TThe Schwarzian Theory - Origins
Thomas G. Mertens a , b ∗ a Department of Physics and Astronomy,Ghent University, Krijgslaan, 281-S9, 9000 Gent, Belgium b Physics Department,Princeton University, Princeton, NJ 08544, USA
Abstract
In this paper we further study the 1d Schwarzian theory, the universal low-energylimit of Sachdev-Ye-Kitaev models, using the link with 2d Liouville theory. Weprovide a path-integral derivation of the structural link between both theories,and study the relation between 3d gravity, 2d Jackiw-Teitelboim gravity, 2dLiouville and the 1d Schwarzian. We then generalize the Schwarzian double-scaling limit to rational models, relevant for SYK-type models with internalsymmetries. We identify the holographic gauge theory as a 2d BF theory andcompute correlators of the holographically dual 1d particle-on-a-group action,decomposing these into diagrammatic building blocks, in a manner very similarto the Schwarzian theory.July 2, 2018 ∗ [email protected] , a r X i v : . [ h e p - t h ] J un ontents N = 1 super-Liouville . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 U (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2.1 Direct evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2.2 Interpretation in terms of N = 2 super-Schwarzian . . . . . . . . . . 295.2.3 Charged Schwarzian from N = 2 Liouville . . . . . . . . . . . . . . . 305.3 Example: SU (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3.1 Partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3.2 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Some relevant formulas for SU (2) k Sachdev-Ye-Kitaev (SYK) models of N Majorana fermions with random all-to-all interac-tions have received a host of attention in the past few years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12],mainly due to the appearance of maximally chaotic behavior [13, 14, 15, 16, 17], suggestinga 2d holographic dual exists. It was realized immediately that the infrared behavior of thesemodels and their relatives is given by the so-called Schwarzian theory, a 1d effective theorywith action given by the Schwarzian derivative of a time reparametrization: S Sch = − C (cid:90) dt { f, t } , (1.1)with { f, t } = f (cid:48)(cid:48)(cid:48) f (cid:48) − f (cid:48)(cid:48) f (cid:48) , the Schwarzian derivative of f . Miraculously, the same action andinterpretation appears when studying 2d Jackiw-Teitelboim (JT) dilaton gravity [18, 19, 20,21, 22, 23, 24, 25, 26], with action: S JT = 116 πG (cid:90) d x √− g Φ (cid:0) R (2) − Λ (cid:1) + S Gibbons-Hawking . (1.2)This leads to the holographic duality between the Schwarzian theory and Jackiw-Teitelboimgravity. UV decorations can be added to both theories if wanted, but this is the minimaltheory on both sides of the duality that contains the universal gravity regime. In [27] wesolved the Schwarzian theory by embedding it in 2d Liouville CFT, fitting nicely with thewell-known piece of lore that Liouville theory encodes the universal 3d gravitational featuresof any 2d holographic CFT.A direct generalization of the SYK model is to consider instead complex fermions. Thesemodels have a U(1) internal symmetry, and the resulting infrared two-point correlator hasthe symmetry [28]: G ( τ , τ ) = (cid:10) ψ † ( τ ) ψ ( τ ) (cid:11) = ( f (cid:48) ( τ ) f (cid:48) ( τ )) ∆ g ( τ ) g ( τ ) G ( f ( τ ) , f ( τ )) , (1.3)for a function f , corresponding to arbitrary conformal transformations, and g , correspondingto arbitrary gauge transformations on the charged fermions. The former is known to be2epresented by a Schwarzian action, whereas the latter is represented by a free 1d particleaction.At large N and low energies, the theory is dominated by quantum fluctuations of just thesetwo fields. In general, the low-energy theory is then S = − C (cid:90) dt (cid:0) { f, t } + a ( ∂ t g ) (cid:1) + S int . (1.4)The interaction term S int will depend on the specific theory at hand. Stanford and Witten[29] obtained this same action by considering the coadjoint orbit action for Virasoro-Kac-Moody systems.Generalizations to non-abelian global (flavor) symmetries of the fermions were studied ine.g. [30, 31, 32].Finally, when considering supersymmetric SYK models with N = 2 supersymmetry, theabove action (with a specific value of a ) arises as the bosonic piece of the N = 2 super-Schwarzian action [33].Our goal here is to understand the structure behind these theories better, and theircorrect bulk descriptions. As a summary, we will find the following diagram of theories(Figure 1), linking four theories through dimensional reduction and holography. The samequadrangle of theories exists for the compact group models as well.
2d Liouville1d Schwarzian3d Gravity2d JT Gravity
HolographyDim. Red.
2d WZW1d particle on group3d Chern-Simons2d BF Theory
Holography
Figure 1: Scheme of theories and their interrelation.Correlation functions of the Schwarzian theory were obtained first in [34, 35] and gener-alized and put in a Liouville context in [27]. We analogously compute correlation functionsfor the compact group models and find a diagram decomposition in perfect analogy withthat of the Schwarzian theory in [27]. For a compact group G , an arbitrary diagram isdecomposed into propagators and vertices: τ τ λ m = e − C λ ( τ − τ ) , λ m λ m Λ M = γ λ m ,λ m , Λ M . (1.5)3here C λ is the Casimir of the irreducible representation λ and m ∈ Ω λ is a weight in therepresentation λ . The vertex function is given essentially by the 3 j -symbol of the compactgroup G : γ λ m ,λ m , Λ M = (cid:18) λ λ Λ m m M (cid:19) . (1.6)The representation labels of each exterior line are summed over. In the Schwarzian the-ory, operator insertions are associated to discrete representations of SL (2 , R ) and externallines to continuous representations, originating from the perfect dichotomy of (normaliz-able) states and (local) vertex operators in Liouville theory. In the rational case here, allrepresentation labels are discrete, related to the state-operator correspondence in rational2d CFT.Our main objective is to demonstrate that the embedding of the Schwarzian theorywithin Liouville theory is not just convenient: it is the most natural way to think about theSchwarzian theory. This will be illustrated by both a field redefinition of Liouville theoryand by immediate generalizations to compact group constructions. To expand our set ofmodels, we also discuss N = 1 and N = 2 supersymmetric Liouville and Schwarzian theo-ries wherever appropriate.The paper is organized as follows. Section 2 contains a path-integral derivation of thelink between Liouville theory and the Schwarzian theory. This was hinted at in [27], butis proven more explicitly here. We use this description of Liouville theory to exhibit moreexplicitly the structural links between these theories in a holographic context in section 3.In section 4 we look at the bulk story for the compact internal symmetries of SYK-typemodels. Section 5 discusses the 1d particle-on-a-group actions and the diagrammatic rulesfor computing correlation functions. We end with some concluding remarks in section 6.The appendices contain some additional technical material.Recently, the papers [36, 37] appeared that also investigate extensions of the Schwarziantheory with additional symmetries. In [27] we provided a prescription for computing Schwarzian correlators through 2d Liou-ville theory on a cylindrical surface between two ZZ-branes. This was based on results in[38, 39] on (the moduli space of) classical solutions of boundary Liouville theory. Herewe will provide a direct Liouville path integral derivation that substantiates our previousprescription. 4 .1 Classical limit of thermodynamics
The Schwarzian limit we will take corresponds to the classical ( (cid:126) →
0) limit of a thermody-namical system. Let us therefore briefly review how this works. For a general theory withfields φ and momenta π φ , the phase space path integral of the thermal partition function isgiven as: Z ( β ) = (cid:90) φ (0)= φ ( (cid:126) β ) [ D φ ] [ D π φ ] e (cid:126) (cid:82) β (cid:126) dτ (cid:82) dx ( iπ φ ˙ φ −H ( φ,π φ ) ) . (2.1)Rescaling β (cid:126) t = τ and taking the classical limit, the p ˙ q -term localizes to configurations with δ ( π φ ˙ φ ) = 0, i.e. static configurations for which ˙ φ = 0 , ˙ π φ = 0. Hence one finds Z ( β ) → (cid:90) [ D φ ] [ D π φ ] e − β (cid:82) dx H ( φ,π φ ) (2.2)which is just the classical partition function for a field configuration.We will take precisely this classical limit in the Liouville phase space path integral in thenext subsection. Liouville theory with a boundary is defined by the Hamiltonian density: H ( φ, π φ ) = 18 πb (cid:18) π φ φ σ e φ − φ σσ (cid:19) . (2.3)with parameters c = 1 + 6 Q and Q = b + b − . The last term integrates to a boundary term.Operator insertions in Liouville are the exponentials V = e (cid:96)φ . Within the older canonicalapproach to Liouville theory, Gervais and Neveu [41, 42, 43, 44] considered a (non-canonical)field redefinition as ( φ, π φ ) → ( A ( σ, τ ) , B ( σ, τ )) with e φ = − A σ B σ ( A − B ) , (2.4) π φ = A σσ A σ − B σσ B σ − A σ + B σ ( A − B ) , (2.5)where A σ = ∂ σ A etc. We want to apply this transformation directly in the path integral.The new functions A and B need to be monotonic (as can be seen from (2.4)): A σ ≥ B σ ≤
0. This transformation is invertible, up to simultaneous SL (2 , R ) transformations on A and B as: A → αA + βγA + δ , B → αB + βγB + δ , (2.6) To be distinguished from the semi-classical limit where β (cid:126) is kept fixed in the limit, see e.g. [40]. α , β , γ and δ can have arbitrary τ -dependence. So we mod out by thistransformation. Note that an SL (2 , R ) transformation preserves the monotonicity proper-ties of A and B . This field redefinition (2.4),(2.5) does not preserve the symplectic measure.We are interested in the large c -regime (small b ), where using this field redefinition, theHamiltonian density (2.3) can be written as H = − c π { A ( σ, τ ) , σ } − c π { B ( σ, τ ) , σ } . (2.7)The Liouville phase-space path integral, with possible insertions of the type e (cid:96)φ , is thentransformed into (cid:10) e (cid:96)φ . . . (cid:11) = (cid:90) mod SL (2 , R ) [ D φ ] [ D π φ ] e (cid:96)φ . . . e c π (cid:82) dσdτ ( iπ φ ˙ φ −H ) = (cid:90) mod SL (2 , R ) [ D A ] [ D B ] Pf( ω ) (cid:18) A σ B σ ( A − B ) (cid:19) (cid:96) . . . e c π (cid:82) dσdτ ( iπ φ ( A,B ) ˙ φ ( A,B )+2 { A,σ } +2 { B,σ } ) . (2.8)The Jacobian factor in the measure is the Pfaffian of the symplectic 2-form ω . Performingthe Gervais-Neveu transformation (2.4),(2.5) on the standard symplectic measure, one finds ω = (cid:90) π dσ δπ φ ∧ δφ = (cid:90) π dσ (cid:18) δA (cid:48)(cid:48) ( σ ) ∧ δA (cid:48) ( σ ) A (cid:48) ( σ ) − δB (cid:48)(cid:48) ( σ ) ∧ δB (cid:48) ( σ ) B (cid:48) ( σ ) (cid:19) + bdy . (2.9)Next we define this theory on a cylindrical surface between two ZZ-branes [45] at σ = 0and σ = π (Figure 2). ZZ ZZ s t s = p T s = 0 Figure 2: Cylindrical surface with ZZ-branes at σ = 0 , π . The τ -coordinate is chosenperiodic with period T .The classical solution of this configuration is well-known [38, 39]: e φ = − f (cid:48) ( u ) f (cid:48) ( v )sin (cid:16) f ( u ) − f ( v )2 (cid:17) . (2.10)6n terms of a single function f that satisfies f ( x +2 π ) = f ( x )+2 π . To implement the bound-ary conditions at the quantum level, it is convenient to perform a thermal reparametrizationof the A and B fields into new fields a and b as A ( σ, τ ) = tan a ( σ, τ )2 , B ( σ, τ ) = tan b ( σ, τ )2 , (2.11)in terms of which (2.4) is rewritten as e φ = − a σ b σ sin (cid:0) a − b (cid:1) . (2.12)The redefinition (2.11) preserves the monotonicity properties a σ ≥ b σ ≤
