On classical and semiclassical properties of the Liouville theory with defects
aa r X i v : . [ h e p - t h ] N ov On classical and semiclassical properties ofthe Liouville theory with defects.May 2015
Hasmik Poghosyan ∗ and Gor Sarkissian , † Yerevan Physics Institute,Alikhanian Br. 2, 0036 YerevanArmenia Department of Physics, Yerevan State University,Alex Manoogian 1, 0025 YerevanArmenia
Abstract
The Lagrangian of the Liouville theory with topological defects is analyzedin detail and general solution of the corresponding defect equations ofmotion is found. We study the heavy and light semiclassical limits of thedefect two-point function found before via the bootstrap program. Weshow that the heavy asymptotic limit is given by the exponential of theLiouville action with defects, evaluated on the solutions with two singularpoints. We demonstrate that the light asymptotic limit is given by thefinite dimensional path integral over solutions of the defect equations ofmotion with a vanishing energy-momentum tensor. ∗ [email protected] † [email protected] ontents Γ functions 36B Volume form on the 3D hyperboloid H +3
36C Action evaluation 37D Defect two-point function 41 Introduction
Defects in two-dimensional conformal field theories can be realized as orientedlines, separating different theories. We are interested in the special class of de-fects, for which the energy-momentum tensor is continuous across the defect [1].Denoting the left- and right- moving energy-momentum tensors of the two theo-ries by T (1) , T (2) , and ¯ T (1) , ¯ T (2) , this condition takes the form: T (1) = T (2) , ¯ T (1) = ¯ T (2) . (1)Inserting a defect in the path integral is equivalent in the operator language tothe insertion of an operator D which maps the Hilbert space of CFT 1 to thatof CFT 2. Condition (1) can be considered as implying that the correspondingoperator D commutes with the Virasoro modes: DL (1) m = L (2) m D and D ¯ L (1) m = ¯ L (2) m D . (2)During the last few years topological defects in the Liouville and Toda fieldtheories attracted some attention due to their relation to the Wilson lines inthe AGT correspondence [2–5] ‡ . Defects in the Liouville field theory have beenconstructed in [7,8]. In these papers defects were constructed as operators on theHilbert space of Liouville theory. To obtain these operators, two-point functionsin the presence of defects were calculated using the conformal bootstrap programfor defects, developed in [7, 9]. It was shown in [7] that there are two families ofdefects: discrete, corresponding to the degenerate fields and labeled by a pair ofpositive integers m and n , with eigenvalues D m,n ( α ) = sin( πmb − (2 α − Q )) sin( πnb (2 α − Q ))sin πb − (2 α − Q ) sin πb (2 α − Q ) , (3)and continuous, labeled by one continuous parameter s with eigenvalues D s ( α ) = cosh(2 πs (2 α − Q ))2 sin πb − (2 α − Q ) sin πb (2 α − Q ) . (4)We denoted here by Q = b + b the background charge, and α labels primaries ofLiouville theory. The defects of the discrete family have a one-dimensional world-volume, and in particular the identity defect D , belongs to the discrete family. ‡ In fact in references [2, 3] the Verlinde loop operators are discussed, but they coincide withtopological defects for the Cardy case [6]. α = η l b and keep η l fixed for b →
0, whereas in the heavy asymptotic limit we take α = η h b and hold η h fixedagain for b → κ entering in the Lagrangian with defect and parameter s labeling the defect oper-ator (4): κ = cosh(2 πsb ) (5)where it is understood that s → ∞ and b → σ = sb fixed.We show that in the light asymptotic limit the defect two-point functions can4e obtained via the path integral over solutions of the defect equations of motionwith vanishing energy-momentum tensor in the large σ limit.We demonstrate that in the heavy asymptotic limit defect two-point functionsare given by the sum of exponentials of the action with defects evaluated on solu-tions with two singular points of the defect equations of motion. To understandbetter the semiclassical origin of the denominator in (4) in the heavy asymptoticlimit, we consider analytic continuation of η to the complex region in the spiritof [34]. We find a discrete family of solutions with two singular points, labelledby two integer numbers N and N . But to fit to semiclassical limit of the defecttwo-point function and to have convergent series we should sum over the saddlepoints with nonnegative N and N for Im(2 η − >
0, and with nonpositive N and N if Im(2 η − <
0. This is an example of the Stokes phenomena [34–37].The paper is organized in the following way.In section 2 we analyze classical Liouville theory with defects. In subsection2.1 we review the general solution of the Liouville equation. In subsection 2.2we present general solution of the defect equations of motion. In subsection 2.3we present the Lagrangian of the product of the Liouville theories on half-planewith the boundary condition specified by a permutation brane. In section 3 wereview defects and permutation branes in quantum Liouville theory. In section4 we review the heavy and light asymptotic semiclassical limits. In section 5 wecalculate the defect two-point function in the light asymptotic limit. In section6 we calculate the defect two-point function in the heavy asymptotic limit. In aseries of appendices we describe some useful technical results.
Let us recall some facts on classical Liouville theory.The action of the Liouville theory is S = 12 πi Z (cid:0) ∂φ ¯ ∂φ + µπe bφ (cid:1) d z . (6)Here we use a complex coordinate z = τ + iσ , and d z ≡ dz ∧ d ¯ z is the volumeform. 5he field φ ( z, ¯ z ) satisfies the classical Liouville equation of motion ∂ ¯ ∂φ = πµbe bφ . (7)The general solution to (7), also derived below, was given by Liouville in termsof two arbitrary functions A ( z ) and B (¯ z ) [38] φ = 12 b log (cid:18) πµb ∂A ( z ) ¯ ∂B (¯ z )( A ( z ) + B (¯ z )) (cid:19) . (8)The solution (8) is invariant if one transforms A and B simultaneously by thefollowing constant M¨obius transformations: A → ζ A + βγA + δ , B → ζ B − β − γB + δ , ζ δ − βγ = 1 . (9)Classical expressions for left and right components of the energy-momentum ten-sor are T = − ( ∂φ ) + b − ∂ φ , (10)¯ T = − ( ¯ ∂φ ) + b − ¯ ∂ φ . (11)Substituting (8) in (10) and (11) we get, that the components of the energy-momentum tensor are given by the Schwarzian derivatives of A ( z ) and B (¯ z ): T = { A ; z } = 12 b (cid:20) A ′′′ A ′ −
32 ( A ′′ ) ( A ′ ) (cid:21) , (12)¯ T = { B ; ¯ z } = 12 b (cid:20) B ′′′ B ′ −
32 ( B ′′ ) ( B ′ ) (cid:21) . (13)The Schwarzian derivative is invariant under arbitrary constant M¨obius transfor-mation: (cid:26) ζ F + βγF + δ ; z (cid:27) = { F ; z } , ζ δ − βγ = 1 . (14)Solutions of the Liouville equation (7) can be described also via linear com-bination of some holomorphic and anti-holomorphic functions. Let us introducethe function V = e − bφ . One can write the Liouville equation (7) as an equationfor V V ∂ ¯ ∂V − ∂V ¯ ∂V = − πµb . (15)Also the left and right components of the energy-momentum tensor (10) and (11)can be written via V ∂ V = − b V T , (16)6 ∂ V = − b V ¯ T . (17)It is straightforward to check that the general solution of eq. (15) is given bylinear combination of two holomorphic a i ( z ), i = 1 ,
