Jhep05(2018)036

JHEP05(2018)036 Springer May 7, 2018 : March 3, 2018 April 23, 2018 April 24, 2018 : : : Revised Published Received Accepted Published for SISSA by https://doi.org/10.1007/JHEP05(2018)036 . 3 1801.09605 The Authors. c 2D Gravity, AdS-CFT Correspondence, Conformal Field Theory, Field The-

In this paper we further study the 1d Schwarzian theory, the universal low- , [email protected] Krijgslaan, 281-S9, 9000 Gent, Belgium Physics Department, Princeton University, Princeton, NJ 08544, U.S.A. E-mail: Department of Physics and Astronomy, Ghent University, Open Access Article funded by SCOAP Keywords: ories in Lower Dimensions ArXiv ePrint: relevant for SYK-type models with internaltheory symmetries. as We a identify 2d the holographic BFon-a-group gauge theory action, and decomposing these compute into correlatorssimilar diagrammatic of building to the blocks, the holographically in Schwarzian dual a theory. manner 1d very particle- Abstract: energy limit of Sachdev-Ye-Kitaevprovide models, a path-integral using derivation the ofthe link the relation structural with between link 2d between 3dSchwarzian. Liouville both gravity, theory. We theories, 2d then and Jackiw-Teitelboim We study generalize gravity, the 2d Schwarzian Liouville double-scaling and limit the to rational 1d models, Thomas G. Mertens The Schwarzian theory — origins JHEP05(2018)036 38 25 38 4 26 21 35 11 37 21 = 2 Liouville = 2 super-Schwarzian 20 14 4 15 4 N – 1 – N k 12 29 14 23 13 2 28 8 24 10 21 33 28 16 24 15 16 = 1 super-Liouville 5.2.2 Interpretation5.2.3 in terms of Charged Schwarzian from 5.3.1 Partition5.3.2 function Correlation functions 5.1.1 From5.1.2 2d WZW to 1d Cylinder particle-on-a-group amplitude 5.2.1 Direct evaluation 3.1.1 Bulk interpretation 3.2.1 Black3.2.2 hole solutions from 3d Fefferman-Graham from 3d N 5.3 Example: SU(2) 5.1 General formalism 5.2 Example: U(1) 4.1 Bulk derivation 4.2 Supersymmetric JT gravity theories 3.1 Liouville with energy injections 3.2 Jackiw-Teitelboim from 3d 3.3 3d embedding 2.2 Gervais-Neveu field2.3 transformation transformation Bäcklund 2.4 2.1 Classical limit of thermodynamics C Partition function of a particleD on Some a relevant group formulas manifold for SU(2) 6 Concluding remarks A Virasoro coadjoint orbits and LiouvilleB branes Lagrangian description of matter sector 4 2d BF theory 5 Correlation functions in group models 3 Classical dynamics of Liouville and 3d gravity Contents 1 Introduction and summary 2 Path integral derivation of Schwarzian correlators JHEP05(2018)036 ] ], 30 29 – (1.3) (1.4) (1.1) (1.2) 21 , correspond- , g )) 2 . τ ( , f ], mainly due to the ) 1 15 τ – ( 1 f ( . G int ) ) 2 1 S Gibbons-Hawking τ τ S ( ( + g g + , 2 ∆ . Miraculously, the same action and } ) f )) g Λ t 2 f, t τ ∂ ( − { ( 0 ], suggesting a 2d holographic dual exists. a f ) dt (2) 20 + 1 R τ – } Z ( 0 16 2 f – 2 – C f, t Φ Majorana fermions with random all-to-all inter- − { g = ( = − N E dt ) √ 2 x Sch Z τ 2 S ( d C ψ ) − Z 1 τ 2 = ( † ]: will depend on the specific theory at hand. Stanford and Wit- S 1 ψ πG 31 D , the Schwarzian derivative of int 16 2 2 S 0 00 f f = ) = 2 3 2 , τ JT − 1 and low energies, the theory is dominated by quantum fluctuations of just S , corresponding to arbitrary conformal transformations, and 0 τ 000 ( f f f ]. N G = 35 – } 33 ] obtained this same action by considering the coadjoint orbit action for Virasoro- f, t { 32 Generalizations to non-abelian global (flavor) symmetries of the fermions were studied At large A direct generalization of the SYK model is to consider instead complex fermions. The interaction term ten [ Kac-Moody systems. in e.g. [ action. these two fields. In general, the low-energy theory is then for a function ing to arbitrary gauge transformations onrepresented the by charged a fermions. Schwarzian action, The former whereas is the known latter to is be represented by a free 1d particle features of any 2d holographic CFT. These models have alator U(1) has internal the symmetry, symmetry and [ the resulting infrared two-point corre- This leads to the holographic dualitygravity. between the UV Schwarzian theory decorations and can Jackiw-Teitelboim theory be added on to both bothwe sides theories solved of if the wanted, the Schwarzian but theorythe duality this by well-known that is embedding piece the it contains of minimal in the lore 2d universal Liouville that gravity CFT, Liouville regime. fitting theory nicely encodes with In the [ universal 3d gravitational interpretation appears when studyingwith 2d action: Jackiw-Teitelboim (JT) dilaton gravity [ Schwarzian derivative of a time reparametrization: with Sachdev-Ye-Kitaev (SYK) models of actions have received aappearance host of of maximally chaotic attention behavior inIt [ the was past realized immediately few thatis years the given [ by infrared the behavior so-called of Schwarzian these theory, models a and 1d their effective relatives theory with action given by the 1 Introduction and summary JHEP05(2018)036 = 2 (1.5) (1.6) ], but N 30 . ] and gener- M 38 2d WZW 2d Λ , , 2 is a weight in the 37 m 2 λ ) and external lines 1d particle on group on particle 1d ,λ = 2 supersymmetry, Ω R 1 , ∈ m -symbol of the compact N 1 j λ , an arbitrary diagram is m γ G Holography . 1 2 = and m m λ 1 2 λ λ Λ M 2 2 λ m M Λ 2d BF Theory BF 2d 1 1 3d Chern-Simons 3d λ m contains a path-integral derivation of the 2 ) arises as the bosonic piece of the = 2 supersymmetric Liouville and Schwarzian a = – 3 – ]. We analogously compute correlation functions N M Λ , 30 , 2 ) 1 ]. For a compact group m τ 2 − 30 ,λ 2 1 τ ( = 1 and m λ 2d Liouville 2d Schwarzian 1d 1 C λ N γ . Scheme of theories and their interrelation. − ]. e 36 = Holography 1 Figure 1 τ . The vertex function is given essentially by the 3 λ 3d Gravity 3d m 2d JT Gravity JT 2d λ is the Casimir of the irreducible representation ), linking four theories through dimensional reduction and holography. The same : λ C 1 G 2 τ The paper is organized as follows. Section Our main objective is to demonstrate that the embedding of the Schwarzian theory Our goal here is to understand the structure behind these theories better, and their Correlation functions of the Schwarzian theory were obtained first in [ Finally, when considering supersymmetric SYK models with Dim. Red. Dim. theory and by immediate generalizationsset of to models, compact we group alsotheories constructions. discuss wherever To appropriate. expand our link between Liouville theory and the Schwarzian theory. This was hinted at in [ to continuous representations, originatingstates and from (local) vertex the operators in perfecttation Liouville theory. labels dichotomy In are of the discrete, rational (normalizable) related case here, to all the represen- state-operator correspondence inwithin rational Liouville 2d theory CFT. isthe not Schwarzian just theory. convenient: it This is will the be most illustrated natural way by to both think a about field redefinition of Liouville group The representation labels of eachoperator exterior insertions line are are associated summed to over. discrete In representations the of Schwarzian SL(2 theory, where representation for the compact groupthat models of and the find a Schwarziandecomposed theory diagram into decomposition in propagators in and [ vertices: perfect analogy with correct bulk descriptions.(figure As a summary, wequadrangle will of theories find exists the for following the diagram compact group of modelsalized theories and as put well. in a Liouville context in [ the above action (withsuper-Schwarzian action a [ specific value of JHEP05(2018)036 . . 3 6 (2.1) (2.2) (2.3) ]. 43 . ) ) φ 0) limit of a thermody- φ,π ( . → ! −H ˙ ~ = 0. Hence one finds φ ) σσ . Within the older canonical φ φ φ φ ˙ `φ iπ π 2 ( φ,π e , ( -term localizes to configurations ˙ − dx q H = p R φ dx = 0 e is kept fixed in the limit, see e.g. [ V dτ R ˙ φ ~ ~ + β β β 0 − 2 σ R e 2 φ 1 ~ ] e φ ] + π φ . The last term integrates to a boundary term. 2 φ D π 2 1 π – 4 – ][ − D b φ

