JHEP02(2021)134 Springer 4 c, February 16, 2021 December 11, 2020 December 28, 2020 : : : , Received Published Accepted and Jakob Salzer 3 b, Published for SISSA by https://doi.org/10.1007/JHEP02(2021)134 [email protected] , Stefan Prohazka 2 b, [email protected] , . 3 Jelle Hartong, 1 a, 2011.13870 The Authors. c 2D Gravity, Classical Theories of Gravity, Space-Time Symmetries

We construct various limits of JT gravity, including Newton-Cartan and Car- , [email protected] ORCID: 0000-0001-7980-5394 ORCID: 0000-0003-0498-0029 ORCID: 0000-0002-3925-3983 ORCID: 0000-0002-9560-344X 1 2 3 4 Peter Guthrie Tait Road, EdinburghCenter EH9 for 3FD, the U.K. FundamentalCambridge, Laws of MA Nature, 02138, Harvard U.S.A. University, E-mail: [email protected] Institute for Theoretical Physics, TUWiedner Hauptstrasse Wien, 8–10/136, A-1040 Vienna,School Austria of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, b c a Open Access Article funded by SCOAP ArXiv ePrint: manifold or some HamiltonianSchwarzian reduction for thereof. JT, After we recoveringaction. show in We that our comment AdS-Carroll formulation on gravity the numerous applications yields and a generalizations. twistedKeywords: warped boundary Abstract: rollian versions ofdimensional dilaton light gravity cone. in Inconnection two with the dimensions boundary scalar, BF as yielding formulation as well boundary our action as boundary the particle conditions a action relate on theory a boundary group on the three- Daniel Grumiller, Limits of JT gravity JHEP02(2021)134 23 11 31 17 19 30 14 32 20 – 1 – 7 26 expanded BF theories 25 17 3 /c 14 4 29 5 20 33 26 6 14 5 /Hyperbolic plane 2 2 3 15 2 6.3 Other kinematical algebras 7.1 Selected applications 7.2 Selected generalizations 5.2 Hamiltonian reduction of boundary action 6.1 Euclidean AdS 6.2 AdS-Carroll 4.1 Metrics and4.2 limits Flat space4.3 dilaton gravity Summary 5.1 Particle on group manifold 3.2 BF on3.3 the light cone Kinematical limits3.4 of BF theories Newton-Cartan dilaton gravity and Carroll dilaton gravity 2.1 BF theories 2.2 Geometric interpretation 3.1 AdS and dS BF theory D Matrix representations A Metric Lie algebras of lowB dimension Lifshitz, Schr¨odingerand 1 C Coadjoint theories and their limits 7 Applications and generalizations 6 Schwarzian-like theories 5 Boundary actions of kinematical BF theories 4 Metric BF theories and their limits 3 Limits of JT gravity Contents 1 Introduction 2 2d gauge theories of gravity JHEP02(2021)134 ]. ], 5 31 ] for ], the ] for a 33 11 30 , , 32 10 gravity. In 2 ]. See [ ] and to higher we discuss two 29 39 6 – – 24 36 ] and implementations we take singular limits ], holographic complex- 3 23 20 , 22 ], holographic renormalization and we review general aspects of the 7 ] and beyond [ ]), relations to random matrix mod- 2 we focus on the subclass of metric BF 15 35 – , 4 ], constant dilaton holography [ 13 34 9 – 2 – we discuss boundary actions and how to perform ], traversable wormholes [ 5 ] features prominently in classical and quantum grav- 4 19 – , 1 18 ] (for reviews see [ 12 ], noncommutative geometry [ 6 ], the attractor mechanism [ 8 -deformations [ ¯ T T ], we conclude with a discussion of possible applications and generalizations. ]. 17 7 ], constructions of the Hartle-Hawking wavefunction [ , 40 This paper is organized as follows. In section In several ways this work mirrors investigations of Chern-Simons theories in 2 + 1 The urge to look for theories beyond the Riemann-Cartan setup is partly motivated For applications or toy models of non-relativistic holography it is of interest to con- 21 16 theories and their limits.a Hamiltonian In reduction section froma the Schwarzian-like action action for byexamples a for imposing particle such certain boundary movingsection constraints. actions, on a first In for group section JT manifold and to then for AdS-Carroll spins [ formulation of 2d gravity as aof gauge JT theory gravity, of among BF-type.to other In section things Newton-Cartan to and Galileantheory Carroll and that dilaton Carrollian does gravity; theories not require that we any we also limit. generalize show In that section there is a light cone the questions addressed andto issues models raised introduced in in this our context work. could potentially bedimensions transposed based onPoincar´eand Lie (A)dS algebras and beyond extended the to Galilei semi-simple [ case, started in [ by the relationin of flat Carrollian space, symmetries andphysics. to partly Eventually, null by some applications surfaces, ofor of our like ultra-relativistic non-relativistic models holography horizons theories may in or serve in the as null condensed spirit gravity of matter infinity duals the for JT/SYK examples correspondence, of and non- many of purpose of our work ismodels, to including show boundary how actions thisour and is construction boundary done allows and conditions. to toof Among discuss address 2d other some dilaton questions aspects applications gravity?” such ofJT or as these gravity?”. “What “Is new is there the a Schwarzian analogue Newton-Cartan for version the AdS-Carroll limit of eralizations of JT gravity, like theNone Callan-Giddings-Harvey-Strominger (CGHS) of model these [ generalizations sorics far (or gave a up corresponding the Cartan assumption formulation). of (pseudo-)Riemannian met- sider singular limits of JT gravity to, say, Carrollian or Galilean spacetimes. The main JT/SYK correspondence [ els [ ity [ of the island proposal toreview on resolve further the aspects black of information two-dimensional loss (2d) problem dilaton [ gravity, including numerous gen- Jackiw-Teitelboim (JT) gravity [ ity as a convenient toy modelones at to a elucidate bare conceptual minimum. problemsCardyology Examples while attempts include keeping implementing [ the ’t technical Hooft’sthermodynamics brick [ wall proposal [ 1 Introduction JHEP02(2021)134 ∗ X ∗ λ (2.2) (2.3) (2.1b) of a Lie given by = ad is a gauge K ∗ I ]; especially E . X IJ µ c λ 45 δ dx = 0 K I λ e K I µ J X A  A J I J A = A with basis JK K c ∧ ∗ A g IJ I − c A I + K is a scalar transforming in the dλ ]), or explicitly, I IJ I c E − I dX 1 2 λ, F = ] (2.1a) X + I . = [ ,A = A K K ∗ λ ∗ E F ]. The structure constants δ X ] is a specific BF theory based on the Lie J λ [ dA X (with the dual  IK BF 43 c A, A , K K [ the adjoint and coadjoint actions are given by = ad e – 3 – L − ] appeared on the arXiv. Where applicable, our 1 2 42 X K ), respectively. Equivalently, in a basis this reads ∗ 2 F · 41 = 0 = + g that is a non-degenerate, symmetric, ad-invariant = λ IJ M X J δ c K Z R E F dA A I ∗ π (ad ∗ e k Y ∈ ). Varying it leads to the equations of motion 2 ∧ X → ≡ ] = J K J −Y g 2.1 F e A ] = ] = X and , I × J I and ad g ) = g ,A ,A λ e · JK ∗ ∗ ( : K K c ∈ e X X Y 1 2 ]) (and hence [ [ IJ ·i K ∗ X c , + BF BF h· IJ = I I A, λ c X,Y L I ] where further details are provided (see also the review [ dA + [ X Shortly after our work [ ] = 44 λ = J δ ] and ad dλ e 1), which features an invariant metric. Some of its limits may lead to Lie I When applicable, upper (lower) signs in equations refer to the Euclidean ( , , ). For I F − I J is defined by the bulk action are defined by [ e (2 δ X,Y = is a dimensionless coupling constant, g so = [ k A = [ ) = J λ δ J e An interesting subclass of BF theories is obtained when the gauge algebra is given The definition of BF theory does not require a trace or invariant metric. This is We follow [ Y invariant metric I e X e ( I by a metric oran (regular) quadratic Liebilinear algebra. form This (in means the that following we the will Lie often algebra omit admits the comma; by ad-invariance we mean and leave invariant the action ( ad different from gauge theories of, e.g.,algebras Chern-Simons or with Yang-Mills type an that invariant are metric. basedand on The Lie gauge transformations are given by where coadjoint representation and thefield Lie with algebra curvature two-form valuedalgebra one-form E ad relevant is section 6 on Schwarz type topological2.1 gauge theories). BF theories BF theory JT gravity in itsalgebra first order formulation [ algebras without metric. Sinceset these subtleties the will stage be by relevant providing for a the rather remaining detailed work, reminder we of BF theories. (Lorentzian) case. Note added. results agree with each other. 2 2d gauge theories of gravity Notation. JHEP02(2021)134 . g ). 1) A , I K 1 δ (2.4) (2.5) Klein − = n JK g IJ g = ISO( 1) as a subgroup 2 dimensional Lie  G spacetime dimen- ]. We provide an − / J and or a non-relativistic n n 39 i A ). There indeed exist J ∧ e F + 1) , I n X I ( A e n h K ). Since it is non-degenerate, = . IJ g . To this end it is convenient = tr( c ), defines a (topological) gauge Carrollian ∈ 1) for dS, and 1 2 IJ g with elements of the Lie algebra 2.4 n, + mBF ] = 0 ∗ exist. This question, studied first g X K L n x, y, z A, dA Klein pair (where = SO(  . + [ J L G X X X 2 Killing vectors. From this follows that one d IJ / is given by g – 4 – LK ]) which has also been used to find generalizations g ], boils down to a classification of so-called = 47 + 1) I = 49 n i X ( , of dimensions 2) for AdS, n without any geometrical meaning. In this subsection we , ,F 1 g ), or metric BF theory ( hX = 0 − 2 dimensional subalgebra of the / 2.1 n F ] (see also [ 1) ] = kinematical spacetime 46 − ,A metric BF theory n ( = SO( X [ n ]. The standard example of metric BF theories are given by simple Lie G = 0 for all Lie algebra elements i X an mBF ] h ), or more explicitly λ, · L ( z, y [ ∗ = [ 1) with kinematical spacetimes X x, , h X 1 that together determine the respective kinematical spacetimes. Note that most ) with λ = structure. The former consists of a degenerate metric whose kernel is spanned by + h − g i , ·i n g , ( ] and answered exhaustively in [ , y ] = ad More generally, one can ask the question which other homogeneous and isotropic As is well-known, (A)dS and Minkowski space, and quotients thereof, are the only The Lagrangian for hX 48 SO( X z, x λ [ algebra of these spacetimes do not exhibitclassified a according metric to of the Lorentzian or existenceGalilean Euclidean of signature an but ultra-relativistic can be a single vector field. A Galilean structure, on the other hand, is defined by a degenerate also isotropic. By this weand mean splits that into the symmetry representations group thereof. contains SO( spaces, i.e., in [ pairs Lorentzian manifolds that aresions both this implies homogeneous the andcan existence isotropic. describe of these In spacetimes,G/ without introducing afor metric, Minkowski, as respectively. the homogeneous In spaces addition to being homogeneous, all these spacetimes are theory based on a gaugeclarify algebra how the fieldsintroducing appearing additional in structure these in actionsto the introduce acquire