2, and two anti-holomorphicfunctions b i (¯ z ), i = 1 , V = p πµb (cid:18) a ( z ) b (¯ z ) − a ( z ) b (¯ z ) (cid:19) , (18)satisfying the condition ( a a ′ − a ′ a )( b b ′ − b ′ b ) = 1 . (19)Usually the fields a i ( z ) and b i (¯ z ), i = 1 , a a ′ − a ′ a = 1 (20)and b b ′ − b ′ b = 1 . (21)It is easy to see that the left and right components of the energy-momentumtensor can be expressed via a i and b i in the very simple form: T = − b ∂ a a = − b ∂ a a (22)and ¯ T = − b ¯ ∂ b b = − b ¯ ∂ b b . (23)The solutions (8) and (18) can be related in the following way. One can solvethe unit Wronskian conditions (20) and (21) via a holomorphic A ( z ) and ananti-holomorphic function B (¯ z ) a = 1 √ ∂A and a = A √ ∂A (24)and b = B √ ¯ ∂B and b = − √ ¯ ∂B . (25)Inserting (24) and (25) in (18) we get (8). Note that the M¨obius transformationsof A and B (9) become linear SL (2 , C ) transformations of a i and b i :˜ a = δa + γa , (26)˜ a = βa + ζ a b = ζ b + βb , (27)˜ b = γb + δb . It is straightforward to check that indeed (18) is invariant under (26) and(27), and both of them keep the unit Wronskian condition.One can also check, that both components of the energy-momentum tensor(22) and (23) are invariant under these transformations as well.We finish this section with a remark which will be important in the partsof this work dealing with the light asymptotic limit. There we will consider ananalytic continuation µ → − µ . At this point it is convenient to write the solution(18) as: V = p − πµb (cid:18) a ( z ) b (¯ z ) + a ( z ) b (¯ z ) (cid:19) . (28)It is easy to check that (28) also solves the Liouville equation, given that a i and b i , i = 1 , Recently in [14] the action of the Liouville theory with topological defects wassuggested: S top − def = 12 πi Z Σ (cid:0) ∂φ ¯ ∂φ + µπe bφ (cid:1) d z + 12 πi Z Σ (cid:0) ∂φ ¯ ∂φ + µπe bφ (cid:1) d z (29)+ Z ∂ Σ (cid:20) − π φ ∂ τ φ + 12 π Λ ∂ τ ( φ − φ ) + µ e ( φ + φ − Λ) b − πb e Λ b (cosh( φ − φ ) b − κ ) (cid:21) dτi . Here Σ is the upper half-plane σ = Im z ≥ is the lower half-plane σ = Im z ≤
0. The defect is located along their common boundary, which is thereal axis σ = 0 parametrized by τ = Re z . Note that Λ( τ ) here is an additionalfield associated with the defect itself. The action (29) yields the following defectequations of motion at σ = 0:12 π ( ∂ − ¯ ∂ ) φ + 12 π ∂ τ φ − π ∂ τ Λ + µb e ( φ + φ − Λ) b − πb e Λ b sinh( φ − φ ) b = 0 , (30) − π ( ∂ − ¯ ∂ ) φ − π ∂ τ φ + 12 π ∂ τ Λ+ µb e ( φ + φ − Λ) b + 1 πb e Λ b sinh( φ − φ ) b = 0 , (31)82 π ∂ τ ( φ − φ ) − µb e ( φ + φ − Λ) b − πb e Λ b (cosh( φ − φ ) b − κ ) = 0 . (32)The last equation is derived from variation of Λ.Using that ∂ τ = ∂ + ¯ ∂ and forming various linear combinations of equations(30)-(32) we can bring them to the form:¯ ∂ ( φ − φ ) = πµbe b ( φ + φ ) e − Λ b , (33) ∂ ( φ − φ ) = 2 b e Λ b (cosh( φ − φ ) b − κ ) . (34) ∂ ( φ + φ ) − ∂ τ Λ = 2 b e Λ b sinh( b ( φ − φ )) . (35)It is shown in [14] that requiring the defect equations of motion to hold for every σ brings additionally to the condition, that Λ is a restriction to the real axis ofa holomorphic field ¯ ∂ Λ = 0 . (36)This condition allows to rewrite (35) in the form ∂ ( φ + φ − Λ) = 2 b e Λ b sinh( b ( φ − φ )) . (37)It is checked in [14] that the system of the defect equations of motion (33)-(37)guarantees that both components of the energy-momentum tensor are continuousacross the defects and therefore describes topological defects: − ( ∂φ ) + b − ∂ φ = − ( ∂φ ) + b − ∂ φ , (38) − ( ¯ ∂φ ) + b − ¯ ∂ φ = − ( ¯ ∂φ ) + b − ¯ ∂ φ . (39)Another interesting consequence of the defect equations of motion, found in [14],is the existence together with the holomorphic field Λ( z ) of an anti-holomorphicfield Ξ: ∂ Ξ = 0 , (40)where Ξ = e − b ( φ + φ ) e b Λ (cosh b ( φ − φ ) − κ ) . (41)or alternatively Ξ = b e − b ( φ + φ ) ∂ ( φ − φ ) . (42)Now we will present the general solution for defect equations of motion (33)-(37). 9e will follow essentially the same strategy which was used in [39] for analyz-ing the boundary Liouville problem. On the one hand since the defect is topolog-ical both components of the energy-momentum tensor are equal being computedin terms of φ or φ . On the other hand each component of the energy-momentumtensor is given by the Schwarzian derivative, which is invariant under the M¨obiustransformation. This naturally leads to the following solution: φ = 12 b log (cid:18) πµb ∂A ¯ ∂B ( A + B ) (cid:19) , (43) φ = 12 b log (cid:18) πµb ∂C ¯ ∂D ( C + D ) (cid:19) , (44)where C = ζ A + βγA + δ and D = ζ ′ B + β ′ γ ′ B + δ ′ . (45)Remembering that φ is invariant under the simultaneous M¨obius transformation(9) of C and D , we can set B = D . Therefore without loosing generality we canlook for a solution in the form: φ = 12 b log (cid:18) πµb ∂A ¯ ∂B ( A + B ) (cid:19) , (46) φ = 12 b log (cid:18) πµb ∂C ¯ ∂B ( C + B ) (cid:19) , (47)where C = ζ A + βγA + δ . (48)Substituting (46) and (47) in (33) we find that it is satisfied with e − Λ b = A − C √ ∂A∂C . (49)Since A and C are holomorphic functions, Λ is holomorphic as well, as it is statedin (36).It is straightforward to check that (37) is satisfied as well with φ , φ and Λgiven by (46), (47) and (49) respectively. And finally inserting (46), (47) and(49) in (34) we see that it is also fulfilled with κ = ζ + δ . (50)10nserting (46), (47) in (42) one can check thatΞ = πµb γB + B ( ζ − δ ) − β ¯ ∂B . (51)Remembering that B is anti-holomorphic we see that Ξ is anti-holomorphic aswell.We can also write the solution of the defect equations of motion using solutionof the Liouville equation in the form (18). Recalling that the M¨obius transfor-mations of the functions A and B become linear SL (2 , C ) transformations of thefunctions a i and b i , which leave the components of the energy-momentum tensor(22) and (23) invariant, we can write the solution (46)-(48) in the form: e − bφ = p πµb (cid:18) a ( z ) b (¯ z ) − a ( z ) b (¯ z ) (cid:19) , (52) e − bφ = p πµb (cid:18) c ( z ) b (¯ z ) − c ( z ) b (¯ z ) (cid:19) , (53)where denoting ~a = ( a , a ), ~c = ( c , c ), and D = δ γβ ζ ! , one has ~c = D~a (54)and 2 κ = Tr D . (55)At this point we would like to make the following remark. Let us considerthe identity defect. It has A = C , and κ = 1. Setting A = C in (49) weobtain e − Λ b = 0. This result can be derived also directly setting φ = φ in (33).Therefore the identity defect does not belong to the family of defects describedby the action (29) and can be derived from them only in the limit Λ → ∞ .This can be understood recalling from appendix D that defects described by(29) have a two-dimensional world-volume in a sense that the values of φ ( τ )and φ ( τ ) at an arbitrary point τ on the defect line are not constrained and thepoint ( φ ( τ ) , φ ( τ )) can take values in the whole plane R . Contrary to this, theidentity defect has a one-dimensional world-volume, since the point ( φ ( τ ) , φ ( τ ))takes values on one-dimensional diagonal φ = φ .11 .3 Lagrangian of the Liouville theory with permutationbranes We can also construct a folded version of the action (29) describing product ofLiouville theories on a half-plane with boundary condition given by permutationbranes: S perm − brane = 12 πi Z Σ (cid:0) ∂φ ¯ ∂φ + µπe bφ + ∂φ ¯ ∂φ + µπe bφ (cid:1) d z (56)+ Z ∂ Σ (cid:20) − π φ ∂ τ φ + 12 π Λ ∂ τ ( φ − φ ) − µ e ( φ + φ − Λ) b + 1 πb e Λ b (cosh( φ − φ ) b − κ ) (cid:21) dτi . Σ denotes here the upper half-plane σ ≥