][ D + 2 φ [ b 1 D πb [ Z 8 = ) β ~ Q → ( φ ) = ) ] appeared that also investigate extensions of the Schwarzian φ β ( and 40 , (0)= Z 2 φ φ, π ( 39 Z Q , the phase space path integral of the thermal partition function φ H π ) = β and taking the classical limit, the ( = 1+6 Let us therefore briefly review how this works. For a general theory with discusses the 1d particle-on-a-group actions and the diagrammatic rules τ Z c 1 5 = t we look at the bulk story for the compact internal symmetries of SYK-type ~ ) = 0, i.e. static configurations for which β ˙ 4 φ ] on (the moduli space of) classical solutions of boundary Liouville theory. Here φ and momenta π ( 42 ] we provided a prescription for computing Schwarzian correlators through 2d Li- φ , δ We will take precisely this classical limit in the Liouville phase space path integral in Recently, the papers [ 30 To be distinguished from the semi-classical limit where 41 1 with parameters Operator insertions in Liouville are the exponentials 2.2 Gervais-Neveu field transformation Liouville theory with a boundary is defined by the Hamiltonian density: which is just the classical partition function for athe field next configuration. subsection. Rescaling with 2.1 Classical limit ofThe Schwarzian thermodynamics limit we will takenamical corresponds system. to the classical ( fields is given as: ouville theory on ain cylindrical [ surface between twowe ZZ-branes. will This provide was a basedprescription. direct on Liouville results path integral derivation that substantiates our previous theory with additional symmetries. 2 Path integral derivation ofIn Schwarzian [ correlators explicitly the structural links betweenIn these section theories in amodels. holographic Section context in section for computing correlation functions.The appendices We end contain some with additional some technical concluding material. remarks in section is proven more explicitly here. We use this description of Liouville theory to exhibit more JHEP05(2018)036 , is . = 0 ) (2.8) (2.9) (2.4) (2.5) (2.6) (2.7) `φ } 0 and σ (2.11) (2.10) e ≥ B,σ { ] at . σ +2 A 48 } . Performing ω )): A,σ + bdy { 2.4 ) )+2 σ ) transformations on ( ]: 0 R A,B , ( , ˙ . 2 φ ) ) δB 42 ) , } σ . To implement the bound- ∧ 2 ( σ, τ 41 0 , σ A,B π ) ( ( ) b B σ φ , , ( iπ ) σ 00 δ )+2 β ( ) σ, τ . ( x B B ( + + 2 δB ] considered a (non-canonical) field B −H f + − ˙ -dependence. So we mod out by this dσdτ { φ ) 47 ), where using this field redefinition, − τ ) φ σ ) = tan R v A γB b αB – π v ( ) ( iπ A π c ( f ) = ( c 0 σ 2 44 24 2 48 − π σ, τ ( f → ) 0 ( ) , ) does not preserve the symplectic measure. u − ( u dσdτ 2 2 +2 f ( ) on the standard symplectic measure, one δA ) ) as } − 0 R σ x . . . e 2.5 σ σσ σ ( f b B π ∧ ` ( B ,B c ,B , σ 2.5 B B 0 – 5 – f ) 48 ) ) ), ( σ − δ β sin A σ 2 )) with − A ( ) 2 ), ( and A + 2 + 00 2.4 σ σ, τ σ, τ ( − B ) transformation preserves the monotonicity proper- ( σ ( . . . e a σσ B 8 σ, τ -regime (small 2.4 a R δA ( A A σ c γA − = A αA − , `φ { can have arbitrary A e φ A ,B ] π = = e δ ( ) → φ c dσ φ φ π 24 e A that satisfies π π ) ) can be written as σ, τ D 0 ) = tan − need to be monotonic (as can be seen from ( and ( f ω Z ][ γ A 2.3 = B φ ( , σ, τ = ( ) D β ] Pf( H [ R A , → , δφ B and α ) D ∧ φ ). A Z ][ φ etc. We want to apply this transformation directly in the path integral. 2 A fields into new fields φ, π A . This field redefinition ( ) D mod SL(2 σ [ B R , B ∂ dσ δπ = (figure π E = 0 Z as: and π Z and σ ... B A A = A = 0. This transformation is invertible, up to simultaneous SL(2 `φ mod SL(2 e σ Next we define this theory on a cylindrical surface between twoThe ZZ-branes classical [ solution of this configuration is well-known [ The Liouville phase-space path integral, with possible insertions of the type We are interested in the large ω ≤ = D and σ in terms of a single function ary conditions at the quantum level,of it is the convenient to perform a thermal reparametrization and The Jacobian factor in thethe measure Gervais-Neveu is transformation the Pfaffian ( finds of the symplectic 2-form then transformed into where the quantities transformation. Note that anties SL(2 of the Hamiltonian density ( where The new functions B A redefinition as ( approach to Liouville theory, Gervais and Neveu [ JHEP05(2018)036 . ) π } = B,σ { (2.12) (2.13) σ 3 . +2 ] , a fixed } } ) requiring C ,σ 0. The ZZ- ) 0), thereby A,σ σ { ( s = ≤ p B 2.12 → = 0 and 2 +2 { σ ˙ π φ p b σ + φ cT T = } 24 ). s iπ ,σ σ ( ) ( σ ) f dτ ( s 0 and a( A T 0 { R [ . More general boundary ≥ π dσ dσ left. σ = π π -coordinate is chosen periodic 0 0 a σ 3 R R τ | = 0 0 = s π C π c ZZ 48 + 2 ) ... e s . The ` b -direction. To obtain a theory with ... e . b(- τ ` 2 , π s = p = 2 ) b . See figure a σ 2 σ p 2 − ) = 0 B b σ a B A σ = - = B )” is left implicit here. In the Schwarzian limit, the σ σ − B s a R simultaneously such that σ A − , A ( sin A A – 6 – 2 at the location of the branes, by ( ( ∞ T ) − + s and their behavior at the branes at ω ) = b ω → t → ∞ φ e ]Pf( c φ and ]Pf( B ) transformation matrix disappears, and it becomes a global gauge a B D R , . We obtain for the Liouville correlator in this limit: ][ D s s = 0 ][ ZZ A p π 2 π A D 2.1 [ p ) D (0) ) + 2 = [ ) + 2 , τ b s π ( (0 Z b ) is rewritten as b ) preserves the monotonicity properties π, τ ( ) Z b s (0) = ) = ) = ) a a( 2.4 π s ( , τ ) = 2.11 b( a -dependence of (0 σ a π, τ ( = → a . = 0 0 = s This double scaling limit is identical to the classical limit of thermodynamics T and, by the monotonicity requirements, -dependence of the SL(2 2 . Cylindrical surface with ZZ-branes at . Left: ZZ-ZZ τ =0 E σ | ... b The Schwarzian limit is defined by taking the small radius limit ( Note that possible quantum renormalization effects (such as theTo Liouville avoid cluttering determinant) the are equations, killed the off “mod in SL(2 2 3 `φ = e a D this limit. arbitrary redundancy. reducing the theory tonon-zero just the action, zero-mode we alongconstant. need the to take discussed earlier in section The redefinition ( boundary state is characterized by conditions and branes are discusses in appendix in terms of which ( Figure 3 Right: the doubling trick allows a description in terms of a single function with period Figure 2 JHEP05(2018)036 . 2 π β 2 β/ f ), so (2.16) (2.14) (2.15) has the → f π, π cT , − . The La- ∼ f and 1 2 . C π β 4 } 2 t ) to ( F,t ) + bdy { b → , dt t π, π = tan π ↔ π − − F R a ( C + bdy ) in these new variables, − ! ) ! . . . e 2.9 ) σ ` ( , σ ( ! 0 δf 2 and then rescale δa ∧ π )) ∧ β ) 2 1 ) t σ C ( σ to implement the boundary conditions 0 − ( ) ( 0 1 → π < σ < f t δf f < σ < π, − δa C 2 − − 2 0 ( ˙ a 1 : 1 mapping from ( ) f , π 1 ) ) β t π 2 , f 1 β ( – 7 – σ t ) 2 ]. Regardless, the final expression for the path in all expressions, one places the branes at a distance ( . f σ ˙ − ( f ( β ( ]. 32 ) for finite temperature. In the process, Liouville 1 2 − A a b − ) 30 ) as a sum of two Schwarzian derivatives, resp. the ( 1.1 ) σ ( σ 0 and that both terms add up, is written as ( 2.7 0 4 sin ) = 2 σ δf ) 2 σ

δa b ) ( σ ∧ right). The symplectic form ( ) ( − σ ) invariant by construction. ∧ f 0 ) ( ω 0 ) measure [ 3 ) R f σ t = a , ( σ ( ˙ ( 00 σ f 00 ] Pf( f 0 everywhere, and / f ], one can then extend this expression to arbitrary times for the δf 1 δa here. To reintroduce ≥ t D

30 [ π

and ) (figure σ ) ]. The boundary term drops out by our choice of boundary conditions, Q its physical dimension and demonstrates that the coupling constant f R etc. Alternatively, one can redefine R σ , dσ ) dσ f , a is the analogue of ( a 51 π R , π π β π , 0 − } = SL(2 50 Z Z / σ 1 SL(2 f S F, t / = = 1 { = tan ω diff S A Z continuous, should be set to 2 diff To summarize, the 1d Lagrangian is the dimensional reduction of the 2d Hamiltonian, As stressed in [ The link between Liouville theoryStanford between and Witten branes showed and that the for a geometric suitable Alekseev- choice of gauge, this becomes the f β 4 ∈ and the 2d local vertex operators become bilocaland operators sets in theThis 1d gives the theory. field dimensions This of is length. the bilocal operators tofor obtain correlators the are most then genericequations obtained in by Euclidean Liouville taking theory. time the Afterwards, configuration.signature. one double can scaling directly Expressions limit Both Wick-rotate these directly ofverified to in Lorentzian by these the several steps known explicit are checks in non-trivial, [ and the correctness of this procedure is operator insertions becometensor bilocal insertions insertions are in writtenholomorphic the in and Schwarzian ( theory. antiholomorphic stressoperators. Liouville tensor. We stress end up This with exhausts a Euclidean the theory non-trivial on Liouville the circle. The theory is reducedgrangian to a Schwarzian system on the circle, with Shatashvili action is made in appendix standard SL(2 integral becomes which is identified with therasoro Alekseev-Shatashvili orbit symplectic [ measure onand the the expression coadjoint Vi- is SL(2 using that on the ZZ-branes, as for f This can be simplified by defining a doubled field JHEP05(2018)036 . . ] 4 2 52 ) and χ (2.18) (2.20) (2.24) (2.21) (2.22) (2.23) (2.17) (2.19) σ A ∂ ]. when we + ( 5 2 . 38 ) , ) ψ σ 37 ( σ ∂ , ( ) δχ ]. Upon taking the σ ∧ ( 51 ) . It will turn out that F dσ , σ σ π ( 0 δφ χ 50 0 R . ), we can make one more ∧ − π δχ 2 c ) . / σ 48 1 2.5 σ − , ψ ) ( ) 0 τ 2 F σ ) = ∂ ), ( ) = ( F , we need to return to the δπ } F , 2 φ χ 2.4 δψ χ σ = , , e ∂ ) ∧ ) ( B, σ σ σ ) σσ = (det { σ π < σ < ) is canonical in field space (see e.g. [ and B χ, π ( σ B + B ≡ − 0 η F ( − < σ < π, σ 0 + 0 ψ + σ 2 F 2 0 − } A 2 , π δB ) π dτη δψ ψ , F − σ ) σ ∧ R ( φ σσ ) to arbitrary (compact) Lie groups. , 0 σ = = ( ) F ) – 8 – A, σ A − ,B R A ln ( { − B σ ) e σ ψ , F dσ ( ) has a harmless symplectic form: ( ( ( ] → σ e φ 00 π ≡ ≡ η ( A B 0 ) 0 dσ ≡ φ D F F Z [ ψ, χ δB ( 2 π π φ σ ( 0 δB ) π R c σ A Z − φ, π ∧ ( π 48 → 0 ) = ) as ) c ) σ 24 σ B σ η ( = ) = φ ( ( 0 − 00 ω F H 2 δA = ) φ, π δB Pf( σ ∧ ( H 0 − ) A σ ) ( σ 00 ). The boundary conditions in terms of these is illustrated in figure ], and we will use this short mnemonic later on in section ( 0 30 δA ) as: 2 δA ) 2.11 σ ∧ ( 0 ψ, χ ) A σ ( 00 δA In these variables, There is a slight variant of this transformation that is better equiped for this purpose, fields using ( To implement the ZZ-boundaryB conditions for Doubling is done in terms of a single field The measure is nowterms innocuous of as an it’s auxiliary fermion field-independent, and can be readily evaluated in The field transformation ( by defining ( or, in terms of thethese variables: Bäcklund field variables correspond toSchwarzian the limit, Alekseev-Shatashvili they fields [ correspond also with the field variables utilized in [ Boundary conditions still needsystem to is be not suited specified for however, the and, doubling when trick. written this way, the proving that the transformation ( and references therein). The Hamiltonian gets transformed into the free-field one: transforming the symplectic measure again into the canonical one: generalize this construction beyond SL(2 2.3 transformation Bäcklund Instead of using thefield Gervais-Neveu redefinition to parametrization get ( a free field theory transformation) (Bäcklund by defining rule we used in [ JHEP05(2018)036 . , ) π 37 . = 2.21 5 which (2.26) (2.27) (2.28) (2.25) σ ψ . s ) 2 = 0 and ) ψ p σ σ = ). ∂ s , ( σ 2 ( ) − ) τ ψ F 2 s t ψ ∂ A( σ ( , iψ dt , in the sense of the above ( ∞ π = 0 (the transformation ( π ], and ultimately arises due − ∞ dτ R β = R = 0 0 = s C 38 ψ )+ , − dσ π π 37 π − − dσ e ), the winding constraint is written ( ) R s π . . . e π π ]. F c B(- − ` 48 π, π Z    , 38 − , p ) = 2 δ β ) by choice) and π 1 37 t = - = ( + + . . . e s ψ π − ` F ( are left, which indeed correspond to shifts in ψ γF αF    dt e e – 9 – F 2 = ) 1 2 ) 1 → t ∞ → 1 t and their behavior at the branes at σ α ) ,τ t ( − 1 F ψ R σ ), with B → σ,τ e ( ) + − ( ψ F π ψ    π, π − ] e and ) ( − ψ 5 dσ e ,τ A F 1 1 D [ σ 1 σ s ( σ − -field for the interval ( ψ ) = R ) ∞ e ψ R p π , = ( = s ψ    F Z ] ) dt e ) ψ s s π ]. A( B( π D − mod SL(2 [ R ) and operator insertions invariant. This leftover gauge symmetry is explicitly distilled -dependence of 38 , σ ) ∞ 37 R 2.27 = 0 (to fix the divergences to , = = 0 0 = s ψ γ Z dσ e . Left: π π mod SL(2 − R These field redefinitions and their 1d Schwarzian result are summarized in figure We remark that this theory exhibits chaotic behavior, even though it looks like a free The gauge symmetry implementation is more subtle now. The original invariance 5 ]. The path integral becomes: is reduced to undoes this redundancy). Onlyleave rescalings the action ( in correlators in [ theory. Within this language, thisto is the explicitly above found constraint in (introducingthe [ a operator 1d insertions. Liouville potential) and the non-local nature of which can be computed explicitly as shown in [ Again taking the double scaling limit reduces this system to the expression: which can be regularized38 and implemented in the theory using a Lagrange multiplier [ defined for the doubled interval ( figure. Defining a doubled as: Figure 4 Right: the doubling trick allows a description in terms of a single function JHEP05(2018)036 (2.30) (2.29) (2.31) (2.32) has been F , ) , 2 ! ) ) ) y σ y,c 2 σ, τ, θ = 1 Liouville theory ψ ( 2 ), and their fermionic 2 ), the general classical ψ N , β 2 ) , θ utilized by Altland, Bagrets 2 − Free-field ( Free-field , θ − Bäcklund ( Bäcklund ψ σ 1 x e 1 ( ). In these variables, super- ψ = B 1 σ, τ, θ ˙ β, ( f . The auxiliary field ψ 2 σ, τ, θ ), 2 ( 2.31 1 ABK ψ ,B i 2 ) ) is written as , θ βD , 1 = 1 super-Liouville theory and the 2 + + ) = the superderivative. x β and , θ φ αβ ( 2 1 e N σ 1 B 2 A ∂ D − 2 σ, τ, θ ψ i ψ ( θ )( 1 B ] for details). These fields are not completely α , α σ, τ, θ + iψ 1 ) − i 53 1 ) and the conjugate momentum as the definition Schwarzian (f) Schwarzian D θ − – 10 – α, D A ( ∂ 1 Gervais-Neveu (A,B) Gervais-Neveu σσ 2.30 = = φ σ, τ, θ i αD Φ ( e − D = A φ ( 2 A e 1 ) 1 2 D f, p → + ), with ], is the dimensional reduction of the transition from Gerveu-Neveu 2 2 σ 2 ) , θ φ 38 2 , − + Liouville ( Liouville x 37 , ψ ( 2 1 φ β 2 ] and we heavily use their results. π , ψ ),