form the a of notion geometric a of interpretation upon for the Galilei andoverview of Carrollian all cases Lie algebras which of we low dimension will that encounter admit2.2 an below invariant metric [ in appendix Geometric interpretation Up until now the BF theory ( δ algebras where one can useLie the algebras matrix beyond trace the to semisimpleexample write ones is that the admit 2 + anhelps 1 to invariant dimensional understand metric, them Poincar´ealgebra. e.g., [ There one exists notable a structure theorem that with equations of motion As the coadjoint and adjoint representations are isomorphic we rewrite the transformation we can use this metric tovia identify elements of the dual h JHEP02(2021)134 B (3.3) (3.1) (3.2) (2.6) in 1 + 1 dimen- g B a P is important as space- ab a  ab h η ˆ Λ X ˆ spanned by the generator Λ 1) in the Lorentzian and − ]. + µ − 52 B , For more details we refer the – = ] = X b i 50 1 b P , P = always a , a P is X [ P B h h = diag(1 ω + ˆ ab Λ are isomorphic. It is the choice which η P ) as temporal (spatial) zweibein component. e P ( can have vastly different geometric properties + H – 5 – H g τ = A b a P P b 0) BF theories can be written in a covariant fashion where a a e  µ > . We will encounter examples illustrating this fact throughout − + = ). They are based on the Lie algebra h ˆ Λ P B i , B ω ] = H , a , the Klein pair dictates the form and use the convention that B P = h h , } B A 1) in the Euclidean case. The most general invariant metric is [ B ) = ( , , 1 P P , , 0) and dS ( 0 H P { . Expanding the Lie algebra-valued one-form of a BF theory based on this < ]. We emphasize again that the choice of subalgebra is the gauge field of the internal symmetry transformation, i.e., the dualized B = 1 and we raise and lower with ˆ = Λ 49 = ( ω = diag(1 g 6= 0 is an overall proportionality factor. a 01 P ab  µ δ The Lie algebras for positive and negative Returning to the problem at hand, we denote the three generators of = For non-homogeneous Carrollian or Galilean spacetimes there does not exists a preferred connection, 1 ab distinction between these two cases.spans As the discussed subalgebra in the previous section, assuming that like the Levi-Civita connection in the case of Lorentzian geometries [ where generator is part of the spin-connection and which is part of the vielbein that provides the we use where η the gauge connection components.(A)dS Then JT we gravity. discuss and provide the kinematical3.1 limits of AdS andThe dS AdS BF ( theory 3 Limits of JTTo gravity set the stage we brieflyNext review we the define well-known a (A)dS novel JTalgebra, BF gravity case but theory in on the BF the underlying formulation. light spacetimes cone, differ which due is to based on different the geometric same interpretations simple of Lie we can now interpret the fieldThe associated to field spin-connection associated to localKlein Lorentz pair thus transformations provides in a the map relativistic from the case. a priori The abstract gauge field to geometric data. this work. sions as denoted by algebra as define, in most cases,spacetimes according a to distinguished the connectionreader curvature to on of [ these this spacetimes connection. times and with classify isomorphic symmetry these depending algebra on the choice of co-metric whose kernel is spanned by a nowhere vanishing one-form. One can furthermore JHEP02(2021)134 ] (in (3.5) (3.6) (3.7) (3.4) 49 , where the and plugging 1 ) this allows us ω P . − 3.2   b b e e ] = P ∧ enforce the 2d torsion ∧ , (right). H ω H a ω [ µ . ). (Dropping the Palatini b 2 b a X a  X =  ( are indicated. i + V + H B a , a H de 2 h de  are depicted in figure a  dS a (left) and dS 2 ).) Thus, JT naturally generalizes to X ) has as most general Lorentz invari- 2 a X P + X 3.4 + H a  and boosts  ab P  ] = ab b , B  P e H b – 6 – X,X , e ( B ∧ [ V ∧ a e a ) e H → ˆ 2 Λ X ) ( for convenience. The fields X − V 2 ( µ a . Upon solving this for the spin connection + P V dω X = a e AdS i  dω P → . Penrose diagrams for AdS ,  X B B a h X B = − X = ˆ JT Λ ] = L dil H , Figure 1 L B [ ) which is the light cone of three dimensional Minkowski space seen as 2d R , (2 sl ] 49 As a reminder the Penrose diagrams ofFor later (A)dS purposes we discuss now briefly how to arrive at arbitrary dilaton gravity being the dilaton field. and invariant metric The Lie algebra is, for1 any + dimension, 1 isomorphic dimensions to also the to one the of one de of Sitter spacetime anti-de [ Sitter). However due to the different choice of algebra manifold. The homogeneous spacee.g., of [ the light cone is based on the following algebra, see, generic dilaton gravity models. The same is3.2 true for its various BF limits on studiedIn the below. addition light cone to thethe previous well-known homogeneous subsection there spaces exists and another their homogeneous BF space theories based mentioned on in the symmetry ant generalization that preservesLagrangian the Palatini condition of vanishing on-shell torsion the that depends on ancondition arbitrary further function generalizes of the dilaton field, X transformations generated by translations models starting from JT. The JT Lagrangian ( where we rescaled constraint for the zweibein it into the action, one is left with the well-known second-order action for JT gravity with of the gauge fieldto and write the the coadjoint Lagrangian scalar. for (A)dS-JT Together gravity with as the metric ( JHEP02(2021)134 (3.8) (3.9) ). (3.10) 2
and τ or Carrollian 1 ∧ 2 ω . 0) B − . ˆ Λ → P − dω e c 1 + ] = X = H P B τ , c + H [
+ e B ∧ ω τ = − H A de M 0), Galilean (ˆ P α (which has timelike asymptotic boundaries) X 2 . + 2 → 2 H + 2 , together with the geometric interpretation of – 7 – ˆ Λ
LC c P e 2 into our theory such that the nonzero commutators X ∧ ] = ˆ M P ω + , H B + [ H H dτ X ˆ Λ is the cosmological constant which is (negative) positive act differently in these two theories (see figures + H B B is introduced so that all contractions involve parameters tending to zero. BF theories allow for a gauge invariant action, irrespective of X P c P = X 2.1 i P = 2 ˆ C ,F X ∓ ] for the explicit construction. hX 49 ] = = H , the corresponding homogeneous spaces differ. In particular, the light cone , h LC B [ L (which has spacelike asymptotic boundaries). Instead of a Penrose diagram we . Two-dimensional future light cone of three-dimensional Minkowski space with vertex 2 0) limits leads again to a well-defined Lie algebra; for a visualization of these limits The light cone theory has the Lagrangian In a sense the light cone is in between AdS The inverse speed of light ˆ → 2 ˆ C The upper (lower)distinction sign is specifies applicable that and for the (A)dS theory spacetimes. is( Euclidean Each (Lorentzian) of in the case flat this ( without compromising the well-definednesswe of introduce the an action additional and generator are theory. given Before by we show this since ‘boosts’ generated by 3.3 Kinematical limitsAs of discussed BF in theories section the existence of an invariant metric on the Lie algebra. This means we can take limits where we used This Lagrangian is equivalent to the JT Lagrangian, but the interpretation is different just depict the lightthe cone three itself generators. in figure subalgebra is a Carrollianspacetime; spacetime, see i.e., [ one can define anand invariant dS Carrollian structure on this Figure 2 removed. Topologically, this is a strip like (A)dS JHEP02(2021)134 (3.12) (3.11) 0); flat M → m c 1 + = P e c + . H τ + ) = 1 B M ( ω

1 ∗

cˆ = → 0 (A)dS-Galilei M

c = M  

was dropped in this diagram and m

ˆ Λ M +

a (Anti-)de Sitter ← P ) = 1

a 0 P e are parametrized as ˆ (

C ∗ + P A

← B

0 ω   – 8 – . Galilei , which shows that the starting point is actually = Poincar´e H 0). Arrows with two lines imply there are two different B A → ) = 1 H ˆ ∗ C ( M ∗ M H   6= 0 is a trivial central extension that before taking any X c + Para-Galilei ∗ (1). However, it becomes a nontrivial central extension in the (A)dS-Carroll for ˆ P u P M 0 leads to the centrally extended Galilean algebra, better known as ⊕ X ) = 1 ) B + ( → R and the gauge connection ∗ 0. We refer to the centrally extended Galilean algebra with nonzero ∗ , B ˆ H ∗ Λ   H (2 → X Static sl X Carroll c + ∗ . To reduce clutter, the central extension B 3 . Kinematical limits of (anti-)de Sitter: non-relativistic/Galilean (ˆ expanded Poincar´ein appendix X = /c 0); ultra-relativistic/Carrollian ( ∗ We now construct the limits of the action, equations of motion, and gauge transfor- The contracted action, equations of motion and gauge symmetries are well defined as The new generator → X ˆ Λ ˆ where the dual basis is defined by long as the Lieinteresting algebra algebras contraction that is do well notand defined. 1 follow We from have a summarized kinematical further limit, possibly like Lifshitz, Schr¨odinger mations explicitly and studycoadjoint scalar their Lorentzian, Galilean and Carrollian invariants. The algebra). Sending the Bargmann algebra. We couldside have of added a each similar of centralalgebras. the extension on other the brackets right leading hand to centrally extended Carrollian and Poincar´e limits could be eliminatedthe by direct a sum shift of Galilean limit ˆ Λ as the extended (A)dS-Galilei algebra (also sometimes referred to as the Newton-Hooke limits, depending on the sign of the cosmological constant. see figure the Euclidean cases and thefrom light any cone limit, algebra are of the not previous represented section, in that this does diagram. not follow Figure 3 ( JHEP02(2021)134 + H H ∗ λ (3.16) (3.17) (3.18) H (3.14a) (3.15a) (3.14b) (3.15b) i + B B . Xe λ ˆ M Λ ) = e ∗ − H λ ∧ ω i M P P . . In the Carrollian  ) τ X is invariant, which is αω B ) (3.13) H 2 Xλ  e M = 0 ˆ + ) ( τ . eλ C ˆ Λ B B ∗ metric F M −  λ − M nor eλ 2 dm M P H B ˆ X τ C − λ P ωλ + ( dX dX ∓ P ( h ) + P X + α ) = 2 P ωλ + ( ∗ τ ˆ ( e C − 2 P ∗ B F ∧ ˆ c i M λ  B P h i ω ω − X e ) dλ 2 . ) H + H ˆ H non-degenerate 2 C − = 0 and ∗ X . e  ) + X dλ B 2 ∗ ∓ 2 F H c i + + P c ( P − i  2 + ˆ de λ P F B + ˆ ) τ i H M λ H + ) M – 9 – ) ∓ B ) are X + ( H B X i 2 αX H = X τλ αX M ) ( ) c is invariant. Only in the Lorentzian/Euclidean case 2 ) H ) + e g 3.13 c e M − − B + ˆ ( + ( ∧ ( eλ H M + ˆ F τ ω M − P Xτ 2 ωλ 0 we get the invariant ‘clock one-form’ + XF P c αX e δ ( ˆ Λ X B 2 P 2 αX + ˆ λ ) ˆ → τλ ( + ˆ C + ( C 2 ] = P c P c ( H ˆ − Λ( ˆ dτ  Λ. Without taking any limit neither ∓ λ F ,A H P = ˆ P ∗ + dX + ( h + τ B X X dλ dX B Xλ [ 2 B H h λ + ) ) ˆ ˆ dλ Λ − δ C e L H h h = ( ∓ − ∧ ∗ h h + + F τ X = = ˆ + Λ ∗ A ∗ B A − ) λ X B δ λ ( , + ad δ dω M F ∗ M λ X = ( = d Since they provide us with additional geometric information, we first investigate the + F P P it justifies our claim tothis define shows that Galilean theories and based