0, and τ parameterizes the boundarylocated at σ = 0. This action gives rise to the boundary equations12 π ( ∂ − ¯ ∂ ) φ + 12 π ∂ τ φ − π ∂ τ Λ − µb e ( φ + φ − Λ) b + 1 πb e Λ b sinh( φ − φ ) b = 0 , (57)12 π ( ∂ − ¯ ∂ ) φ − π ∂ τ φ + 12 π ∂ τ Λ − µb e ( φ + φ − Λ) b − πb e Λ b sinh( φ − φ ) b = 0 . (58)12 π ∂ τ ( φ − φ ) + µb e ( φ + φ − Λ) b + 1 πb e Λ b (cosh( φ − φ ) b − κ ) = 0 . (59)Again using that ∂ τ = ∂ + ¯ ∂ and forming various linear combinations, one canbring the system (57)-(59) to the form ∂φ − ¯ ∂φ = πµbe b ( φ + φ ) e − Λ b , (60) ∂φ − ¯ ∂φ = − b e Λ b (cosh( φ − φ ) b − κ ) , (61) ∂φ + ¯ ∂φ − ∂ τ Λ = − b e Λ b sinh( b ( φ − φ )) . (62)One can check that equations (60)-(62) imply the permutation brane conditions: T (1) = ¯ T (2) | σ =0 , (63)¯ T (1) = T (2) | σ =0 or using (10) and (11) − ( ∂φ ) + b − ∂ φ = − ( ¯ ∂φ ) + b − ¯ ∂ φ , (64)12 ( ¯ ∂φ ) + b − ¯ ∂ φ = − ( ∂φ ) + b − ∂ φ . (65)To solve equations (60)-(62) we will use the same strategy as before, withthe only difference that now the M¨obius transformation relates holomorphic andantiholomorphic functions: φ = 12 b log (cid:18) πµb ∂A ¯ ∂B ( A + B ) (cid:19) , (66) φ = 12 b log (cid:18) πµb ∂B ¯ ∂C ( C + B ) (cid:19) , (67)and C = ζ A + βγA + δ . (68)The expressions (66)-(68) solve equation (60) with the Λ given by the relation e − Λ b = C − A √ ∂A ¯ ∂C . (69)It is straightforward to see that the expressions (66)-(68) together with the Λgiven by (69) solve also eq. (62).Finally inserting φ , φ and Λ given by (66), (67) and (69) respectively in eq.(61) we get that it is satisfied as well with the following κκ = ζ + δ . (70) Liouville field theory is a conformal field theory enjoying the Virasoro algebra[ L m , L n ] = ( m − n ) L m + n + c L
12 ( n − n ) δ n, − m , (71)with the central charge c L = 1 + 6 Q . (72)Primary fields V α in this theory, which are associated with exponential fields e αϕ , have conformal dimensions∆ α = α ( Q − α ) . (73)13he fields V α and V Q − α have the same conformal dimensions and represent thesame primary field, i.e. they are proportional to each other: V α = S ( α ) V Q − α , (74)with the reflection function S ( α ) = ( πµγ ( b )) b − ( Q − α ) b Γ(1 − b ( Q − α ))Γ( − b − ( Q − α ))Γ( b ( Q − α ))Γ(1 + b − ( Q − α )) . (75)Two-point functions of Liouville theory are given by the reflection function(75): h V α ( z , ¯ z ) V α ( z , ¯ z ) i = S ( α )( z − z ) α (¯ z − ¯ z ) α . (76)Introducing ZZ function [40]: W ( α ) = − / ( πµγ ( b )) − ( Q − α )2 b π ( Q − α )Γ(1 − b ( Q − α ))Γ(1 − b − ( Q − α )) , (77)the two-point function can be compactly written as S ( α ) = W ( Q − α ) W ( α ) . (78)Another useful property of ZZ function is W ( Q − α ) W ( α ) = − √ πb − (2 α − Q ) sin πb (2 α − Q ) . (79)The spectrum of the Liouville theory has the form H = Z ∞ dP R Q + iP ⊗ R Q + iP , (80)where R α is the highest weight representation with respect to the Virasoro alge-bra. Let us recall the form of continuous family of defects and permutation branesin the Liouville field theory computed in [7, 8] using appropriate generalizationof the Cardy-Lewellen equation [9]. The details can be found in appendix D.Topological defects are intertwining operators X commuting with the Virasorogenerators [ L n , X ] = [ ¯ L n , X ] = 0 . (81)14uch operators have the form X = Z Q + i R dα D ( α ) P α , (82)where P α are projectors on a subspace R α ⊗ R α : P α = X N,M ( | α, N i ⊗ | α, M i )( h α, N | ⊗ h α, M | ) . (83)Here | α, N i and | α, M i are vectors of orthonormal bases of the left and rightcopies of R α respectively. The eigenvalues D ( α ) can be determined via the two-point functions computed in the presence of a defect X h V α ( z , ¯ z ) XV α ( z , ¯ z ) i = D ( α ) S ( α )( z − z ) α (¯ z − ¯ z ) α (84)It is shown in [7] that h V α ( z , ¯ z ) X s V α ( z , ¯ z ) i = − W ( α ) 2 / cosh(2 πs (2 α − Q ))( z − z ) α (¯ z − ¯ z ) α (85)and therefore for D s ( α ) one can write using (78) and (79) D s ( α ) = − / cosh(2 πs (2 α − Q )) S ( α ) W ( α ) = cosh(2 πs (2 α − Q ))2 sin πb − (2 α − Q ) sin πb (2 α − Q ) . (86)The parameter s is a continuous parameter labeling the defect. The defects canbe characterized also by the value of the two-point function of the degeneratefield V − b/ in the presence of a defect. It is a function A ( b ) of b . It is shown in [7]that the parameter s is related to the function A ( b ) by the equation:2 cosh 2 πbs = A ( b ) (cid:18) W ( − b/ W (0) (cid:19) . (87)The permutation branes boundary states | B i P on product L × L of two Liouvilletheories satisfy the gluing condition [41]:( L (1) n − ¯ L (2) − n ) | B i P = 0 , (88)( L (2) n − ¯ L (1) − n ) | B i P = 0 . Comparing the gluing conditions (88) and (81) one can see that topological defectsrelated to permutation branes by folding trick, consisting of exchanging left andright components of the second copy, and hence these branes are characterizedby the same two-point functions (85) with z and ¯ z exchanged h V (1) α ( z , ¯ z ) V (2) α ( z , ¯ z ) i P = − W ( α ) 2 / cosh(2 πs (2 α − Q ))( z − ¯ z ) α (¯ z − z ) α . (89)15 Semiclassical limits
Let us consider the action (6) for the rescaled variable ϕ = 2 bφS = 18 πib Z (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z , (90)where λ = πµb .This form shows that b plays in the Liouville theory the role of the Planckconstant, and one can study semiclassical limit taking the limit b →