0. The ZZ-boundary state is characterized by φ → ∞ at the location of the branes, by (2.12) requiring a = b | σ =0 and, by the monotonicity requirements, a = b + 2 π | σ = π . More general boundaryconditions and branes are discusses in appendix A. See Figure 3 left. s = 0 s = p a( s )b( s ) s s = 0 s = p a( s )b(- s ) ss = - p p p Figure 3: Left: σ -dependence of a and b and their behavior at the branes at σ = 0 and σ = π . Right: The doubling trick allows a description in terms of a single function f ( σ ).The Schwarzian limit is defined by taking the small radius limit ( T → τ -direction. To obtain a theory withnon-zero action, we need to take c → + ∞ simultaneously such that cT π = C , a fixedconstant. This double scaling limit is identical to the classical limit of thermodynamics Note that possible quantum renormalization effects (such as the Liouville determinant) are killed off inthis limit. (cid:10) e (cid:96)φ . . . (cid:11) ZZ-ZZ = (cid:90) a (0 , τ ) = b (0 , τ ) a ( π, τ ) = b ( π, τ ) + 2 π [ D A ] [ D B ] Pf( ω ) (cid:18) A σ B σ ( A − B ) (cid:19) (cid:96) . . . e c π (cid:82) π dσ (cid:82) T dτ ( iπ φ ˙ φ +2 { A,σ } +2 { B,σ } ) → (cid:90) a (0) = b (0) a ( π ) = b ( π ) + 2 π [ D A ] [ D B ] Pf( ω ) (cid:18) A σ B σ ( A − B ) (cid:19) (cid:96) . . . e C (cid:82) π dσ [ { A ( σ ) ,σ } + { B ( σ ) ,σ } ] . (2.13)This can be simplified by defining a doubled field f to implement the boundary conditionson the ZZ-branes, as f ( σ ) = (cid:26) a ( σ ) , < σ < π,b ( − σ ) , − π < σ < , (2.14)for f continuous, f σ ≥ f a 1 : 1 mapping from ( − π, π ) to ( − π, π ), so f ∈ diff S /SL (2 , R ) (Figure 3 right). The symplectic form (2.9) in these new variables,using that f σ = a σ and f σ = − b σ and that both terms add up, is written as ω = (cid:90) π dσ (cid:32) δa (cid:48)(cid:48) ( σ ) ∧ δa (cid:48) ( σ ) a (cid:48) ( σ ) − (cid:18) πβ (cid:19) δa (cid:48) ( σ ) ∧ δa ( σ ) (cid:33) − ( a ↔ b ) + bdy , = (cid:90) π − π dσ (cid:32) δf (cid:48)(cid:48) ( σ ) ∧ δf (cid:48) ( σ ) f (cid:48) ( σ ) − (cid:18) πβ (cid:19) δf (cid:48) ( σ ) ∧ δf ( σ ) (cid:33) + bdy , (2.15)which is identified with the Alekseev-Shatashvili symplectic measure on the coadjoint Vi-rasoro orbit [47, 48]. The boundary term drops out by our choice of boundary conditions,and the expression is SL (2 , R ) invariant by construction.The link between Liouville theory between branes and the geometric Alekseev-Shatashviliaction is made in appendix A.Stanford and Witten showed that for a suitable choice of gauge, this becomes the stan-dard SL (2 , R ) (cid:81) t / ˙ f ( t ) measure [29]. Regardless, the final expression for the path integral To avoid cluttering the equations, the ”mod SL (2 , R )” is left implicit here. In the Schwarzian limit, thearbitrary τ -dependence of the SL (2 , R ) transformation matrix disappears, and it becomes a global gaugeredundancy. β should be set to 2 π here. To reintroduce β in all expressions, one places the branes at a distance β/ A = tan πβ a etc. Alternatively, one can redefine C → C πβ and then rescale t → t πβ and f → f πβ .This gives the field f its physical dimension and demonstrates that the coupling constant C ∼ cT has thedimensions of length. (cid:90) diff S /SL (2 , R ) [ D f ] Pf( ω ) (cid:32) ˙ f ( t ) ˙ f ( − t )4 sin (cid:0) ( f ( t ) − f ( − t )) (cid:1) (cid:33) (cid:96) . . . e C (cid:82) π − π dt { F,t } . (2.16)The theory is reduced to a Schwarzian system on the circle, with F = tan f . The La-grangian { F, t } is the analogue of (1.1) for finite temperature. In the process, Liouville op-erator insertions become bilocal insertions in the Schwarzian theory. Liouville stress tensorinsertions are written in (2.7) as a sum of two Schwarzian derivatives, resp. the holomorphicand antiholomorphic stress tensor. This exhausts the non-trivial Liouville operators. Weend up with a Euclidean theory on the circle.As stressed in [27], one can then extend this expression to arbitrary times for the bilocaloperators to obtain the most generic Euclidean time configuration. Expressions for corre-lators are then obtained by taking the double scaling limit directly in the known equationsin Liouville theory. Afterwards, one can directly Wick-rotate these to Lorentzian signature.Both of these steps are non-trivial, and the correctness of this procedure is verified by severalexplicit checks in [27].To summarize, the 1d Lagrangian is the dimensional reduction of the 2d Hamiltonian,and the 2d local vertex operators become bilocal operators in the 1d theory. This is therule we used in [27], and we will use this short mnemonic later on in section 5 when wegeneralize this construction beyond SL (2 , R ) to arbitrary (compact) Lie groups. Instead of using the Gervais-Neveu parametrization (2.4),(2.5), we can make one more fieldredefinition to get a free field theory (B¨acklund transformation) by defining φ F ≡ ln ( − A σ B σ ) , (2.17) π F ≡ A σσ A σ − B σσ B σ , (2.18)transforming the symplectic measure again into the canonical one: δA (cid:48)(cid:48) ( σ ) ∧ δA (cid:48) ( σ ) A (cid:48) ( σ ) − δB (cid:48)(cid:48) ( σ ) ∧ δB (cid:48) ( σ ) B (cid:48) ( σ ) = δπ F ( σ ) ∧ δφ F ( σ ) , (2.19)proving that the transformation ( φ, π φ ) → ( φ F , π F ) is canonical in field space (see e.g. [49]and references therein). The Hamiltonian gets transformed into the free-field one: H = c π (cid:90) π dσ (cid:18) π F ∂ σ φ F ) (cid:19) . (2.20)9oundary conditions still need to be specified however, and, when written this way, thesystem is not suited for the doubling trick.There is a slight variant of this transformation that is better equiped for this purpose,by defining ( ψ, χ ) as: A σ ≡ e ψ , B σ ≡ − e χ , (2.21)or, in terms of the B¨acklund variables: φ F = ψ + χ, π F = ψ σ − χ σ . It will turn out thatthese field variables correspond to the Alekseev-Shatashvili fields [47, 48]. Upon taking theSchwarzian limit, they correspond also with the field variables utilized in [34, 35].In these variables, H = − c π (cid:82) π dσ ( { A, σ } + { B, σ } ) = c π (cid:82) π dσ (( ∂ σ ψ ) + ( ∂ σ χ ) ). Thefield transformation ( φ, π φ ) → ( ψ, χ ) has a harmless symplectic form: δA (cid:48)(cid:48) ( σ ) ∧ δA (cid:48) ( σ ) A (cid:48) ( σ ) − δB (cid:48)(cid:48) ( σ ) ∧ δB (cid:48) ( σ ) B (cid:48) ( σ ) = δψ (cid:48) ( σ ) ∧ δψ ( σ ) − δχ (cid:48) ( σ ) ∧ δχ ( σ ) . (2.22)The measure is now innocuous as it’s field-independent, and can be readily evaluated interms of an auxiliary fermion η asPf( ω ) = (cid:90) [ D η ] e − (cid:82) dτη (cid:48) η = (det ∂ τ ) / . (2.23)To implement the ZZ-boundary conditions for ψ and χ , we need to return to the A and B fields using (2.11). The boundary conditions in terms of these is illustrated in Figure 4.Doubling is done in terms of a single field F s = 0 s = p A( s )B( s ) s s = 0 s = p A( s )B(- s ) ss = - p Figure 4: Left: σ -dependence of A and B and their behavior at the branes at σ = 0 and σ = π . Right: The doubling trick allows a description in terms of a single function F ( σ ). F ( σ ) = (cid:26) A ( σ ) , < σ < π,B ( − σ ) , − π < σ < , (2.24)10efined for the doubled interval ( − π, π ), with F ( π ) = F ( − π ) + ∞ , in the sense of the abovefigure. Defining a doubled ψ -field for the interval ( − π, π ), the winding constraint is writtenas: F ( π ) = F ( − π ) + ∞ → (cid:90) π − π dσ e ψ = ∞ , (2.25)which can be regularized and implemented in the theory using a Lagrange multiplier [34, 35].The path integral becomes: (cid:90) mod SL (2 , R ) (cid:82) π − π dσ e ψ = ∞ [ D ψ ] e ψ ( σ ,τ ) e ψ ( − σ ,τ ) (cid:16)(cid:82) σ − σ dσ e ψ ( σ,τ ) (cid:17) (cid:96) . . . e c π (cid:82) π − π dσ (cid:82) dτ ( iψ σ ψ τ − ( ∂ σ ψ ) ) . (2.27)Again taking the double scaling limit reduces this system to the expression: (cid:90) mod SL (2 , R ) (cid:82) π − π dt e ψ = ∞ [ D ψ ] e ψ ( t ) e ψ ( − t ) (cid:16)(cid:82) t − t dt e ψ (cid:17) (cid:96) . . . e − C (cid:82) π − π dt ( ∂tψ )22 , (2.28)which can be computed explicitly as shown in [34, 35].We remark that this theory exhibits chaotic behavior, even though it looks like a free theory.Within this language, this is explicitly found in [34, 35], and ultimately arises due to theabove constraint (introducing a 1d Liouville potential) and the non-local nature of theoperator insertions.These field redefinitions and their 1d Schwarzian result are summarized in Figure (5). N = 1 super-Liouville The preceding discussion can be generalized to N = 1 super-Liouville theory and the N = 1super-Schwarzian. We will be more sketchy in this paragraph, some details are left to the The gauge symmetry implementation is more subtle now. The original invariance F → αF + βγF + δ , (2.26)is reduced to γ = 0 (to fix the divergences to σ = ± π by choice) and β = 0 (the transformation (2.21)undoes this redundancy). Only rescalings F → α F are left, which indeed correspond to shifts in ψ whichleave the action (2.27) and operator insertions invariant. This leftover gauge symmetry is explicitly distilledin correlators in [34, 35].Also, quantum renormalization effects should be taken into account when considering the 2d system asdiscussed in [47, 48]. iouville ( f, p ) Gervais-Neveu (A,B) Bäcklund ( y,c )Schwarzian (f) Free-field ( y ) ABKZZ-ZZthermal
Figure 5: Liouville theory in 2d in its different incarnations, and the resulting 1d theory onefinds upon taking the double scaling (classical) limit. The redefinition ˙ f = e ψ utilized byAltland, Bagrets and Kamenev (ABK) [34, 35], is the dimensional reduction of the transitionfrom Gerveu-Neveu variables to B¨acklund variables.reader. The analogous treatment of Gervais and Neveu for N = 1 Liouville theory appearedin [50, 51, 52] and we heavily use their results. N = 1 super-Liouville theory is defined by the Hamiltonian density H = 1 πb (cid:18) π φ φ σ e φ − φ σσ − iψ ψ e φ + i ψ ψ σ − ψ ψ σ ) (cid:19) , (2.29)for a scalar φ and two Majorana-Weyl fermions ψ and ψ . The auxiliary field F hasbeen eliminated by its equations of motion. In superspace ( σ, τ, θ , θ ), the general classicalsuper-Liouville solution for the superfield Φ( σ, τ, θ , θ ) is written as e Φ = ( D α )( D β ) A − B − αβ , (2.30)in terms of superholomorphic bosonic functions A ( x + , θ ), B ( x − , θ ), and their fermionicpartners α ( x + , θ ), β ( x − , θ ), with D i = ∂ θ i + θ i ∂ σ the superderivative.As before, this can be generalized to an off-shell field redefinition in the phase space pathintegral: ( φ, π φ , ψ , ψ ) → ( A ( σ, τ, θ ) , α ( σ, τ, θ ) , B ( σ, τ, θ ) , β ( σ, τ, θ ) , (2.31)utilizing the off-shell generalization of (2.30) and the conjugate momentum as the definitionof the non-canonical field redefinition (see [50] for details). These fields are not completelyindependent, but satisfy D A = αD α, D B = βD β, (2.32)making the transformation a super-reparametrization, and reducing the number of real com-ponents from eight to four, matching the l.h.s. of (2.31). In these variables, super-Liouvilletheory is naturally interpreted as the theory of all super-reparametrizations, generalizing12his statement from previous sections.To rewrite the theory in terms of these variables, consider first the differential equation D i x = V i x, (2.33)for a fermionic function V i ( σ, τ, θ i ). For e.g. i = 1, one checks that this equation is solved for x = ( Dα ) − , A ( Dα ) − , α ( Dα ) − with V i equal to (minus) the super-Schwarzian derivative,and A and α linked by (2.32). Indeed, evaluating the above for e.g. x = ( Dα ) − givesexplicitly V = D (( Dα ) − ) Dα = − D αDα + 2 D αD α ( Dα ) = − Sch( α, A ; σ, θ ) . (2.34)Analogous formulas hold for V in terms of β and B .It was then demonstrated in [50] that the Hamiltonian density can be written as H = c π ( U + U ) , (2.35)where U i is the bosonic ( ∼ θ i ) component of V i : V ( σ, τ, θ ) = Λ ( σ, τ ) + U ( σ, τ ) θ , V ( σ, τ, θ ) = Λ ( σ, τ ) + U ( σ, τ ) θ . (2.36)The bosonic pieces of V i thus become the Hamiltonian density in real space (after integratingover θ ). The fermionic parts (the Λ’s) in (2.36) are interpreted as the supercharge densities.ZZ-brane boundary conditions at σ = 0 , π require that Φ → ∞ at those locations, whichmeans by (2.30), next to the bosonic conditions on A and B , that α = ± β | σ =0 ,π . This againallows us to recombine A and B into a single reparametrization F , and α and β into η , thesuperpartner of F . For the latter, one needs to choose (cid:103) N S (opposite) boundary conditionson the branes such that α = β on one end and α = − β on the other. This leads to anantiperiodic fermionic field η on the doubled circle, which indeed corresponds to a thermalsystem. It is possible to choose other fermionic boundary conditions at the ZZ-branes, butthis only leads to the N = 0 Schwarzian as discussed in [27].Super-Liouville vertex operators e α Φ become bilocal super-Schwarzian operators of the form(2.30), given by arbitrary super-reparametrizations of the classical Liouville solution. Here we analyze some aspects of the classical dynamics of 2d Liouville and 3d AdS gravitywith the dimensional reduction to the 1d Schwarzian and 2d Jackiw-Teitelboim gravity in13ind. The larger goal is to demonstrate the structural links between 2d Liouville theory,3d gravity, the Schwarzian theory, and JT gravity. The next section generalizes this furtherto other theories.