55 φ 1 ]. – 2 , θ ZZ-ZZ 1 and two Majorana-Weyl fermions thermal 51 53 , φ, π πb + ( φ x 50 2d 1d ( = = 1 super-Liouville α . Liouville theory in 2d in its different incarnations, and the resulting 1d theory one finds H = 1 super-Liouville theory is defined by the Hamiltonian density N As before, this can be generalized to an off-shell field redefinition in the phase space N = 1 super-Schwarzian. We will be more sketchy in this paragraph, some details are left Also, quantum renormalization effects should be taken into account when considering the 2d system as components from eight toLiouville theory four, is matching naturally the interpretederalizing as l.h.s. this the of statement theory from of ( previous all super-reparametrizations, sections. gen- discussed in [ utilizing the off-shell generalization of ( of the non-canonical field redefinitionindependent, (see but [ satisfy making the transformation a super-reparametrization, and reducing the number of real partners path integral: eliminated by its equationssuper-Liouville solution of for motion. the superfield Φ( In superspace ( in terms of superholomorphic bosonic functions for a scalar 2.4 The preceding discussionN can be generalizedto to the reader. Theappeared analogous in treatment [ of Gervais and Neveu for Figure 5 upon taking the double scaling (classical)and limit. Kamenev The (ABK) redefinition [ variables to variables. Bäcklund JHEP05(2018)036 . β ,π =0 gives σ and (2.33) (2.34) (2.36) (2.35) | 1 β α − ) = . Dα 2 . , and α ) θ ]. 1 ) F = ( 30 x σ, θ at those locations, σ, τ on the other. This ; ( (opposite) boundary 2 , that β U B − α, A g NS → ∞ ) + = and Sch( α σ, τ − A ( 2 = α 2 ) are interpreted as the supercharge , 2 ) = Λ ) ) . 2 2 αD U B 2.36 3 Dα require that Φ x, ( + i D : 1 2 σ, τ, θ V i and ( , π U V 2 = 1, one checks that this equation is solved for ( on one end and = + β into a single reparametrization V i π x c α – 11 – β on the doubled circle, which indeed corresponds = 0 equal to (minus) the super-Schwarzian derivative, 3 i become bilocal super-Schwarzian operators of the 4 B 12 = 0 Schwarzian as discussed in [ , i η Dα D σ = 1 Φ D V θ = α N ) − e α and ] that the Hamiltonian density can be written as H = A 53 with σ, τ ). For e.g. ( in terms of 1 i 1 − Dα 2 U ) ). Indeed, evaluating the above for e.g. ) ) component of V . For the latter, one needs to choose 1 i θ − F ) + σ, τ, θ Dα ) thus become the Hamiltonian density in real space (after inte- 2.32 ( ( i ∼ i V σ, τ V Dα , α ), next to the bosonic conditions on ( 1 1 (( − 3 ) D 2.30 Dα = ) = Λ ( 1 linked by ( 1 ). The fermionic parts (the Λ’s) in ( θ V α ,A 1 ), given by arbitrary super-reparametrizations of the classical Liouville solution. − is the bosonic ( σ, τ, θ ) ( i and 1 U 2.30 V , the superpartner of Dα A η Super-Liouville vertex operators ZZ-brane boundary conditions at It was then demonstrated in [ To rewrite the theory in terms of these variables, consider first the differential equation = ( Here we analyze some aspectswith of the the dimensional classical reduction dynamics to ofmind. the 2d The Liouville 1d larger and Schwarzian and 3d goal3d AdS 2d is gravity, the gravity Jackiw-Teitelboim to Schwarzian gravity theory, demonstrate in and the JTto gravity. structural other The links theories. next between section 2d generalizes this Liouville further theory, form ( 3 Classical dynamics of Liouville and 3d gravity into conditions on the branesleads such to that an antiperiodicto fermionic a field thermal system.ZZ-branes, but It this is only possible leads to to choose the other fermionic boundary conditions at the grating over densities. which means by ( This again allows us to recombine where The bosonic pieces of Analogous formulas hold for and explicitly for a fermionic function x JHEP05(2018)036 ) 6 x (3.4) (3.5) (3.6) (3.1) (3.2) (3.3) . , ) . However slice equals ∞ , t = 0 −− +

T − − ), the Liouville + → ]; the Schwarzian , x 2.4 ) and . 27 ): − – ,T , thus we relabel the )) x t, x, r σ 2.7 } ++ ( 25 ( T r t, x B , σ 0 ( ) π as we do not include ZZ-branes −− . c σ, τ dx T T 2 24 ( −− . This corresponds to swapping t ), the total change in boundary Z B − T ) + { − + → π ) = c t, x σ t, r, x ( − 24 ++ . ) = x } 2 T − ( ++ ) , t and σ T ) are most useful. ) B ( −− B x ) = ) = B σ − dx t, x , A → ( A σ, τ A σ, τ ,T ( ( Z τ B ( – 12 –

{ σσ ∼ + −− T + ) = φ , x e } ) , t t, x + ) = ) fields as ,T ) ( x } B ( σσ , σ, τ t, x is expected to describe the universal gravitational features A , σ ( ( ) A c A 00 dxT π { T c ( σ, τ ) on a region where energy is conserved, all functions become Z ( 24 is transformed from the Poincarépatch into an arbitrary frame. dx A − 3.2 { − ) = Z t π are holomorphic resp. antiholomorphic functions and the Liouville ( dx c ) = d dt 24 + E + . B x π σ − ( dx c φ 24 e ++ and τ ) = − T = A = = 2 σ, τ ) ( t ds x ( ], we analyzed the Schwarzian theory at the classical level in 2d Jackiw-Teitelboim dt ++ When evaluating ( The lightcone stress tensor components are given by equation ( Liouville theory at large T 27 We take here a more general situation than in the previous section dE 6 where but consider instead an infinite plane. This equation is not thatzero-mode, powerful in it general. becomes the However, when classicalequation reducing Schwarzian is equation to just of the motion energy spatial [ (= conservation. holomorphic and this just reduces to the uniformizing coordinate identification: Within a holographic theoryenergy with equals bulk the coordinates net ( bulk inwards flux from the boundary: its time coordinate identifiedLiouville with coordinates the to reflect Liouville this: spatialthe we roles coordinate set of time and space in Liouville theory. The total energy on a constant- leading to Energy conservation would ordinarily resultthis in is holomorphicity for violated ifenergy the into system the is system. not closed, We as allow happens for when this one possibility would here. inject additional The Schwarzian theory has On-shell, metric of holographic CFTs,exponential and is it related to is the this ( regime we discuss here. As in ( In [ (JT) gravity by allowing energy injectionsthe from matter the energy boundary. determines We a demonstratedshow preferred there how that coordinate that frame analysis close directly tothis generalizes the purpose, to boundary. the the Here higher Gervais-Neveu we variables dimensional ( Liouville theory. For 3.1 Liouville with energy injections JHEP05(2018)036 and (3.7) (3.8) (3.10) ) into ) ] that (3.11) A 7 , u 56 , . − 2 − A, B 2 ( z X dz ,X 3 2 − + ∂ + ) (3.9) 2 + / 3 X 3 − z X ( ) leads to a radial 2 + − dx ∂ O 2 X + 3.9 z − ∂ dx − ) + + , in ( − t=0 − ) E>0 E=0 solution ( X ) x X . We consider the region after + ( − 3 t 2 − ∂ x − x + = ∂ 4 ( was generated. Right: classical ( x x ) determined by solving + ). Hence the functions z − - x − X ∂X , L ) x x + T ) 2 ( T ( ∂ + < t < t π + 8 x c 1 B du x z ( t L ( 12 left). It has been shown in [ + = + + 6 L ) = − u ) = 2 ∂X 2 − 2 z ) and u x p dX ( x , u z ( + + − x − ( 2 = 2 X – 13 – z L 1 X 2 dX 2 − 2 − X t=t t=t ≡ ∂ − +

