ${\tau }^{\prime }$ through $q^{\prime }=e^{2\mathit{\pi i}{\tau }^{\prime }}$ and the ambiguity in $\left(a,b\right)$ does not affect $q^{\prime }$.

This line of reasoning therefore leads to the candidate partition function

$Z_{\text{thermal}}$ 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 ${\left(L_{-k}\right)}^{n_k}\dots {\left(L_{-3}\right)}^{n_3}{\left(L_{-2}\right)}^{n_2}$ can be applied to global AdS, (along with their $\overline{L}$ partners which have been suppressed) where the $n_m$ are nonnegative integers.⁷ Including also the classical action $I=\mathit{\beta M}=-\frac{\mathit{i\pi c}}{12}\left(\tau -\overline{\tau }\right)$ gives

where the Dedekind eta function is

Plugging into (5.5) gives a candidate partition function that is modular invariant by construction. Unfortunately, the result is unsatisfactory because the sum over $\left(c,d\right)$ 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