Carrollianconnected on gravitational via theories. the dualities same upon Additionally algebra exchanging,Carroll. can e.g., be time This geometrically and should different, spacemay be but as be are in viewed compact the as and case a the of time local Galilei direction and statement, non-compact. since globally the spatial direction limit the spatial zweibein component we can define the invariant Lorentzian/Euclidean This is a manifestation of the fact that we are looking beyond Lorentzian geometries, and Expectedly, these transformations are independent ofpendence curvature, of as the evident parameter from theirfamiliar inde- from Lorentzian geometry where noexists. distinguished invariant In vector the field Galilean or one-form limit ˆ invariants of the local boosts All contraction parameters havedefined. positive The exponent equations and of consequently motion of the ( limits are well- The action is invariant underλ the gauge transformations parametrized by with curvature The Lagrangian is given by JHEP02(2021)134 ]. on 55 and 6= 0) de H . The ˆ (3.20d) Λ ω in terms ω = 0 can be de is closed and so τ ) for the special cases that spans the kernel v 3.13 ]. The remaining two equations . The fact that ] the undetermined components τ 57 2 [ α ˆ 56 c is to ensure that the curvatures are m − . = 1 without loss of generality. In this . This leads to an expression involving ω e m α = 0= 0= 0= 0 (3.19a) (3.19b) (3.19c) (3.19d) = 0= 0= 0= 0 (3.20a) (3.20b) (3.20c) and e e e along the vector = e e e τ de and ∧ ∧ ∧ dτ , ˆ C ∧ ∧ ∧ ee τ τ τ ω τ ω 2 ω ˆ Λ αω ˆ c – 10 – Λ + ∓ − , which is thus an independent field. This is not + + ˆ − e connection), since the curvature equations do not de are both nonzero the equations setting the dm dω dτ dm dω m ˆ C along = 0, setting , is then either set equal to the volume form (for c ω and dω c , i.e., ω in terms of the NC fields in the Lagrangian. The role of ω ] for a general discussion of 2-dimensional Carrollian geometries). The P = 0 can also be rephrased as saying that the intrinsic torsion is zero [ ˆ Λ = 0), corresponding to the usual 1+1 dimensional maximally symmetric 54 X = 0. When ˆ , = 0 we are dealing with the following curvature equations de ˆ ˆ 53 = 0 equations of motion read C C F . τ (see [ 2 α were shown to correspond to Lagrange multipliers enforcing the constraint that the ˆ c We next consider the case ˆ When We will study the equations of motion of the Lagrangian ( ee ω − = 0 or curvatures equal to zero are the zero torsion conditions in the Cartan description of ˆ We interpret these equations in the languageand of third Newton-Cartan (NC) equation geometry. are Theboost the second connection curvature constraints imposedfix to the be NC able geometry. tothese fully The Newton-Cartan solve top spaces for admit equation absolute the states time that (provided there the are clock no one-form closed time circles extrinsic curvature vanishes. In ourfrom setting varying the vanishing ofCarroll the boost extrinsic curvature invariant. results case the Unlike in the Lorentzianof case, the for vielbeine Carrollianfix (and geometries the possibly we vielbein the cannot componentuncommon solve of for for Carrollian geometries.of For example in [ interpreted as vanishing extrinsicthe curvature. two vectors To that see are this,the dual to one Lie the evaluates derivative 1-forms the ofof 2-form the Carrollian metric vanishing of which correspond to acurvature constant U(1) curvature gauge 2-dimensional field Carrollian on geometry with it a that zero is given by Euclidean/Lorentzian spaces. Therem is also a zero curvature abelian gauge field, namely c P Euclidean/Lorentzian geometries. Theabelian zero curvature torsion of equationsor can to be zero solved (for for JHEP02(2021)134 (3.21) (3.23) ]. The should ) (or its . V 37 ) , e 3.5 ) (3.24) , in general. ∧ 36 , dt ω t, x 34 ( = H + , τ c H P = 0= 0 (3.22c) (3.22d) ) = 0) = 0 (3.22a) (3.22b) X X M e dm H H ( M = = M by a redefinition of the dX and implies that torsion X H H X M , H + P X X,X X,X X = 0 defines a NC geometry. ( ( τ X ) + c P τ e V τ V X ∧ + − ∓ ) ω (1) field the curvature is not constant. The P M u ) with ˆ ∓ X dX t, x c ( ω X ω X de 3.13 ( X X  − P H P = = does depend on X ) change, but we nevertheless display all for ]) there is a ‘constant dilaton’ and a ‘linear dX X + dX V 58 3.20 – 11 – dτ H P .X X X ˆ Λ is based on the extended (A)dS Galilei algebra, also + ). The last equation on the left controls the curvature of e ), after removing the = 0 = 0 = 0 = 0 ∧ 3.5 e e e that are non-linear in τ = 0 = = const τ 3.4 ∧ ∧ ∧ ) ∧ M M V H τ τ in the third term can be exchanged by a field redefinition, so ω X X ω ) ) H H ω + ∓ ˆ Λ = 0 leads to the Bargmann algebra which does not admit an for the NC limit of JT and an arbitrary function more generally. X,X ) with all contraction parameters set to unity is identical to the ( de dm X V X,X X,X ( ( 3.13 + ∝ V V H ) X H X ∂ only, just like in ( ∂ denote a distinction between a Lorentzian or Euclidean version of NC dilaton + X dω . As discussed above, the Lagrangian ( + X H is exact). The last equation fixes the boost curvature which can be viewed as the X,X = ( dω not dτ τ signs in front of V ∓ constant dilaton : linear dilaton : Only two of the curvature equations ( We propose that this generalization is given by the NC dilaton gravity Lagrangian NCdil L As in ordinary dilatondilaton’ gravity sector. (see, e.g., [ no longer vanishes, i.e., the NCThus, clock if 1-form one no requires longer NC locallydepend dilaton is on gravity given to by maintain torsionlessnessthe the geometry. potential For functions equations on the right hand side are the NC version of the dilaton equations of motion. The first equation on the left only changes if gravity. We nevertheless keep them to show they doconvenience not on matter. the left handof side motion below, on together the with right the hand remaining side. half of the equations where The they do Lagrangian of the JTgenerator model ( It is then fairtorsionful to generalization ask below what that is equation)? the NC version of more generic dilaton gravity ( called (extended) Newton-Hooke algebra,metric. which in The 1 case +invariant 1 with metric. dimensions admits an invariant 3.4 Newton-Cartan dilatonThe gravity and Lagrangian Carroll ( dilaton gravity 1+1 dimensional version of the equation2d for we NC can gravity. formulate It a is Lagrangiandimensions interesting theory to the of point NC Chern-Simons out gravity formulation that (coupled in of torelated NC scalars) to whereas gravity a in requires generator 2+1 an that additional1 is + connection 1 not dimensional contained case within with the nonzero Bargmann algebra [ so that JHEP02(2021)134 + B B λ (3.28) (3.25) (3.29) (3.27) (3.26a) (3.26b) (3.26d) , where + . dt A M · ) λ ξ ) = 0. As in c H = ) provided the t, x λ ,X . c . 3.22 dt . P ) X = Φ( B λ ) H ( M λ P dy . M )]. First we solve the m V X ) t, x X y X V λ ( ( + X,X − Φ( ( H V V + x H λ V B ∂ X P H = λ X . Most of the remaining diffeo- P V λ X = Z (1) transformations ∂ V u X − ∓ dx − ) = = = 0= (3.26c) ) := = ) H P M X , and time-dependent shifts generated t, x , and X ( ) is solved as follows. The penultimate δX dτ 0 ( ( B δX δX δX dt ω w x dX ∗ λ K ) dV + 3.22 = ) = 1. Finally, there is another abelian gauge x t, x e ). The latter two can be used to fix to zero ) = t → ( t, x – 12 – ( g x = Φ( ) t, x , which partly gauge fixes diffeomorphisms. Next, ) ) H K ) ) and boosts accompanied by compensating spatial H , after setting to zero all integration functions by dt H t ) = 1. X Φ( ( eλ x e ( ) = 2 x eλ and t f = ∂ w ( ∧ − 0 = const. The other three equations combined yield the − ∝ x τ B t B = P τ X,X ξ = P 0 ( t , Galilean boosts ( M + 2 ∗ τλ eλ P ξ ∂ dx m τλ λ V τλ t ( /m ∓ ( X ∓ + X = 1 V that we exploit to fix the mass 1-form as 1 2 V ∂ H P H are determined as roots of the potential, ) is invariant under a non-linear modification of the symme- → = − = 0 by M X X − c t H  λ ∂ ωλ ωλ ∂ B = X H = λ dτ 3.21 −  − − M X H P M B M dt e X dω and X ∗ dλ dλ dλ dλ = c ) solves all the equations of motion on the left side of ( − − − − we are back to the NC case with cosmological constant and obtain the τ X . is a 2d vector field, establishes that gauge transformations reduce on-shell to = = = = 3.27 ξ δe δτ ) δω δm 3.15 = const (1) transformations with The right half of the equations of motion ( Our 1-forms read The general solution of NC dilaton gravity in the linear dilaton sector is obtained as The Lagrangian ( , where u V M M one trivially yields non-linear (Casimir) relation If expected confining potential,residual Φ gauge fixing. The result ( Newton potential Φ obeys the second order partial differential equation symmetry generated by Φ is interpreted as Newtontrivial potential. coordinate At shifts, this stageby the residual gauge transformationsdiffeomorphisms are with integration functions encountered below. follows [for simplicity weclock restrict 1-form to equation thewe torsionless gauge case fix boost invariancemorphism by invariance demanding is fixed by setting Choosing a field-dependent parametrization ofλ the gauge transformations, diffeomorphisms generated by tries ( ordinary dilaton gravity, curvature istorsion constant does for not constant vanish dilaton but solutions; is by also contrast, constant for general constant dilaton solutions We defined our orientation by The constants JHEP02(2021)134 . ) 0 = X ) + m B ( . λ X 0 V ) = 0, (3.32) (3.33) (3.34) ( c P m ) where w ) can be t ( ,X 0 c ) (3.35) x (the same is -dependence 3.29 X ( + X V t, x ( V 0 X H de . , ∂ c P P ) (3.30) P − X X x/m X , related to the mass X ( 0 = = = w = m -dependence of the new P H 0 ) + , dω e P X X X m ∗ ∧ X − ω when plugged into the second = + H -dependence on the other hand ) . The V P X H dτ X ( ). Finally, the relation ( t, x H , also on the mass parameter t c ( ( Φ = ) (3.31) X X 0 , obtaining the solution above with X X f X 2 x X + ∂ m ( ) integrates to = = = 0 = e w − /X P M X X ∧ λ X 3.22 τ ) = can be gauge fixed to zero with our remaining ) = – 13 – ) and, via x ) = 1 P X 0 . The discussion of equations of motion, constant x ) ). Integrating this equation yields Φ = X Φ( 1 ( Φ( − 3.3 V t, x X,X ( 3.22 ( H = = 0, yielding V M X P X + X = 0 = H H are again determined as roots of the potential, and thus controls the extrinsic curvature or intrinsic torsion of X X (1) field by a redefinition of c P 0 u X dω x de X m ). = V = . It contains one relevant constant of motion, P determines the curvature of the geometry with linear x and X X ∂ c Car-dil V − ) to gauge fix ). We exploit now the residual gauge transformation generated by X t t L ( ( . = P H g X X de  ∗ ) = t ( x linear dilaton : constant dilaton : ), where the integration function Φ Along the same lines we also propose general Carrollian