∓ , u t ) X + = ) and + X 2 2 + x x 2 ∓ + z 2 ∂ 2 − , x - ( ∂ 2 x ) x 2 z ( ds O dx ). It is clearest to demonstrate this in a region where no , z ) A − X where a non-zero boundary − X − − − 2 ) + x , x ∂ 1 2 , x X ( 2 ) = + − + − z x ++ -- ∂ x + − 2 ( x u T T L t > t + x ( X + + X + 2 + ∂ X 8 = dx ), is precisely the required frame. Setting ) ) + + X x x ( 3.10 ( + . Then L . Left: classical injection of bulk energy between X = = T 2 The full bulk diffeomorphism is given by ds X indeed correspond to the boundary reparametrization that, upon extending into the 7 bulk using ( given B is found by extending the transformation into the bulk, with the chiral functions to the Banados metric ( The above can be interpreteda as a new diffeomorphism preferred from vacuum frameadditional PoincaréAdS ( matter falls inthe (or general is bulk extracted) diffeomorphism (figure that brings the PoincaréAdS Figure 6 the injection takes place injection of a translationally symmetric pulse into the bulk. 3.1.1 Bulk interpretation JHEP05(2018)036 . 3 L G 3 2 (3.14) (3.15) (3.12) (3.13) (3.16) = c 8 , , 0 gravity is the 2 after the pulse: 2 0, equal to (half) Λ < z ) ambiguity, which dz = 0. This was indeed B − R t t > + , , . (2) − ]. 0 0 and R . This is the Schwarzian dx 27 3 SL(2 A 2 + 0 to match 3d gravity with × Φ λG = 0 for ) dx g , → , t > , t > 2 ) for R tx ∼ − , 3 T − 2 L √ G dϕ + − L 4 x G 3.11 x x 2 + Φ d 2 L πE πE c c − 2 to set 2 Z , but 12 12 z λ 2 3 + q q + −− /G ν T π 2 1 2 2 z πλG = dx 0 0 ∼ − µ – 14 – 16 tanh tanh 9 ++ C dx ]. This is done by considering the 3d ansatz = T + πE πE c c 0 is of course the BTZ black hole frame. (2) µν 57 2 − ). g , t < , t < 12 12 Λ + − ) directly leaves a SL(2 dx = x t > x p x p − ( − 2       L 3.11 (3) ds L R + ) = ) = 2 + to obtain a finite limit with + − G x x dx − ( ( ∞ + A √ B ) representing a fluctuating holographic boundary caused by matter + L x − 3 = → d 2 ,X right). This requires + Z ds 6 λL 3 X ( . u 1 πG 2 2 L 16 − a mass scale. This yields directly λ This 3d perspective on the bulk is very useful, and we here mention some aspects that The Schwarzian coupling constant As an explicit example, consider a translationally invariant injection of matter through Note that these functions are not strictlyΛ = holomorphic, due to the jump at 8 9 for arbitrary chiral functions allowed in regions where energy is not conserved. 3.2.1 Black holeAt solutions the from level 3d ofBanados classical metric: solutions, the general vacuum solution of 3d Λ 2d Liouville theory at largeSo central we charge, choose with Brown-Henneauxdouble central scaling charge limit from the bulk perspective. become easier to understand when embedding the theory in 3d. which is indeed JT gravity. It has been knownyields for 2d Jackiw-Teitelboim a gravity long [ time that a spherical dimensional reduction of 3dwith gravity The resulting Banados metric at 3.2 Jackiw-Teitelboim from 3d injections. Note that solvingis ( fixed by boundary (gluing) conditions, just as ina the pulse 2d (figure case [ the energy injected. One can then immediately solve ( trajectory JHEP05(2018)036 . ]. τ 58 (3.20) (3.18) (3.19) (3.17) ], and 27 , ]) leads to -zero-mode, 24 28 [ ϕ – of the dilaton , is equivalent to 24 ) 2 , a constant, as − a/ L x dϕ a/ ∼ 2 = − . Asymptotically, the 2 2 ∼ ) + − − x 2 L . , ( x 2 2 a µ = − dϕ dϕ q ) 2 + + ρ x x ( L ( 4 , + a µ 2 coth 2 µa r + Φ dϕ 2 z 2 µa j − a dρ √ 2 dx i ρ + = coth − dx + ρ ij µa 2 2 2 dx h a z dt + + – 15 – ) = 2 dx ν µa ) and associated dilaton field Φ dx − − ij µ x ≈ − 2 h 2 ρ − dx − 4( ds + , is just the standard Fefferman-Graham asymptotic ex- µν dx . The resulting 3d space is a non-rotating BTZ black hole, g x − ϕ ( + ϕ a µ = = in 2 2 q µdx a ds ds sinh a , the metric becomes 4 − x − 2 + ), only constant Schwarzian solutions survive the reduction, as this is the + = x , combined with an asymptotically Poincarémetric. Here we demonstrate that, 2 ], JT gravity is defined by enforcing an asymptotic value Φ 3.11 = ds = t 27 z – One starts with 3d gravity in the bulk, with periodically identified Euclidean time By ( Performing a spherical dimension reduction requires 24 at 2 and the Schwarzian limit. Its boundary contains 2done Liouville obtains theory. 2d JT Instead gravityand reducing in are to the only the bulk. linked angular through These this two 2d higher-dimensional theories story. are Finally living dimensionally in reducing distinct regions imposing Fefferman-Graham gauge in 3d. 3.3 3d embedding Armed with the above embedding ofwithin the 3d Schwarzian theory gravity, we within can Liouville and now JT relate gravity four different theories through dimensional reduction above 3d metric behaves as which, upon absorbing pansion. Hence imposing Fefferman-Graham gauge in 2d and Φ which is of the form of a spherical dimensional reduction: giving the 2d JT black hole metric Performing the purely radial transformation setting Poincaréboundary conditions directly in 3d. The 3d BTZ metric can be written as black hole spacetimes as the only solutions, perfectly analogous to the3.2.2 2d CGHS models [ Fefferman-Graham fromIn 3d [ Φ upon embedding in 3d, both of these conditions follow from just imposing asymptotically it should be independentdimensionally of reducing to a 2d JT black hole. generic 3d metric outside matter.of a And given any mass. 2d vacuum Indeed, metric directly in solving JT the theory vacuum is JT a equations black (as hole in [ JHEP05(2018)036 9 limit. c . The propagation of 8 and is repeated in figure t 1 = f JT gravity / BF theory BF / gravity JT Schwarzian / particle on group on particle / Schwarzian – 16 – r ZZ ZZ = ZZ ] that the Schwarzian theory is holographically dual to Jackiw- 27 – 25 3d Gravity / CS / Gravity 3d ). Liouville / WZW / Liouville 7 . Link between four theories through dimensional reduction, both for the gravity sector, . Left: cylindrical surface bounded by ZZ-branes. Middle: the exponential map transforms As we will demonstrate starting from the next section, an analogous story holds for We can omit the ZZ-branes if we realize that their entire goal in life is to combine left- ZZ 4 2d BF theory 4.1 Bulk derivation It was suggested inTeitelboim [ gravity. Within JT gravity, the Schwarzian appears as follows. The dilaton restricting to the angulardimensionally zero-mode reducing leads the to boundary 2d theoryThe BF leads resulting theory to scheme in the of 1d afor models particle convenience. different was on region. already a shown Further group in manifold. figure and right moving sectors into one periodicinto field, a thereby (chiral) transforming torus. the cylindrical Thisjust surface equivalence the is identity module also along demonstrated the in smaller figure circle is a consequence ofgroup taking the theory: large Chern-Simons (CS) in 3d reduces to 2d WZW on the boundary. Instead to a torus with only one chirality. Liouville theory leads to thetheory Schwarzian (figure theory as the angular zero-mode of the boundary Figure 8 this into an annularsemicircles region are in identified as the shown. upper Right: half performing plane. the doubling The trick ZZ-branes (method are of images) on leads the real axis and the as for the groupholographic boundary. theory Reducing sector. to the The angular zero-mode interior gives of a the 2d bulk torus and is a the 1d boundary 3d line. bulk. The torus itself is the Figure 7 JHEP05(2018)036 ] ]. 60 26 (4.1) ] and [ 59 2d WZW 2d 1d particle on group on particle 1d s 0 d Holography z=0 = t 0 A , e k A z= = 2 JT supergravity in [ j ∂ i ]. Consider the 2d BF theory obtained N 2d BF Theory BF 2d A 26 3d Chern-Simons 3d ijk + - J J x 3 d = 1 and – 17 – 3 M N Z t z=0 left). One can directly deduce the Schwarzian action ∼ 10 e CS S 2d Liouville 2d Schwarzian 1d z= right). The correct bulk theory that describes this situation . Scheme of theories and their interrelation. we extend the argument (in the bosonic case) to include an -- T 10 ++ B T Holography Figure 9 3d Gravity 3d 2d JT Gravity JT 2d . Left: injecting energy in JT gravity leads to a preferred coordinate frame at each The argument we present is a dimensional reduction of the 3d Chern-Simons story and The gauge theory variant of this story is readily formulated: we need a preferred gauge Dim. Red. Dim. the direct analog of theas Schwarzian a argument dimensional of reduction [ from 3d CS theory: transformation on the boundaryinto curve the at system each (see instant, figure determinedis 2d by the BF theory. injected charge from the bulk 2d JT dilatonThis gravity argument theory has from been the Gibbons-Hawking generalizedrespectively. boundary to term In [ appendix arbitrary matter sector. field blows up nearKeeping the AdS fixed boundary, its with asymptotics,instant, a depending coefficient requires depending on performing on thefluctuating a the injected boundary matter coordinate / curve sector. extracted (figure transformation energy at from each the system. This results in a Figure 10 time, resulting in atransformation fluctuating at boundary each time. line. Right: injecting charge leads to a preferred gauge JHEP05(2018)036 , one (4.6) (4.7) (4.8) (4.9) (4.2) (4.3) (4.4) (4.5) (4.10) (4.11) , solves 3.2 ∂M again, which | C = 0 0 A g δ = 1. v . 0 , , . ) now becomes: , ν µ ) 0 dtχA 0 J J A . 4.2 σ, µ 2 g gives the equations of motion: , µν ˙ ∂M σ = ˙ χ χδA I xA . σ. 2 ∂M dt = 2 1 2 | dtχδ µ 2 − d ˙ σ 0 ∂ 0 χ and + A µ ∂M M δχ ∂M = = I Z on-shell µ δA 0 ∇ I = – 18 – tt µ 1 2 A A = 1 2 δS T ( = 0 in the bulk. So we parametrize the solution as A xχF , = 2 10 v χ = = F d in the Chern-Simons action, by analogy with section is. ∂M S : S = 0 I π Q k M k g matter 4 ), the full action ( Z δ 1 2 ∞ sets S F 4.7 = χ = + S r . Varying w.r.t. ) over B 4.2 = 0. One obtains: φ that defines the specific theory. We choose to find a finite limit. The resulting 2d action is proportional to some ∂ v k χ ∼ and φ χ A ∼ φ A Path integrating ( Sending in charge through a matter field requires the additional term Reintroducing the correct prefactor 10 and the total boundary energy is needs to set is not quantized even though the original The total boundary charge is defined as Using the boundary condition ( These can be cancelled by constraining: for a parameter and the boundary terms at which is the charge analoguefield of given the in energy-momentum appendix matter source for the gravitational just like 3d CS theory. Restrictingthis the problem, gauge but transformations to creates satisfy dynamical degrees of freedom at the boundary. This action is not gauge-invariant, but changes as with JHEP05(2018)036 . was ] for 5.2 = 0. (4.15) (4.12) (4.13) (4.14) σ 61 F ), which 4.11 , with = 2 JT super- g µ ], see also [ ∂ N 1 23 − g , ) = − ) CS theory just gives us , J µ R 0 , ) particle-on-a-group action A − ), or by directly varying the below). R χA + , , . So the full theory reduces to J with contribution ( 2 4.5 Tr ( ) ) BF theory [ 4.2 ∂M g dt | t R CS 0 ∂ , dt σ 1 H A and ( ∂M − evolves due to matter charge; I + g σ = ∂M 1 2 σ I χ = ˙ grav Tr( ]. Operator insertions on the other hand , + − r χ H dt 30 J = = χF r = ∂M to change as well asymptotically to keep fixed – 19 – Tr I ¨ H σ 0 x 1 2 2 A d dt σ J = M transforms in the adjoint representation), up to the S Z ∂M I χ = − , one obtains S , after integrating by parts, one finds the boundary term: σ = ). Either by using 4.7 matter S matter 0. The only influence of the CS theory on the gravity part is in the S ≡ below. ) BF theory, which is the first-order formalism equivalent of dimensionally CS µν 5 R , T 3d bulk gravity coupledbecause to 3d CSdefinition of theory the total leads Hamiltonian: toprovides just decoupled a shift equations in of the energy. motion This will indeed be observed below in section One can write Jackiw-Teitelboim itself asrecent developments. an SL(2 In fact, dimensionallythe reducing SL(2 SL(2 reducing the Ricci scalar directly. Andis indeed, the equivalent SL(2 to theare Schwarzian not action so [ simple. which is the action ofin a section particle on a group manifold, to be studied more extensively which is gauge-invariantboundary ( term again.The The boundary equations condition of is motion againthe require chosen boundary as action: gravity would fix the relative coefficient (see section Non-abelian generalization is straightforward. The non-abelian BF theory is This procedure is independent of the gravity (Schwarzian) part. Some Comments: As charge is sent in, one requires • • • • which determines how thepure gauge gauge in transformation the bulk but becomes physical on the boundary. the boundary condition ( boundary action in terms of representing the net inward flux of charge. For the matter action JHEP05(2018)036 2) | (1) = 2 q u ( . (4.18) (4.19) (4.20) (4.21) (4.16) (4.17) N osp χF R ⊕ in terms of 2) | J and supercon- . p , 3 ( i 2 η E ˜ , ψ + osp a = 1 φD P + a and one additional η (1) BF theory , α u = 2 ˜ Q ˜ , φDψ , η (2). = 2). In particular the sl = 1 ) + Q ⊕ ˜ N χF, ψ and ) p , ∧ + ( A ) 2) ( , 2) algebra whose explicit form can be | ψ so | ) R F ∧ 3 (2 E (2 + χB, a η ξB, A ]: ], JT gravity itself can be written as an ηF F + + + osp ( osp + 23 (2) bosonic generators and 2 fermionic generators. 