In [24], we analyzed the Schwarzian theory at the classical level in 2d Jackiw-Teitelboim(JT) gravity by allowing energy injections from the boundary. We demonstrated there thatthe matter energy determines a preferred coordinate frame close to the boundary. Here weshow how that analysis directly generalizes to the higher dimensional Liouville theory. Forthis purpose, the Gervais-Neveu variables ( A , B ) are most useful.Liouville theory at large c is expected to describe the universal gravitational features ofholographic CFTs, and it is this regime we discuss here. As in (2.4), the Liouville exponentialis related to the ( A , B ) fields as e φ ∼ A σ B σ ( A − B ) . (3.1)On-shell, A and B are holomorphic resp. antiholomorphic functions and the Liouville metric ds = e φ dx + dx − is transformed from the Poincar´e patch into an arbitrary frame. The lightcone stress tensor components are given by equation (2.7): T ++ ( σ, τ ) = − c π { A ( σ, τ ) , σ } , T −− ( σ, τ ) = − c π { B ( σ, τ ) , σ } , T + − = 0 , (3.2)leading to T ( σ, τ ) = T σσ ( σ, τ ) = T ++ + T −− . (3.3)Energy conservation would ordinarily result in holomorphicity for T ++ and T −− . Howeverthis is violated if the system is not closed, as happens when one would inject additionalenergy into the system. We allow for this possibility here. The Schwarzian theory hasits time coordinate identified with the Liouville spatial coordinate σ , thus we relabel theLiouville coordinates to reflect this: we set τ → x and σ → t . This corresponds to swappingthe roles of time and space in Liouville theory. The total energy on a constant- t slice equals E ( t ) = (cid:90) dxT σσ ( t, x ) = (cid:90) dx ( T ++ ( t, x ) + T −− ( t, x )) . (3.4)Within a holographic theory with bulk coordinates ( t, r, x ), the total change in boundaryenergy equals the net bulk inwards flux from the boundary: dE ( t ) dt = − c π ddt (cid:90) dx ( { A ( t, x ) , t } + { B ( t, x ) , t } ) = − (cid:90) dx T r ( t, x, r → + ∞ ) . (3.5) We take here a more general situation than in the previous section 2 as we do not include ZZ-branesbut consider instead an infinite plane. x ) zero-mode, it becomes the classical Schwarzian equation of motion [22, 23, 24]; theSchwarzian equation is just energy conservation.When evaluating (3.2) on a region where energy is conserved, all functions become holo-morphic and this just reduces to the uniformizing coordinate identification: T ++ ( x + ) = − c π (cid:8) A ( x + ) , x + (cid:9) , T −− ( x − ) = − c π (cid:8) B ( x − ) , x − (cid:9) , (3.6)where x ± = τ ± σ . The above can be interpreted as a diffeomorphism from vacuum Poincar´e AdS ( A, B ) intoa new preferred frame ( x + , x − ). It is clearest to demonstrate this in a region where noadditional matter falls in (or is extracted) (Figure 6 left). It has been shown in [53] that T ++ T-- t x + x - xu t x + x - xu t=0E=0E>0t=t t=t Figure 6: Left: classical injection of bulk energy between t < t < t . We consider theregion after the injection takes place t > t where a non-zero boundary T ±± was generated.Right: classical injection of a translationally symmetric pulse into the bulk.the general bulk diffeomorphism that brings the Poincar´e AdS solution ( X + , X − , u ) ds = − dX + dX − + du u , (3.7)to the Banados metric ( x + , x − , z ) ds = L + ( x + ) dx + L − ( x − ) dx − + (cid:18) − z + z L + ( x + ) L − ( x − ) (cid:19) dx + dx − + dz z , (3.8)15s found by extending the transformation X ± = X ± ( x ± ) + O ( z ) , u = z (cid:112) ∂X + ( x + ) ∂X − ( x − ) + O ( z ) (3.9)into the bulk, with the chiral functions X ± ( x ± ) and L ± ( x ± ) determined by solving − (cid:8) X ± , x ± (cid:9) ≡ L ± ( x ± ) = 12 πc T ±± ( x ± ) , (3.11)given T ±± . Then X + ( x + ) = A ( x + ) and X − ( x − ) = B ( x − ). Hence the functions A and B indeed correspond to the boundary reparametrization that, upon extending into the bulk us-ing (3.10), is precisely the required frame. Setting z = (cid:15) in (3.9) leads to a radial trajectory u ( X + , X − ) representing a fluctuating holographic boundary caused by matter injections.Note that solving (3.11) directly leaves a SL (2 , R ) × SL (2 , R ) ambiguity, which is fixed byboundary (gluing) conditions, just as in the 2d case [24].As an explicit example, consider a translationally invariant injection of matter througha pulse (Figure 6 right). This requires T ++ = T −− to set T tx = 0 for t >
0, equal to (half)the energy injected. One can then immediately solve (3.11) for A and B after the pulse: A ( x + ) = (cid:40) x + , t < (cid:112) c πE tanh (cid:16)(cid:113) πEc x + (cid:17) , t > , (3.12) B ( x − ) = (cid:40) x − , t < (cid:112) c πE tanh (cid:16)(cid:113) πEc x − (cid:17) , t > . (3.13)The resulting Banados metric at t > It has been known for a long time that a spherical dimensional reduction of 3d gravity yields2d Jackiw-Teitelboim gravity [54]. This is done by considering the 3d ansatz ds = g (2) µν dx µ dx ν + λ − Φ dϕ , (3.14) The full bulk diffeomorphism is given by X ± = X ± ( x ± ) + 2 z ∂ ± X ± ∂ ∓ X ∓ ∂ + X + ∂ − X − − z ∂ X + ∂ − X − , u = z (4 ∂ + X + ∂ − X − ) / ∂ + X + ∂ − X − − z ∂ X + ∂ − X − . (3.10) Note that these functions are not strictly holomorphic, due to the jump at t = 0. This was indeedallowed in regions where energy is not conserved. λ a mass scale. This yields directly πG (cid:90) d x √− G (cid:0) R (3) − Λ (cid:1) = 2 π πλG (cid:90) d x √− g Φ (cid:0) R (2) − Λ (cid:1) , (3.15)which is indeed JT gravity.The Schwarzian coupling constant C ∼ /G , but G L → c = L G . Sowe choose λL → + ∞ to obtain a finite limit with G ∼ λG . This is the Schwarzian doublescaling limit from the bulk perspective.This 3d perspective on the bulk is very useful, and we here mention some aspects thatbecome easier to understand when embedding the theory in 3d. At the level of classical solutions, the general vacuum solution of 3d Λ < ds = L + dx + L − dx − + (cid:18) − z + z L + L − (cid:19) dx + dx − + dz z , (3.16)for arbitrary chiral functions L ± ( x ± ).Performing a spherical dimension reduction requires L + = L − = L , a constant, as it shouldbe independent of ϕ . The resulting 3d space is a non-rotating BTZ black hole, dimensionallyreducing to a 2d JT black hole.By (3.11), only constant Schwarzian solutions survive the reduction, as this is the generic3d metric outside matter. And any 2d vacuum metric in JT theory is a black hole of a givenmass. Indeed, directly solving the vacuum JT equations (as in [21, 22, 23, 24, 25]) leadsto black hole spacetimes as the only solutions, perfectly analogous to the 2d CGHS models[55]. In [21, 22, 23, 24], JT gravity is defined by enforcing an asymptotic value Φ ∼ a/(cid:15) of thedilaton Φ at z = (cid:15) , combined with an asymptotically Poincar´e metric. Here we demonstratethat, upon embedding in 3d, both of these conditions follow from just imposing asymptot-ically Poincar´e boundary conditions directly in 3d. The 3d BTZ metric can be writtenas ds = − ρ − µa ) dt a + dρ ρ − µa + ρ dϕ . (3.17) Λ = − L . ρ = √ µa coth (cid:0)(cid:112) µa ( x + − x − ) (cid:1) [21, 24], andsetting t = x + + x − , the metric becomes ds = − µdx + dx − a sinh (cid:0)(cid:112) µa ( x + − x − ) (cid:1) + µa coth (cid:18)(cid:114) µa ( x + − x − ) (cid:19) dϕ , (3.18)which is of the form of a spherical dimensional reduction: ds = g µν dx µ dx ν = h ij dx i dx j + Φ ( x ) dϕ , (3.19)giving the 2d JT black hole metric h ij and associated dilaton field Φ . Asymptotically, theabove 3d metric behaves as ds ≈ − dx + dx − z + a dϕ z , (3.20)which, upon absorbing a in ϕ , is just the standard Fefferman-Graham asymptotic expansion.Hence imposing Fefferman-Graham gauge in 2d and Φ ∼ a/(cid:15) is equivalent to imposingFefferman-Graham gauge in 3d. Armed with the above embedding of the Schwarzian theory within Liouville and JT gravitywithin 3d gravity, we can now relate four different theories through dimensional reductionand the Schwarzian limit.One starts with 3d gravity in the bulk, with periodically identified Euclidean time τ . Itsboundary contains 2d Liouville theory. Instead reducing to the angular ϕ -zero-mode, oneobtains 2d JT gravity in the bulk. These two 2d theories are living in distinct regionsand are only linked through this higher-dimensional story. Finally dimensionally reducingLiouville theory leads to the Schwarzian theory as the angular zero-mode of the boundarytheory (Figure 7).We can omit the ZZ-branes if we realize that their entire goal in life is to combine left-and right moving sectors into one periodic field, thereby transforming the cylindrical surfaceinto a (chiral) torus. This equivalence is also demonstrated in Figure 8. The propagationof just the identity module along the smaller circle is a consequence of taking the large c limit.As we will demonstrate starting from the next section, an analogous story holds forgroup theory: Chern-Simons (CS) in 3d reduces to 2d WZW on the boundary. Insteadrestricting to the angular zero-mode leads to 2d BF theory in a different region. Furtherdimensionally reducing the boundary theory leads to the 1d particle on a group manifold.The resulting scheme of models was already shown in Figure 1 and is repeated in Figure 9for convenience. 18 iouville / WZW3d Gravity / CS Schwarzian / particle on groupJT gravity / BF theory tf r Figure 7: Link between four theories through dimensional reduction, both for the gravitysector, as for the group theory sector. The interior of the torus is the 3d bulk. The torusitself is the holographic boundary. Reducing to the angular zero-mode gives a 2d bulk anda 1d boundary line. ZZ ZZ ZZ ZZ = =
Figure 8: Left: cylindrical surface bounded by ZZ-branes. Middle: The exponential maptransforms this into an annular region in the upper half plane. The ZZ-branes are on thereal axis and the semicircles are identified as shown. Right: Performing the doubling trick(method of images) leads to a torus with only one chirality.
2d Liouville1d Schwarzian3d Gravity2d JT Gravity
HolographyDim. Red.
2d WZW1d particle on group3d Chern-Simons2d BF Theory
Holography
Figure 9: Scheme of theories and their interrelation.
It was suggested in [22, 23, 24] that the Schwarzian theory is holographically dual to Jackiw-Teitelboim gravity. Within JT gravity, the Schwarzian appears as follows. The dilaton field19lows up near the AdS boundary, with a coefficient depending on the matter sector. Keep-ing fixed its asymptotics, requires performing a coordinate transformation at each instant,depending on the injected / extracted energy from the system. This results in a fluctuatingboundary curve (Figure 10 left). One can directly deduce the Schwarzian action from the z=0 z= e T ++ T -- z=0 J - J + z= e A d s t t Figure 10: Left: injecting energy in JT gravity leads to a preferred coordinate frame at eachtime, resulting in a fluctuating boundary line. Right: injecting charge leads to a preferredgauge transformation at each time.bulk 2d JT dilaton gravity theory from the Gibbons-Hawking boundary term [23]. Thisargument has been generalized to N = 1 and N = 2 JT supergravity in [56] and [57]respectively. In appendix B we extend the argument (in the bosonic case) to include anarbitrary matter sector.The gauge theory variant of this story is readily formulated: we need a preferred gaugetransformation on the boundary curve at each instant, determined by the injected chargeinto the system (see Figure 10 right). The correct bulk theory that describes this situationis 2d BF theory.The argument we present is a dimensional reduction of the 3d Chern-Simons story and thedirect analog of the Schwarzian argument of [23]. Consider the 2d BF theory obtained as adimensional reduction from 3d CS theory: S CS ∼ (cid:90) M d x(cid:15) ijk A i ∂ j A k , (4.1)with A φ ∼ χ and ∂ φ = 0. One obtains: S = (cid:90) M d xχF + 12 (cid:73) ∂M dtχA . (4.2) Reintroducing the correct prefactor k π in the Chern-Simons action, by analogy with section 3.2, oneneeds to set A φ ∼ χk to find a finite limit. The resulting 2d action is proportional to some C again, whichis not quantized even though the original k is. δ g S = 12 (cid:73) ∂M dtχδ g A , (4.3)just like 3d CS theory. Restricting the gauge transformations to satisfy δ g A = 0 | ∂M , solvesthis problem, but creates dynamical degrees of freedom at the boundary.Sending in charge through a matter field requires the additional term S matter = (cid:90) M d xA µ J µ , (4.4)which is the charge analogue of the energy-momentum matter source for the gravitationalfield given in appendix B. Varying w.r.t. A µ and χ gives the equations of motion: F = 0 , ∇ µ χ = (cid:15) µν J ν , (4.5)and the boundary terms at r = + ∞ :12 (cid:73) ∂M ( A δχ − χδA ) . (4.6)These can be cancelled by constraining: v χ = A | ∂M , (4.7)for a parameter v that defines the specific theory. We choose v = 1.Path integrating (4.2) over χ sets F = 0 in the bulk. So we parametrize the solution as A µ = ∂ µ σ. (4.8)Using the boundary condition (4.7), the full action (4.2) now becomes: S = 12 (cid:73) ∂M dt ˙ σ . (4.9)The total boundary charge is defined as Q = δS on-shell δA = ˙ σ, (4.10)and the total boundary energy is T tt = ˙ σ . (4.11)21or the matter action S matter , after integrating by parts, one finds the boundary term: S matter = − (cid:73) ∂M dt σ J r = − (cid:73) ∂M dt σ ( J + − J − ) , (4.12)representing the net inward flux of charge.As charge is sent in, one requires A to change as well asymptotically to keep fixed theboundary condition (4.7). Either by using χ = ˙ σ and (4.5), or by directly varying theboundary action in terms of σ , one obtains¨ σ = J r , (4.13)which determines how the gauge transformation σ evolves due to matter charge; σ was puregauge in the bulk but becomes physical on the boundary.Some Comments: • This procedure is independent of the gravity (Schwarzian) part. N = 2 JT supergrav-ity would fix the relative coefficient (see section 4.2 below). • Non-abelian generalization is straightforward. The non-abelian BF theory is S = (cid:90) M d x Tr χF + 12 (cid:73) ∂M dt Tr χA , (4.14)which is gauge-invariant ( χ transforms in the adjoint representation), up to the bound-ary term again. The equations of motion require A µ = g − ∂ µ g , with F = 0. Theboundary condition is again chosen as χ = A | ∂M . So the full theory reduces to theboundary action: S = 12 (cid:73) ∂M dt Tr( g − ∂ t g ) , (4.15)which is the action of a particle on a group manifold, to be studied more extensivelyin section 5 below. • One can write Jackiw-Teitelboim itself as an SL (2 , R ) BF theory [20], see also [58] forrecent developments. In fact, dimensionally reducing SL (2 , R ) CS theory just gives usthe SL (2 , R ) BF theory, which is the first-order formalism equivalent of dimensionallyreducing the Ricci scalar directly. And indeed, the SL (2 , R ) particle-on-a-group actionis equivalent to the Schwarzian action [27]. Operator insertions on the other hand arenot so simple. •
3d bulk gravity coupled to 3d CS theory leads to decoupled equations of motionbecause T CSµν ≡
0. The only influence of the CS theory on the gravity part is in thedefinition of the total Hamiltonian: H = H grav + H CS with contribution (4.11), whichprovides just a shift in the energy. This will indeed be observed below in section 5.2.22 .2 Supersymmetric JT gravity theories The identification of the non-interacting gauge sector as a 2d BF theory can also be under-stood from supersymmetry as will be illustrated here. Pure 3d gravity can be written asa sl (2) ⊕ sl (2) Chern-Simons theory. Similarly, Achucarro, Townsend and Witten demon-strated a long time ago that ( p , q ) 3d supergravity can be written as a osp ( p | ⊕ osp ( q | p = q leads to a osp ( p |
2) 2d BF theory.And indeed, as known since a long time [20], JT gravity itself can be written as an sl (2) BFtheory: S JT = (cid:90) Tr( ηF ) , (4.16)with A = e a P a + ωJ , field strength F = dA + A ∧ A and η = η a P a + η J in terms of zweibein e a ( a = 1 ,
2) and spin connection ω .Supersymmetric generalization is now straightforward, as one just generalizes the gaugegroup from sl (2) to either osp (1 |
2) ( N = 1) or osp (2 |
2) ( N = 2). In particular the N = 2JT supergravity action may be written as [61][62]: S N =2 JT = (cid:90) STr( E F ) , (4.17)in terms of the field strength F = d A + A ∧ A , with the dilaton superfield E and supercon-nection A , expanded into the osp (2 |
2) generators as: E = η a P a + η J + φ α Q α + ˜ φ α ˜ Q α + χB, a = 1 , , α = 1 , , (4.18) A = e a P a + ωJ + ψ α Q α + ˜ ψ α ˜ Q α + ξB, (4.19)for three sl (2) generators P a , J , four fermionic generators Q ± , ˜ Q ± and one additional u (1)generator B . These eight generators satisfy an osp (2 |
2) algebra whose explicit form can befound in the literature. For simplicity, we set the cosmological constant zero here, as this does not influence thestructure of the theory. In components, the action is S N =2 JT = (cid:90) (cid:104) η a De a + η R + ηψ ∧ ˜ ψ + χ ( F + ψ ∧ ˜ ψ ) + ˜ φDψ + φD ˜ ψ (cid:105) . (4.20)The piece coming from just the bosons is then S N =2 JT (cid:51) (cid:90) η a De a + η R + χF, (4.21) The osp (1 |
2) BF theory would have just the 3 sl (2) bosonic generators and 2 fermionic generators.Generally, the osp ( p |
2) 2d BF theory has the bosonic algebra so ( p ) ⊕ sl (2). u (1) BF theory (cid:82) χF .Studying the N = 2 theory on its own would be interesting as this couples the gravitationaland gauge sectors in the bulk. This is left for future work. We focus now on the boundary theories of the 3d Chern-Simons and 2d BF models. Wewill provide a prescription for computing correlation functions of the 1d particle-on-a-grouptheory, following the logic used in the Schwarzian theory in [27] and in section 2. We start byproviding a general formalism starting from 2d Wess-Zumino-Witten (WZW) rational CFTand performing a double-scaling limit. Our main interest is again in computing the cylinderamplitude between vacuum branes. After that, we consider U (1) and SU (2) as two examplesthat will allow us to write down the generic correlation function using diagrammatic rules. Consider the 2d WZW system with path integral (cid:104) F ( g ( z, ¯ z )) (cid:105) = 1 Z (cid:90) [ D g ] F ( g ( z, ¯ z )) e − k π (cid:82) d z Tr( g − ∂gg − ¯ ∂g )+ k Γ , (5.1)for g ∈ G , integer level k , and with Γ the Wess-Zumino term which will not be needed. Anoperator F ( g ) is inserted, with F a scalar-valued function on the group. As well-known,this theory enjoys invariance under a local group transformation g → g ( z ) gg (¯ z ).Just as in Liouville theory, we focus on the moduli space of classical solutions of thistheory to deduce the link between the 2d and 1d operators. This system has the classicalsolution g ( z, ¯ z ) = f ( z ) ¯ f (¯ z ), with f and ¯ f local group elements as well.Inserting a brane at z = ¯ z (or u = v in Lorentzian signature) imposes reflecting boundaryconditions: J ( z ) = ¯ J (¯ z ) ⇒ − ∂gg − = g − ¯ ∂g, (5.2)which, when translated into a condition on f , requires ¯ f = f − . This boundary conditionprojects the symmetry onto its diagonal subgroup; the condition (5.2) is preserved underthe group transformation provided g = g − . In terms of f , the symmetry transformationis now f → g f .At the second boundary brane at σ = π , where u = τ + π, v = τ − π , one has g = f ( τ + π ) f − ( τ − π ) which satisfies the boundary condition if f is 2 π -periodic: f ( x + 2 π ) = f ( x ).24ence, after implementing the boundary conditions, the system is characterized by a single2 π -periodic function f .Just as with the Schwarzian theory, we imagine performing a change of field variables from g to f . The transformation g ( z, ¯ z ) = f ( z ) f − (¯ z ) has, in analogy with (2.4), a redundancyin description: f ∼ f γ for γ ∈ G any global group element. One can then identify a localWZW operator F ( g ( z, ¯ z )) with a bilocal 1d operator as z → t and ¯ z → t .Dimensionally reducing as in the Liouville/Schwarzian case, the WZW action itself imme-diately reduces to the particle-on-a-group action, the Wess-Zumino term Γ vanishes upondimensional reduction.Hence the rational generalization of the Schwarzian story requires us to compute the 1dpath integral over the group:1 Z (cid:90) G local /G global f ( t + 2 π ) = f ( t ) [ D f ] F (cid:0) f ( t ) f − ( t ) (cid:1) e − kT π (cid:82) π − π dt Tr( f − ∂ t f ) . (5.3)The periodicity of 2 π can be changed into β by rescaling the