0, in such away that λ is kept fixed.Let us consider correlation functions in the path integral formalism: D V α ( z , ¯ z ) · · · V α n ( z n , ¯ z n ) E = Z D ϕ e − S n Y i =1 exp (cid:18) α i ϕ ( z i , ¯ z i ) b (cid:19) . (91)We would like to calculate this integral in the semiclassical limit b → α i scales with b . Since S scales like b − , for operators to affect the saddle point, we should take α i = η i /b ,with η i fixed. The conformal weights ∆ α = η (1 − η ) /b scale like b − as well.This is the heavy asymptotic limit. Another choice of the operator scaling willbe discussed in the next subsection.We see from (91) that in the semiclasscial limit the correlation function isgiven by e − S cl where, at least naively, in a sense which will be clarified below, S cl is the action S = 18 πib Z (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z + n X i =1 η i b ϕ ( z i , ¯ z i ) , (92)evaluated on the solution of its equation of motion: ∂ ¯ ∂ϕ = 2 λe ϕ − π n X i =1 η i δ ( z − z i ) . (93)Assuming that in the vicinity of the insertion point z i , one can ignore the expo-nential term we get that in the neighborhood of the point z i ϕ has the followingbehavior ϕ ( z, ¯ z ) = − η i log | z − z i | + X i as z → z i . (94)16ne can insert this solution back into the equation of motion to check, ifindeed the exponential term is subleading. We find, that this happens whenRe η i < . (95)This constraint is known as Seiberg bound [19]. It is the semiclassial version of thequantum condition (74) stating that V α and V Q − α represent the same quantumoperator. Either α or Q − α always obey the Seiberg bound.Remembering that in the Liouville theory we have also a background charge atinfinity, conditions (94) should be complemented by the behavior at the infinity: ϕ ( z, ¯ z ) = − | z | as | z | → ∞ . (96)Since the energy-momentum tensor in the presence of primary fields acquires aquadratic singularity,the functions a j , j = 1 ,
2, should solve the equation ∂ a j + b T a j = 0 , (97)where b T = n X i =1 (cid:18) η i (1 − η i )( z − z i ) + c i ( z − z i ) (cid:19) (98)and c i are the so called accessory parameters.If one tries naively to evaluate the action (92) on a solution obeying (94), onefinds that it diverges. Therefore we should consider a regularized action. It wasconstructed in [20]: b S reg = 18 πi Z D −∪ i d i (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z + 12 π I ∂D ϕdθ + 2 log R (99) − n X i =1 (cid:18) η i π I ∂d i ϕdθ i + 2 η i log ǫ i (cid:19) . Here D is a disc of radius R , d i is a disc of radius ǫ i around z i . It was shownin [20] that the action (99) satisfies the equation ∂∂η i b S reg = − X i , (100)where X i is defined by the boundary condition (94).The Polyakov conjecture proved in [42] states, that the action (99) obeys alsothe relation: ∂∂z i b S reg = − c i . (101)17et us write down a regularized version of the action with a defect.First of all let us write it in terms of λ = πµb , ϕ = 2 bφ , ϕ = 2 bφ , and˜Λ = 2 b Λ: b S top − def = 18 πi Z Σ (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z + 18 πi Z Σ (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z (102)+ Z ∂ Σ (cid:20) − π ϕ ∂ τ ϕ + 18 π ˜Λ ∂ τ ( ϕ − ϕ ) + λ π e ( ϕ + ϕ − ˜Λ) / − π e ˜Λ / (cid:18) cosh (cid:18) ϕ − ϕ (cid:19) − κ (cid:19)(cid:21) dτi . Since we consider here only insertion of the bulk field, and do not considerinsertion of the defect or boundary fields, the regularized action takes the form: b S top − def = 18 πi Z Σ R −∪ i d i (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z (103) − n X i =1 (cid:18) η i π I ∂d i ϕ dθ i + 2 η i log ǫ i (cid:19) + 12 π Z s R ϕ dθ + log R + 18 πi Z Σ R −∪ j d j (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z − n + m X j = n +1 η j π I ∂d j ϕ dθ j + 2 η j log ǫ j ! + 12 π Z s R ϕ dθ + log R + Z R − R (cid:20) − π ϕ ∂ τ ϕ + 18 π ˜Λ ∂ τ ( ϕ − ϕ ) + λ π e ( ϕ + ϕ − ˜Λ) / − π e ˜Λ / (cid:18) cosh (cid:18) ϕ − ϕ (cid:19) − κ (cid:19)(cid:21) dτi , where Σ Ri is a half-disc of the radius R and s Ri is a semicircle of the radius R inthe half-plane Σ i , i = 1 , Another limit is the so called light asymptotic limit. Here we take α = bη . (104)In this limit the operator insertions have no influence and the components of theenergy-momentum tensor are (anti-) holomorphic and regular functions every-where on sphere and hence vanish. Eq. (16) and (17) imply that V ≡ e − bφ shouldbe at most of first degree in z and ¯ z , thus leading to the solutions § : V ( z, ¯ z ; R ) = √− λ ( sz ¯ z + tz + u ¯ z + v ) , R = s tu v ! , (105) § It is shown in [22] that to have solution in light limit one needs to perform analyticalcontinuation µ → − µ . R = sv − ut = 1 . (106)Therefore the path integral in the light limit becomes a finite-dimensional in-tegral over parameters ( s, t, u, v ) which besides constraint (106) may satisfy someadditional constraints like reality and defect/boundary condition. The reality of V requires the matrix R to be Hermitian. A way to parameterize the Hermitianmatrices R is R = X − X X + iX X − iX X + X ! , (107)where the constraint X − X − X − X = 1, makes clear that the moduli spaceof the real solutions of the Liouville equation (7) with the vanishing energy-momentum tensor is a three-dimensional hyperboloid H +3 . Hence, for example inthe bulk Liouville theory, the correlation function in the light asymptotic limittakes the form D V bη ( z , ¯ z ) · · · V bη n ( z n , ¯ z n ) E → e − S l Z H +3 dR n Y i =1 V − η i ( z i , ¯ z i ; R ) , (108)where S l is the value of the action on these solutions. The action S l is independenton R , since the derivative of S l by any element of R vanishes, thanks to (105)being solution of the equations of motion: ∂S l ∂R ij = δS l δφ ∂φ∂R ij = 0 (109)To avoid calculation of S l and some overall factors in the integration measure, itis more convenient, as suggested in [23], to compute the ratio D V bη ( z , ¯ z ) · · · V bη n ( z n , ¯ z n ) E h V (0) i . (110)Therefore defining D V bη ( z , ¯ z ) · · · V bη n ( z n , ¯ z n ) E light ≡ Z M dR n Y i =1 V − η i ( z i , ¯ z i ; R ) , (111)where M is the moduli space of solutions with a vanishing energy-momentumtensor satisfying the corresponding boundary conditions in question, we can write D V bη ( z , ¯ z ) · · · V bη n ( z n , ¯ z n ) E h V (0) i → D V bη ( z , ¯ z ) · · · V bη n ( z n , ¯ z n ) E light h V (0) i light . (112)19he moduli space M for the Liouville theory with a boundary was studied in [23].It was found that in the boundary Liouville problem M is a subspace of H +3 with X set to the boundary cosmological constant. In the next section we willconstruct M for the Liouville problem with defects. Let us now specialize to the light asymptotic limit rules to the defects. We shouldfind solutions for φ and φ in the form (105) satisfying the defect equations ofmotion. We find it convenient to use in this section a new constant ˜ λ ≡ − λ = − πµb . One can check that expressions V ( z, ¯ z ; R ) ≡ e − bφ = p ˜ λ ( s z ¯ z + t z + u ¯ z + v ) , (113) R = s t u v ! , det R = 1 , R † = R and V ( z, ¯ z ; R ) ≡ e − bφ = p ˜ λ ( s z ¯ z + t z + u ¯ z + v ) , (114) R = s t u v ! , det R = 1 , R † = R satisfy the defect equations of motion (33)-(37) with2 κ = Tr (cid:0) R R − (cid:1) = s v + s v − u t − u t (115)and e − b Λ = z ( s t − s t ) + z ( s v − s v + u t − u t ) + u v − u v . (116)Let us show that the relation (115) results from the general formula (55).Note that one can write the solution (113) in the general form (28) V ( z, ¯ z ; R ) = p ˜ λ ( s z ¯ z + t z + u ¯ z + v ) = p ˜ λ [ z ( s ¯ z + t )+( u ¯ z + v )] (117)with a = z , a = 1 , (118) b = s ¯ z + t , b = u ¯ z + v . φ by rotatingthe pair a , a by a SL (2 , C ) matrix D = ζ βγ δ ! , namely taking˜ a = ζ z + β , (119)˜ a = γz + δ and keeping the same b and b as in (118). Using (119) we get that φ is givenby R = DR . Recalling that according to (55) 2 κ = Tr D we arrive to (115).We would like to mention also a folded version of the defect solution, obeyingthe permutation brane boundary conditions. One can see that the expressions(113) and (114) satisfy the permutation branes boundary conditions (60)-(62)with 2 κ = Tr( R T R − ) = s v + s v − t t − u u (120)and e − b Λ = τ ( s t − s u ) + τ ( s v − s v + t t − u u ) + t v − u v . (121)Note that equations (120) and (121) are in fact a folded version of the corre-sponding defect expressions (115) and (116) derived by exchanging u ↔ t , asa result of the z ↔ ¯ z exchange. The relation (120) can be justified again usingthe general formalism developed in section 2.3.In the parameterization (107) for the Hermitian matrices R and R R = X − X X + iX X − iX X + X ! , R = Y − Y Y + iY Y − iY Y + Y ! , (122)the defect parameter (115) is equal to the Minkowski inner product of the vectors X µ and Y µ κ = X µ Y µ = X Y − X Y − X Y − X Y . (123)Using that X , Y ≥ X µ and Y µ both have the unit Minkowski norm,it is easy to show that X µ Y µ ≥
1, with equality when X µ = Y µ [43]. It meansthat the real solutions of the defect equations of motion with vanishing energy-momentum tensor exist only for κ ≥
1. The border at κ = 1 is expected. At thispoint R = R and we have the identity defect which has e − b Λ = 0, which reflectsthat the identity defect does not belong to the family of two-dimensional defectdescribed by the action (29). It may happen that the semiclassical limit for other21alues of κ can be obtained using complex solutions of the defect equations ofmotion. Here we will consider only the values of κ greater than 1.We are in a position to write the two-point correlation function in the presenceof a defect: h V α ( z , ¯ z ) XV α ( z , ¯ z ) i light = (124) Z H +3 × H +3 dR dR δ (cid:16) Tr (cid:0) R R − (cid:1) − κ (cid:17) V − η ( z , ¯ z ; R ) V − η ( z , ¯ z ; R ) . Here dR i , i = 1 , H +3 . Thisexpression allows to establish conformal invariance of a defect two-point function.Let us perform the transformation R → LR L † and R → LR L † , (125)where L is a SL (2 , C ) matrix: L = m nk l ! . Note the transformation rule ofthe functions V − η ( z, ¯ z ; R ) under L : V − η ( z, ¯ z ; LRL † ) = 1 | nz + l | η V − η (cid:18) mz + knz + l , c.c ; R (cid:19) . (126)Performing the change of the integration variables (125), using that the δ -functionarguments is invariant under (125) and the transformation rule (126), we obtain h V α ( z , ¯ z ) XV α ( z , ¯ z ) i light = (127)1 | nz + l | η | nz + l | η D V α (cid:18) mz + knz + l , c.c. (cid:19) XV α (cid:18) mz + knz + l , c.c. (cid:19) E light , which is the standard consequence of the conformal invariance, when we remem-ber that in the light asymptotic limit ∆ ηb → η . This calculation shows thatthe invariance of the defect parameter κ under (125) is related to the conformalinvariance of the defect two-point function.Using conformal invariance we can set z to ∞ and z to 0 to derive: h V bη ( z , ¯ z ) XV bη ( z , ¯ z ) i light = ˜ λ − η ( z − z ) η (¯ z − ¯ z ) η (128) × Z H +3 × H +3 dR dR δ (cid:16) Tr (cid:0) R R − (cid:1) − κ (cid:17) ( R ) − η ( R ) − η . To calculate this integral we express the Hermitian matrices R and R asproducts R = gg † , R = ˜ g ˜ g † , g, ˜ g ∈ SL (2 , C ) , (129)22mplying that V = p ˜ λ (cid:16) | g z + g | + | g z + g | (cid:17) , (130) V = p ˜ λ (cid:16) | ˜ g z + ˜ g | + | ˜ g z + ˜ g | (cid:17) . (131)At the next step we will parametrize ˜ g as a product of matrices g and U :˜ g = gU, (132)where U is an SL (2 , C ) matrix U = u u u u ! , u u − u u = 1 . (133)Inserting (129) and (132) in (115) we obtain2 κ = Tr U U † . (134)This can be understood noting that the solutions (130) and (131) correspondto a i ( z ) = g i z + g i , ˜ a i ( z ) = ˜ g i z + ˜ g i , i = 1 , , (135) b i (¯ z ) = ¯ g i ¯ z + ¯ g i , ˜ b i (¯ z ) = ¯˜ g i ¯ z + ¯˜ g i , i = 1 , . (136)It is obvious that ˜ a i = X j =1 a j u ji , (137)˜ b i = X j =1 b j ¯ u ji . (138)We see that passing from g to ˜ g = gU brings to the simultaneous rotations of a i and b i , i = 1 ,
2, by matrices U and ¯ U . Therefore the defect parameter κ is equalto the trace of the product U U † . In these variables the integral (128) simplifiesand reads h V bη ( z , ¯ z ) XV bη ( z , ¯ z ) i light = ˜ λ − η ( z − z ) η (¯ z − ¯ z ) η (139) × Z dR dU δ ( | u | + | u | + | u | + | u | − κ )( R ) − η ( R ) − η , where dR and dU are the corresponding integration measures which will beelaborated below. 23sing SU (2) freedom in the choice of g we can adopt the parameterization g = ρ − w ρ ! , ρ ∈ R , w ∈ C (140)and R = ρ − + | w | ρ w ρ ¯ w ρ ! . (141)Parameterizing ˜ g in the same way˜ g = ρ − w ρ ! , ρ ∈ R , w ∈ C , (142)we find that the elements of the matrix U = g − ˜ g satisfy the relations: u = 0 , (143) u = u − ≡ u , u ∈ R ,ρ = ρ u ,w = ρ − u + w u . (144)Eq. (143) implies R = ρ − u − + | ρ − u + w u | ρ u ( ρ − u + w u ) ρ u ( ρ − ¯ u + ¯ w u ) ρ u ! . (145)Using the volume form on the 3D hyperboloid H +3 computed in appendix B (211),one obtains for the integration measure dR dR = 4 ρ dρ d w udud u . (146)Now the integral (139) takes the form h V bη ( z , ¯ z ) XV bη ( z , ¯ z ) i light = 4˜ λ − η ( z − z ) η (¯ z − ¯ z ) η (147) × Z ρ dρ d w udud u δ (cid:18) u + 1 u + | u | − κ (cid:19) ρ − + | w | ) η ρ η u η . We see that the delta function in the integrand of (147) can be different fromzero only for κ > u and then over u we obtain h V bη ( z , ¯ z ) XV bη ( z , ¯ z ) i light = (148)2 π ˜ λ − η (cid:16) ( κ + √ κ − − η − ( κ − √ κ − − η (cid:17) (1 − η )( z − z ) η (¯ z − ¯ z ) η × Z ρ dρ d w ρ − + | w | ) η ρ η . Performing the integral over w one gets Z ρ dρ d w ρ − + | w | ) η ρ η = 12 η − Z dρ ρ = 12 η − δ (0) . (149)The integral with respect to w converges if 2 η >