dilaton gravity To give one example we choose Our solution ) = ξ t t t ( ( ∂ 0 0 ˙ The constants and curvature is again constanttrue in for the constant dilaton sector, leads to non-vanishing the geometry as discussedand in linear section dilaton sectors is analogous to the NC case above. Here we eliminated the potential term corresponding to constant curvature. A non-trivial which is just theThis Newton-potential shows in that three one spatialsuitably, can dimensions which obtain is for higher-dimensional again an Newton identical object to potentials of how by things mass choosing work in usual 2d dilaton gravity. depends only on the spatial coordinate shows that the label ‘linearspatial dilaton coordinate sector’ is indeedof justified, as the the solution. dilaton In is the linear chosen in gauge the the Newton potential solved for x ∓ equation of motionΦ on the rightresidual ( gauge transformation generated by The last equation of motion on the right ( JHEP02(2021)134 . (4.1) C ˆ Λ (4.2) 2 B ˆ c ˆ 0) or Carrollian Λ µ − = → i c 1 P ] = , P P , = h H [ c ) (4.3) ) is the center of the Lie algebra g g ( ( Z Z ] by one without adding any element ˆ Λ ) g 2 , ˆ 0), Galilean (ˆ g C H 3.1 2 µ c where → ] + dim  g g ˆ , Λ = ] = ˆ g P i , – 14 – H B , [ H h = dim [ g dim ]). As long as we do not take any limit, the center is trivial and 2 P ˆ 2 C 39 2 ˆ C ˆ c ) leads to a degenerate metric. In case we want to end up with a µ ∓ 6= 2+0 and no invariant metric is possible. The addition of a nontrivial  4.2 ] = = H i , B B [ , B h 0) limit of ( In the following sections we show that the addition of central extensions is also sufficient → ˆ C 4.2 Flat space dilatonA gravity metric BF formulationsuitably centrally for extended. the We show Poincar´ealgebra(A)dS now at is how the to only level obtain possible of the the when resulting action. theory the as algebra a limit is of but two on the right hand side, balancing theto equation equip again, the Lie 3 + algebras 1 andcan = theories be (2 with + an obtained 1) invariant + metric from (0invariant and + metric a show 1). is how limit. these based theories on Another so option called to coadjoint obtain Lie algebras algebras as with discussed a in non-degenerate appendix for the existence of a(see, metric e.g., on section a 3.2 Lie in3 algebra [ = 3 + 0. However,to taking the any center, i.e., limit 3 reducescentral dim [ element adds one element to the dimension of the Lie algebra (on the left hand side), algebra and metric.( Taking either the flatmetric ( BF theory there exists a necessary condition with the invariant metric parametrized such that any limit of the Lie algebra is well defined on the level of the Lie 4.1 Metrics and limits Starting point is the decomposed (A)dS algebra ( when the limit has thealgebras chance and to limits lead can to be anothertheories metric generalized and BF to theory. observe also Then obtain the weconditions Poincar´e,Carrollian and relation in show Galilean the of how next BF the the section we latter provideat a to the summary NC of end the theory. of various theories this Before we section. have discussing unveiled boundary For metric BFalgebra. theories In we that demand casemore taking the subtle limits, since existence especially the contracted of of Lieof the an algebras the invariant do metric invariant metric. not of necessarily metric theadditional inherit Since on the Lie many structure non-degeneracy algebra, interesting the plays is theories gauge a are rˆolefor based our on setup metric Lie of algebras boundary and conditions, this we explain first 4 Metric BF theories and their limits JHEP02(2021)134 (4.7) (4.4) (4.6) (4.5a) (4.5b) , which is ˆ Λ remains / 1 ) 2 such that the µ M ]). χ . + . µ = 61 ) , i M B = ( M 60 ab , + 2 B χ ab h B  µη µ ˆ Λ (3). They encompass the ), which yields the CGHS = = − . so ] = ( i i b 2.4 b 1 M P ab P , ,  , B a a h ] (see also [ P P [ h ] = 59 2) and 0, assuming b , P , → (1 µ a P )) ˆ − so Λ [ = ' 3.2 i M 1) , , B – 15 – h ) and add an additional generator (2 so 3.1 ]. ' b 39 ) 2 P b µ R a . , b ˆ Λ ˆ  Λ + P 3 2 (2 b − µ a µ sl  ab = = − ] = a i i µη P B M , , , ) has a higher dimensional generalization, called Maxwell alge- ] = . We now take the flat limit = a B B M makes explicit that all, but the sphere, are based on the same Lie [ ˆ h h µ Λ P i 4.6 , b 1 B P [ 7→ , is still a trivial central extension, but it is introduced such that after the a 0 it is nontrivial and leads to an invariant metric for (centrally extended) µ P M h ]. This is also true for the extended Poincar´e(Euclidean) algebra which arise → . Table ˆ Λ ] in the formulation of Cangemi and Jackiw [ 63 , 31 3.2 62 The table starts by providing the necessary information to construct the theories based The Carrollian and Galilean limits can be done analogously to the flat limit above. The algebra ( It remains to show that we can also take the limit at the level of the action, for which We start with the (A)dS algebra ( 6= 0. This means we can write down a metric BF theory ( best read together with figure on the simple Liewell algebras known (A)dS BFsection theories, their Euclidean cousins and the light cone BF theory of They lead to (A)dS CarrollCGHS and model (A)dS via Galilei geometric theories, dualities respectively, and as are summarized related in4.3 to table the Summary We have summarized interesting homogeneous spaces and Lie algebras in table bra, which emergesfields in [ the studywhen of considering particles a charged infield. particle classical in From homogeneous 1+1 the dimensionsmetric electromagnetic point generalization in of of a view Poincar´e[ constant of electric metric (magnetic) Lie algebras the Maxwell algebra is the natural This is the (non-semisimple)µ extended Poincar´ealgebra withmodel non-degenerate [ metric for with invariant metric where we used finite, yielding Poincar´e.The well-defined limit on thea Lie limit algebra of implies the upon equations substitution of of motion. the fields we introduce the shifted invariant metric (cf. ( algebra is given by At this point flat limit JHEP02(2021)134 µ µ µ µ µ µ − − − (1) u ======i i i i i i M M M M M M , , , , , , H P B H P B h h h h h h (1)), see and their u µ µ µ µ µ µ HP P P H H P P P g µ χ χ µ χ χ − χ χ − χ − χ µ − µ χ µ µ − − µ µ ⊕ ======i i i i i i i i i i i i i i i i i i i i i i i P P P M P H M H P M P P P P P P P P P P P P P , , , , , , , , , , , , , , , , , , , , , , , (1) P H P P P H H H P P P P P P P P P P P P P P P h h h h h h h h h h h h h h h h h h h h h h h u . µ µ µ µ µ µ µ BP HP HP BH BH BP BP H ⊕ µ χ χ − χ χ µ χ χ − χ χ − − − χ µ µ µ µ µ − − µ ) ======i i i i i i i i i i i i i i i i i i i i i i i i R Invariant metric 4.2 H P P Z P H Z H P Z P H H H H H H H H H H H H H , , , , , , , , , , , , , , , , , , , , , , , , , H B H H H B B B B B B H H H H H H H H H H H H H denote nontrivial central h h h h h h h h h h h h h h h h h h h h h h h h (2 c sl µ µ µ BH H H B B B B B µ χ χ µ χ χ µ χ χ µ χ µ µ χ − µ µ χ µ µ − µ − µ ======i i i i i i i i i i i i i i i i i i i i i i i i . They allow for one nontriv- B H H B H B P B B H B B B B B B B B B P B B B B (1) ( , , , , , , , , , , , , , , , , , , , , , , , , B B H B H B P B B H B B B B B B B B B B B B B B u h h h h h h h h h h h h h h h h h h h h h h h h 3.3 ⊕                         ) ND? ), underlying Lie algebras R B B B B B P B B , , B B Z Z − − B B M B B − − M − B B − − g (2 ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ]. Superscripts sl P P P P P P P P P P P P P P P P P P P . , , , , , , , , , , , , , , , , , , , H H H H H H H H H H H H H H H H H H H [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ 65 , Z M H H M H H H H M H H H H H H H H H 2.2 64 ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = P P P P P P P P P P P P P P P P P P P , , , , , , , , , , , , , , , , , , , B B B B B B B B B B B B B B B B B B B [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ – 16 – P P B P P M P P M P P M P P P P M − − − P − P − ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = ] = H H H H H H H H H H H H H H H H H H H , , , , , , , , , , , , , , , , , , , B B B B B B B B B B B B B B B B B B B [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ (1)’s correspond to central extensions that are trivial c 0 0 c 0 0 c 0 0 u c 0 0 c 0 0 c 0 0 cc cc cc II II II VI VI VI VII VII VII VIII VIII Nonzero commutation relations cc cc cc c c c ) VIII 1) 1) VI 1) 1) VI 1) 1) VI c c c 2) 1) VIII 1) , , , , , , R , , , , (1 (1 (1 (1 (1 (1 (2) (2) VII (2) (2) VII (2) (2) VII (1 (2 (2 (3) IX (2 Abelian I Heisenberg Heisenberg II Heisenberg Heisenberg II Heisenberg Heisenberg II iso iso iso iso iso iso iso iso iso iso iso iso sl so so so so ˆ Λ 0) ˆ ˆ Λ Λ ˆ ˆ ˆ ˆ ˆ C, C C Λ Λ → ˆ ˆ ˆ ˆ ˆ ˆ Limit ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ c, c, c, c, c, C, C, c c C C Λ Λ c c C C Λ Λ (1). The two u ⊕ one can show that all the theories below the first horizontal dividing line (1) , introduced in order to have a non-degenerate metric, decouples on-shell. u Z 4.2 . ⊕ . Overview of homogeneous spaces/Klein pairs ( ) 4.2 R Taking a second limit, e.g., first the flat and then the Carrollian, leads to the Car- Taking either one of the flat/Carrollian/Galilean limits of the (A)dS theory we arrive , (2 Static ˆ Doubly extended para-Galilei Para-Galilei ˆ Doubly extended Galilei Galilei ˆ Doubly extended Carroll Carroll Extended dS-Galilei dS-Galilei ˆ Extended AdS-Carroll AdS-Carroll Extended Poincar´e Poincar´e Extended AdS-Galilei AdS-Galilei ˆ Extended dS-Carroll dS-Carroll Extended Euclidean Euclidean Light cone Anti-de Sitter (AdS) Hyperbolic de Sitter (dS) Sphere Homogeneous space can be obtainedsl by contraction andbefore taking taking limits quotients but startingscribes become a from non-trivial standard afterwards. the NC/Carrollgenerator Nevertheless, parent structure the theory since theory the still additional de- field associated to the ial central extension thatextended renders theories the can invariant also metric be non-degenerate. obtained from These a (centrally) limit,roll, shown Galilei in and section para-Galileifrom theories. a For these degenerate theories toin doubly a section centrally non-degenerate extending invariant leads metric. Using the procedure described that we disallow anphysical interpretation, exchange as of described in time section and space)at and Poincar´e/(A)dSCarrollian/(A)dS Galilean therefore theories. in These BFalgebras their theories that are geometric do based not on and admit Lie an invariant metric, see section extensions. Rows incentrally extended red once are (twice)section Lorentzian and versions arise of as the limits previous of row, rowsalgebra; in however, green they (blue) are differ as homogeneous spaces (under the additional assumption Table 1 invariant metric (‘ND?’ indicatesnames Non-Degeneracy). and according The to real Bianchi’s Lie classification algebras [ are indicated by their JHEP02(2021)134 . 