time coordinate as t → πβ t ,which can alternatively be achieved by placing the branes at β/ global group f → f γ , but are notinvariant under local transformations. This immediately generalizes the Schwarzian cosetdiff S /SL (2 , R ) to the generic rational case as the right coset G local /G global . Taking intoaccount the periodicity of f , this integration space is also written as the right coset of theloop group: LG/G , which is known to be a symplectic manifold. The resulting partitionfunction could then be computed using the Duistermaat-Heckman (DH) theorem just as inthe Schwarzian case [63]. Note that the transformation f → g f , g ∈ G , is a symmetry ofthe action: it is the remnant of the WZW symmetry in 1d as remarked above. But it is notnecessarily a symmetry of operator insertions and it isn’t a gauge redundancy.We did not work out the measure [ D f ] explicitly as in section 2, but by general argu-ments this has to be the standard √ G measure of the group metric: ds = G µν dx µ dx ν =Tr [ g − dg ⊗ g − dg ].The double scaling limit we take is T → k → ∞ with the product kT ∼ C held fixedproportional to a coupling constant C . We will be more specific about this below in section5.3. The coupling constant C allows us to explore the semi-classical regime of (5.3) at C → + ∞ .Structurally the particle-on-a-group action is very similar to the Schwarzian action. TheLagrangian L and Hamiltonian H can be written as a particle moving on the group manifold As for the Schwarzian case, reintroducing β makes the constant C have dimensions of length. Thequantization of the level k is immaterial in the double scaling limit. L = C G µν ˙ x µ ˙ x ν , H = 12 C G µν p µ p ν , (5.4)making it clear that this action has H = L . The classical equations of motion are ∂ t ( f − ˙ f ) =0, identifying conserved currents J ( t ) = J a ( t ) τ a = f − ˙ f with Casimir equal to the Hamil-tonian (up to an irrelevant prefactor):Cas ≡ Tr( J ( t ) J ( t )) = J a J b Tr( τ a τ b ) = Tr( f − ∂ t f ) ∼ H = L. (5.5)The quantization of a particle on a group manifold is in principle well-known (see e.g.[64]). Consider for instance the partition function (without operator insertions), and ignorefirst the modding f ∼ f γ we wrote in (5.3). Then this is manifestly the path integralrewriting of the Lorentzian partition function Tr e − βH . As mentioned above, the theory isinvariant under G × G as f ( t ) → g f ( t ) g . Using operator methods, this can be used to provethat each energy-eigenvalue, with irrep label j , has a degeneracy of (dim j) . As an example,the SU (2) group manifold is just the three-sphere S , which has SO (4) (cid:39) SU (2) × SU (2)isometry, meaning an organization of the energy spectrum in (2 j + 1) degenerate states.This can indeed also be seen explicitly for SU (2) in [65], and in the general case in [64, 63],both with operator methods and path integral methods. Thus Z = (cid:88) j (dim j) e − βC j . (5.6)Reintroducing the gauge-invariance f ∼ f γ in (5.3) merely requires gauge fixing the thermalpath integral, which yields an overall factor of the (finite) group volume (vol G ) − , which isincluded in the zero-temperatore entropy S and dismissed. As mentioned above, this doeshowever allow one to prove one-loop exactness of the path integral through the DH formula.The above expression is indeed what we will obtain in section 5.3 below for SU (2), and isreadily generalized beyond that. We provide some more explicit formulas in appendix C. Just as to get to the Schwarzian from Liouville in section 2, we place two vacuum branesand consider the WZW amplitude on a cylinder between these vacuum branes (as earlier inFigure 2): (cid:104) brane | e − ˜ T H cl | brane (cid:105) , (5.7)with ˜ T = 2 π /T , the length of the cylinder in the closed channel when the circumference isfixed to 2 π . As well-understood, a boundary state | a (cid:105) can be expanded into Ishibashi statesas | a (cid:105) = (cid:88) i S ia (cid:112) S i | ˆ i (cid:105)(cid:105) . (5.8)26he sum ranges over all integrable representations of the Kac-Moody algebra ˆ g , which in the k → + ∞ limit becomes just all irreducible representations of the Lie algebra g . In the limitof interest where the length of the cylinder becomes much longer than its circumference, theIshibashi states are themselves dominated by their zero-mode ( n = 0) states | ˆ i (cid:105)(cid:105) = (cid:88) m i ,n | i, m i , n (cid:105) ⊗ | i, m i , n (cid:105) → (cid:88) m i | i, m i , n = 0 (cid:105) ⊗ | i, m i , n = 0 (cid:105) . (5.9)The Kac-Moody algebra reduces to the zero-mode Lie algebra. One can thus write for (5.7): (cid:88) i,j (cid:112) S ∗ i S j (cid:88) m i , m j (cid:104) i, m i | δ ij e − βC j | j, m j (cid:105) , (5.10)in terms of the modular S -matrix and the Casimirs C i of the irreps. Including operatorinsertions in the middle, requires splitting the evolution into separate pieces and insertingcomplete sets of primaries around each such insertion. For instance, the two-point functionof this system can be written as: (cid:88) i,j (cid:112) S ∗ i S j e − C i τ e − C j ( β − τ ) (cid:88) m i , m j (cid:104) i, m i | F ( g ) | j, m j (cid:105) . (5.11)The matrix element can e.g. be computed in configuration space as (cid:104) i, m i | F ( g ) | j, m j (cid:105) = (cid:90) dg (cid:104) i, m i | g (cid:105) F ( g ) (cid:104) g | j, m j (cid:105) , (5.12)which is the method we utilized for the Schwarzian theory in [27].In the next two subsections we will consider the two simplest examples. The gener-alization to arbitrary compact groups will be obvious at the end. We will end up with adiagrammatic decomposition of the general correlator, analogously as in the Schwarzian case[27]. Just as in that case, we remark that the resulting expression is non-perturbative inthe coupling constant C : the diagrams just represent convenient packaging of the buildingblocks of the general expressions. U (1) As a first example, let’s take U (1). We start with a direct evaluation of its correlators follow-ing the preceding discussion. Afterwards we will embed the theory into N = 2 Liouville andfind the same answer. The latter serves as a further consistency check on the Schwarzianlimit from supersymmetric versions of Liouville theory. All states obtained by acting with J a − n on a primary state have non-trivial dependence on τ , and aresubdominant in the T → .2.1 Direct evaluation Consider a free boson field φ in 2d with action S = (cid:82) dudv∂ u φ∂ v φ . The classical solution isgiven by φ ( u, v ) = σ ( u ) + ¯ σ ( v ) . (5.13)Perfect reflection at u = v and u − v = 2 π requires σ = − ¯ σ and σ ( u + 2 π ) = σ ( u ).Natural vertex operators are the exponentials: V Q = e iQφ ( u,v ) = e iQσ ( u ) e − iQσ ( v ) . (5.14)The classical moduli space is parametrized by a real periodic function σ , so the Schwarzian1d limit entails: (cid:90) [ D φ ] V Q . . . e − S → (cid:90) [ D σ ] e iQσ ( t ) e − iQσ ( t ) . . . e − (cid:82) dt ( ∂ t σ ) . (5.15)In this particular case, the bilocal operator is just a product of two local operators.Of course the resulting theory is free and immediately solvable. Consider e.g. a two-pointcorrelator: (cid:10) e iQσ ( t ) e − iQσ ( t ) (cid:11) . (5.16)The classical equation of motion for σ , including the operator insertions, is solved analo-gously as in the semi-classical regime of Liouville theory (and written here in Lorentziansignature): ¨ σ = Qδ ( t − t ) − Qδ ( t − t ) , (5.17)hence ˙ σ increases by Q at t and decreases again to its original value at t . Thus theoperators inject and extract charge, and ˙ σ represents the total charge in the system, asfound earlier from the bulk perspective in section 4. The Gaussian path integral is readilycomputed as: 1 Z (cid:90) [ D σ ] e iQ ( σ − σ ) e − (cid:82) dt ˙ σ = (cid:114) β π (cid:90) dqe − q τ e − ( q − Q )24 ( β − τ ) . (5.18)If the integral on the r.h.s. is truly an integral ranging from −∞ to + ∞ , one obtains: e − Q τ ( β − τ )4 β , (5.19)which at β → + ∞ asymptotes to → e − Q τ . This, as we show below in (5.66), is the generalresult for any non-abelian group as well, with Casimir Q /
4. This two-point function is ofthe shape as in Figure 11. 28 .0 0.2 0.4 0.6 0.8 1.0 t Figure 11: Two-point function of U (1) theory in units where β = 1. N = 2 super-Schwarzian The U (1)-sector is relevant for e.g. the N = 2 super-Schwarzian. This is because it con-tains, in addition to the fermionic superpartners, also an additional bosonic field σ thatis identified with the above U (1)-sector. Here we demonstrate this directly. In the nextparagraphs we will identify it from its N = 2 Liouville ancestor.The bosonic piece of the super-Schwarzian action is the Schwarzian plus a free bosonfield σ [33]: S = C (cid:90) dt (cid:0) − { f, t } + 2 ˙ σ (cid:1) . (5.20)The relative coefficient was fixed by N = 2 supersymmetry. An N = 2 super-reparametrizationof the invariant super-distance is given by the following expression:1 τ − τ − θ ¯ θ − θ ¯ θ → D ¯ θ ¯ θ (cid:48) D θ θ (cid:48) τ (cid:48) − τ (cid:48) − θ (cid:48) ¯ θ (cid:48) − θ (cid:48) ¯ θ (cid:48) . (5.21)For a purely bosonic reparametrization, τ (cid:48) = f ( τ ) , θ (cid:48) = ρ ( τ ) θ, ¯ θ (cid:48) = ¯ ρ ( τ )¯ θ, with ρ ¯ ρ = ˙ f , ρ/ ¯ ρ = e iσ , (5.22)the bosonic piece of (5.21) is given by e i ( − σ + σ ) (cid:113) ˙ f ˙ f ( f − f ) . (5.23)This can be viewed as a simultaneous reparametrization f ( τ ) and gauge transformation g ( τ ) ≡ e iσ ( τ ) on the charged 1d operator O → e iσ O , as given in (1.3).29 .2.3 Charged Schwarzian from N = 2 Liouville
It is possible to obtain this theory directly from N = 2 Liouville theory. The N = 2supersymmetric generalization of Liouville theory consists of the Liouville field φ , the su-perpartners ψ ± and ¯ ψ ± and a compact boson Y , forming the full supersymmetric multiplet.The central charge is c = 3 + 3 Q = 3 + 3 /b . Details can be found in the literature, butwill not be needed here. Take this theory on the cylinder bounded by two ZZ-branes and consider imposingantiperiodic boundary conditions in N = 2 Liouville along the small circle (NS-sector)(Figure 12). opposite ZZ ZZ antiperiodic NS opposite ZZ ZZ periodic NS ~ Figure 12: Left: Cylinder with (cid:103)
N S boundary conditions around the small circle, leading tosupersymmetric quantum mechanics in 1d. Right: Cylinder with