65 a STr( α Tr( α χ , , with the dilaton superfield sl ˜ . ˜ Q Q Z dA Z + 64 De ω α α a – 20 – ˜ ˜ φ ˜ ψ ψ = η = ) supplemented with a = A ∧ A + = 1) or ∧ + Z =2 F JT α + α 4.16 N S N JT 2) generators as: 3 ηψ Q | Q A S α d α + ) 3d supergravity can be written as a (2 φ =2 2) ( ψ q | = N R JT , + , four fermionic generators 3 + osp (1 p S F η J ,J ]. Dimensionally reducing these (super)gravity theories for the 3 2) 2d BF theory. a + η | osp ωJ P p 63 a ( , + + 11 a a 62 De , field strength osp a P P = 2 theory on its own would be interesting as this couples the gravi- η a a 2) and spin connection h η e ωJ , N 2) 2d BF theory has the bosonic algebra | = = + Z p ( = 1 a E A = (2) to either a P 2) BF theory would have just the 3 osp ( | . These eight generators satisfy an sl a (2) Chern-Simons theory. Similarly, Achucarro, Townsend and Witten demon- (2) generators leads to a =2 e (1 a B , expanded into the sl sl e N q JT = A osp S ⊕ = A p Studying the For simplicity, we set the cosmological constant zero here, as this does not influence Supersymmetric generalization is now straightforward, as one just generalizes the gauge And indeed, as known since a long time [ (2) The sl 11 (2) BF theory: which is indeed bosonic JT gravity ( tational and gauge sectors in the bulk. This isGenerally, left the for future work. The piece coming from just the bosons is then found in the literature. the structure of the theory. In components, the action is for three generator in terms of the fieldnection strength with zweibein group from JT supergravity action may be written as [ case sl The identification of the non-interactingstood gauge sector from as supersymmetry a 2d asa BF will theory be can illustrated also be here.strated under- a Pure long 3d time gravity agoChern-Simons can that theory be [ ( written as 4.2 Supersymmetric JT gravity theories JHEP05(2018)036 ). + x τ ( ( (5.2) (5.1) f f . = 2 ) = ). t g π ), a redun- z (¯ 2 → . We start by 2.4 + 2 z , 2 gg x ) Γ ( k z f ( 1 )+ and ¯ , one has g ) is preserved under π ¯ 1 ∂g t 1 − → − 5.2 τ → g ∂gg z = . This boundary condition 1 1 − -periodic: g − ¯ ] and in section ∂g, π f 1 , the symmetry transformation π, v Tr( 30 − z f = g is 2 + 2 d ¯ τ f f = R = π 1 ) has, in analogy with ( k group element. One can then identify z − 16 u (¯ − 1 e − ∂gg )) f ) ¯ z in Lorentzian signature) imposes reflecting global , requires z . In terms of ( f z, v , where 1 ( f π local group elements as well. – 21 – − 2 g any ⇒ − ( g = ¯ f = F ) = G = u ] ¯ z σ ) g 1 ∈ z g z, and (¯ D a scalar-valued function on the group. As well-known, ( γ [ ¯ (or J g f F z Z )) with a bilocal 1d operator as for ¯ z ) = 1 Z = ¯ z z, ( fγ ( = z J g ), with ( i z ∼ , and with Γ the Wess-Zumino term which will not be needed. An (¯ F )) k ¯ . f ¯ z f ) f z z, ( ( f g ( . F h f ) = ) is inserted, with 1 ¯ z ) which satisfies the boundary condition if g g . The transformation ( π z, f ( F , integer level → g − G to f τ ( g ∈ 1 Dimensionally reducing as in the Liouville/Schwarzian case, the WZW action itself Just as with the Schwarzian theory, we imagine performing a change of field variables At the second boundary brane at Inserting a brane at Just as in Liouville theory, we focus on the moduli space of classical solutions of this − g -periodic function f ) π dancy in description: a local WZW operator immediately reduces to theupon particle-on-a-group dimensional action, reduction. the Wess-Zumino term Γ vanishes π Hence, after implementing the boundary2 conditions, the system is characterized by a single from which, when translated into aprojects condition the on symmetry ontothe its group diagonal transformation subgroup; provided theis condition now ( theory to deduce thesolution link between the 2d and 1d operators. Thisboundary system conditions: has the classical for operator this theory enjoys invariance under a local group transformation 5.1.1 From 2d WZWConsider to the 1d 2d particle-on-a-group WZW system with path integral amplitude between vacuum branes. After that,that we will consider U(1) allow and us SU(2) to as two write examples down the generic correlation function5.1 using diagrammatic rules. General formalism We focus now onwill the provide boundary a prescription theories for of computingtheory, correlation following the the functions 3d logic of used the Chern-Simons in 1d the andproviding particle-on-a-group Schwarzian a theory 2d general in BF [ formalism starting models. fromand 2d performing We Wess-Zumino-Witten a (WZW) double-scaling rational limit. CFT Our main interest is again in computing the cylinder 5 Correlation functions in group models JHEP05(2018)036 , t = π β ν 2 held (5.4) (5.5) (5.3) dx C → µ , which t 3 ∼ dx S µν kT , but are not . Taking into G , is a symmetry fγ = L. G . 2 global → 2 = ∈ ) ds f f 1 t /G with Casimir equal to , but by general argu- H g ∂ , has a degeneracy of ˙ 1 , j 2 f − ∼ 1 f local 2 apart. Both the action f 1 2 − ) G g . As mentioned above, the group , f have dimensions of length. The β/ Tr( f ν t dt p → βH = C ∂ with the product µ π 1 − π a p f − e − R τ global ) f µν π t ( kT G 16 → ∞ a . Using operator methods, this can ). Then this is manifestly the path − 2 C J 1 k e g 2 ) 5.3 t ) ) = Tr( ( = ) = 2 b . The classical equations of motion are t f t τ ( ( 1 L a 1 by rescaling the time coordinate as g 0 and J . We will be more specific about this below τ − allows us to explore the semi-classical regime β C = f ] explicitly as in section makes the constant ,H → → – 22 – ) ν f Tr( C 1 β ˙ b ) can be written as a particle moving on the group T t x measure of the group metric: H D t ( J µ ( a ˙ f x H f G J we wrote in ( µν √ F as G ] f )) = 2 fγ C G t ) D ( t [ ( × = ∼ J f ) , this integration space is also written as the right coset of the t global L f G f ( ]. Note that the transformation ) = Z π /G J 66 is immaterial in the double scaling limit. + 2 can be changed into local t Tr( k ( . G π f . and Hamiltonian , which is known to be a symplectic manifold. The resulting partition ≡ ∞ 1 The coupling constant Z L + dg 1 Cas ) to the generic rational case as the right coset 12 − . → R LG/G g , C ⊗ 5.3 . As an example, the SU(2) group manifold is just the three-sphere ) = 0, identifying conserved currents ˙ dg ]). Consider for instance the partition function (without operator insertions), and 2 SL(2 ) at f 1 / 1 67 − 1 The quantization of a particle on a group manifold is in principle well-known (see Structurally the particle-on-a-group action is very similar to the Schwarzian action. We did not work out the measure [ The double scaling limit we take is Hence the rational generalization of the Schwarzian story requires us to compute the − g 5.3 As for the Schwarzian case, reintroducing S f ( 12 t theory is invariant under be used to(dim prove j) that each energy-eigenvalue, with irrepquantization label of the level e.g. [ ignore first theintegral modding rewriting of the Lorentzian partition function Tr making it clear∂ that this actionthe Hamiltonian has (up to an irrelevant prefactor): in section of ( The Lagrangian manifold as ments this has toTr be the standard fixed proportional to a coupling constant loop group: function could then bein computed the using Schwarzian case the [ of Duistermaat-Heckman the (DH) action: theorem it justnot is as necessarily the remnant a of symmetry the of WZW operator symmetry insertions in and 1d it as isn’t remarked a above. gauge But redundancy. it is The periodicity of 2 which can alternatively beand achieved the by operator placing insertions theinvariant are branes under left at local invariant transformations. underdiff the This immediately generalizes theaccount Schwarzian the coset periodicity of 1d path integral over the group: JHEP05(2018)036 ): . C 5.7 (5.8) (5.9) (5.6) (5.7) (5.11) (5.10) , and are ], and in τ , which is 13 1 68 . − ) i . In the limit , which in the g G ˆ g = 0 . , n i i = 0) states j m n m i, below for SU(2), and is , j, i | j ) 5.3 i ⊗ | g m ( , we place two vacuum branes j, F can be expanded into Ishibashi | = 0 2 | i j , of the irreps. Including operator i . i a i , n j m | 0 βC i . C − i, βC ii m h e i ˆ − j | ij i, e | i 0 δ ) merely requires gauge fixing the thermal m brane , 2 i | a | i S i i X S cl m m 5.3 X q m ) and dismissed. As mentioned above, this does τ ˜ TH i, i in ( h − 0 − X – 23 – β j (dim j) S e ( | j i → m fγ j = , 0 C i X X i , n − ∼ on a primary state have non-trivial dependence on m i a e | = j n τ f 0 m i a − brane Z S C J i, h i − ∗ 0 e S j 0 i ⊗ | p S i -matrix and the Casimirs ∗ , n 0 ], both with operator methods and path integral methods. Thus i S i,j S X m 67 p , SU(2) isometry, meaning an organization of the energy spectrum in 0 limit. i, | 66 × i,j → ,n X i T X m , the length of the cylinder in the closed channel when the circumference = /T . As well-understood, a boundary state SU(2) 2 ii π i π ˆ ' | ): limit becomes just all irreducible representations of the Lie algebra degenerate states. This can indeed also be seen explicitly for SU(2) in [ 2 = 2 2 ∞ ˜ T + All states obtained by acting with +1) j → 13 of this system can be written as: subdominant in the in terms of theinsertions modular in the middle, requirescomplete sets splitting of the primaries evolution around into each separate such pieces insertion. and For instance, inserting the two-point function The Kac-Moody algebra reduces to the zero-mode Lie algebra. One can thus write for ( The sum ranges over all integrable representationsk of the Kac-Moody algebra of interest where thethe length Ishibashi of states the are cylinder themselves becomes dominated by much their longer zero-mode than ( its circumference, with is fixed to 2 states as 5.1.2 Cylinder amplitude Just as to getand to consider the the Schwarzian from WZWin Liouville amplitude figure in on section a cylinder between these vacuum branes (as earlier path integral, which yields an overallincluded factor in of the the zero-temperatore (finite) entropy grouphowever volume allow (vol one to prove one-loopThe exactness of above the expression path is integral through indeedreadily the what generalized DH formula. we beyond will that. obtain We in provide section some more explicit formulas in appendix (2 the general case in [ Reintroducing the gauge-invariance has SO(4) JHEP05(2018)036 = 2 (5.15) (5.16) (5.12) (5.13) (5.14) (5.17) N ). u ( . Thus the 2 σ t . 2 ) ) = σ π t , so the Schwarzian ∂ , ( σ i dt + 2 j R . The classical solution u m ( 1 2 φ ]. σ v − j, | 30 φ∂ g h u . and ) . . . e ) v ) , g ( ¯ σ 2 ) ( t 2 ( − t F . iQσ dudv∂ ) . i − = − v iQσ g R e E ( t | − ) ) σ ( i σ 2 u e t = ( ) ( 1 m t Qδ S ( ) + ¯ i, iQσ iQσ h − u e ( − iQσ ) represents the total charge in the system, as e σ 1 e dg = ) requires t ] 1 σ ) – 24 – t σ ( π Z − ) = u,v D t ( [ , including the operator insertions, is solved analo- = ( iQσ = 2 σ i e u, v Z j iQφ D v ( Qδ e φ m − → = = j, u S | ¨ and decreases again to its original value at σ Q ) − in 2d with action 1 V g t ( φ and : the diagrams just represent convenient packaging of the build- F . . . e at | v C i Q = Q m V ] u i, φ h D [ Z increases by ]. Just as in that case, we remark that the resulting expression is non-perturbative σ 30 Of course the resulting theory is free and immediately solvable. Consider e.g. a two- Natural vertex operators are the exponentials: In the next two subsections we will consider the two simplest examples. The gener- gously as in thesignature): semi-classical regime of Liouville theory (and written here inhence Lorentzian ˙ operators inject and extract charge, and ˙ In this particular case, the bilocal operator is justpoint a correlator: product of two local operators. The classical equation of motion for The classical moduli space is1d parametrized limit by a entails: real periodic function Perfect reflection at 5.2.1 Direct evaluation Consider a free bosonis field given by 5.2 Example: U(1) As a first example,following let’s the take preceding U(1).Liouville discussion. and We find the start Afterwards same withSchwarzian we answer. limit a will The from latter direct supersymmetric embed serves evaluation versions as the of of a theory Liouville further its theory. consistency into correlators check on the alization to arbitrary compactdiagrammatic groups decomposition will of be the obviouscase at [ general the correlator, end. analogouslyin We as the will coupling in end constant the uping Schwarzian with blocks a of the general expressions. which is the method we utilized for the Schwarzian theory in [ The matrix element can e.g. be computed in configuration space as JHEP05(2018)036 that (5.18) (5.23) (5.20) (5.21) (5.22) (5.19) σ = 2 super- , . ), is the general ) N iσ τ 2 , one obtains: = 1. − e 5.66 β β ∞ . ( = 2 0 1 ) ¯ θ ¯ ρ Q 0 4 2 − θ to + q ( 0 2 − θ − t 2 0 2 e 1.0 θ ˙ ¯ θ −∞ τ f, ρ/ 0 1 2 D 4 . q 4. This two-point function is of θ 0 1 = / ¯ θ − 2 2 − 1 ¯ ρ ¯ θ 0 σ 0.8 2 Q ρ τ D dqe . 2 + 2 ˙ − ˙ . The Gaussian path integral is readily Z f 0 1 } , 1 ˙ 4 τ π ) f β 0.6 ) τ with 4 2 f, t − q f β β ) r ( 4 2 = 2 supersymmetry. An τ − ¯ σ → θ, − { 2 = 2 super-Schwarzian. This is because it con- = ) 1 Q + . This, as we show below in ( = 2 super-Schwarzian – 25 – 0.4 = 2 Liouville ancestor. τ 1 2 f τ − N ˙ ( σ ( σ N 2 dt 4 e 1 ρ − Q N N dt ¯ θ ( i Z 2 − R = ¯ e θ e 0 − C 0.2 ¯ θ e − ) → 2 = 2 σ ¯ θ S 1 − θ, 1 1 θ ) σ 0.0 τ ( 0.96 0.94 0.92 0.90 1.00 0.98 ( − ρ iQ 2 ) is given by e . τ ] = . Two-point function of U(1) theory in units where σ − 0 11 5.21 1 D [ τ asymptotes to , θ Z ) ∞ τ 1 Z ( + f Figure 11 → = ]: 0 β τ 36 [ σ The bosonic piece of the super-Schwarzian action is the Schwarzian plus a free boson the bosonic piece of ( reparametrization of the invariant super-distance is given by the following expression: For a purely bosonic reparametrization, field The relative coefficient was fixed by 5.2.2 Interpretation inThe terms U(1)-sector of is relevanttains, for in e.g. addition the tois the identified fermionic with superpartners, theparagraphs also above we an U(1)-sector. will additional identify Here it bosonic we from field its demonstrate this directly. In the next which at result for any non-abelian groupthe as shape well, as with in Casimir figure computed as: If the integral on the r.h.s. is truly an integral ranging from found earlier from the bulk perspective in section JHEP05(2018)036 ], ). is Y 69 = 2 = 2 Q 5.20 (5.24) (5.25) (5.26) (5.27) N N , the su- in ( φ σ ] to those of [ have weight: 70 P , Q ZZ y and U(1) charge ). , 4) / ` 2 bk 2 1.3 Q + → = 2 ) and gauge transformation 2 k τ ( boundary conditions around the 2 ( P NS β f opposite Q − antiperiodic e . − NS can be repeated when adding the 2 2 = 2 Liouville theory. The ` 2 → Q 2 , 2 , as given in ( 2 N b 2 ) 2 ˙ b O Y ZZ + iσ 3 − τ, z dt , forming the full supersymmetric multiplet. 2 e ]. To go from the conventions of [ ( η and Liouville momentum sector are of the form: 2 . Details can be found in the literature, but ` 2 Y P 2 69 00 Z Q θ /b + 2 – 26 – C = 2 Liouville NS Q O → = 2 Liouville along the small circle (NS-sector) 2 y ∆ = b 1 = 2 ZZ N 8 Q 2 N 2 S , to match with the super-Schwarzian field b Y = 3 + 3 + . σ 2 Q 2 2 ∆ = boundary conditions around the small circle, leading to super- 2 2 i P P e Q with charge q = 2 `φ → i g NS e 2 ~ Y NS periodic opposite ) = P = P,Q | 14 τ, z = 3 + 3 and a compact boson `,Q ( and 2 V c 2 ¯ ψ b NS P,Q 2 ZZ ch → and 2 b on the charged 1d operator . This leads to the additional 1d action in the Schwarzian limit: ψ ) Y τ ( . Left: cylinder with ). character for a primary with Liouville momentum iσ e 12 ≡ NS Take this theory on the cylinder bounded by two ZZ-branesThis and leads consider to imposing the removal of all fermionic degrees of freedom in the 1d theory, and Two convention schemes exist: we follow that of [ ) τ 14 ( The given by: one needs to set whereas Liouville states leading to the identification The required building blocksLiouville of primary vertex our operators story in are the readily available in the literature. retains only the Liouville field(leading itself to (leading the to U(1) thefree Schwarzian) theory). and boson the The compact analysis boson of section will not be needed here. antiperiodic boundary conditions(figure in 5.2.3 Charged SchwarzianIt from is possiblesupersymmetric to generalization obtain of this Liouville theoryperpartners theory directly consists of from theThe Liouville central field charge is symmetric quantum mechanics insmall 1d. circle, leading Right: to a cylinder removal with of all fermions uponThis dimensional reduction. can be viewedg as a simultaneous reparametrization Figure 12 JHEP05(2018)036 ) 4) τ / 2 − ) β (5.36) (5.34) (5.28) (5.29) (5.30) (5.31) (5.32) (5.33) (5.35) ( Q 2 E − − . q -matrix as: e i ( . τ S 1 . − 4) E 2 Y ZZ / − q 2 | 2 E i e ) 2 , πk. 2 e ) p Q Q , q φ y π 2 dE e − k ( . q βE 4 4 ∞ ( sinh 2 i h / / − 4) . + 2 2 2 2 e ) − q / q 4 πk 2 2 , q Z 4 Q − / , 3 q 2 / ) 2 y 1 E E b 4)sinh(2 k 2 q φ, Y 2 4) / − | √ ( p / Q 2 i . dE 2 − i 2 ) q 4 φ, Y E ,q K ` Q − 4 ( / ∞ → 2 E / − 2 + φ, Y k p + 2 4 1 4) 2 Γ( 1 h i E q ψ 2 / p Q k / ) Eψ -integral is the same as in bosonic