3 A A ]). (5.1) (5.2) (5.3) 49 ]. In a and 67 X , 66 ) since 3 = 0 + 3. . i 4.3 , δA is in general too strict. . hX A M X i ) on a manifold with bound- ∂ , Z . Yet another generalization are 2.4 hX ) + C df i . M ∂ is assumed to be fixed on the bound- M , δA and Z ] ∂ | 2 1 X df B A A, = i − + [ M – 17 – ,F ∂ | X d hX df M X Z i − h ] = ,F X X δ h A, ( [ I M Z Imposing this boundary condition requires the addition of a boundary = 4 ]. δI 69 , ) such that the full action reads 68 , 2.4 ] this boundary condition was interpreted as a Yang-Mills theory with a position-dependent 17 , 69 13 The variation of the action of a metric BF theory ( Most of our theories arise from limits of centrally extended (A)dS, but there are further Taking all three limits leads to the static case, yielding an abelian algebra, which always This is related to theIn imposition of [ either Dirichlet or Neumann conditions on the metric (and not both). 3 4 coupling constant that is localized near the boundary. Note that we implicitlyas used elements the of invariant theary metric [ Lie here algebra. interm order The to to one-form ( write both and the theory is topologicalthe without any connection dynamics on components theputting as boundary. Dirichlet If zweibein we boundary are and to conditionsInstead spin-connection interpret of on for Dirichlet all boundary a components conditions gravitational we of impose theory, ary reads A possible choice ofWith boundary this choice conditions no is further boundary to term take is needed Dirichlet for boundary a well-defined conditions variational on principle In the case of Chern-Simons theoriesreduce in to three Wess-Zumino-Witten dimensions (WZW) it models is on well-knownsimilar that manifolds way, these it with theories can boundaries be [ shown thatreduce BF to theories with the a action particular of choice of a boundary particle condition on the group manifold of the chosen gauge group. 5 Boundary actions ofIn kinematical this BF section theories weintroduced discuss in boundary the previous actions sections, associated restricted with to the metric5.1 BF kinematical theories. BF theories Particle on group manifold interesting spaces on which one cancoadjoint base theories, Lifshitz, Schr¨odinger,1/c expanded that Poincar´e,and wetheories discuss based in on appendix thea remaining limit kinematical (we homogeneous do spaces not that discuss do the not cases follow of from torsional Galilean and S17-S20 of table 1 in [ allows for an invariantFor metric, the static and and fulfills the thespace para-Galilei case is necessary the trivial, condition group i.e., ( action theyquotient of do by the not them, boosts act leading on at to the all homogeneous an and Aristotelian leave algebra points that unaltered. is It abelian is in then 1+1 natural dimensions. to JHEP02(2021)134 ) and 5.3 (5.8) (5.9) (5.4) (5.5) (5.6) (5.7) ˜ h (5.10) g ∈ ), the action ( ˜ A, A . The gauge modes ρ, t G i i g g t t ∂ ∂ G 1 1 factorizes − − ∈ ). g g, g 2.5 A, g t h ˜ h, h ] ∂ 1 1 − being the coordinate along the bound- − ) ) and should therefore be thought of as . A,B . g [ t f ) ˜ h g t Q Q ρ 1 5.5 d ∂ ( X i ] where the two-dimensional boundary actions also − 1 b , ) = ) − t f = ( and 71 g } , t ( hX ) with ∂ g B A ˜ 70 Q – 18 – ( = ˜ = Q = with constant ρ, t dt ,Q A C ˜ g A ) and assuming the orientation ( M Q ∂ { 5.2 Z i 1 1 2 − ). This action is invariant under two copies of the global ˜ hgh . We assume that the homogeneous space on which the − gg t 5.2 7→ ] = g ) and is a consequence of arriving at this action from a 2d bulk g ˜ → ∞ [ A, ∂ h I 5.7 1 ρ ) one can integrate out the dilaton field enforcing the constraint − ) 5.3 f t ∂ are the Lie algebra generators of the corresponding group elements = ( A ˜ A ˜ Q and Taken together, the above action is the one-dimensional equivalent to the WZW are elements of the gauge group of the BF theory. In order to simplify the discussion ˜ 5 A g Using the boundary condition ( Starting from ( We introduce a coordinate system ( Compare this to the three dimensional cases [ 5 = 0 which is solved locally by thus become physical at the boundary due to the explicit breaking of gauge invariance correspond to a redundancy in our reduction ( , respectively. The right charges have the Poisson bracket ˜ extrinsic to the action ( action. action appearing at the boundary of a CS theoryinherit in a three gauge dimensions. symmetry corresponding to the global symmetry group. with a similar expression forh the left charges. Note,being however, gauged, that i.e., the left all transformations the left charges should be set to zero on-shell. This requirement is with corresponding left and right charges where h by the boundary conditionsymmetry ( group, i.e., under the transformation becomes which is the actiong for a particle moving on the group manifold of where ˜ we assume that close to the boundary the group element ˜ gravitational theory is definedcan has be only either one a boundary circle component, or a the line. topologyF of which that is conserved on-shell using the right hand sideary of that ( is located at The integrand of the boundary term is recognized as the quadratic Casimir JHEP02(2021)134 3 ]. ]. 72 75 .A – g (5.11) (5.12) 72 . Assum- g t ∂ 1 ] or the BMS − ) which can be g 70 The structure of 5.2 6 . A can be understood in . has the form Q 2 g g . t ∈ i ∂ . 1 6 − ,K ] g 2 γ, K , γ ) based on the Lie algebra 1 ] on three-dimensional flat space. ) current algebra to the Virasoro ) we calculate the Poisson bracket γ [ R 5.7 ) to constraints on 77 h , 5.10 ). (2 + 5.5 γ ] sl has vanishing Poisson brackets with all 2 ,γ 1 = 0 K γ [ i = Φ – 19 – on-shell to some value determined by the fixed γ, K } γ 2 γ Q Φ i − h , g 1 t γ ∂ 1 gravity yields Brown-Henneaux boundary conditions [ Φ − { 3 1 with Brown-Henneaux boundary conditions [ − ) 3 ). γ, g f t ]. 5.7 ∂ ≡ h ( 78 γ ]. Following this procedure one finds the Alekseev-Shatashvili action Φ and by the flatness condition ( ] for the boundary conditions of [ 76 . (By fixed we imply that 71 A K be the set of right charges of the action ( gravity several inequivalent boundary conditions were identified, e.g., [ A 3 Q Using the Poisson bracket for the right charges ( Let Boundary conditions on spin-connection and vielbein translate to conditions on the In deriving the action we have only used the boundary condition ( In the case of WZW models based on simple algebras, the question of consistent sets of constraints has 6 of the constraints been analyzed in detail in [ The constraint thus setsalgebra the element charge functions on phase space; we assume the same for ary conditions that could bewill (and be for useful some for applications the have to two be) examples relaxed; that however, we they generic provide constraint in compatible section with the algebra structure on gauge-connection ing that the boundary conditionsaction, are we consistent with can the view algebraicthe the structure algebra former of puts as the restrictions particle constraints onboundary on the conditions. the set In right of what charges possible follows we consistent make constraints some and specific thus assumptions possible about the bound- geometric action [ Both of these theoriesnothing have but a the flavor of conservation of hydrodynamicsflat the in space stress-energy case). the tensor sense As (ora that reviewed its similar below, their Carrollian way. the dynamics analogue Schwarzian is action in for the AdS algebra, which applied to AdS For AdS From the point ofcurrent view constraints [ of theas boundary boundary WZW theory for model AdS these boundary conditions act as 5.2 Hamiltonian reduction ofFor many boundary applications action we arerather not impose interested in (physically or the loosest geometricallyexample set motivated) is restrictions of the on boundary Drinfeld-Sokolov the conditions, reduction fields. but of A the famous imposed for any metric BF theory.to But interpret this is these not BF the endmight theories of impose as the additional story (non- asymptotic as boundary or ultimatelyvielbein conditions we ultra-relativistic) want on and theories the spin-connection. metric of In or, gravity otherof equivalently, on where words, the we one boundary are action looking for ( a (Hamiltonian) reduction JHEP02(2021)134 2 is K even- second ) unless 5.12 . Γ. The nature of the 6.3 ]. ⊂ ) by looking for , i.e., the JT model in 2 79 Γ] 5.7 , or the subalgebra is abelian, . The constant element g ⊥ Γ ) together with a system of second ∈ 5.7 K Γ where [Γ ∈ 2 , γ 1 γ basis there is no obvious choice, but changing works, we first review the construction of the P – 20 – 5 , . B of the gauge algebra , . Other kinematical algebras where our procedure H = 0 Γ ) be second-class. i 6 6.2 ) to Schwarzian-like actions. Γ ) lead to the action ( 5.7 5.12 K, First-class constraints are the hallmark of gauge symmetry 5.2 h 7 instructs to look for two-dimensional non-abelian subalgebras in section 5.2 /Hyperbolic plane 2 2 . The main new example is the construction of the boundary action for 6.1 . Constraints with Poisson brackets that do not vanish on the constraint surface are called To show how our proposal in section In summary, boundary conditions on the fields of the BF theory compatibleIn with the the above we spelled out a purely algebraic way to arrive at consistent bound- This means we have found a way to determine consistent boundary conditions com- We note first that a new constraint is generated on the right hand side of ( . From this follows immediately that second class constraints come always in even numbers. We use here the standard terminology to refer to constraints whose Poisson brackets vanish on-shell as 7 The discussion in section of the Euclidean AdS algebra. In the first class class in section (extended) AdS-Carroll does not work without modifications are briefly mentioned6.1 in section Euclidean AdS In this section wekinematical algebras consider and look metric for boundary BFthat conditions, reduce theories as the discussed based particle in the action on previous ( section, the various (1+1)-dimensional Schwarzian action for a BF theory based on the symmetry algebra of Euclidean AdS find boundary conditions for BF theoriesgeometric with picture Carrollian/Galilean interpretation is where often the apply not this as procedure clear to two as examples. in the relativistic6 case. In the Schwarzian-like next theories section we for gravitational theories arechooses conceived. boundary conditions Rather, such oneas that starts solutions. they from allow a This anthe bulk is interesting second perspective class order indeed perspective, of and the and bulkthe case we geometries above will for see criteria. the below that BF But these theory the boundary on purely conditions satisfy AdS algebraic point of view presented here allows also to universal boundary condition ( class constraints. The latter can be solved directlyary in the condition action for [ a metric BF theory. This is usually not the way boundary conditions currently interested in, we do notwe want to demand introduce that (further) the gauge symmetries. system in Therefore, ( patible with thedimensional algebraic non-abelian structure subalgebras ofsubsequently the chosen boundary such that action ( all the generators belong to aconstraints subalgebra, is i.e., now determined bythe the constraints second are term. first-class. If in a system andrender require the additional first-class constraints constraints in second-class. the For form the of type gauge-fixing of conditions boundary that conditions we are JHEP02(2021)134 } ]. , 0 80 L (6.8) (6.4) (6.5) (6.6) (6.7) (6.1) (6.2) (6.3) , has to + L K { ) for some 1 out of the alge- 2 0 . It can be shifted α preserving the form − L + − P i L g which is the group for which − t = λ ∂ } 0 1 1 ) read + − . P 2 { 0 + / , g ) is completely captured by the L (hyperbolic in table = dt ) 5.11 0 α 0 R 2 2 ρ L λ , 000