1. Having computed theintegral under this assumption, we can define it away from this region by analyticcontinuation. The integral with respect to ρ diverges. This divergence wasanalyzed in [19] and related to the infinite volume of the dilation group. It givesrise in fact to the δ (0) which appears in the two-point function of coincident fieldsof the continuous spectrum. We get a finite result taking the ratio h V bη ( z , ¯ z ) XV bη ( z , ¯ z ) i light h V ( z , ¯ z ) XV ( z , ¯ z ) i light = ˜ λ − η sinh 2 πσ (1 − η )(1 − η ) ( z − z ) η (¯ z − ¯ z ) η sinh 2 πσ . (150)Here we set κ = cosh 2 πσ .Using the properties of the Γ functions collected in appendix A one can cal-culate the light asymptotic limit of the ZZ function (77): W − α = ηb W − α =0 → ˜ λ − η − η , (151)and setting s = σb and α = ηb we also obtaincosh 2 πs (2 α − Q )cosh 2 πsQ → e − πη | σ | . (152)Hence, recalling (85) we get in the light asymptotic limit for the defect two-pointfunction derived via the bootstrap program h V bη ( z , ¯ z ) XV bη ( z , ¯ z ) ih V ( z , ¯ z ) XV ( z , ¯ z ) i → ˜ λ − η (2 η − e − πη | σ | ( z − z ) η (¯ z − ¯ z ) η . (153)In the limit of large σ we get full agreement between (150) and (153). Itmay happen that inclusion of one-loop determinants could make this agreement25xact for all values of σ . The study of this point is left for future work. It isinteresting to note, that in boundary conformal Toda field theory the agreementbetween the light asymptotic limit of boundary one-point function with the pathintegral calculations was also reached in [23] in the limit of the large boundarycosmological constant. In this section we consider the heavy asymptotic limit of two-point functions inthe presence of defects (85). Now we should find asymptotic behaviour of theinverse ZZ function (77) and of the factor cosh(2 πs (2 α − Q )) in the limit b → α = ηb , and s = σb . In the heavy asymptotic limit we should keep onlyterms having the form ∼ e /b .To understand the semiclassical origin of the denominator in (86) we findvery useful to consider, in the spirit of [34], analytic continuation of the Liouvilletheory with a complex η and complex saddle points.Taking η to satisfy the Seiberg bound (95) Re η < , and using properties ofΓ functions collected in appendix A, we obtain W − α = ηb → C ( b, η ) λ − η b π (cid:0) η − b (cid:1) exp (cid:18) η − b h log(1 − η ) − i(cid:19) . (154)where C ( b, η ) = − − / b Γ(2 η )(2 η − (155)= exp (cid:18) −
34 log 2 + iπ + log Γ(2 η ) − − η ) + 3 log b (cid:19) We see that all the terms in (155) are negligible compare to terms growing like ∼ e /b in the limit b →
0, and therefore C ( b, η ) can be omitted. The importanceof the term π ( η − b ) is explained in [34]. It was shown that this term in thesemiclassical interpretation arises as a sum over some “instanton” like sectors.As a preparation to this point we will expand this term in two ways as suggestedin [34]. Denoting y = e iπ (2 η − /b one can write1sin π (cid:0) η − b (cid:1) = 2 iy − y − = 2 i ∞ X k =0 y − (2 k +1) = − i ∞ X k =0 y k +1 . (156)26ne expansion is valid for | y | > | y | <
1. So either way, there is aset T of integers with 1sin π (cid:0) η − b (cid:1) = ± i X M ∈ T e iπ ( M ∓ / η − /b , (157)where T consists of nonnegative integers if Im(2 η − /b > η − /b < T in (157) can be understood as sum of saddle points in the minisu-perspace approximation keeping only constant mode of φ . In this approximationthe Liouville path integral becomes the integral representation of the Γ( x ) func-tion [34]: Γ( x ) = Z ∞−∞ dφ exp( − S ) , (158)where the minisuperspace action is S = − xφ + e φ . (159)The steepest descent analysis of the Γ( x ) function asymptotic behaviour for thelarge negative x , was carried out in [44]. It is based on the lengthy and carefulanalysis of the integration contours of the integral representation of the Γ( x )function (158), along which it converges in quadrants Re x <
0, Im x > x <
0, Im x <
0. In the physical literature it is reviewed in [34, 37]. In thisway we obtain the factor πx in (205) as a sum over the saddle points of theaction (159).Setting α = ηb and s = σb we easily obtain:cosh 2 πs (2 α − Q ) → e πb | σ | (1 − η ) . (160)Now we are in a position to write down the limiting form of the defectscorrelation functions.Inserting (154), (160) in (85) we can write in the heavy asymptotic limit h V α ( z , ¯ z ) XV α ( z , ¯ z ) i ∼ ( z − z ) − η (1 − η ) /b (¯ z − ¯ z ) − η (1 − η ) /b (161) × λ − ηb π (cid:0) η − b (cid:1) exp (cid:18) η − b h log(1 − η ) − i(cid:19) e πb | σ | (1 − η ) . Using also (157) we get h V α ( z , ¯ z ) XV α ( z , ¯ z ) i ∼ X M ,M ∈ T exp (cid:0) − S def M ,M (cid:1) , (162)27here b S def M ,M = − iπ ( M + M ∓ η −
1) + 4 η (1 − η ) log | z − z | (163) − (1 − η ) log λ − (4 η −
2) log(1 − η ) + (4 η − − π | σ | (1 − η ) . It is instructive to compare the heavy asymptotic limit of the defect two-pointfunction with the corresponding limit of the usual two-point function, computedin [34] h V α ( z , ¯ z ) V α ( z , ¯ z ) i ∼ | z − z | − η (1 − η ) /b (164) × λ (1 − η ) /b π (2 η − /b exp (cid:18) η − b [log(1 − η ) − (cid:19) . The relation of (161) to (164) naturally gives the heavy asymptotic limit of theeigenvalues D s ( α ) of the defect operator: D s ( α ) = h V α ( z , ¯ z ) XV α ( z , ¯ z ) ih V α ( z , ¯ z ) V α ( z , ¯ z ) i → e πb | σ | (1 − η ) sin π (cid:0) η − b (cid:1) . (165)Of course (165) can be also easily derived directly from (86) in the heavy asymp-totic limit. According to the general prescription of the semiclassical heavy asymptotic limit,we should find solutions of the Liouville equation, satisfying the defect equationsof motion and possessing the logarithmic singularities (94) at points z and z .The form of the solution of the defect equations of motion (46) and (47) impliesthat we should find functions A ( z ), C ( z ) and B (¯ z ) in such a way that φ has alogarithmic singularity at the point z and φ has a logarithmic singularity at thepoint z . Since the energy-momentum tensor is continuous across a defect thisimplies that we should find solutions possessing two singular points. Two-pointsolutions are well known ( see for example [34]) and we can build from them theAnsatz satisfying the defect equations of motion.To build the solution with the required singularities one should take a function A ( z ) which is smooth and holomorphic away from z and z . Let us take A ( z ) as A ( z ) = e ν ( z − z ) η − ( z − z ) − η . (166)28ne has also a = 1 √ ∂A = e − ν p ( z − z )(2 η −
1) ( z − z ) − η ( z − z ) η , (167) a = A √ ∂A = e ν p ( z − z )(2 η −
1) ( z − z ) η ( z − z ) − η . (168)Inserting (167) or (168) in (22) we obtain the energy-momentum tensor b T = η (1 − η )( z − z ) + η (1 − η )( z − z ) − η (1 − η ) z − z (cid:18) z − z − z − z (cid:19) , (169)which indeed possesses two singular points (98), with accessory parameters c = − c = 2 η (1 − η ) z − z . (170)The anti-holomorphic part is: B (¯ z ) = − (¯ z − ¯ z ) − η (¯ z − ¯ z ) η − , (171) b = B √ ¯ ∂B = 1 p (¯ z − ¯ z )(2 η −
1) (¯ z − ¯ z ) − η (¯ z − ¯ z ) η , (172) b = − √ ¯ ∂B = 1 p (¯ z − ¯ z )(2 η −
1) (¯ z − ¯ z ) η (¯ z − ¯ z ) − η . (173)Let us take the holomorphic part for φ as C ( z ) = e ν ( z − z ) η − ( z − z ) − η = e ν − ν ) A ( z ) , (174)and the antiholomorphic part again given by (171). Using (50) one gets κ = cosh( ν − ν ) . (175)Inserting (166), (174) and (171) in (46) and (47) we obtain: e − ϕ = λ (2 η − | z − z | (cid:16) e ν | z − z | η | z − z | − η (176) − e − ν | z − z | − η | z − z | η (cid:17) ,e − ϕ = λ (2 η − | z − z | (cid:16) e ν | z − z | η | z − z | − η (177) − e − ν | z − z | − η | z − z | η (cid:17) .
