# Jackiw-Teitelboim (JT) gravity JT gravity is a solvable toy model of quantum gravity. Its (Euclidean) action is given by$I[g, \phi]=-\frac{1}{16 \pi G_N} \int_{\mathcal{M}} \sqrt{g}\, \phi(R+2)-\frac{1}{8 \pi G_N} \oint_{\partial \mathcal{M}} \sqrt{h}\, \phi(K-1),$where $\phi$ is a scalar field called the dilaton. To study higher topologies, it is useful to also introduce a topological term $-S_0\,\chi$ where $\chi$ is the Euler character, related to the genus of the surface $g$ and the number of boundaries $n$ by $\chi=2-2g-n$. The dilaton field appears linearly in the action, so it plays the role of a Lagrange multiplier. In a path integral, this field can be integrated out (along some contour) so that the resulting path integral is a gravitational path integral with the constraint on the metric that $R=-2$ everywhere. This greatly simplifies the problem of doing the [[0555 Gravitational path integral|gravitational path integral]]. With this simplification, the only degrees of freedom for a fixed topology lie on the boundary/boundaries. It is remarkable that JT gravity is [[0471 String-matrix duality|dual]] to an [[0154 Ensemble averaging|ensemble]] of [[0197 Matrix model|random matrix models]]. ## Path integral The path integral for JT is done in a few steps. Consider the case with one boundary first, i.e., $\langle Z(\beta)\rangle$. Now proceed as follows: (1) cut the geometry along some geodesic circle with length $b$, which can be done for any genus $g\ge1$; (2) path integrate the Schwarzian modes for the component with the boundary and integrate over the moduli of the Riemann surface with $b$ fixed; (3) integrate over the length of the geodesic circle $b$. With one boundary, the partition function is computed to be$\langle Z(\beta)\rangle \simeq e^{S_0} Z_{\text {Sch}}^{\text {disk }}(\beta)+\sum_{g=1}^{\infty} e^{(1-2 g) S_0} \int_0^{\infty} b \,d b\, V_{g, 1}(b) Z_{\mathrm{Sch}}^{\text {trumpet }}(\beta, b),$where $V_{g,n}(b)$ is the [[0617 Weil-Petersson volume|Weil-Petersson volume]]. Higher-point correlation functions are computed using pair-of-pants decomposition. ## Refs - original - [[1985#Jackiw]] - [[1983#Teitelboim]] - modern studies - [[2014#Almheiri, Polchinski]] - [[2016#Maldacena, Stanford, Yang]] - Schwarzian description - [[2016#Jensen]] - [[2016#Maldacena, Stanford, Yang]] - [[2016#Engelsoy, Mertens, Verlinde]] - reviews - [[2022#Mertens, Turiaci (Review)]] - boundary particle formulation - [[2018#Kitaev, Suh]] - [[2018#Yang]] - [[2019#Suh]] ## Quantisation - canonical quantisation - without matter: quantum particle in an exponential well [[2018#Harlow, Jafferis]] - with non-trivial matter CFT: solved by Weyl transformation to flat space - Euclidean path integral - get an average over quantum systems - alternative boundary conditions - [[Henneaux1985]] - [[Navarro-SalasNavarroAldaya1992]] - [[ConstantinidisPiguetPerez2008]] - bulk Hilbert space - [[2024#Iliesiu, Levine, Lin, Maxfield, Mezei]] ## Generalisations - end-of-the-world branes - [[2019#Blommaert, Mertens, Verschelde]]: eigenbranes (FZZT boundaries) - [[2020#Goel, Iliesiu, Kruthoff, Yang]]: energy branes, $\alpha$-branes etc - [[2021#Gao, Jafferis, Kolchmeyer]]: dynamical branes as EFT - JT supergravity - [[JohnsonRossoSvesko2021]]@[](https://arxiv.org/pdf/2102.02227.pdf) - [[2021#Rosso, Turiaci]] - [[2023#Turiaci, Witten]] - with dynamic EOW branes - [[2019#Blommaert, Mertens, Verschelde]]: FZZT-type branes called eigenbranes - [[2021#Gao, Jafferis, Kolchmeyer]]: EOW brane as effective model of a modified version of SSS - [[2023#Belaey, Mariani, Mertens]]: branes in JT supergravity - with higher derivative corrections (in 4d then reduced to 2d) - [[2021#Banerjee, Mandal, Rudra, Saha]] - adding quadratic curvature to the action (and performing the Gaussian integral) - [[BelokurovShavgulidze2022]][](https://arxiv.org/pdf/2206.05172.pdf) - JT with conical defects - [[2019#Mertens, Turiaci]] - [[2020#Maxfield, Turiaci]]: dimensional reduction from 3D Seifert manifolds and sharp defects - [[MeffordSuzuki2020]]@[pdf](https://arxiv.org/pdf/2011.04695.pdf) - [[2020#Witten (June, b)]]: sharp defects and corresponding deformations in the matrix potential - [[2020#Turiaci, Usatyuk, Weng]]: defects with general defect angle $\in(0,2\pi)$ - [[2023#Eberhardt, Turiaci]]: [[0617 Weil-Petersson volume|Weil-Petersson]] measure and recursion - [[2023#Lin, Usatyuk]] - JT with fermions - [[BanksDraperZhang2022]][](https://arxiv.org/abs/2205.07382) - JT with gauge field - [[2019#Kapec, Mahajan, Stanford]] - [[2019#Iliesiu]] - JT with higher spin and SUSY - [[2023#Griguolo, Guerrini, Panerai, Papalini, Seminara]] - include higher topologies - see [[2019#Saad, Shenker, Stanford]] - non-perturbative - [[JohnsonRosso2020]] - [[2023#Eynard, Garcia-Failde, Gregori, Lewanski, Schippa]] - [[2023#Johnson]] - very high genus - [[Kimura2021]][](https://arxiv.org/pdf/2106.11856.pdf): - cut-off JT and Liouville CFT${}_1$ - [[2024#Ferrari]] - JT with matter - [[2022#Jafferis, Kolchmeyer, Mukhametzhanov, Sonner (Short)]] - [[2022#Jafferis, Kolchmeyer, Mukhametzhanov, Sonner (Long)]] - JT with matter on conical defect geometries - [[2021#Iliesiu, Mezei, Sarosi]] - [[2023#Boruch, Iliesiu, Yan]] - [[2023#Lin, Usatyuk]] - [[0545 de Sitter quantum gravity|dS]] JT - canonical quantisation - [[2024#Alonso-Monsalve, Harlow, Jefferson]] - [[2024#Held, Maxfield]] - [[2025#Dey, Nanda, Roy, Sake, Trivedi]] - misc papers - classifying BC: [[GoelIliesiuKruthoffYang2020]] [pdf](https://arxiv.org/pdf/2010.12592.pdf) - dS JT and [[0167 Butterfly velocity|butterfly velocity]] [pdf](https://arxiv.org/pdf/2010.14539.pdf) - more general 2d dilaton gravity [Narayan](https://arxiv.org/pdf/2010.12955.pdf) - limits of JT [](https://arxiv.org/pdf/2011.13870.pdf) - a non-ensemble version by including a set of branes: [[2020#Blommaert]] - microstates: [[Johnson2022]][](https://arxiv.org/pdf/2201.11942.pdf) ## Calculations and properties - low-temperature entropy - [[JanssenMirbabayi2021]][](https://arxiv.org/abs/2103.03896) - JT with matter correlators - [[2017#Mertens, Turiaci, Verlinde]]: disk - [[2018#Yang]] - [[2019#Saad]]: genus-1, 2-point, 1-boundary (eq.4.21) - [[2020#Blommaert]] - [[2021#Iliesiu, Mezei, Sarosi]]: (eq.3.9) - [[2022#Jafferis, Kolchmeyer, Mukhametzhanov, Sonner (Long)]]: disk, trumpet, three boundary - [[2024#Iliesiu, Levine, Lin, Maxfield, Mezei]] ## Related - [[0471 String-matrix duality]]