# 3D gravity In three dimensions, for the theory of pure Einstein gravity (i.e. without matter) with or without a cosmological constant, any solution is locally the same (dS, Minkowski, or AdS), and there are no propagating degrees of freedom. In what follows, we will use the notation that the total spacetime dimension is $d+1$ before specialising to $d=2$. To see this, first notice that, the Riemann curvature tensor in general can be decomposed as$R_{\mu \nu \alpha \beta}=W_{\mu \nu \alpha \beta}+\frac{2}{d-1}\left(g_{\alpha[\mu} R_{\nu] \beta}+R_{\alpha[\mu} g_{\nu] \beta}\right)-\frac{2}{d(d-1)} R g_{\alpha[\mu} g_{\nu] \beta}.$In flat space, $R_{ab}=0$ when there is no stress tensor, so the Weyl tensor $W_{\mu\nu\rho\sigma}$ carries all information about the curvature of the geometry. From this, we say that the Weyl tensor is the "gravitational part" and contains the graviton degrees of freedom. In 3-dimensions ($d=2$), by simple counting, we see that $R_{\mu\nu}$ has 6 degrees of freedom which equals the number of independent degrees of freedom in $R_{abcd}$. This means that the Weyl tensor does not have any degrees of freedom, i.e. there are no propagating gravitons in pure 3D gravity! Now, Einstein's equation without matter is given by$R_{ab}-\frac{1}{2} Rg_{ab}+\Lambda g_{ab}=0,$from which we get (upon taking the trace)$R=-\frac{d(d+1)}{L^2}, \quad \Lambda=-\frac{d(d-1)}{2L^2}.$I.e., the curvature is constant everywhere. Substitution back into the vacuum Einstein equation then gives$R_{ab}=\frac{2\Lambda}{d-1}g_{ab}.$In 3d, we can then substitute this into the expression for the Riemann tensor, now without the Weyl tensor:$R_{\alpha \beta \mu \nu} \approx \Lambda\left(g_{\alpha \mu}g_{\beta \nu}-g_{\alpha \nu} g_{\beta \mu}\right),$i.e., the Riemann tensor is completely fixed in terms of the metric, up to a constant that is fixed if we fix the cosmological constant. The only solution to this equation (locally) is given by the Minkowski/dS/AdS metric. Let us now focus on the case of negative cosmological constant for the main part of the this note, though we will turn to positive cosmological constant in a later section. Now, one shouldn't conclude that 3d gravity is trivial! Even though all solutions are locally AdS, there are globally very different solutions: solutions with different genera and boundaries. Gravitons can also exist "on the AdS boundary", which are called boundary gravitons; black hole solutions also exist. The fact that all solutions are all locally AdS, it is possible to obtain them by [[0099 Quotient method in AdS3|taking quotients]] of global AdS$_3$. 3D gravity in AdS has an infinite dimensional [[0060 Asymptotic symmetry|asymptotic symmetry group]], which is summarised in [[0085 Asymptotic symmetry of AdS3|asymptotic symmetry of AdS3]]. This agrees with the conformal group of [[0003 2D CFT|2d CFT]]. The simplicity and nontriviality together make 3d gravity a great testing ground for ideas in quantum gravity. Despite the great lot of progress that has been made, the quantum theory of 3D pure gravity is far from being understood. The gravitational path integral has various divergences which make it seemingly pathological or at least hard to interpret. An early attempt to fully quantise 3D gravity exploits a rewriting of the 3d gravity theory as a $\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$ [[0089 Chern-Simons theory|Chern-Simons]] theory. The two theories are classically equivalent, but unfortunately the quantum theory of CS theory does not lead to a meaningful or well-behaved theory of quantum gravity. In particular, points in the phase space of the CS theory can map to degenerate metrics in the gravity theory. More recently, two important developments have improved our understanding of 3d quantum gravity. The first one is the interpretation that the [[0555 Gravitational path integral|gravitational path integral]] computes an [[0154 Ensemble averaging|averaged]] quantity over some ensemble of boundary theories/observables. The second one is the formulation of 3D gravity as a topological quantum field theory called [[0596 Virasoro TQFT|Virasoro TQFT]]. ## Refs on classical 3D gravity - CS formulation of 3D gravity (any sign of $\Lambda$) - [[1988#Witten]] - [[2009#Skenderis, van Rees]] - dual states - special about AdS - [[1990#Mess]] - all locally AdS solutions can be obtained by [[0099 Quotient method in AdS3]] - not true for positive curvature - causal properties - [[Barbot200509]] - [[Barbot200510]] - review - [[VanRees2006]] - [[Rsc0004 CompereFiorucci Ch2 3d gravity]] - calculation of geodesic lengths and entanglement issues - [[2014#Maxfield]] - Euclidean solutions - see [[2009#Skenderis, van Rees]] - non-orientable solutions - [[2007#Yin]] - in a box - [[KrausMontenMyers2021]][](https://arxiv.org/pdf/2103.13398.pdf) - helpful for [[0170 