29t is easy to see that ϕ and ϕ given by (176) and (177) have the requiredsingularity (94) around z and z respectively. In fact each of the functions ϕ or ϕ given by (176) and (177) coincides with the solution describing a saddlepoint for a two-point function considered in [34]. But in [34] this solution wasconsidered on a full plane with the same parameter ν everywhere, whereas hereeach of them is considered on a corresponding half-plane, namely in (176) z belongs to the upper half-plane Σ , and in (177) z belongs to the lower half-planeΣ , and we should also remember that, z ∈ Σ and z ∈ Σ . The defect iscreated by the choice of different parameters ν and ν , ν = ν .From (176) and (177) we obtain ϕ = 4 iπN − log λ + 2 log(1 − η ) (178) − (cid:18) e ν | z − z | η | z − z | − η | z − z | − e − ν | z − z | − η | z − z | η | z − z | (cid:19) ,ϕ = 4 iπN − log λ + 2 log(1 − η ) (179) − (cid:18) − e ν | z − z | η | z − z | − η | z − z | + e − ν | z − z | − η | z − z | η | z − z | (cid:19) . Here N and N are integer. The possibility to add the term 4 iπN j , j = 1 , φ j → φ j + 2 πiN j /b , or multiplying by 2 b ,under ϕ j → ϕ j + 4 πiN j , j = 1 ,
2. Note that the bulk Liouville equation (7)is invariant under the symmetry ϕ j → ϕ j + 2 πiN j , and it is broken to ϕ j → ϕ j + 4 πiN j by the exponential terms of the defect action (29).To evaluate the action on the solutions (176), (177), we will use the strategyused in [20]. Namely we will write the system of differential equations which thisaction should satisfy. The first equation is (100), which given that η = η = η ,reads b ∂S defcl ∂η = − X − X . (180)where X i is defined in (94). The leading terms of ϕ around z are ϕ → − η log | z − z | + X , (181)where X = 4 πiN − log λ + 2 log(1 − η ) − (2 − η ) log | z − z | − ν . (182)30imilarly the leading terms of ϕ around z are ϕ → − η log | z − z | + X , (183)where X = 4 πiN − log λ + 2 log(1 − η ) − (2 − η ) log | z − z | + 2 ν . (184)Inserting (182) and (184) in (180) one obtains b ∂S defcl ∂η = − πi (2 N + 2 N )+2 log λ − − η )+(4 − η ) log | z − z | +2( ν − ν ) . (185)We would like to emphasize yet another difference from the calculation of theheavy asymptotic limit of the two-point function in [34]. In the case of the usualtwo-point function the integers N and N are equal since we have one continuousfunction φ . Here they can be different since we have two different functions ϕ and ϕ .The action with defect (103) implies also b ∂S defcl ∂κ = 1 iπ Z ∂ Σ e Λ b dτ . (186)Inserting (166) and (174) in eq. (49) one obtains e Λ b = 12 sinh( ν − ν ) (2 η − z − z )( z − z )( z − z ) . (187)Using that 1 i Z ∂ Σ dz ( z − z )( z − z ) = 2 π ( z − z ) , (188)we obtain b ∂S defcl ∂κ = 2 η − ν − ν ) . (189)Integrating equations (185) and (189) we obtain: b S def N ,N = − iπ (2 N + 2 N ) η + 4 η (1 − η ) log | z − z | (190)+2 η log λ − (4 η −
2) log(1 − η ) + 4 η − ( ν − ν )(1 − η ) + C , where C is a constant. To derive the penultimate term we should remember therelation (175). To fix the constant term we can directly compute the action (103)for the Ansatz (178)-(179) with η = 0: 31 = 4 iπN − log λ − log (cid:18) e ν | z − z | | z − z | − e − ν | z − z | | z − z | (cid:19) , (191) ϕ = 4 iπN − log λ − log (cid:18) e ν | z − z | | z − z | − e − ν | z − z | | z − z | (cid:19) . (192)Evaluation of the action (103) on the Ansatz (191), (192) is lengthy andexplained in appendix C. The result is b S = 2 iπ ( N + N ) − log λ − − ( ν − ν ) . (193)Comparing (193) with (190) fixes the constant C : C = 2 iπ ( N + N ) − log λ − . (194)Inserting this value of C in (190) we indeed obtain (163) if we set N = M , (195) N = M ∓ , (196)and 2 πσ = ν − ν . (197)Some comments are in order at this point:1. The action (190) satisfies the Polyakov relation (101) with the accessoryparameters defined in (170): b ∂S defcl ∂z i = ( − ) i +1 η (1 − η ) z − z , i = 1 , . (198)2. In eq. (157) M takes nonnegative integer values if Im(2 η − /b >
0, andnonpositive integer values if Im(2 η − /b <
0. Therefore N also runsover nonnegative or nonpositive integer values depending on the sign ofIm(2 η − /b , and N takes values { , , . . . } , when Im(2 η − /b > N takes values {− , − , . . . } , when Im(2 η − /b <
0. The fact that for thedifferent values of the parameter η we should take contribution of differentsets of the saddle points is known as the Stokes phenomena [34–37], and32as studied in detail for two- and three-point correlation functions of theLiouville field theory in [34]. Recall that it is caused by the fact that thesum (157) converges for the different values of M depending on the sign ofIm(2 η − /b . The values of parameters at which the jump of the set ofthe contributing saddle point occurs define a (anti-) Stokes line. We havea Stokes line if at some values of parameters the imaginary parts of theactions for two saddle points, say a and b , coincide: Im S a = Im S b . Wehave an anti-Stokes line if at some values of parameters the real parts ofthe actions for two saddle points, say a and b , coincide: Re S a = Re S b .Crossing these lines, a jump in the set of the contributing saddle point mayoccur. For the Stokes lines it is caused by the fact that there is a steepestdescent contour connecting two saddle points. For the anti-Stokes line it isimplied by the coincidence of the magnitudes of the amplitudes e S a and e S b for the different saddle points. From (163) or (190) we see that Re S def N ,N are the same for all N and N if Im(2 η −
1) = 0. The line Im(2 η −
1) = 0is the anti-Stokes line at which indeed we observe a jump in the set of thecontributing saddle points.3. The discussion above of the differences between the calculation of two-pointfunction with and without defect suggests nice interpretation of the defectoperator. We have seen that there exist two sources of discontinuity givingrise to the corresponding terms in the defect operators. The heavy asymp-totic limit of D ( α ) (165) has an exponential in the numerator and sinefunction in the denominator. The exponential term in the numerator aswe have seen originates from the discontinuity created by the choice of thedifferent parameters ν and ν . The correspondence between the N i and M i parameters makes clear that the different logarithmic branch solutions,given by N and N , are responsible for the quadratic sin π (cid:0) η − b (cid:1) term inthe (161). On the other hand, as we have mentioned before, in the heavyasymptotic limit the calculation of the usual two-point function one has N = N , and it reflects the presence of the term sin π (cid:0) η − b (cid:1) in the denom-inator of (164) in the first degree. Therefore the denominator sin π (cid:0) η − b (cid:1) in D ( α ) reflects the possibility of the choice of different logarithmic brancheswith N = N in the solution of the defect equations of motion. The finalquantum expression (86) results from the quantum corrections restoring b ↔ b − duality of the Liouville theory.33et us analyze in the heavy asymptotic limit also the relation (87) between pa-rameter s and A ( b ) 2 cosh 2 πbs = A ( b ) (cid:18) W ( − b/ W (0) (cid:19) . (199)It is easy to compute that lim b → W( − b / − √ λ . (200)Setting that s = σb , we get cosh 2 πσ = 2 A (0) λ . (201)This implies that the parameter κ is proportional to A (0): κ = 2 A (0) λ . (202)Note that in the light asymptotic limit as well as in the heavy asymptotic limitwe get the same relation between σ and κκ = cosh 2 πσ . (203) The methods developed in this paper can be applied