h 6.4 ) that the fixed element = A ρ λ . + − = + ) − | e + + 0 ρ L 5.12 ) g HL t L t ), we find the two choices Γ = φ − − L is related to the mass of spacetime [ ( 0 y ∂ λ e − − ) = SL(2 1 λ ( e L 1 2 L 0 R B D.1 − ρ O L , − L g 0 − + 2 y = ρ e + 0 e ) e t − ¯ L + ( X – 21 – L λ 1 4 ρ y 0 L + e e α -dependent group element + α t + 6= 0 the constraints ( 2 = ) = = 2 1) = PSL(2 = t + , ( + g 0 L dρ X | α L ρ i − . g e λ g t = t δ ∂ HL , that is ultimately inconsequential for the reduced action. for the 2 ∂ = 1 5.2 1 + t 8 + − with ds − α g A B ), these constraints reduce to those implied by the well-known 0 0 , g although the latter element has no influence on the constraints. L L 0 = − ) and the cosmological constant set to unity. In the second order − α ρ L and + in order to have non-zero inner product with L h 0 L + 4.1 + = α + L ) transforms with an infinitesimal Schwarzian derivative, and L − t basis . Taking the latter pair (the other one leads to the same conclusions upon ( φ = exp( + 0 } L L α b 0 − L L = , , is single valued in the first order formulation. 0 ]) are given by − 2 ) upon introducing the constraints implied by the above boundary conditions. L K L ) is an arbitrary function and the dilaton field has the asymptotic behavior , t { 80 ( + 5.7 , L L 68 , the field t We turn now to the derivation of the boundary action that follows from the particle Under gauge transformations generated by This parametrization is valid for SO(2 8 A action ( Using Gauss parametrization globally AdS which corresponds to thetransformations. transformation The of zero-mode a of stress the function tensor under infinitesimal conformal We have therefore reconstructed thebraic above considerations boundary of conditions section on AdS of formulation these boundary conditions translate to the metric where In the first ordere.g., formulation [ these boundary conditions in the highest-weight gauge (cf., with the generators ( Assuming that the radial dependencegroup of element the connection ( boundary conditions for the connectionparticular on Euclidean choice AdS of by elements of Choosing which yields with commutation relations explicitly given byand ( Γ = redefinition of the radialbe coordinate) proportional we to find from ( to the JHEP02(2021)134 , ) R 6 , (6.9) (6.15) (6.10) (6.12) (6.16) ) we can 5.2 has to obey g , that the Schwarzian 5.1 ). As argued in section . 5.8  2 ) . 0 = 1 (6.11) in ( } has not been worked out in detail, it is h ( ˜ h bc 2 ) reads 2 } y, t 3.2 t π − . ( { β  2 ) (6.14) 1 L ) t y, t } ) yields t − ( 9 ad { + ( ) g β 2 f } y, t 5.7 = 0 (6.13) t dt πh { ∂ } case. ( ) =  h, t Z − β { 2 dt ¯ y, t – 22 – X  { + t ) = Z t t ∂ b dt ]. d ( ( 1 2 = tan ] = the function g + L y + 83 Z y [ g , I ¯ ] = cy ay X 82 y [ ] for the dS ) we have not assumed anything about the topology of the I 7→ ] = 81 y h [ 6.10 I ) lead to two algebraic equations that can be solved in terms of the 6.3 . Going once around the circle, the group element β to the leading order of the dilaton so that we find + f t t ∂ ∼ denotes the Schwarzian derivative. Using the boundary condition ( t ) can be equipped with the above boundary conditions that lead to a Schwarzian ·} R , , {· . Plugging the solutions into the action ( (2 y From the point of view taken in this section, all homogeneous spaces based on the alge- In deriving the action ( In terms of the group elements sl While the case of the two-dimensional light cone of section 9 bra action on the respective boundaries,the modulo bulk possible spacetime; subtleties see concerning e.g., the [ topology of highly suggestive in light of the results [ under which the Schwarzian action reads in order to beachieved by single-valued. the field It redefinition is straightforward to see that this identification can be boundary. This is thereforei.e., the the appropriate zero temperature resulttime. if result The since the finite the boundary temperature boundaryperiodicity is result is coordinate taken obtained corresponds to if be to the Euclidean boundary a is line, taken to be a circle with Since the equation of motion of the Schwarzian action is just we see, as inaction the just higher-dimensional encodes cases mass mentioned conservation. in section that correspond to groupthis multiplication transformation from should the be left, thought of as a gauge symmetry. thus reproducing the Schwarzian action.transformations The action is invariant under the finite PSL(2 field where further relate the two constraints ( JHEP02(2021)134 . ¯ X that = (6.20) (6.17) (6.18) M (6.19a) f M t λ ∂ , the alge- = 1 in the + 1 P . κ arbitrary such P ) M λ 00 M γ λ M ), and the form of + P Y + ρ − 1 become second-class. + is naturally related to L T P case above, boundary 0 M c B − , with − ) where we set λ ] at these specific values 2 M λ L ω dρ ( 1) enforces the torsion con- M , 2 1 85 5.2 + = exp( γ + 0 + (1 P b − respectively. We choose the + and X + 1 dt L e + iso ) 0 + P } it is unclear whether a two- ] that studied three-dimensional L + M λ P + M λ , L 1 38 = and uncovered a warped Virasoro γ and gauge field as H − 0 L c = + e K { + 2 X 1 1) 0 λ in order to look for consist boundary = , P L ) B t (1 + A and ( 5.2 λ  T iso } H = P + , M = L + − – 23 – 1 δ L L  { ρ X L − + e (1) (6.19b) ) B t 00 + O ( λ X L + + ) the state-dependent functions transform as + + P L T − + 0 + Z ¯ L X 6.19 λ ρ ρ + e e + H in ( 0 0 = ( = T ]. The fact that the symmetry algebra X A + A 2 X λ = ) is then required to be of the form 85 , = X 84 and so forth. In contrast to the (Euclidean) AdS 5.11 T is non-zero and the constraints associated to δ 0 i e in ( ,K K − (1) level are zero so that the only non-trivial cocycle is the twist, L h u Under gauge transformations with generator From the form of the algebra presented in table warped Virasoro symmetries was also foundChern-Simons in theory the work based [ onsymmetry two as copies asymptotic of symmetrybe algebra. related We to therefore some expect versionof of the the the central boundary warped charges. action Schwarzian We to action will of find [ this expectation confirmed below. This transformation law is theand hallmark of twisted warpedconventions symmetry. of Both [ central charge where the radial dependence is capturedthe by dilaton the field group follows element from the universal boundary condition ( preserve the form of that This translates to boundary conditions reminiscent of highest-weight gauge one finds two subalgebraslatter spanned as by the subalgebraleads the again to generators identical ofelement conclusions which upon are redefinition imposed of the as radial constraints coordinate). (the The former fixed to the algebraic algorithmconditions. presented in section dimensional non-abelian algebra exists, but after the basis change where the components ofstraint the for former arecondition labelled for such this that spacetime have not been discussed in the literature. We therefore turn We turn our attention tobra a of non-Lorentzian extended spacetime. AdS-Carroll Asalgebra. (AdSC) apparent from is The table isomorphic interpretation tospace of the is centrally-extended the different. 2d generators, We Poincar´e expand however, the when (co)adjoint viewed vector as a homogeneous 6.2 AdS-Carroll JHEP02(2021)134 . goes → ∞ (6.28) (6.25) (6.26) (6.27) (6.21) (6.23) (6.24) n ρ ). When ap- 6.22 and rescaling both ) (6.22) ρ ρ − ∂ e . dρ
− T 0 t ) + 0 M ∂ γ y ( dt ρ = diverges while the vector − T + e M (log | q ∂ − . = Z M ) leads to ρ g , one finds that the pull-back of the z t 1 t M γ 1 − n e ∂ ∂ dx e 1 due to the second-class nature of the 6.9 1 P + + P + − = − ρ = 00 φ g Ω e e z n dx n y, z − ρ − = → L e = 0 which one may regard as the vacuum 0 − ) n n 0 φ − , y e 2 – 24 – q M dt e + denotes the vector field lying in its kernel. This 2 L ) ) dρ Ω y ρ = n (log e 0 2 − = z e = 1 → T 6.19 2 = 0 = dρ q + 2 q g M | ds ) dt = g , the degenerate metric t + q ∂ Z dρ ρ 1 e ¯ + − . X 3 g of the BF theory is subsequently interpreted as a null-coordinate → ∞ dt = = ( t ρ 0 I T e ). The boundary of the null-surface is obtained in the limit algebraically in terms of ) in the case = ( P = 0 has induced degenerate metric and normal vector of the form q 6.19 , φ 6.22 − − x φ ) the zweibein is the degenerate metric and q 6.19 We turn now to the boundary action that can be obtained from the above boundary We address one more interesting feature of the AdSC geometry ( In order to obtain some geometric insight into the boundary condition, we can recover allow to solve constraints. Plugging the solutions into the action ( conditions. Parametrizing the group element as the current constraints implied by ( of AdSC spacetimes aprecisely conformal the class same boundary ofshare structure vectors these as together geometries 2d with the asymptoticallythe the same flat same local zero conformal spacetimes. symmetry metric. boundary group, So structure. This i.e., not the is only Poincar´egroup, but also to zero. But uponquantities introducing by the the boundary conformal defining factor functionAdSC Ω structure = to the boundary Ω = 0 yields Since Ω is defined only up to a non-vanishing factor, one finds as the boundary structure configurations of ( The boundary coordinate along the boundary of AdS proaching the boundary the null-surface coinciding with ( where AdSC spacetime can bespace. regarded More precisely, as starting a from Poincar´epatch null coordinates surface in three embedded dimensions in three-dimensional AdS from ( and thus the Carrollian structure JHEP02(2021)134 1) , (2 (6.29) (6.30) (6.31) so . As in the c 1) ].