TTbar]] - higher derivative - [[BuenoCanoLlorensMorenoVanDerVelde2022]][](https://arxiv.org/pdf/2201.07266.pdf) - particles/defects - [[2024#Li (Z)]]: spinning particles ## Refs on quantum gravity in 3D - classical 3D gravity and Chern-Simons - [[1986#Achucarro, Townsend]] - [[1988#Witten]] - 3D gravity partition functions - [[2000#Dijkgraaf, Maldacena, Moore, Verlinde]] - [[2007#Witten (Jun)]] - spectral density - [[2007#Maloney, Witten]] - [[2014#Kelly, Maloney]] - [[2019#Benjamin, Ooguri, Shao, Wang]]: points out that the density of states is negative near the edge of the spectrum - [[2020#Maxfield, Turiaci]]: off shell topologies - adding conical defects (particles) - [[2019#Benjamin, Ooguri, Shao, Wang]] - [[2020#Benjamin, Collier, Maloney]] - one-loop determinant - [[2008#Giombi, Maloney, Yin]] - one-loop determinant in flat space - [[2015#Barnich, Gonzalez, Maloney, Oblak]] - vanishing at 2- and 3-loop: [[2023#Leston, Goya, Perez-Nadal, Passaglia, Giribet]] - boundary modes - [[2018#Cotler, Jensen]] - higher spin gravity - [[AldayBaeBenjaminJorge-Diaz2020]] - fake partition functions and topological recursion - [[2022#Eberhardt]] - 3D gravity and universal OPE statistics from 2D CFT bootstrap - [[2016#Chang, Lin]]: horizon and universality - [[2019#Collier, Maloney, Maxfield, Tsiares]]: universal OPE coefficients - [[2022#Chandra, Collier, Hartman, Maloney]]: 3d gravity solutions and 2d CFT universal OPE expressions - [[0596 Virasoro TQFT|Virasoro TQFT]] formulation - [[2023#Collier, Eberhardt, Zhang]] - [[2024#Collier, Eberhardt, Zhang]] - ensemble average and coarse graining - [[2020#Belin, de Boer]] - [[2020#Cotler, Jensen (Jun)]]: [[0159 Torus wormhole|torus wormhole]] - [[2021#Belin, de Boer, Liska]] - [[2022#Chandra, Collier, Hartman, Maloney]] - [[2023#Belin, de Boer, Jafferis, Nayak, Sonner]] - [[2024#de Boer, Liska, Post]] - [[2024#Pelliconi, Sonner, Verlinde]]: conical defects - see also: [[0600 A tensor model for AdS3]] - [[2025#de Boer, Kames-King, Post]]: off-shell statistics - Turaev-Viro formulation - [[2025#Hartman (Jul, a)]] - [[2025#Hartman (Jul, b)]] ## 3D gravity vs Chern-Simons theory - gravity phase space on $\Sigma$ = (Teichmüller space on $\Sigma$)$^2$ - not all initial data for CS theory correspond to smooth geometries of the gravity theory when time-evolved; but if restricting the Teichmüller subspace for the gauge field configurations, they do correspond to regular metrics - Hilbert space - quantisation of Teichmüller space = Liouville conformal blocks - $\mathcal{H}_{\text {gravity }}=\mathcal{H}_{\Sigma} \otimes \overline{\mathcal{H}}_{\Sigma}$, where $\mathcal{H}_{\Sigma}$ = Liouville conformal blocks on $\Sigma$ ## Canonical quantisation - 3D gravity is renormalisable if no matter field is present ## Lorentzian solutions - time-symmetric ones without conical defects - [[1999#Brill]]: arbitrary genus, arbitrary number of asymptotic regions - [[AminneborgBengtssonBrillHolstPeldan1997]]: arbitrary genus, one asymptotic region - seen from outside the horizon - only [[0086 Banados-Teitelboim-Zanelli black hole]] - see [[Rsc0004 CompereFiorucci Ch2 3d gravity]] - non-time-symmetric ones without conical defects - just add rotations - with conical defects - [[THooft1993]] particles/conical defects - [[Marschull1999]] particles forming black holes ## Positive cosmological constant Defining$\mathcal{A}^a=\omega^a+\frac{i}{\ell_{\mathrm{dS}}} e^a, \quad \bar{\mathcal{A}}^a=\omega^a-\frac{i}{\ell_{\mathrm{dS}}} e^a,$the 3D gravity action can be put into the form of a [[0089 Chern-Simons theory|CS gauge theory]] with Lie algebra $\mathfrak{s l}(2, \mathbb{C})$, action given by$S=\frac{k}{4 \pi} \int \operatorname{tr}\left(\mathcal{A} \wedge \mathrm{~d} \mathcal{A}+\frac{2}{3} \mathcal{A} \wedge \mathcal{A} \wedge \mathcal{A}\right)+\frac{\bar{k}}{4 \pi} \int \operatorname{tr}\left(\bar{\mathcal{A}} \wedge \mathrm{~d} \bar{\mathcal{A}}+\frac{2}{3} \bar{\mathcal{A}} \wedge \bar{\mathcal{A}} \wedge \bar{\mathcal{A}}\right).$The level $k$ is pure imaginary$k=\frac{i \ell_{\mathrm{dS}}}{4 G_{\mathrm{N}}} \in i \mathbb{R}_{+}.$ ## Finite regions and Conformal Turaev-Viro Theory The Laplace transform between fixed-length and fixed-angle path integrals:$Z_A(M, \gamma(\mathsf{P}^{\prime}))=\sqrt{2} \int_{\mathbb{R}_{+}^n} d \mathsf{P}\left(\Pi_{i=1}^n e^{4 \pi i P_i^{\prime} P_i}\right) Z_L(M, \gamma(\mathsf{P})).$Relation to lengths and angles:$\ell_i=4 \pi b P_i,\quad \psi_i=2 \pi i b P_i.$Relation to VTQFT:$Z_L(M, \gamma(\mathsf{P}))=\left|Z_{\mathrm{vir}}\left(M_E, \Gamma(\mathsf{P})\right)\right|^2.$Relation to CTV:$Z_A(M, \gamma(\mathsf{P}))=Z_{\mathrm{CTV}}\left(M_E, \Gamma(\mathsf{P})\right).$Another relation between VTQFT and CTV via chain-mail links:$Z_{\mathrm{CTV}}\left(M_E, \Gamma(\mathrm{P})\right)=Z_{\mathrm{Vir}}\left(M_E, \Gamma_{C H}(\mathrm{P})\right).$