N S boundary conditionsaround the small circle, leading to a removal of all fermions upon dimensional reduction.This leads to the removal of all fermionic degrees of freedom in the 1d theory, and retainsonly the Liouville field itself (leading to the Schwarzian) and the compact boson Y (leadingto the U (1) theory). The analysis of section 2 can be repeated when adding the free boson Y . This leads to the additional 1d action in the Schwarzian limit: S = C (cid:90) dt ˙ Y , (5.24)leading to the identification Y = 2 σ to match with the super-Schwarzian field σ in (5.20).The required building blocks of our story are readily available in the literature. N = 2Liouville primary vertex operators in the N S sector are of the form: V (cid:96),Q = e (cid:96)φ e i Q Y , ∆ = (cid:96) − b (cid:0) (cid:96) − Q (cid:1) → (cid:96) , (5.25)whereas Liouville states | P, Q (cid:105) with charge Q and Liouville momentum P have weight:∆ = 18 b + P b Q . (5.26) Two convention schemes exist: we follow that of [66]. To go from the conventions of [67] to those of[66], one needs to set b → b and 2 P → P . N S character for a primary with Liouville momentum P = 2 bk and U (1) charge Q isgiven by: ch NSP,Q ( τ, z ) = q P + b Q y Q θ ( τ, z ) η → e − β ( k + Q / y Q , (5.27)in the large τ -limit. The ZZ-brane wavefunction is determined by the modular S -matrixas: | Ψ ZZ ( P, Q ) | = S P,Q = b π Pb ) sinh(2 πbP ) (cid:12)(cid:12) cosh π ( bP + ib Q ) (cid:12)(cid:12) → b πk sinh 2 πk. (5.28)The total vacuum character then has the small T -behavior: χ ( τ = iT ) → (cid:90) dQ dk k sinh 2 πke − β ( k + Q / y Q = (cid:90) dQ (cid:90) + ∞ Q/ dE sinh 2 π (cid:112) E − Q / e − βE y Q , (5.29)hence the density of states is identified as ρ ( E, Q ) = sinh 2 π (cid:112) E − Q / . (5.30)The lack of a ∼ / √ E divergence as E → (cid:104)O (cid:96),Q ( τ , τ ) (cid:105) = (cid:104) ZZ | e − Hτ e (cid:96)φ e i Q Y e − H ( β − τ ) | ZZ (cid:105) = (cid:90) + ∞ q / dE (cid:90) + ∞ q / dE e − E τ e − E ( β − τ ) × (cid:104) ZZ | k , q (cid:105) (cid:90) dφdY (cid:104) k , q | φ, Y (cid:105) e (cid:96)φ e i Q Y (cid:104) φ, Y | k , q (cid:105) (cid:104) k , q | ZZ (cid:105) . (5.31)Let’s compute this explicitly. The ZZ-brane wavefunction is given by ψ ZZ ( E, q ) = (cid:104)
E, q | ZZ (cid:105) = 2 πib (cid:112) E − q / i (cid:112) E − q / . (5.32)The minisuperspace limit of bulk N = 2 Liouville theory leads to a removal of all fermions,and the result is the Schr¨odinger equation:( − ∂ φ − ∂ Y + e φ ) ψ ( φ, Y ) = Eψ ( φ, Y ) , (5.33)with E the energy, solved by ψ E,q ( φ, Y ) = (cid:104) φ, Y | E, q (cid:105) = 2Γ( − i (cid:112) E − q / K i √ E − q / ( e φ ) e i q Y . (5.34)31he basic integral we need to compute is (cid:90) dY (cid:90) dφe (cid:96)φ e i Q Y ψ ∗ k ,q ( φ, Y ) ψ k ,q ( φ, Y ) . (5.35)The Y -integral just gives δ ( Q − q + q ) and the φ -integral is the same as in bosonic Liouville[27]. So we end up with (cid:90) dq (cid:90) + ∞ q / dE (cid:90) + ∞ ( q − Q ) / dE e − E τ e − E ( β − τ ) sinh(2 π (cid:112) E − q /
4) sinh(2 π (cid:112) E − ( q − Q ) / × Γ( (cid:96)/ ± i (cid:112) E − q / ± i (cid:112) E − ( q − Q ) / (cid:96) ) . (5.36)Shifting the energy variables by the charge, then leads to: (cid:104)O (cid:96),Q ( τ , τ ) (cid:105) = (cid:90) dq (cid:90) + ∞ dE (cid:90) + ∞ dE e − ( E + q / τ e − ( E +( q − Q ) / β − τ ) × sinh(2 π (cid:112) E ) sinh(2 π (cid:112) E ) Γ( (cid:96)/ ± iE ± iE )Γ( (cid:96) ) , (5.37)where now the energy variables E and E are only the energies of the Schwarzian subsystem,not the total energy. Factorization is now manifest, and the q -integral agrees indeed with(5.18). One can write a diagrammatic decomposition of a general correlator, as done in [27]. Thetwo-point correlator for instance is given diagrammatically as: A ( k i , q, (cid:96), Q, τ i ) = k , qτ τ k , q − Q(cid:96), Q , (5.38)where each line contains also a conserved charge, next to the Schwarzian SL (2 , R )-labels. The computation in this section is done for C = 1 / .3 Example: SU (2) The vacuum character for SU (2) k on a cylinder of circumference T and length π , transformsunder an S -transformation as: χ (cid:18) iT π (cid:19) = k/ (cid:88) j =0 S j χ j (cid:18) i πT (cid:19) , S j = (cid:114) k + 2 sin (cid:18) π (2 j + 1) k + 2 (cid:19) , j = 0 , , . . . k , (5.39)which can be evaluated in the T → χ j (cid:18) i πT (cid:19) → (2 j + 1) e − π T h j = (2 j + 1) e − ˜ T (2 h j ) = (2 j + 1) e − π T ( k +2) j ( j +1) , (5.40)where h j = j ( j +1) k +2 . The second equality expresses the character in terms of the closed channelwith length ˜ T = 2 π /T . Keeping fixed T ( k + 2) = 4 π /β , this becomes (2 j + 1) e − βC j withthe Casimir C j = j ( j + 1). The analogue of the Schwarzian double scaling limit is here thatthe level k → + ∞ as T →
0. The vacuum character (5.39) finally becomes: Z ( β ) = lim T → χ (cid:18) iT π (cid:19) = (cid:88) j √ π ( k + 2) / (2 j + 1) e − βC j = (cid:88) j S (2 j + 1) e − βC j , (5.41)which, up to normalization constants, is a discrete quantum system with Hamiltonian =Casimir, and with the dimension of the irreps as density of states: ρ ( j, m ) = dim j = 2 j + 1.Note that the sum ranges over both integers and half-integers.As in the Schwarzian theory, the prefactor can be written in terms of a ground state entropyas e S , and requires regularization by taking finite k . In this case, the prefactor is just S which goes to zero as k → ∞ . This prefactor will cancel in correlation functions and ishence irrelevant for our computations; we drop it from here on.At low temperatures, only the vacuum contributes and Z →
1. At high temperatures, thesum can be replaced by an integral and Z → √ πβ / . Alternatively, the expression (5.41) isreadily Poisson-resummed.For a general Kac-Moody algebra ˆ g , it is well-known that the S j elements in the modular S -matrix carry information about the quantum dimension d j of the integrable representationˆ j , and this reduces to the dimension in the classical ( k → ∞ ) limit: d j = S j S → dim j. (5.42)33t is instructive to recompute Z ( β ) from the closed channel: (cid:104) brane | e − ˜ T H | brane (cid:105) = (cid:88) i,j (cid:112) S ∗ i S j (cid:104)(cid:104) ˆ i | e − ˜ T H | ˆ j (cid:105)(cid:105) → S (cid:88) i,j dim i dim j δ ij e − βC j , (5.43)using the Ishibashi states in the k → ∞ limit (5.9): | ˆ j (cid:105)(cid:105) → j (cid:88) m = − j | j, m (cid:105) . (5.44) Next we proceed by computing correlators of the SU (2) theory. Instead of evaluatingconfiguration space integrals, we will compute the matrix element (5.12) directly using grouptheory as follows. General operator insertions F ( g ) are all built from the field g ( z ), so we canorganize them into tensor operators O J,M ¯ M transforming in an irreducible representation of G , essentially by using the Peter-Weyl theorem. In the double scaling limit, one finds thebi-local operators: F ( g ) → F (cid:0) f ( t ) f − ( t ) (cid:1) = (cid:88) J,M, ¯ M c J,M, ¯ M O J,M ¯ M . (5.45)The resulting elementary bilocal operator O J,M ¯ M will turn out to be identifiable with thematrix element: O J,M ¯ M ≡ (cid:2) f ( t ) f − ( t ) (cid:3) M ¯ M = [ R J ( f ( t ))] Mα (cid:2) R J ( f − ( t )) (cid:3) α ¯ M , (5.46)for the group element f in the spin- J representation R J . For a general operator O J,M ¯ M transforming both in the holomorphic and antiholomorphic sector as a tensor operator, adoubled Wigner-Eckart theorem holds: (cid:104) j m ¯ m | O J,M ¯ M | j m ¯ m (cid:105) = C j m ,j m ; J − M C j ¯ m ,j ¯ m ; J − ¯ M A j j J , (5.47)in terms of two Clebsch-Gordan (CG) coefficients and a reduced matrix element A j j J . Notethat a reordering of the arguments of the CG coefficients has been performed, resulting insome j -dependent factors that are absorbed into the reduced matrix element, see appendixD for details. The appearance of two Clebsch-Gordan coefficients will be crucial in whatfollows.To determine the reduced matrix element A j j J , one can evaluate this expression for anychoice of the m ’s. 34e will determine it below for SU (2), and conjecture that for a general group G forirreducible representations λ , λ and λ , it equals A λ λ λ = (cid:114) dim λ dim λ dim λ . (5.48)The SU (2) k OPE coefficient was written down in [69], and is for the case m = ¯ m = j and m = ¯ m = J (cid:104) j m ¯ m | O J,M ¯ M | j m ¯ m (cid:105) = (cid:10) Φ j m ¯ m Φ J,M ¯ M Φ j m ¯ m (cid:11) = D j j J , (5.49)for fusing j and j into J . In the large k limit, this is given explicitly as D j j J → (cid:112) (2 j + 1)(2 j + 1)(2 J + 1) Γ(2 j + 1)Γ(2 J + 1)Γ( j + j + J + 2)Γ( j + J − j + 1) . (5.50)On the other hand, the SU (2) Clebsch-Gordan coefficient for combining j and j into J equals C j m ; j j ; J − J = (cid:115) (2 J + 1)Γ(2 J + 1)Γ(2 j + 1)Γ( j + j + J + 2)Γ( J + j − j + 1) ( − ) J − j − j δ (cid:80) i m i . (5.51)Some details of these computations are given in appendix D. We obtain the ratio D Jj j C j m ; j j ; J − J = (cid:114) (2 j + 1)(2 j + 1)2 J + 1 , (5.52)identifying the reduced matrix element in (5.47) as A j j J = (cid:114) (2 j + 1)(2 j + 1)2 J + 1 , (5.53)which indeed suggests the general form (5.48).The matrix element in the double scaling limit (and with the normalization (5.44)) isthen written by the Wigner-Eckart theorem as (cid:104)(cid:104) j | O J,M ¯ M | j (cid:105)(cid:105) → (cid:88) m ,m C j m ,j m ; J − M A j j J δ M ¯ M , (5.54)35n terms of the Clebsch-Gordon coefficients and the reduced matrix element. Only operatorsthat are left-right symmetric can connect the two Ishibashi states, yielding the Kronecker-delta. The sum over CG coefficients squared is just the fusion coefficient: (cid:88) m ,m C j m ,j m ; J − M = N Jj j . (5.55)It equals 1 by unitarity of the CG-matrix, and can connect only states satisfying the triangleinequality. The formula (5.54) is a classical limit of a formula recently derived by Cardyin [70] (derived there for diagonal minimal models) where the Ishibashi matrix element iswritten as (cid:104)(cid:104) j | e − τH O J,MM e − τH | j (cid:105)(cid:105) = (cid:16) π τ (cid:17) ∆ J ( S ) / (cid:115) S j S j S J N Jj j , (5.56)for a (diagonal) primary operator O J,MM . The Euclidean propagators e − τH and the firstfactors on the r.h.s. can be viewed as regularization artifacts of the Ishibashi states torender them normalizable. We conjecture this formula and its classical limit hold for anyrational CFT. In any case, we have illustrated it explicitly for SU (2) k which is the relevantsymmetry group for e.g. N = 4 super-Schwarzian systems (see e.g. [71]).The normalization of the intermediate operator O J,M ¯ M has been fixed above by the 2dCFT state-operator correspondence in (5.49). There is however a more convenient normal-ization for the 1d theory, by taking the operator and the SL (2)-field Φ J,M ¯ M to be insteadrelated as O J,M ¯ M = 1 √ dim J Φ J,M ¯ M , (5.57)which we now adopt.Higher-point functions can now be deduced analogously, and we arrive at a diagram-matic decomposition of a general correlation function, where one sums over all intermediaterepresentation labels using (cid:88) j i ,m i dim j i A ( j i , m i ; τ i ) , (5.58)and where the momentum amplitude A ( j i , m i ; τ i ) is computed using the Feynman rules: τ τ jm = e − C j ( τ − τ ) , j m j m JM = γ j m ,j m ,JM . (5.59)36he vertex is essentially the Clebsch-Gordan coefficient, but it can be written more sym-metrically in terms of the 3 j -symbol: (cid:18) j j j m m m (cid:19) = ( − ) j − j − m √ j + 1 C j m ,j m ; j − m , (5.60)as γ j m ,j m ,JM = (cid:18) j j Jm m M (cid:19) . (5.61)The CG coefficients determine the fusion of the representations at each double vertex.Hence, we obtain finally for the 1d two-point function ( O J,M := O J,MM ): (cid:104)O J,M (cid:105) = (cid:88) j ,j ,m ,m dim j dim j A ( j i , m i , J, M, τ i ) , (5.62)with the amplitude A diagrammatically: A ( j i , m i , J, M, τ i ) = j m τ τ j m J M (5.63)Combining everything we arrive at: (cid:104)O