, 0 ) 41 0 In the parametriza- y 10 ] found essentially the ) will therefore lead to ]. (log 86 M 5.9 γ 86 ( , − 85 b y , i.e., spacetimes in the second Q 00 z cc + ) are given by − d 0 ) + 0 6.19 y z . 7→ = 0 (log 0 we should look for even-dimensional non- z L t ). Starting with the doubly extended Heisen- ∂ 1 2 and Heisenberg 5.2 5.7 – 25 – = c = T L (2) t corresponding to e.g., AdS-Galilei, it is straightfor- ∂ c iso y c z (2) + a iso 0 7→ ) 0 y y , in order to cover all kinematical algebras with invariant metric. 1 this result agrees with the zero-temperature version of [ as representatives for all kinematical spacetimes based on 2 (log iu e − = = y T . Disregarding the sphere and the static spacetime it remains to discuss c ) are a priori also applicable to those theories albeit with different geometrical 1) ), the two free functions in the connection ( , 6.19 (1 and AdS-Carroll 6.26 iso In the case of the algebra According to the algorithm of section As a final comment we remind the reader that the AdSC theory discussed here exhibits 2 Upon redefining 10 first-class constraints and gauge symmetry in the boundary action. ward to show that noalgebra, two-dimensional and non-abelian thus subalgebra the exists fields, unless to one become allows complex. the algebra does not allow for anythat such all subalgebra. two-dimensional subalgebras The algebra aretwo-dimensional is abelian. nilpotent so Furthermore, from that the which center follows any ofthat four-dimensional the commutes subalgebra algebra with necessarily is all contains ofthe an its algebraic remaining element generators. structure Boundary and conditions act that derive by from restricting the charges and fourth block of table commutative subalgebras that we canthe use dynamics to of write the point downberg particle boundary algebra action conditions that ( that corresponds reduce to e.g., flat Carroll spacetime, we find that the form of the 6.3 Other kinematicalIn algebras the last twoAdS sections we constructed boundaryand actions for hyperbolicspacetimes space/Euclidean with symmetry algebras ditions ( interpretation of generators andsame free action, but fields. withspace Indeed, complex and found fields, the an as work explicitSYK boundary relation [ model. to action a in scaling limit the of case the of effective action (Euclidean) of flat the complex that correspond toSchwarzian the case discussed global above,transformation symmetry arising these group due transformations to of the should redundancy AdSC, be in i.e., interpreted the reduction. asthe ISO(1 same gauge algebra as dS-Galilei and 2d Minkowski space. Consequently, the boundary con- Finally, this action is invariant under the finite transformations tion ( and the equations of motion are equivalent to their conservation We recognize this as a form of the twisted warped action of [ JHEP02(2021)134 ) has the limit of JT , and point 2 3.21 ]. 7.1 88 , 87 ] for a selected list of 58 ) (or its torsionful gener- 3.5 . . ) was easy enough to construct, 7.2 6.2 3.8 needs to be modified to construct boundary ). 5 The NC dilaton gravity action ( ] for the four-dimensional case. Thus, likely ap- 3.5 – 26 – 83 We were slightly less explicit concerning Carroll dilaton ; the Schwarzian analogue for the AdS-Carroll ). 3.4 The light cone Lagrangian ( 5.7 ) are in the context of three-dimensional asymptotically flat gravity. 3.8 ), but clearly the same techniques that we used to study NC dilaton 3.33 ] the light cone theory might be interesting in relation to asymptotically flat 39 Carroll dilaton gravity. gravity ( gravity can beprevious applied item apply there. to thisrelationship case, Similar to with the other remarks additional field concerning interesting theories option on applications to null explore as manifolds, the see, in e.g., the [ exact solubility of thedimensional equations gravity of models motion, — theliterature we possibility of have to provided ordinary accommodate only 2dmodels) one higher- dilaton it example. gravity seems (see Given likelywaiting the that the to there vast table be are in applied. several [ interesting NC dilaton gravity models Newton-Cartan dilaton gravity. same status for NCalization gravity below as that generic equation) dilatonnumerous gravity for similarities ( Riemann-Cartan — gravity. the While existence we of have constant shown and linear dilaton sectors, the but the physical interpretationin of [ the latter remainsspacetimes in to three be dimensions; cf. explored. [ plications of As ( advertised Moreover, it could be interestingthe to light study cone a dilaton theory, analogous gravity-inspired generalization to of ( Light cone theory. Rather than summarizing our results in past tense, we address potential applications We leave the construction of alternative boundary actions for these cases for future This means that the procedure in section • • • 7.1 Selected applications Without claiming to be complete,by here order is of a appearance: list of selected applications of our results, listed gravity is the twisted warped action, discussed in section of various limits ofout JT some gravity interesting and and their viable dilaton generalizations in generalizations section in section 7 Applications and generalizations The main conclusionanswers of to our the work questions isgravity, discussed posed that in in section it the is introduction: done there — is in a particular, NC we version have of provided 2d dilaton actions for these examples.at all, The i.e., simplest to ‘modification’action stick is with is to always the given not most by general enforce ( boundary any conditions. constraints Inwork. this case the boundary JHEP02(2021)134 , ]. ). 13 6.2 5.7 , largely also lead 1 C , B ]. Of course, taking 86 we provided in section 2 ] for reviews) could be transposed to There is a long-term relationship be- 15 – to other models of interest. Such appli- ). 13 5.2 6.28 Possibly the largest set of potential applications – 27 – The theories mentioned in appendices The loosest set of boundary conditions for any BF-type model We were even less explicit regarding several other entries in table . Some of these models may be useful for applications in the context of 3.4 conditions. For metric BFoperations theories of the holographic relation reduction should ofat be three-dimensional the straightforward, Chern-Simons since boundary to the commutes 2dholographic with WZW reduction dimensional of reduction 2dIndeed and BF many yields to of the a a findings corresponding corresponding and boundary tools theory, used for see, 2 e.g., + 1 [ dimensional Chern-Simons, like, Relation to three-dimensionaltween models. three- and two-dimensionalduction gravity, since of the the former. latterthere arises But the by are relation dimensional numerous is re- similarities, deeperlations particular than as in that. Chern-Simons the Also andSchwarz at respective BF-theories a type gauge — technical and theoretic level both can formu- are feature topological non-trivial gauge edge theories modes, of depending on the boundary JT/SYK correspondence associated withapplications chaos, to quantum relation gravity, etc. to (seethese random [ limiting matrix cases. models, We arediscoveries convinced in that the this near route future. can lead to numerous exciting is to generalize thethat JT/SYK our correspondence models to emerge interestingsimilar from limit limiting on a cases. the limit dual quantum of Theexpectation mechanics model. is JT fact the A suggests flat concrete space/cSYK that example correspondence realizinglimits discussed one this in is can [ just also the first implement step a in a much bigger picture. Many of the developments in the JT/SYK-like correspondences. which led to a twisted warped actionLifshitz, ( Schr¨odingeret al. to BF theories, and thussection there will be ‘dilatonLifshitz gravity’ versions or thereof, Schr¨odingerholography. analogous to always leads to aSince boundary for many action gravity-inspired describing applications somethingary a conditions like particle is Brown-Henneaux preferred, bound- on it a willthe group be boundary rewarding manifold action to described ( apply the incations Hamiltonian section will reduction be of analogous to the example of AdS-Carroll because we do notthe have static a case. good proposal Nevertheless,same for such techniques applications applications of, may as exist, say, forJT-like the and and NC para-Galilei if dilaton or they dilaton gravity-like models do, gravity that again can build the Boundary be upon actions. these applied homogeneous to spaces. construct and solve The rest. Besides the rather direct applications above there are also several exciting potential • • • • • applications that will require —the in items some by likelihood cases substantial of — substantial further progress): input (here we order JHEP02(2021)134 ], ]), (1). 96 45 u ], we , ]. The 40 101 49 – ]. 8 99 (1). Also Guica’s u deformations and certain JT ] (see also refs. therein), who discussed 95 ]) are based on boost eigenstates, rather The limits of JT gravity that we considered Theories without light cone may have a hard 97 As homogeneous spaces Minkowski and AdS- – 28 – Einstein gravity with Comp´ere-Song-Strominger -deformations upon dimensional reduction have Quantization of 2d NC dilaton gravity seems a fea- ¯ 3 T T ]. As rˆolemodel we point here to a recent example by ], both on the gravity side and the SYK-side. Analogous 94 , 73 suggest that there might be a deeper relationship that remains 93 ] break/deform Lorentz symmetries and require an extra 98 6.2 deformations of JT? ], can be applied to BF theories. However, for WZW models without JT 92 – 89 deformations [ Given that the morea nice standard interpretation inspeculate that terms a of similar relationship flowlimits could equations hold of between JT in gravity. 2d Addressingof dilaton this the last gravity second point [ and would third require a item better in understanding this list. than the more conventional momentum eigenstates.described exchange. This therefore mirrors the above Relation to break/deform Lorentz symmetries and typically introduceJT an extra findings of section to be explored.cal Two constant, observations that triggered gives ourit AdS its interest: might ‘boxlike’ inherit 1.) properties,celestial similar is Since still amplitudes advantageous the nonzero holographic (for for cosmologi- features. a AdS-Carroll review 2.) see [ Recent works on our models is possible and could lead toMinkowski unexpected and insights. AdS-Carroll. Carroll are both based onboosts the and Poincar´ealgebra and spatial are translations, connected a via relationship an that exchange is of true in any dimension [ Quantization and holography. sible endeavor, due to theand quantum might integrability provide of an BF-theories interesting (seeand avenue the to holography. review Galilean [ While and this Carrollianinformation quantum leads loss gravity to problem various in puzzles, any like of is these theories?, there we something emphasize like that an quantizing all highly entropic objects correspondingand to regardless finite of temperature theirentropy states, geometric and, see interpretation e.g. whenever it [ is possible,first of step provide in interest a this to direction Cardy-inspired understandenergy could microstate derived their be from a counting. the thermodynamical analysis on-shell A starting action, with analogous the to free 2d dilaton gravity [ boundary conditions [ applications and relations to higher-dimensionalvarious limiting limits theories of should exist JT for discussed the in our work. Thermodynamics and Cardyology. time of defining black hole-like entities. Nevertheless, some of these theories do feature invariant metric the situationmodels is not based well on understood Liean algebras in invariant admit general; metric a for [ Sugawara exampleChaturvedi, construction Papadimitriou, WZW only Song when and Yuthe possessing [ dimensional reduction of AdS e.g., in [ • • • • JHEP02(2021)134 ) ]) is 5.2 (7.1) 5.11 104 ) was enforced is now as target 5.12 ] known as Poisson- I J A X 105 in the constraint ( ∧ I γ A ) and is first order in deriva- K and I X A ( K IJ P 1 2 The algorithm explained in section + A variant of the previous item is to include I ]) and take various limits analogous to the , subject to the non-linear Jacobi identities dA I JI – 29 – Chern-Simons models in 3d are rigid, i.e., their 103 , P X Our focus was on gravity variables, zweibein and − 68 ] = = I ,A IJ ] and refs. therein). Similarly, one can extend to higher and connection 1-forms I ) = 0. In this languages, BF theories (with or without P I ]. The Poisson-sigma model Lagrangian X 102 [ X 106 PSM I, J, K L + cycl( JK P L ∂ IL arbitrary Poisson-tensor P metric) are merely specialwe cases have of taken PSMs can withcretly, be linear we rephrased Poisson have as tensor, already corresponding and introduced limits the specific limits of PSMs the when Poisson providing tensor. the action Se- for 2d features again scalars tives, just likespace BF coordinates theories, spanning but a the Poisson manifold. interpretation of The the main new ingredient is an most general consistent deformation (inanother the Chern-Simons sense model of with Barnich thestory and same applies Henneaux number to [ of 2d gauge BFconsistent symmetries. deformation theories, of A but 2d similar with BF ansigma is interesting a model non-linear extension: (PSM) gauge [ the theory most [ general boundary actions (see [ spin gravity (along thepresent lines work. of In all [ theseof generalizations the one underlying would theory. still keep theNon-linear topological nature gauge symmetries. new conserved charges and additional boundarythe degrees topological of freedom, nature but of will the maintain theory. Supersymmetry and/or higher spins. supersymmetry and consider various limits of 2d supergravity theories and their and for purely theoretical reasons,boundary to conditions. get a better understanding ofAdditional the topological landscape of fields. connection, but for some applicationsfields. it Such can a generalization be is of straightforward interest in the to BF-formulation add and non-abelian will lead gauge to imposed a number ofwere assumptions. assumed to be In state-independent, particular, andto the be constraint algebra second ( class.of Neither them of it these assumptions isA is possible mandatory, full to so classification by construct relaxing of a either all whole menagerie possibilities of could new be boundary worthwhile, conditions. for novel applications More general boundary conditions. Most likely there are further potential applications that are missing in our lists above, • • • • but let us move on to our final7.2 point, generalizations. Selected generalizations Here is a list of five classes of generalizations that we consider promising and viable: JHEP02(2021)134 . 