( ) E Z 2 Y 2 p β φ ) q − q 2 Q π πib p − = i Q − 2 ) = πbP e π − E 2 i -behavior: φ, Y 1 ( 2 `φ ib T Γ(1 + 2 1 p E πke E e ,q π ZZ φ, Y sinh 2 i + | 1 p p = ( ∗ k i i ) 0 is an indication of the lack of supersymme- sinh(2 i ψ τ ψ ) − ) sinh(2 ) dE bP – 27 – − sinh 2 τ ) and the Y ( φ, Y φ b P 2 β → 2 2 − Γ( 2 ∞ ( | ZZ 2 Q π q e π β ) in the ZZ-cylinder amplitude, we get: 1 i | `/ + H ( E e Q/ 2 = − , q + + ) = sinh 2 Z Γ( E = 2 Liouville theory leads to a removal of all fermions, `φ 1 e i 1 5.25 2 Y − cosh k E, q Y sinh( q × h h e ∂

dQ dk k N 2 Q dQ τ dφe i b E, q 2 1 E,Q − − e ( Z | E Z ) = 2 Z φ ρ = `φ − Q ∂ ( e e → = dφdY 2 δ − φ, Y ) dY E, q ( h P,Q Hτ 0 Z ( dE − S iT i divergence as Z e 4 1 ZZ / | ) = = = 2 E ψ , q ) 2 1 τ | √ Q ( k ∞ ZZ ) / − 0 h φ, Y | + 1 q ( χ ( = Z ∼ P,Q ZZ E,q i 1 -limit. The ZZ-brane wavefunction is determined by the modular ( ) 2 ψ ]. So we end up with 2 τ ZZ dE × h 30 , τ Ψ 4 1 ∞ | / τ the energy, solved by + 2 ( -integral just gives ]. q E Z Y 71 `,Q The basic integral we need to compute is Inserting one vertex operator ( dq hO Z The Liouville [ with The minisuperspace limit of bulk and the result is the Schrödingerequation: Let’s compute this explicitly. The ZZ-brane wavefunction is given by The lack of a try [ hence the density of states is identified as The total vacuum character then has the small in the large JHEP05(2018)036 ]. + j , 30 2 k (5.40) (5.37) (5.39) (5.41) (5.38) , ... j )-labels. ) 1 τ R , βC − , , transforms , 1 2 β − π , e 2 4)( +1) / j 2 ( , = 0 ) finally becomes: ) j ) Q + 1) 2 2 − +2) j π q k 5.39 , this becomes (2 iE 4 ( , j -integral agrees indeed (2 ) indeed yields the correct T +( q /β 2 and length − 2 00 E 1 ) e ( π 5.20 S ` T , − iE e 1 j Γ( + 1) τ τ + 1) + 2 X j 4) j k 2 / (2 2 = Q q π `/ j + − + 2) = 4 , q 1 = (2 Γ( 1 ], which in ( βC E ) ) k `, Q k ( , q j ( 2 − 2 sin 30 − h e k T E e 2 (2 2 2 [ ˜ T / p 2 + 2 2 − π τ dE e k + 1) = 1 ∞ j are only the energies of the Schwarzian subsys- 0. The vacuum character ( r + C – 28 – (2 2 ) = 0 + 1) i 2 = Z → E j / ) sinh(2 j 1 3 1 0 T π E + 1). The analogue of the Schwarzian double scaling 2 0 limit using dE ). and = (2 as j + 2) √ p ( 1 j on a cylinder of circumference ∞ → . Keeping fixed j π h k ,S 5.18 ∞ + E k , q, `, Q, τ ( 2 0 i T π T + /T = k Z 4 2 ( j π j − 2 T π X 2 → dq e C i sinh(2 A k = Z × = 2 j + 1) χ = ˜ T j j i π 0 ) iT 2 (2 S . The second equality expresses the character in terms of the closed 2 2 , τ → =0 0 +1) 1 k/ j +2 X j τ χ k ( ( 0 j = π 15 → T -transformation as: 2 `,Q T ). = i S π with the Casimir hO j iT 2 j h j 5.18 χ ) = lim βC 0 One can write a diagrammatic decomposition of a general correlator, as done in [ β The computation in this section is done for ( − χ e 15 Z prefactor in the action to agree with ( where channel with length 1) limit is here that the level which can be evaluated in the 5.3.1 Partition function The vacuum character for SU(2) under an where each line contains also a conserved charge, next5.3 to the Schwarzian Example: SL(2 SU(2) where now the energy variables tem, not the totalwith energy. ( Factorization is now manifest, and the The two-point correlator for instance is given diagrammatically as: Shifting the energy variables by the charge, then leads to: JHEP05(2018)036 ) ¯ M +1. , 5.41 j j J,M (5.44) (5.42) (5.43) (5.47) (5.45) (5.46) O βC − e ij δ , ) limit: , ) directly using elements in the ¯ J α M of the integrable 2 j j ) = dim j = 2 1 0 j j )) → ∞ 5.12 d S 2 A . t k ( ¯ ¯ j, m M 1 M ( − − ρ dim i dim j 1. At high temperatures, J f ; J,M ( 2 i,j → J ¯ O X m 2 ¯ . In this case, the prefactor is R ) are all built from the field transforming in an irreducible M Z ,j 00 k g 1 ( S ¯ ¯ M . For a general operator m F 1 J,M, J Mα j c R . J,M C ¯ ))] . Alternatively, the expression ( j. M i 1 ii → O 2 π M t / j ˆ ( | 3 − √ will turn out to be identifiable with the X ): 2 β f J J,M, ; j, m dim ( ¯ | ˜ 2 TH M J j 5.9 = m → − − R 2 e → j | ,j J,M = ) Z i ˆ X 1 2 j m hh = [ O t 0 00 – 29 – m ( , it is well-known that the 1 j S S limit ( ¯ 1 j ˆ g 0 M representation − C S . This prefactor will cancel in correlation functions = M i f ∗ ii → 0 J ) j = ) j ˆ 1 S | d 2 t i → ∞ t ) from the closed channel: ( 2 ( p → ∞ β f k 1 ¯ ( m k − 2 i,j Z f X F ) m 1 2 = t j → | ( i in the spin- f 0 ¯ ) M g f ( ≡ F J,M , essentially by using the Peter-Weyl theorem. In the double scaling ¯ brane M | G | O 1 , and this reduces to the dimension in the classical ( J,M ˜ TH ¯ j ˆ m , and requires regularization by taking finite − 1 O 0 e S m | e -matrix carry information about the quantum dimension 1 0 j S h which goes to zero as 00 brane S At low temperatures, only the vacuum contributes and For a general Kac-Moody algebra As in the Schwarzian theory, the prefactor can be written in terms of a ground state h ), so we can organize them into tensor operators z ( for the group element transforming both in thedoubled holomorphic Wigner-Eckart and theorem antiholomorphic holds: sector as a tensor operator, a The resulting elementary bilocal operatormatrix element: group theory asg follows. Generalrepresentation operator of insertions limit, one finds the bi-local operators: 5.3.2 Correlation functions Next we proceedconfiguration by space computing integrals, correlators we of will the compute SU(2) the theory. matrix element Instead ( of evaluating using the Ishibashi states in the representation It is instructive to recompute and is hence irrelevant for our computations; we dropthe it sum from can here be on. replacedis by readily an Poisson-resummed. integral and modular Casimir, and with the dimension ofNote the that irreps the as sum density of ranges states: over both integers andentropy half-integers. as just which, up to normalization constants, is a discrete quantum system with Hamiltonian = JHEP05(2018)036 . 2 J J j 2 for j )) is 1 = j G (5.52) (5.53) (5.48) (5.49) (5.55) (5.51) (5.54) (5.50) 2 into A m 2 5.44 j = ¯ . . 2 i , and m m J i 1 2 1 + 1) j j j P 1 δ D j , 2 j ¯ M − = − 1