J,M (cid:105) = 1 Z ( β ) (cid:88) j ,j ,m ,m dim j dim j e − C j τ e − C j ( β − τ ) (cid:18) j j Jm m M (cid:19) , (5.64)which, for the particular case of the two-point function, can be written fully in terms of theinteger fusion coefficients N Jj j : (cid:104)O J,M (cid:105) → Z ( β ) (cid:88) j ,j dim j dim j e − C j τ e − C j ( β − τ ) N Jj j dim J . (5.65)This immediate simplification only occurs for the two-point function.Just as for U (1), this correlator is finite as τ →
0. The qualitative shape of the correlator issimilar to the U (1)-case. Some examples are drawn in Figure 13. Our choice of normalization(5.57) ensures that (cid:104)O J,M ( τ = 0) (cid:105) = 1. As a check, some simplifying limits can be taken.At zero temperature, C j = 0, so j = 0, and J = j . So (cid:104)O J, M (cid:105) β →∞ → e − C J τ . (5.66)37 .2 0.4 0.6 0.8 1.0 t Figure 13: Two-point function of SU (2) theory (5.65) in units where β = 1 for several valuesof J : J = 0 (red), J = 1 (orange), J = 3 / J = 2 (green), J = 5 (blue), J = 7(black).When J = 0 (insertion of the identity operator), j = j and one finds (cid:104)O , (cid:105) = 1, confirmingthe overall normalization of (5.65).The partition function Z ( β ) itself (5.41) is also directly computed using the Feynmandiagram decomposition: Z ( β ) = (cid:88) j,m dim j e − C j β = (cid:88) j (dim j) e − C j β = (5.67)The time-ordered four-point function is drawn as A (cid:0) j i , m i , J i , M i , τ i (cid:1) = j m j m J M J M j m j m τ τ τ τ (5.68)and is given by the expression: (cid:104)O J ,M O J ,M (cid:105) = (cid:88) j i ,m i dim j dim j dim j A ( j i , m i , J i , M i , τ i )= 1 Z ( β ) (cid:88) j i ,m i e − C j ( τ − τ ) e − C j ( τ − τ ) e − C j ( β − τ + τ − τ + τ ) (5.69) × dim j dim j dim j (cid:18) j j J m m M (cid:19) (cid:18) j j J − m m M (cid:19) . Note that as β → ∞ , this four-point function factorizes in two zero-temperature two-pointfunction, coming from the clustering principle, and the dependence on only two independent38ime differences, just as happens in the Schwarzian case [27].This construction is immediately generalized to arbitrary compact groups G , and leadsto the rules as given in section 1.The braiding and fusion matrices, which are given by q -deformed 6 j -symbols of the group G [72], become the classical 6 j -symbol of the group G . As emphasized for the Schwarziancase in [27], this quantity is used to swap the operator ordering and reach specific out-of-timeordered (OTO) correlators of interest, dual to shockwave interactions in the gravitationalcase [73]. For the Schwarzian theory, we find the precise semi-classical (large C ) shockwaveexpressions of [14, 23] starting from the exact OTO correlators in [74]. We leave a moredetailed discussion to future work. In this work, we presented more evidence and extensions to the link between 2d Liouvilletheory and the 1d Schwarzian theory. We believe this is the most natural way to look atthe Schwarzian theory. The first half of this paper focussed on the Liouville path integraldirectly, where we emphasized the relevance of the parametrization of Gervais and Neveuin this context.We further extended the AdS argument for preferred coordinate frames of [21, 23, 24] tothe case of gauge theories and preferred gauge transformations.In the second half of this work, we demonstrated that the Schwarzian limit is only a special(irrational) case of the simpler case of rational compact models. All of these geometric the-ories have the property that the Hamiltonian, Lagrangian and Casimir coincide, and thatlocal operators in 2d CFT become bilocal operators in 1d QM in a double-scaling limit. Weproduced correlation functions from the 2d WZW perspective, although our analysis wasnot entirely rigorous as we used the generalization of the prescription of [27]. It would bean improvement to complement this with a path-integral analysis as in section 2 for therational theories as well, including the measure in the path integral. This is left to futurework.Nonetheless, we deduced expressions for time-ordered correlators and provided diagram-matic rules. Out-of-time-ordered correlators can also be studied and require introducing 6 j symbols to swap internal lines in diagrams. It would be particularly interesting to link thisto results on OTO-correlators in rational 2d CFT, as in e.g. [75].These theories also seem to be related to group field theories, utilized in the spinfoam for-mulation of LQG, which in turn seem to be related to the tensor models of e.g. [76, 77].A very interesting extension to study deeper would be to understand N = 2 Liouville theoryin the (cid:103) N S -sector, which allows one to connect to the 1d N = 2 super-Schwarzian theory.39he latter contains non-trivial interactions between the Y -boson and the Liouville field φ itself. However, technical obstructions appear to be present when analyzing the mini-superspace regime and performing the φ -integrals directly in coordinate space. We hope tocome back to this problem in the future.The structure present in the rational theories, suggests the Schwarzian three-point vertex γ (cid:96) ( k , k ) should also be interpretable as a 3 j symbol of SL (2 , R ) with 1 discrete and 2continuous representations. If this can be made more explicit, then the generalizations tothe supersymmetric Schwarzian correlators can be conjectured to hold in terms of 3 j and6 j symbols of OSp(1 |
2) and OSp(2 |
2) for N = 1 and N = 2 super-Schwarzian theoriesrespectively, without resorting to the coordinate space evaluation of the Liouville integralsas mentioned above.We will make the link between SL (2 , R ) BF theory and the Schwarzian explicit in upcomingwork, using a complementary bulk holographic perspective on bilocal correlation functionsin terms of boundary-anchored Wilson lines in BF theory. This was already hinted at in[78].A further question is whether anything can be learned for 4d gauge theories, as 2d boundaryLiouville/Toda CFT was demonstrated in an AGT context in [79] to be linked to (a certainsubclass of) these. Taking the double scaling limit should have an analogue in 4d gaugetheories.One of the holographic successes of the Schwarzian theory is a correct prediction of theBekenstein-Hawking entropy of the JT black holes [23, 24]. Within the Liouville frame-work, it arises fully from the modular S -matrix as S BH = log S p . On the other hand, itwas found in [80] that the topological entanglement entropy in 2d irrational Virasoro CFTmatches the Bekenstein-Hawking entropy for 3d BTZ black holes: S BH = log S p +0 S p − . Itwould be interesting to utilize the 2d/1d perspective to shed more light on some of thepuzzles that appear in 3d gravity and its relation to 2d Liouville dynamics. Acknowledgements
I am deeply grateful to A. Blommaert, N. Callebaut, H. T. Lam, G. J. Turiaci andH. L. Verlinde for numerous discussions and questions that greatly benefitted this work.The author acknowledges financial support from the Research Foundation-Flanders (FWOVlaanderen).
A Virasoro coadjoint orbits and Liouville branes
It is instructive to generalize the construction in section 2 to more general branes, andmake the link with the Alekseev-Shatashvili geometric action and the coadjoint orbits ofthe Virasoro group more explicit.W.l.g. we keep the left brane fixed as a ZZ-brane. We first present the 2d case, and discuss40he Schwarzian limit at the very end only. We can now list the possible generalizations inFigure 14.
ZZ FZZT q ZZ ZZ
ZZ ZZ / SL n (2, R )diff S / SL(2, R )diff S / U(1)diff S Figure 14: Other brane configurations. Top: the ZZ-ZZ system. Middle: the ZZ-ZZ ,n system. Bottom: the ZZ-FZZT system.In section 2, if one replaces the brane at σ = π by a ZZ ,n brane [45], one can use thesame definition of a and b , but now take the boundary condition as a = b + 2 πn | σ = π for n >
1. This gives singularities in the Liouville field φ also in between both branes. Setting f → nf , the periodicity of f is restored, and one obtains again a diff S /SL (2 , R ) theory.Gervais and Neveu [41, 42, 43, 44] considered boundary conditions that in modern parlancewould be called FZZT branes [46]. One can directly implement an FZZT brane at σ = π byrequiring instead a = b + 2 πθ | σ = π , with r = cos ( πθ ) and ∂ σ φ = − re φ (cid:12)(cid:12) σ = π as the boundarycondition. This boundary condition breaks the SL (2 , R ) redundancy to U (1). The classicalsolution corresponding to these situations is: e φ = − θ f (cid:48) ( u ) f (cid:48) ( v )sin (cid:16) θ f ( u ) − f ( v )2 (cid:17) , (A.1)where one sets θ = n ∈ N to find the ZZ ,n brane again. Either of these alternative boundary conditions can be absorbed back into the action byrescaling f → θf . The only effect is a change πβ → πβ θ in (2.15) and in the Hamiltonian interms of f . After doing this, the field f is again a circle diffeomorphism as before. Strictly speaking, we should set θ → iθ to describe the genuine FZZT-branes, where the equation r = cosh ( πθ ) becomes the standard FZZT relation; FZZT-branes correspond to hyperbolic orbits, whereaswe described the elliptic orbits instead. Our choice of notation follows [38].