1 VII, which correspond ' 1) , (1 ). In order to construct bound- su ' 3.33 2) ) [see particularly the non-linear gauge , (1 3.21 so ' ], cf., our summary table – 30 – 1) ]. While ambitious, it could pay off to add matter , 65 , 30 (2 64 so For one and two dimensional Lie algebras solely the ' ) R , (2 ) = 2. sl ), NC dilaton gravity ( g Topological gauge theories like BF or PSM have numerous techni- )] and Carroll dilaton gravity ( 3.5 3.26 Three dimensional Lie algebras that admit an invariant metric are either ] for a review). The Roman numerals refer to the name of specific Lie algebras ) = 1 and dim( ) = 3. to theories like NC or1-loop Carroll effects, dilaton backreactions, gravity etc. and address some questions concerning cal advantages, but miss anphysical important degrees aspect of of freedom. physics,of namely A keeping locally the possible propagating model compromise simplematter and between degrees physically the rich of conflicting is freedom. to desires Some couplecontext consequences BF of of theories 2d such (or dilaton a PSMs) gravity coupling to in are [ reviewed in the symmetries ( ary actions for alldone these in models generic it PSMs, willworthwhile. which be is necessary why to this understand particular how generalizationAdding this seems matter. can eminently be dilaton gravity ( 107 g g DG was supported by the Austrian Science Fund (FWF), projects P 30822-N27 and While further generalizations not envisaged here are conceivable, the lists above provide • dim( abelian Lie algebras admit an invariant metric. dim( abelian (denoted by BianchiThe as simple I) one or is simple either and amount to three distinct cases in total. A Metric Lie algebrasWe of summarize low dimension low-dimensionale.g., real [ Lie algebrasaccording that to Bianchi’s admit classification an [ invariant metric (see, is it Holographic?”. The workFellowship of “Non-Lorentzian JH Geometry is in supported Holography” bysupported (grant the by number Royal the J-4135 UF160197). Society Erwin-Schr¨odingerfellowship of University JS the Research and Austrian was Science by Fund the (FWF) NSF grant 1707938. P 32581-N27. DuringERC the Advanced start Grant “High-Spin-Grav” of and thisT.1025.14 by project FNRS-Belgium and (Convention Convention the FRFC IISN research PDR Research 4.4503.15). of Project Grant SP SP (RPG-2019-218) was “What was supported is supported Non-Relativistic by by Quantum the Gravity the and Leverhulme Trust Acknowledgments We thank Arash Ranjbar,discussions. Dieter Van den We Bleeken wantmetric and Lie to Jos´eFigueroa-O’Farrill for algebras. thank useful Jos´eFigueroa-O’Farrill for valuable advice concerning already a plethora of possibilities for future research. JHEP02(2021)134 ] M 110 (B.2) of our (B.3c) (B.1a) (B.3a) (B.3b) (which ). The 2) [ c 0 cc 1 expanded n, II /c ' = VI ]), ( ) translations c P cc 1) 109 , (1 iso . P B H C − ) and space ( ] = H P ] = 2 ] = ] = 2 , ). They serve in this work as a H B D D R [ , , , , D D C [ [ [ (3 ), time ( so IX leading to the sphere (or Euclidean D × ' ] (see also section 4 in [ R χ . or Poincar´ealgebra (2) 108 = c 0 su i D 1) [ ) and – 31 – expanded BF theories , ' D R , n, h , ]. The higher-dimensional algebras do not play any C = VII /c (3) c (2 39 so sl (2) × H iso z R M P B ] = ] = ] = ] = H P P P , , , , D B D C [ [ [ [ . 1 Beyond dimension five one can use the classifications of metric Lie algebras For five dimensional Lie algebras there are six Lie algebras that admit an There exist five metric Lie algebras of dimension four. One of them is ]. Kinematical Lie algebras and their relation to metric Lie algebras in any . Even including this central extension leaves the bilinear form degenerate. Z 5. 111 = 2 this algebra can be enhanced to the algebra Schr¨odinger by adding mass ] = > z 3) [ C and underlies the doubly extended Carrollian and Galilean theories. ) ) = 5. ) = 4. , g g g H n, 1 For The Lifshitz algebra is spanned by dilatations ( This algebra has the samesion bilinear [ form as the Lifshitz case, but admits one central exten- This algebra has no nontrivial central extensions. and special conformal transformations with the most general (degenerate) invariant bilinear form given by B Lifshitz, Schr¨odingerand 1 Here we provide all the necessaryBF data theories. to None construct of Lifshitz, them Schr¨odingerand 1 BF (even theories. with central extensions taken into account) lead to metric that have been obtainedand for ( signature ( dimension have also been studiedrˆolein in the [ present work. invariant metric, five of whichThe are remaining trivial unique central indecomposabletable extensions of metric the Lie four algebra dimensional is ones. dim( Heisenberg (centrally) extended Euclidean are isomorphic to theircentrally (A)dS-Carrollian extended and Poincar´ealgebra is (A)dS-GalileanEuclidean also cousins, case known cf., is as sometimes table Maxwell referred or to Boidol asdim( algebra oscillator algebra. and the de Sitter), see table dim( the abelian Lie algebra.simple three Additionally, dimensional there cases arestarting the point for trivial contractions central to extensions the of remaining the two metric two solvable algebras given by the to the (A)dS and light cone cases or JHEP02(2021)134 e m ]). , τ 39 (B.4) (B.6) (B.7) (B.8c) (B.8a) (B.5a) (B.8b) (B.5b) ]) which and any of 115 2 – M Z with a level (1) + 113 P n . The terms with . (1) 4 n P Z ] = (1) H + H B , χ χ ] = ] = m (1) H , = = B (1) [ B i i 1 have been modded out. (1) , H B B , (1) [ [ we end up with the Lie algebra H B h ) , n > n ( (1) τ B 1 ··· (1) h Z P n + . + 2 expanded Poincar´ealgebra (till level 1) P 3 6 − ] = e Z Z c (1) /c ) P − − n (1) + ( H τ H , (1) τ ] = ] = ] = B . Substituting this expansion into the Cartan BH HB [ =0 ∞ 1 – 32 – + X n χ χ (1) (1) (1) − H H B c ω = = = , , , B B P P i i τ [ [ [ H ). We have checked that the addition of = , (1) it is possible that the bracket of a level B B A 4.3 h (1) , 2 H H − h c (1) 3 6 H Z Z ] = P ) is the algebra where all levels ), cf., ( , ] = ] = ] = B (and analog for the remaining generators). Since the structure [ P , B.4 B.4 (1) (1) n B P P 2 (1) [ , , − B BB H B c [ [ χ χ H ] = = P = i i 112 expanded algebra in 1+1 dimensions allows for 6 nontrivial central extensions B ) P M , n (1) ] = ( dependence is analytic so that we can Taylor expand B B H /c H h 5 , , c ] = ] = Z B B H P [ h , , to B H [ [ The 1 The nonzero commutation relations of the 1 1 Z , C Coadjoint theories andFor their completeness limits we brieflystays, mention basically another by class construction, of non-degeneratetinctive under theories feature that limits. where the These algebraic the structure theories invariant generalizes have metric to the generic dis- dimension (see, e.g., [ A necessary condition forcentral extensions a to non-degenerate ( invariantthe other metric central would extensions be does not to lead only to add such a two non-degenerate invariant metric. M elements constants can also depend on generator gives a generatorsuperscript of (1) a denote level generators thatwhere of is one next strictly quotients to out larger leading allalgebra. than order Lie The (‘level algebra algebra 1’). elements ( of The level level bigger 0 than algebra, 0 is again the Galilei where we assume onlyconnection and even defining powers a Lie of algebra generator for each where the dots denoteetc. possibly depend other on elements the inthat speed the the Lie of algebra. light (as When is the the 1-forms case for the Poincar´ealgebra) we can assume These algebras appear in thehave context recently of found Lie applications algebra inexplained expansions the as (see, context e.g., follows. [ of Take the general Cartan relativity. connection The 1-form, notation say can be are given by [ with the most general degenerate invariant bilinear form JHEP02(2021)134 and (C.4) (C.2) ij (C.3c) (C.3a) (C.3b) ]. (3) so that its part described ]). . so 119 αβ – 116 ] = 0 (C.1) h = 1 β i ∗ ∗ ∗ ) is degenerate under ∗ 117 β α e H B P δ , 2 2 ∗ , α ˆ ˆ 4.2 B C C ). There are two curious P ∗ = 2 h e  ∓ c i [ ]. In particular, the three- 2.4 β ∗ 45 e ] = ] = ] = ˆ , ∗ ∗ ∗ α P P H e , , , h B H P they are defined on the vector space [ [ [ γ g matrix with entry 1 in the slot ∗ e n β × αγ n c , which are well defined by construction (a = 1 ∗ − i g ∗ ), which corresponds to the H – 33 – ] = , , respectively) by the commutation relations 4.2 β H α ∗ h ∗ ) we have the nonzero commutators e e , α denote the 4.1 e ) [ ∗ and ∗ n αβ ( P ij H ∗ 2 α h ]. Given a Lie algebra ˆ ˆ c Λ X P e ˆ = 46 − Λ − is not central in general. i ∗ β g ] = ] = ] = e γ ∗ ∗ ∗ , e H B B α induces the limits of by γ , , , e = 1 h g B H P g [ [ [ αβ i c ∗ B (spanned by , ) that guarantees the existence of the non-degenerate invariant metric. The ∗ B ] = ] for details). Second, a comment in relation to 2+1 dimensions and Chern- h g β ˙ e + 39 C.4 , g α e . In the following, let [ 5 is some arbitrary (possibly degenerate or zero) invariant symmetric bilinear form on Having provided the Lie algebra and the invariant metric it is now an easy exercise to We discuss all limits and theories at once. Starting with (A)dS this means that addi- They are based on so called coadjoint Lie algebras, which are a subcase of the already (3). For further recent works based on these algebras we refer to [ αβ . The extension of D Matrix representations We collect in thissection appendix matrix representationszero that everywhere are else. useful in the calculations of Simons theories: theas coadjoint 2 + Lie 1 algebrasdimensional dimensional are Poincar´ealgebra can BF precisely be theories, regarded the see,Chern-Simons as ones theory e.g., the that coadjoint can section algebra be can 6.2.2.so of equivalently be [ regarded written as a (2+1)-dimensional BF theory of write down the actionfeatures by just of inserting the into BFalgebras the and theories Lagrangian their based invariant ( metric on isare not coadjoint the constrained algebras Lie to that 1+1 algebras. aresection interesting dimensions 7 candidates First, and of for as [ the such generalization these existence to higher of dimension the (see As already discussed, thelimits; part it of is ( the invariantlimit metric of given the by algebra ( fact that generalizes under certain circumstances to double extensions [ and in addition to theabove, invariant we metric have ( by construction tionally to commutation relations ( which is non-degenerate dueh to the second term.g The other part of the invariant metric mentioned double extension [ direct sum and the invariant metric JHEP02(2021)134 ]. , (D.7) (D.3) (D.4) (D.5) (D.6) Phys. , SPIRE . IN [ ]. (5) 34 X + SPIRE . (5) IN 23 ]. [ (3) ) (D.2) 13 X (1982) 3517 (2002) 4029 X (2) 22 = 2 26 with commutation rela- 19 X 1 (D.1) H − SPIRE , 0 − IN = M , . 6.2 [ 2 1 1 (1985) 343 + . (2) (5) 11 15 − M − The solitons in some geometrical − ]. X X ( = = = 1 252 ] = 1 2 i ]. i = 2 − 0 M L = SPIRE Phys. Rev. D M L , (3) , (1990) 134 22 m, n , 0 , P IN + 0 h L X [ L L SPIRE [ h black holes from counting microstates 341 = (5) 14 IN used in section Class. Quant. Grav. n P D X ][ , M + , − − (2) 12 m P Nucl. Phys. B 2 , Classical and Quantum Properties of L – 34 – = X , ) − ]. (1979) 572 n − Z L (3) 23 ,  = 2 − = 40 L X + is conveniently represented as i = 2 L  − m − SPIRE = (5) i 25 Nucl. Phys. B L L IN with commutation relations , − 6.1 , X − ), which permits any use, distribution and reproduction in Entropy of [ L ] = L 0 Liouville Field Theory ]. + , P ] = ( hep-th/9810251 + L L , + n , h [ L  L − (5) 13 h (2) , 21 L L [ SPIRE , X m X (3) 12 L + IN [ (1983) 41 = L = X ][ CC-BY 4.0 P + = Gravitation and Hamiltonian Structure in Two Space-Time Dimensions L 126 Theor. Math. Phys. This article is distributed under the terms of the Creative Commons + , L (1999) 081501 Lower Dimensional Gravity (5) 35 ) basis X R 59 , − (2 sl (5) 12 X hep-th/0203038 Rev. D Noncommutative gravity in two[ dimensions Phys. Lett. B Two-dimensional Black Holes field theories M. Cadoni and S. Mignemi, S. Cacciatori, A.H. Chamseddine, D. Klemm, L. Martucci, W.A. Sabra and D. Zanon, C. Teitelboim, R. Jackiw, R.B. Mann, A. Shiekh and L. Tarasov, B.M. Barbashov, V.V. Nesterenko and A.M. Chervyakov, E. D’Hoker and R. 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