M J j 2 δ + 1) − ¯ + m J J 2 2 J ) 2 , j j m 1 − j 2 ( j . We obtain the ratio A , . Φ + 1) . 2 ¯ D j M M 2 2 1 + 1) j J − + 1)Γ(2 j + 2)Γ( λ ], and is for the case J 1 + 1) 2 ; N + 1 J,M j J j 2 72 2 , one can evaluate this expression for λ + 1) j Φ m J = − + J 1 dim 2 2 2 2 Γ(2 2 2 + 1 ¯ ,j j j + 1)(2 m 1 j j 1 M 1 1 1 j λ J dim − m j m ) as 2 + + 1 A J 1 2 ; j + 1)(2 j (2 1 2 J ). j 1 C Φ m dim j 5.47 2

r 2 + 1)Γ(2 Γ( limit, this is given explicitly as – 30 – ,j (2 ,m 5.48 1 = = r J 1 k , it equals X m i r m J λ + 2)Γ( 1 2 = 2 + 1) j − = ¯ J λ m C J J ; 2 2 2 J 2 2 λ + and j 2 j 1 j m ii → 2 1 ,m 2 + 1)Γ(2 λ j 1 J 2 j 2 2 1 ; j j X j j 1 λ A | J D m | A , + m + 1)(2 ¯ 1 (2 ¯ 1 M 2 M 1 2 j λ j j . In the large C J,M J Γ( J,M s | O | O 1 + 1)(2 1 into ’s. = j ¯ m 1 2 hh j 1 J m j J − OPE coefficient was written down in [ m (2 J 1 ; k = j 2 -dependent factors that are absorbed into the reduced matrix element, see p h j and j 3 2 for details. The appearance of two Clebsch-Gordan coefficients will be crucial j 1 ; m → 1 j D m J = ¯ 1 2 1 j j j 3 C D m The matrix element in the double scaling limit (and with the normalization ( The SU(2) We will determine it below for SU(2), and conjecture that for a general group To determine the reduced matrix element in terms of the Clebsch-Gordon coefficients andthat the are reduced left-right matrix element. symmetric can Onlydelta. operators connect The the sum two Ishibashi over states, CG yielding coefficients the squared Kronecker- is just the fusion coefficient: which indeed suggests the general form ( then written by the Wigner-Eckart theorem as identifying the reduced matrix element in ( Some details of these computations are given in appendix On the other hand,equals the SU(2) Clebsch-Gordan coefficient for combining and for fusing any choice of the irreducible representations Note that a reordering ofing the in arguments of some the CGappendix coefficients has been performed,in result- what follows. in terms of two Clebsch-Gordan (CG) coefficients and a reduced matrix element JHEP05(2018)036 to be (5.62) (5.58) (5.59) (5.60) (5.61) (5.57) (5.56) ¯ M J,M . and the first ): , τH ,JM 2 j 2 − 1 J e j m , J,MM 2 ]). ) N which is the relevant i ,j O 2 1 74 k , j 0 m 3 J 0 S 1 := j m 1 S j 0 γ − 3 S j , J, M, τ J,M ; has been fixed above by the i 2 = s . O 1 2 2 m ¯ 2 M / 2 , m m m 1 i , 1 2 ,j ) j j j , 1 ¯ 0 ( 0 ) M J J,M M i m 2 S 1 τ O ( 2 j A ; 2 i J,M j J C m 2 JM Φ 3 ∆ 1 1 , m ). There is however a more convenient ) is computed using the Feynman rules: m i j i m j τ − τ π ( 2 + 1 2 ; 1 dim j j i 5.49 3 A dim J − 1 j 1 i = j 2 √ , m = – 31 – ) . The Euclidean propagators i √ j ii − = ( , 2 ( ,JM dim j j ) 2 dim j A ¯ 2 | 1 M i = m τ J,MM 2 ,m − ,m O 1 ,j τH 2 i J,M X 1 τ j 3 − 3 ( ,m O m j X e j 2 m 1 2 C j ,j ) is a classical limit of a formula recently derived by Cardy 2 1 γ 2 j − -symbol: j m = 4 super-Schwarzian systems (see e.g. [ e j J,MM = 1 5.54 1 N i O j m = τH J,M − 1 e τ hO | 1 j hh jm 2 ] (derived there for diagonal minimal models) where the Ishibashi matrix element is τ Hence, we obtain finally for the 1d two-point function ( Higher-point functions can now be deduced analogously, and we arrive at a diagram- The normalization of the intermediate operator 73 The CG coefficients determine the fusion of the representations at each double vertex. as The vertex is essentiallymetrically the in Clebsch-Gordan terms coefficient, of but the it 3 can be written more sym- and where the momentum amplitude which we now adopt. matic decomposition of a general correlationrepresentation function, labels where using one sums over all intermediate 2d CFT state-operatornormalization correspondence for in the ( 1dinstead theory, related by as taking the operator and the SL(2)-field Φ for a (diagonal) primaryfactors operator on the r.h.s.render them can normalizable. be Werational viewed conjecture CFT. as this In formula regularization any and case, artifactssymmetry we its group have of classical illustrated for limit the it e.g. hold explicitly Ishibashi for for states SU(2) any to inequality. The formula ( in [ written as It equals 1 by unitarity of the CG-matrix, and can connect only states satisfying the triangle JHEP05(2018)036 = 1, (5.67) (5.64) (5.65) (5.66) (5.63) i 0 , 0 hO , 2 . Our choice of = 7 (black). . J J M 13 2 j 1 2 J 2 j j m N = 1 for several values of dim J 1 ) . So 1 β τ 1 j m j 1 − τ β t and one finds ( = ) = 5 (blue), 1.0 2 j τ 2 = J J j C − β 1 2 β − j ( e m m 2 C 0. The qualitative shape of the = τ j 1 2 . JM − 1 j j 0.8 C τ j 1 e J − j C → 2 e C − τ − e τ = 0, and 1 ) in units where 2 e j = 1. As a check, some simplifying limits τ 2 2 C 0.6 j = 2 (green), i − → 5.65 J e (dim j) ). 2 ) is also directly computed using the Feynman dim j j = 0) →∞ X – 32 – 1 0.4 β τ ) = 5.65 i i ( = 0, so = 5.41 dim j 2 M β j 1 j J,M J, dim j C C 2 (yellow), 2 : − / 0.2 hO hO ,j 2 e 1 j , J, M, τ X j 1 i dim j J j ) itself ( = 3 ) 2 β N β J , m ( 1 ,m ( i 1 j dim j Z 0.8 0.6 0.4 0.2 0.0 1.0 Z ( ,m 2 X 2 j,m X A ,j diagrammatically: 1 i → j 2 ) A ) ensures that ) = β J,M 1 ( β ( Z = 1 (orange), hO Z 5.57 J = i . Two-point function of SU(2) theory ( = 0 (insertion of the identity operator), J,M J hO = 0 (red), The partition function Just as for U(1), this correlator is finite as J : diagram decomposition: When confirming the overall normalization of ( This immediate simplification only occurs for the two-point function. correlator is similar to thenormalization U(1)-case. ( Some examples arecan drawn in be figure taken. At zero temperature, which, for the particularthe case integer of fusion the coefficients two-point function, can be written fully in terms of Combining everything we arrive at: Figure 13 J with the amplitude JHEP05(2018)036 ] ]. 77 27 , (5.68) (5.69) 26 , 24 . , and leads 2 G ) 1 2 τ 2 -symbols of the + J M 4 j τ 2 2 − j 3 m τ ) 3 i + 3 2 3 m τ , τ j m i − 3 − j 1 4 . As emphasized for the β τ τ ( 3 ,M G j i 2 -deformed 6 C q ]. ,J − 1 2 1 1 2 i 1 e 30 ) J M M m m M 3 1 2 1 2 , m τ j j J J 3 i 3 − j j 4 m ( τ 4 ( 1 1 2 2 3 A j j τ τ 3 m C 3 m − 3 j e 3 ) 1 τ dim j − = – 33 – 2 ] starting from the exact OTO correlators in [ 2 τ dim j ( 1 -symbol of the group i 2 j 26 j , C , τ i dim j − 17 argument for preferred coordinate frames of [ e 1 i dim j 2 ,M . i 1 ,m i X 1 j ,J i dim j ) i ]. For the Schwarzian theory, we find the precise semi-classical β 1 dim j ( , m ,m i 76 i X Z j j × ], this quantity is used to swap the operator ordering and reach 4 = = 30 A i , this four-point function factorizes in two zero-temperature two-point 2 ,M 2 J → ∞ O β 1 ], become the classical 6 ,M 75 1 [ J ) shockwave expressions of [ G C hO In the second half of this work, we demonstrated that the Schwarzian limit is only a spe- We further extended the AdS This construction is immediately generalized to arbitrary compact groups The braiding and fusion matrices, which are given by cial (irrational) case of thetheories simpler have the case property of that rational the Hamiltonian, compactlocal Lagrangian models. operators and in Casimir All coincide, 2d of and CFT these that produced become geometric bilocal correlation operators functions in from 1d QM the in 2d a WZW double-scaling limit. perspective, We although our analysis was theory and the 1dthe Schwarzian Schwarzian theory. theory. We The believe firstdirectly, this where half is we of the emphasized this most the paperin natural relevance focussed this way of context. on to the the look parametrization Liouville at of path Gervais integral and Neveu to the case of gauge theories and preferred gauge transformations. We leave a more detailed discussion to future work. 6 Concluding remarks In this work, we presented more evidence and extensions to the link between 2d Liouville group Schwarzian case in [ specific out-of-time ordered (OTO)in correlators the of gravitational interest, case dual [ (large to shockwave interactions function, coming from the clustering principle,time and the differences, dependence just on only as two happens independent in the Schwarzian caseto [ the rules as given in section Note that as and is given by the expression: The time-ordered four-point function is drawn as JHEP05(2018)036 j ]. 80 , 79 = 2 Liouville . It would be for the ratio- ] to be linked ]. − 2 p 0 N 82 ]. It would be an 78 S + 30 p 0 ) with 1 discrete and S = 2 super-Schwarzian R , -boson and the Liouville N Y = log = 2 super-Schwarzian theories BH S N . On the other hand, it was found ]. Within the Liouville framework, p 0 S 27 , symbol of SL(2 26 j = log = 1 and -integrals directly in coordinate space. We N φ ) BF theory and the Schwarzian explicit in BH S R , – 34 – 2) for | -matrix as S 2) and OSp(2 | -sector, which allows one to connect to the 1d ) should also be interpretable as a 3 g 2 NS , k 1 symbols to swap internal lines in diagrams. It would be particularly interesting k ( j ]. ` itself. However, technical obstructions appear to be present when analyzing the symbols of OSp(1 γ 81 ] that the topological entanglement entropy in 2d irrational Virasoro CFT matches j φ One of the holographic successes of the Schwarzian theory is a correct prediction of the We will make the link between SL(2 A further question is whether anything can be learned for 4d gauge theories, as 2d The structure present in the rational theories, suggests the Schwarzian three-point These theories also seem to be related to groupA field very theories, interesting utilized extension in to the study spinfoam deeper would be to understand Nonetheless, we deduced expressions for time-ordered correlators and provided dia- 83 Acknowledgments I am deeply gratefulVerlinde to for A. numerous discussions Blommaert, and N. questions Callebaut, that H. greatly benefitted T. this Lam, work. G. The J. au- Turiaci and H. L. it arises fully from the modular in [ the Bekenstein-Hawking entropy for 3dinteresting BTZ to black utilize holes: the 2d/1dappear perspective in to 3d shed gravity more and light its on relation some to of 2d the Liouville puzzles dynamics. that boundary Liouville/Toda CFT wasto demonstrated (a in certain an subclass AGTin context of) 4d in these. gauge [ theories. Taking the double scaling limit shouldBekenstein-Hawking entropy have of the an JT analogue black holes [ upcoming work, using a complementary bulkfunctions in holographic terms perspective of on boundary-anchored bilocal Wilson linesat correlation in in BF [ theory. This was already hinted 2 continuous representations. Ifto this the can supersymmetric be Schwarzianand made correlators 6 more can explicit, be thenrespectively, conjectured the without to generalizations resorting to hold the inas coordinate terms mentioned space above. of evaluation of 3 the Liouville integrals theory. The latter contains non-trivialfield interactions between the mini-superspace regime and performinghope the to come back to this problem in thevertex future. to link this to results on OTO-correlators in rationalformulation 2d of CFT, LQG, as which in in turn e.g. seem [ to be relatedtheory to in the the tensor models of e.g. [ improvement to complement this withnal a path-integral theories analysis as as well, in including section the measure in the pathgrammatic integral. rules. This is Out-of-time-ordered leftducing to correlators 6 future can work. also be studied and require intro- not entirely rigorous as we used the generalization of the prescription of [ JHEP05(2018)036 ) π π R = , σ = | (A.1) σ as the system. πn SL(2 π ,n / = 1 1 + 2 σ

b S φ ], one can use = re a 48 − = φ σ ) redundancy to U(1). ) ) ∂ brane [ R R R , ,n (2, 1 ]. n 41 also in between both branes. to more general branes, and / U(1)/ / SL(2, / ) and 1 1 1 / SL / , 1 1 φ 2 2 πθ ( 16 ) 2 diff S diff S diff by a ZZ diff S diff v ) ( v f π ( 0 2 − ) q f = u ) = cos ( 1,1 u 1,n f σ ( 2 0 θ FZZT ZZ r ZZ f brane again. – 35 – sin ,n 2 1 to describe the genuine FZZT-branes, where the equation θ , with 2 is restored, and one obtains again a diff π iθ − = f σ | → = ]. One can directly implement an FZZT brane at ] considered boundary conditions that in modern parlance θ φ πθ 49 47 e , but now take the boundary condition as – b + 2 44 . b to find the ZZ and = 14 N a ZZ ZZ ZZ a ∈ n = , if one replaces the brane at , the periodicity of θ 2 nf ) becomes the standard FZZT relation; FZZT-branes correspond to hyperbolic orbits, whereas . Other brane configurations. Top: the ZZ-ZZ system. Middle: the ZZ-ZZ → πθ ( 1. This gives singularities in the Liouville field 2 f W.l.g. we keep the left brane fixed as aIn section ZZ-brane. We first present the 2d case, n > Strictly speaking, we should set = cosh 16 2 where one sets r we described the elliptic orbits instead. Our choice of notation follows [ boundary condition. ThisThe boundary classical condition solution corresponding breaks to the these SL(2 situations is: for Setting theory. Gervais and Neveu [ would be called FZZTby branes [ requiring instead and discuss thegeneralizations Schwarzian in limit figure at the very endthe only. same definition We of can now list the possible A Virasoro coadjoint orbits andIt Liouville is branes instructive tomake generalize the link the with constructionthe the in Virasoro Alekseev-Shatashvili group section geometric more action explicit. and the coadjoint orbits of Bottom: the ZZ-FZZT system. thor acknowledges financial support from thederen). Research Foundation-Flanders (FWO Vlaan- Figure 14 JHEP05(2018)036 . = 14 (A.4) (A.5) (A.2) (A.3) -term ˙ . For φ θ φ . ) and the π ! , 2.8 # 0 , dσ πδφ, ω ) , σ π ˙ 0 0 ff b 2 R 0 θf ( b h ), whereas for the = τ , − ( ≡ tan α # is determined only up a 0 , which indeed is what ! ) ) 2 ) and in the Hamiltonian α τ τ π ( ( c δ df f β ) + 0 00 0 12 f f 2.15 b )+ )+ is as before the uniformizing , h + S σ, τ − σ,τ σ,τ ( F − ( ( 2 in ( # 0 e F F ! F θ ] ) ) 2 − ˙ τ τ π f ff β ( ( 2 0 0 000 γ → α D 0 00 b f [ f f ]. On the other hand, it is known since f ) → ] more explicit as follows. The −