41n all of these cases, we can make the link between the Liouville action in (2.8) and thegeometric action of Alekseev and Shatashvili [47, 48] more explicit as follows. The π φ ˙ φ -termin (2.8) is precisely the canonical 1-form α integrated over time, with α = (cid:82) π dσ πδφ, ω = dα . Given the symplectic 2-form ω , and ignoring global issues, α is determined only up toan exact form df , which integrates to zero as we take periodic boundary conditions in time.Explicitly, and after doubling, the geometric action is given by: S geom = (cid:90) dτ α = (cid:90) dτ (cid:90) π − π dσ (cid:34) c π ˙ ff (cid:48) (cid:32) f (cid:48)(cid:48)(cid:48) f (cid:48) − (cid:18) f (cid:48)(cid:48) f (cid:48) (cid:19) (cid:33) − b ˙ f f (cid:48) (cid:35) , (A.2)with α = (cid:90) π − π dσ (cid:34) c π dff (cid:48) (cid:32) f (cid:48)(cid:48)(cid:48) f (cid:48) − (cid:18) f (cid:48)(cid:48) f (cid:48) (cid:19) (cid:33) − b df f (cid:48) (cid:35) , (A.3)and ω = dα given by equation (2.15) as can be explicitly checked, and the coadjoint orbitparameter b = − (cid:16) πβ c π θ (cid:17) in terms of the FFZT brane parameter θ . For the diff( S ) /U (1)orbit, one mods out by F ( σ, τ ) → F ( σ, τ ) + a ( τ ), whereas for the diff( S ) /SL (2 , R ) orbit,one mods out by F ( σ, τ ) → α ( τ ) F ( σ,τ )+ β ( τ ) γ ( τ ) F ( σ,τ )+ δ ( τ ) , which indeed is what we find from the results ofsection 2 above. The function F is as before the uniformizing coordinate, related to f by F = tan θf .This result demonstrates the equivalence of Liouville between branes and the coadjoint orbitaction (including the Hamiltonian term) for the different orbits depicted in Figure 14. Atthe level of the partition function, this also follows directly since both of these evaluateto the same Virasoro character. Indeed, computing characters of irreps of the algebra isprecisely the goal of the coadjoint orbit construction: χ h ( T ) = (cid:90) [ D f ] e − (cid:82) T ( i d − ω + Hdτ ) = (cid:90) [ D f ] e − S , h ≡ h ( b ) , (A.4)with S = (cid:90) dτ (cid:90) π − π dσ (cid:32) i (cid:34) c π ˙ ff (cid:48) (cid:32) f (cid:48)(cid:48)(cid:48) f (cid:48) − (cid:18) f (cid:48)(cid:48) f (cid:48) (cid:19) (cid:33) − b ˙ f f (cid:48) (cid:35) + c π (cid:26) tan θf , σ (cid:27)(cid:33) . (A.5)This complicated expression can be transformed into the Floreanini-Jackiw path integralfor a chiral boson, yielding indeed a single character [48]. On the other hand, it is knownsince the introduction of ZZ-branes in [45] that the cylinder amplitude of Liouville betweenthese ZZ-branes is computing the Virasoro vacuum character. As mentioned above, chang-ing branes changes the character computed.It is in any case reassuring to see this equality directly within the path integral.42he above procedure has the additional benefit that one now also has a dictionarybetween operator insertions in Liouville and operators in the Alekseev-Shatashvili geometricaction theory (A.4),(A.5). Explicitly, one has the correspondence e (cid:96)φ ( σ,τ ) ↔ (cid:32) − θ f (cid:48) ( σ, τ ) f (cid:48) ( − σ, τ )sin (cid:0) θ ( f ( σ, τ ) − f ( − σ, τ )) (cid:1) (cid:33) (cid:96) . (A.6)This correspondence is fully at the level of the 2d theories, and can be viewed as an inter-esting conclusion in its own right.Finally taking the Schwarzian limit of interest, we need | b | → ∞ such that b T = − (cid:18) πβ Cθ (cid:19) ⇒ θ = β π √− b TC . (A.7)As discussed in the main text, the above geometric action (A.2) (the p ˙ q -term in the La-grangian) disappears in this limit, and only the Hamiltonian density (the Schwarzian deriva-tive) remains. As the Hamiltonian is itself the generator of a U (1)-symmetry, Stanford andWitten applied the Duistermaat-Heckman theorem to prove the one-loop exactness of theresulting 1d partition function [29]. This one-loop exactness fails for correlation functionshowever and one has to resort to other methods, by using the correspondence (A.6) in theSchwarzian limit, where the τ -dependence drops out in (A.6) and one recovers (2.16) when θ = 1 to find the ZZ-ZZ system again.When changing the branes, the resulting 1d theories are all pathological as thermal systems,except the ZZ-ZZ system that is studied here. B Lagrangian description of matter sector
A general matter sector in the Poincar´e upper half plane, is given by S m = (cid:90) df dz L m ( q, ∂ f q ) , (B.1) The geometric action is identified in [81] as a Berry phase associated to a closed path in the Virasorogroup. Their holographic interpretation in AdS /CFT in terms of precession of inertial frames agrees withtheir absence in the dimensionally reduced 2d JT gravity, dual to the Schwarzian.
43n terms of a canonical variable q . Consider now the coupling to the dynamical boundaryas: S = − (cid:90) dt { f, t } + (cid:90) dtdz f (cid:48) L m ( q, ∂ f q ) . (B.2)The two sectors only interact through the dynamical time variable f ( t ). As a sanity check,the matter equations of motion are given by f (cid:48) ∂ L m ∂q = ∂ t (cid:18) f (cid:48) ∂ L m ∂∂ f q f (cid:48) (cid:19) ⇒ ∂ L m ∂q = ∂ f (cid:18) ∂ L m ∂∂ f q (cid:19) , (B.3)and are not influenced by the coupling to the dynamical boundary, as this is only a timereparametrization that indeed should not affect matter equations.Varying (B.2) w.r.t. f yields: δS = (cid:90) dt (cid:20) { f, t } (cid:48) f (cid:48) δf − (cid:90) dx (cid:18) −L m + ∂ f q ∂ L m ∂∂ f q (cid:19) δf (cid:48) (cid:21) = (cid:90) dt δf (cid:20) { f, t } (cid:48) f (cid:48) + H (cid:48) m (cid:21) . (B.4)The second term is found by writing ∂ f q = q (cid:48) f (cid:48) and using δ f (cid:48) = − f (cid:48) δf (cid:48) . This leads to { f, t } (cid:48) + f (cid:48) H (cid:48) m = 0 , (B.5)which matches the first derivative of eq. (3.16) of [24]. C Partition function of a particle on a group manifold
Using the normalized eigenfunctions ψ abj ( g ) = (cid:113) d j vol G ( D j ) ab ( g ) with D j ( g ) the representationmatrices of the representation j , the Euclidean propagator from the point g to g with e φ = g ( g ) − is given by the formula [64]: K ( g , g ; t ) = (cid:88) j,a,b ψ ab ∗ j ( g ) ψ abj ( g ) e − C j t = 1vol G (cid:88) j dim j χ j ( φ ) e − C j t . (C.1) In path integral language, integrating the Jackiw-Teitelboim action (1.2) over Φ fixed the AdS metric;the Gibbons-Hawking boundary term then reduces to the Schwarzian action.We chose here to perform the time reparametrization throughout the 2d bulk; the z -dependence of thefluctuating boundary is O ( (cid:15) ) and can be ignored here. Useful property: (cid:16) f (cid:48)(cid:48) f (cid:48) (cid:17) (cid:48) f (cid:48) (cid:48) = { f, t } (cid:48) f (cid:48) . (B.6) χ j ( φ ) = Tr (cid:104) D j ( g ) D † j ( g ) (cid:105) . Setting t = β and φ = 0, corresponding tothe sum over all based loops on G , one finds χ j (0) = dim j and K ( g , g ; β ) = 1vol G (cid:88) j (dim j) e − C j β , (C.2)which is indeed the path integral over LG/G with the (vol G ) − factor coming from theright coset. This factor is absorbed into a contribution to the zero-temperature entropy S and dismissed. The ordinary partition function of a particle on a group only contains thepath integration over LG and is indeed Z = (cid:90) dg K ( g , g ; β ) = (cid:88) j (dim j) e − C j β . (C.3) D Some relevant formulas for SU (2) k The three-point function of the SU (2) k WZW model can be found in e.g. [69] as (cid:104) Φ j ,m , ¯ m (0)Φ j ,m , ¯ m (1)Φ j ,m , ¯ m ( ∞ ) (cid:105) = δ (cid:32)(cid:88) i m i (cid:33) δ (cid:32)(cid:88) i ¯ m i (cid:33) D j j j . (D.1)In the special case that m = ¯ m = j and m = ¯ m = j , one has explicitly: D j j j = Γ( j + j − j + 1)Γ( j + j − j + 1)Γ(2 j + 1) (cid:118)(cid:117)(cid:117)(cid:116) γ (cid:0) k +2 (cid:1) γ (cid:0) j +1 k +2 (cid:1) γ (cid:0) j +1 k +2 (cid:1) γ (cid:0) j +1 k +2 (cid:1) × P ( j + j + j + 1) P ( j + j − j ) P ( j + j − j ) P ( j + j − j ) P (2 j ) P (2 j ) P (2 j ) (D.2)with γ ( x ) = Γ( x )Γ(1 − x ) , P ( j ) = j (cid:89) m =1 γ (cid:18) mk + 2 (cid:19) . (D.3)In the large k regime, we get γ (cid:16) αk (cid:17) → kα , P ( j ) → j (cid:89) m =1 km = k Γ( j + 1) , (D.4)and hence D j j J → (cid:112) (2 j + 1)(2 j + 1)(2 J + 1) Γ(2 j + 1)Γ(2 J + 1)Γ( j + j + J + 2)Γ( j + J − j + 1) . (D.5)45n the other hand, we can write (cid:10) Φ j ,m , ¯ m (0)Φ J,M, ¯ M (1)Φ j ,m , ¯ m ( ∞ ) (cid:11) = (cid:104) j , − m | Φ J,M | j , m (cid:105) = C JM,j m ; j − m ˜ A j j J = C j m ,j m ; J − M A j j J (D.6)The second line uses the standard form of the Wigner-Eckart theorem by combining j with J into j . In the last equality we rearranged the Clebsch-Gordan coefficients usingthe symmetry of the 3 j -symbols; this reordering produces extra j -dependent factors thatare absorbed into a new reduced matrix element A j j J . The Clebsch-Gordan coefficient forcombining j and j into J is: C j m ,j ,j ; J − J = (cid:115) (2 J + 1)Γ( J + j − j + 1)Γ( J + j − j + 1)Γ( j + j − J + 1)Γ( j + j + J + 2) × ( − ) J − j − j (cid:112) Γ(2 J + 1)Γ(2 j + 1)Γ( j + j − J + 1)Γ( j + J − j + 1) × (cid:88) k ( − ) k k !( j + j − J − k )!( j + J − j − k )!( − k )! k !( J + j − j + k )! , (D.7)with m = − m − m = − j + J . 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