→ Z 51 π 0 ˙ β ) ! 2 2 51 f σ, τ f = , 2 ( ) − π F 50 σ, τ c 0 00 0 ( 000 48 is again a circle diﬀeomorphism as before. f f f Hdτ F integrated over time, with f – 36 – " f + α

ω 2 1 ) as can be explicitly checked, and the coadjoint dσ in terms of the FFZT brane parameter 0 . − f π df − 2 π θf above. The function 2 , and ignoring global issues, i d − ] that the cylinder amplitude of Liouville between these π 0 ( 2.15 000 Z ω 2 c θ f T 0 f 48 48 R π c dτ "

− 24 = tan 0 e ˙ π Z f ] β f dσ 2 F f π π = π c D − [ − by 48 Z " f Z dτα = i = , which integrates to zero as we take periodic boundary conditions in

. The only effect is a change α Z 0 ) = df b θf = dσ T ) orbit, one mods out by ( π π h R given by equation ( → − , χ U(1) orbit, one mods out by . After doing this, the field Z f geom / f ) S dα 1 dτ SL(2 S / = Z ) ) is precisely the canonical 1-form 1 ω = It is in any caseThe reassuring above procedure to has see the this additional equality benefit directly that within one the now path also integral. has a dictionary be- This result demonstrates the equivalence of Liouville between branes and the coadjoint In all of these cases, we can make the link between the Liouville action in ( Either of these alternative boundary conditions can be absorbed back into the action S 2.8 . Given the symplectic 2-form S ZZ-branes is computing thebranes Virasoro changes the vacuum character character. computed. As mentioned above, changing tween operator insertions in Liouville and operators in the Alekseev-Shatashvili geometric This complicated expression can be transformeda into the chiral Floreanini-Jackiw boson, path yielding integral for indeedthe a introduction single of character ZZ-branes [ in [ with orbit action (including the HamiltonianAt term) the for level the of different the orbitsto partition depicted the function, in this same figure also Virasoro followsprecisely character. directly the since goal Indeed, both of computing of the these characters coadjoint evaluate of orbit construction: irreps of the algebra is orbit parameter the diff( diff( we find from thecoordinate, results related of to section with and dα to an exact form time. Explicitly, and after doubling, the geometric action is given by: by rescaling in terms of geometric action of Alekseev and Shatashviliin [ ( JHEP05(2018)036 ) ) A.6 2.16 (B.1) (B.2) (B.3) (A.6) (A.7) metric; 2 -dependence of the -term in the La- z ˙ q , fixed the AdS p . such that 2 ` ). As a sanity check, t q ! ( m f 2 f . L ) and one recovers ( . ∂ T ) (the ∂∂ ) )) 0 ) over Φ | → ∞ q b 0 A.6 f C ) b A.2 f 1.2 − | σ, τ ∂ √ − q, ∂ σ, τ ( ( = π , f β − ) m 2 ( q m 0 L − f 0 f L = ∂q ) ) ∂ θ q, ∂ ( σ, τ σ, τ ( ( m dtdz f 0 in terms of precession of inertial frames agrees with f ⇒ L ⇒ ( f 2 Z θ 2 + 2 – 37 – dfdz } 0 /CFT sin 1 3 f 2 Z ]. This one-loop exactness fails for correlation func- θ q f, t Cθ ] as a Berry phase associated to a closed path in the Virasoro m 2 f { = π 32 -dependence drops out in ( L 84 β − 2 τ ∂ ∂∂ m dt . Consider now the coupling to the dynamical boundary 0

S q f Z − t − = ↔ ∂ = 2 = ). Explicitly, one has the correspondence T S ) 0 ) and can be ignored here. b m ( σ,τ A.5 L ( ∂q O ∂ `φ 0 ), ( e f As the Hamiltonian is itself the generator of a U(1)-symmetry, Stanford A.4 17 = 1 to find the ZZ-ZZ system again. θ When changing the branes, the resulting 1d theories are all pathological as thermal Finally taking the Schwarzian limit of interest, we need The geometric action is identified in [ In path integral language, integrating the Jackiw-Teitelboim action ( 18 We chose here to perform the time reparametrization throughout the 2d bulk; the 18 17 group. Their holographic interpretation intheir AdS absence in the dimensionally reduced 2d JTthe gravity, dual Gibbons-Hawking to boundary the term Schwarzian. then reduces to thefluctuating Schwarzian action. boundary is The two sectors only interactthe through the matter dynamical equations time of variable motion are given by in terms of a canonicalas: variable B Lagrangian description ofA matter general sector matter sector in the Poincaréupper half plane, is given by in the Schwarzian limit, where the when systems, except the ZZ-ZZ system that is studied here. As discussed in thegrangian) main disappears in text, this limit, the and abovetive) only the geometric remains. Hamiltonian density action (the ( Schwarzian deriva- and Witten applied the Duistermaat-Heckmanthe theorem resulting to 1d prove partition thetions one-loop function exactness however [ and of one has to resort to other methods, by using the correspondence ( This correspondence isinteresting fully conclusion at in its the own level right. of the 2d theories, and can be viewed as an action theory ( JHEP05(2018)036 0 1 g S (B.6) to (B.4) (B.5) (C.2) (C.3) (C.1) (D.1) 0 . g 0 m . t . j H ] as 2 j C 3 1 ) the represen- + 72 j − j g e 0 ( D ) } j 0 φ . This leads to 0 ! ( f D i j f, t ¯ δf = 0, corresponding to { χ m 2 factor coming from the 0 1 φ . f i β 1 X j , − − ) with C β )

dim j j g and dt δf − = ( δ G C e 0 j β − ab 2 1 Z ! f X ) e i j δ = 2 = m G t D ( 19 1 , 0 . i ]. 0 G X vol j } (dim j) δf 0 d 27 = 0

f vol j (dim j) = f, t δ X { 0 m t q j (0) = dim j and q j m f and using j with the (vol = k X H ]: = C . Setting L 0 χ 0 0 0 i i − ) = ∂ f ∂∂ q f ) ) 67 G e    β 0 ) = q ) 0 1 ; + g – 38 – g f ∞ 1 = 0 ( ( 0 ( 0 g ∂ 0 vol † j 00 3 } WZW model can be found in e.g. [ ( q f f LG/G f ab j , g , the Euclidean propagator from the point ¯ f m + D 0 k ab j , j ψ ) ∂ g 3 f, t 1 ψ m ( ) =    { ) g ,m β 0 ( , one finds 3 K ; j g j 0 −L 0 ( G D ∗ dg , g h ab j 0 (1)Φ g ψ dx Z 2 ( yields: ¯ m and is indeed , = Z K f 2 X j,a,b ) = Tr Z − ,m φ 2 LG ( j j ) = δf t is given by the formula [ 0 χ ; } 1 1 0 ) w.r.t. (0)Φ − f 1 , g ) f, t 0 0 ¯ m { B.2 , g g 1 ( ( 1 ,m g K dt 1 j = Z Φ h φ e = Varying ( Useful property: 19 δS D Some relevant formulasThe for three-point SU(2) function of the SU(2) and dismissed. The ordinarypath partition integration function over of a particle on a group only contains the which is indeed theright path coset. integral This over factor is absorbed into a contribution to the zero-temperature entropy with the character the sum over all based loops on C Partition function of aUsing particle the on normalized a eigenfunctions tation group matrices manifold of thewith representation The second term is found by writing which matches the first derivative of eq. (3.16) of [ reparametrization that indeed should not affect matter equations. and are not influenced by the coupling to the dynamical boundary, as this is only a time JHEP05(2018)036 2 j (D.6) (D.7) (D.8) (D.2) (D.3) (D.4) (D.5) +1) , 2 )! j +1 . k +2 3 ) . j k − 2 2 2 + j j J i 1 − 2 +1) j + 1) − J 1 J + γ j 2 2 J 1 3 1 j − j − j , m j 1 j 1 2 − j j J 2 j ) j ˜ 2 − + +1 A | A j 1 +2 +2 2 + , 1 − 1 j J k k ( M j 2 + m J + 1) -dependent factors that ( . − 1 J,M + +1)Γ( − j !( J j 1 P γ ; J 2 Φ k + 1) j J γ ) 2 k ; j + 1) | 1 2 )! j m 1 j − 1 2 k m j 2 + 2 m 2 Γ( ,j m j +1 − − 1 + 1) +2 k 1 +1)Γ( − − 3 j k , one has explicitly: m 2 + = 1 , 2 j )!( + 1)Γ(2 2 ) 1 j 3 + 2)Γ( . The Clebsch-Gordan coefficient 1 j 1 2 2 JM,j j j 3 k j j J 2 j γ k j + k +2) J m 2 h C C j − γ − j ) + 2 = 2 1 J (2 2 =1 j j + j j = = = − 3 =1 j J j ( Y v u u u t ( P m Γ(2 + 2 Y A

) + m j P m 2 − ) 2 +1)Γ( + 1)Γ(2 ) j J j 3 J 2 + = ¯ ∞ ) = J j j + + 1) → ( (2 1 j + 3 1 2 j 3 ( + 2)Γ( − 1 j P j ) ¯ m m j ) , 2 j J Γ( 1 2 – 39 – j ( Γ( +1)Γ( − j )!( 2 + 2 ,m ,P k + j (2 j 2 +1)Γ(2 2 and + 1)Γ(2 ) j 1 + 1) j − P − J x j 2 J + ) ( 1 j J J + 1 j x − (2 ,P P j 1 Γ(2 − (1)Φ = + j Γ( . It simpliﬁes to + 1) k α 2 ¯ 2 ), which permits any use, distribution and reproduction in M j p J J 1 is: Γ(1 Γ( 2 j + 1) m j + 1)(2 + + J → 3 − 1 s J,M, 2 + 1)Γ( = ¯ j 2 1 j j Γ(2 j ) = j 2 2 = !( − x + j +1)Γ( − k ( k J into α m J 2 -symbols; this reordering produces extra (0)Φ ) J γ − j − j 2 1 = k − j J 3 ¯ (2 ; CC-BY 4.0 γ m 3 + 1)(2 ( X + j , 2 1 1 1 ,j m × × s j + j 2 This article is distributed under the terms of the Creative Commons ,m ( and ,j 1 1 = − (2 1 j j P 1 2 J j . In the last equality we rearranged the Clebsch-Gordan coeﬃcients using m regime, we get Φ p − 1 Γ( 1

m × j j J k ; − C 2 = → ,j 2 2 = j J ,j into 3 2 1 1 1 j j 1 j j m J m D D 1 j C Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. with are absorbed into afor new combining reduced matrix element The second line useswith the standard formthe of symmetry the of the Wigner-Eckart theorem 3 by combining On the other hand, we can write and hence with In the large In the special case that JHEP05(2018)036 , . 04 11 JHEP D 94 , (2017) (2016) 10 07 (2017) 148 (2017) 092 JHEP JHEP , , 12 05 (2014) 067 ]. , (2016) 106 , talk given at the , KITP seminar, Phys. Rev. JHEP JHEP 03 , , , 08 ]. (2015) 132 JHEP SPIRE JHEP , , 10 November 2014 (2016) 110 IN [ 05 JHEP , . 12 JHEP SPIRE , IN ][ JHEP , , talk at KITP, University of , talk at KITP, University of JHEP ) ) , 1 2 arXiv:1505.08108 Space-time in the SYK model 12 February 2015 . . , A bound on chaos ]. ]. 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