to other theories with defects,like N = 1 superconformal Liouville theory, conformal and superconformal Todatheories.The Lagrangian of the N = 1 Liouville theory with defects is constructedin [14] using the technique of the type II integrable defects. The defect two-point functions in superconformal Liouville theory can as well be constructedvia the bootstrap program [45]. It is interesting to use the methods of thispaper to construct solutions for superconformal Liouville field theory of the defectequations of motion and study the light and heavy asymptotic limits.The defect operators in conformal Toda field theory are constructed in [5, 8].It is possible using methods of this paper together with the technique of type IIdefects to construct the Lagrangian of conformal Toda field theory with topolog-ical defects and compare with semiclassical limits of defect two-point functions.This program can as well be generalized to superconformal Toda field theory.34et us mention also other interesting problems where methods developed inthis paper can be applied.One of the most important problems regarding non-rational conformal fieldtheories is to find for them a relation to a three-dimensional topological field the-ory description similar to that of the rational ones. This is still a rather difficultand poorly studied problem. The first step was done in [46], where the classicalphase space of the Chern-Simons gauge theory with SL (2 , R ) gauge group hasbeen studied and shown to coincide with the Teichm¨uller space of Riemann sur-faces. It is established by now [47], that the Hilbert space of states obtained byquantizing the Teichm¨uller space is isomorphic to the space of conformal blocksof Liouville theory. The methods and solutions derived in this paper can be use-ful to elaborate on the relation between Chern-Simons gauge theory, Teichm¨ullerspace of Riemann surfaces, and Liouville field theory including also defects.Defects appear in many areas in String theory as well as in condensed matter.In particular they play an important role in the entropy entanglement prob-lems [48]. The methods of semiclassical calculations of the defect two-point func-tions developed here can be used also in that areas. As we mentioned in theintroduction heavy and light asymptotic limits appear in many instances of AGTand AdS/CFT correspondences. The insights gained in the study of these limitsin the presence of defects can be useful to incorporate defects in these problems. Acknowledgments
The work of H.P. was partially supported by the Armenian SCS grant 13-1C132and by the Armenian-Russian SCS grant-2013. The work of G.S. was partiallysupported by the Armenian SCS grant 13-1C278 and ICTP Network NET68.We would like to thank Rubik Poghossian and Shmuel Elitzur for many usefuldiscussions. We thank also the JHEP referee for his valuable comments andsuggestions which contributed to the substantial improvement of the text of thepaper. 35
Properties of Γ functions The limiting behavior of the terms with Γ functions can be calculated using theapproximation Γ( x ) ∼ e x log x − x + O (log x ) . (204)for x with large positive real part.For x with negative real part using the formulaΓ( x )Γ( − x ) = − πx sin πx , (205)one can bring problem to the previous case.We also need the well-known behavior of the Γ( x ) function for x around zero:Γ( x ) ∼ x . (206) B Volume form on the 3D hyperboloid H +3The 3D hyperboloid H +3 is a pseudo-sphere X − X − X − X = 1 (207)in the ambient Minkowski space with the metric: ds = − dX + dX + dX + dX . (208)In the parametrization (141), one has X − X = 1 ρ + | w | , (209) X + X = ρ ,X + iX = ρw ,X − iX = ρ ¯ w . Substituting (209) in (208) one obtains ds = 4 dρ ρ + ρ d (cid:18) wρ (cid:19) d (cid:18) ¯ wρ (cid:19) . (210)The corresponding volume form is √ det Gdρd w = 2 ρdρd w . (211)36 Action evaluation
The solutions (191) and (192) have the form: ϕ = 4 iπN − log λ − Z , (212) ϕ = 4 iπN − log λ − Z , where Z = s z ¯ z + t z + u ¯ z + v , (213) Z = s z ¯ z + t z + u ¯ z + v , with s j = ± ν j | z − z | , u j = ± e − ν j z − e ν j z | z − z | , (214) t j = ± e − ν j ¯ z − e ν j ¯ z | z − z | , v j = ± e ν j | z | − e − ν j | z | | z − z | , j = 1 , . where we take upper signs for ν j positive and lower signs for ν j negative. Thischoice of signs makes s j ≥
0. Note that s j v j − u j t j = − j = 1 , . (215)It is useful to introduce also real and imaginary parts of u i and t i : t j = m j + in j , u j = m j − in j , j = 1 , . (216)The function ˜Λ can be found setting η = 0 in (187) e − ˜Λ / = 2 sinh( ν − ν ) z − z ( z − z )( z − z ) . (217)Before starting the calculations one should examine the zeros of Z and Z . Itis easy to see, that Z j as a quadratic form, vanishes on a circle C j with thecenter (cid:16) − m j s j , n j s j (cid:17) and the radius s j , j = 1 ,
2. Since we have the topologicaldefect, as long as the discs confined by C and C do not overlap, we can avoidsingularities moving the defect to the safe region between C and C . Rememberthat the defect is located along the horizontal axis, and ϕ and ϕ are consideredon the upper and lower half-planes respectively. Therefore Z has no zeros if C is located in the lower half-plane and Z has no zeros if C is located in the upperhalf-plane. This happens, when n < − n > . (219)These constraints enable us to avoid the singularities.Check when these constraints are satisfied. Writing z = x + iy , and z = x + iy , we get from eq. (214): n j = ± (cid:18) e ν j y − e − ν j y | z − z | (cid:19) . (220)Recalling that y >
0, and y <
0, and that we should take upper signs forpositive ν j and lower sign for negative ν j , we see that we obtain negative n andpositive n taking ν > , and n = e ν y − e − ν y | z − z | , (221) ν < , and n = e − ν y − e ν y | z − z | . (222)Taking | ν j | big enough we can always satisfy the condition | n j | >
1. This meansalso that we take in (214) the upper sign for j = 1 and the lower sign j = 2.Let us now insert the solution (191) and (192) in the action (103). We willevaluate each term in the R → ∞ limit. Start by computing the bulk part. Thebulk Lagrangians can be written as a total derivative:18 πi (cid:0) ∂ϕ j ¯ ∂ϕ j + 4 λe ϕ j (cid:1) = ∂ ¯ z K jz − ∂ z K j ¯ z , j = 1 , , (223)where K jz = 14 πi (cid:18) − s j z + u j )( s j | z | + t j z + u j ¯ z + v j ) + s j log( s j | z | + t j z + u j ¯ z + v j )( s j z + u j ) (cid:19) , (224)and K j ¯ z = ¯ K jz . We see that under the conditions (218) and (219) the denominatorsin (224) have no singular points.The integral over the r.h.s. of (223) reduces to the contour integral: Z Σ Rj (cid:0) ∂ ¯ z K jz − ∂ z K j ¯ z (cid:1) d z = Z R dτ ( K jz + K j ¯ z ) + Z s Rj ( iK jz z − iK j ¯ z ¯ z ) dθ . (225)The integral over the semi-circle of the big radius R is evaluated to yield Z s Rj ( iK jz z − iK j ¯ z ¯ z ) dθ = 12 log s j R . (226)38n the other hand the regularizing terms in the action produce12 π Z s R ϕ dθ + log R = − log (cid:0) s R (cid:1) + log R + 2 iπN −
12 log λ , (227)12 π Z s R ϕ dθ + log R = − log (cid:0) s R (cid:1) + log R + 2 iπN −
12 log λ . (228)The integral over the real axis of the first term in K j gives − π Z dτ n j ( s j τ + u j )( s j τ + t j )( s j τ + 2 m j τ + v j ) = − n j q n j − n j ) (229)and of the second produces12 π Z dτ n j s j log( s j τ + 2 m j τ + v j )( s j τ + u j )( s j τ + t j ) = − log h sgn( n j ) (cid:16) n j − q n j − (cid:17)i −
12 sgn( n j ) log s j . (230)Here we introduced the sign function sgn( x ) ≡ x | x | .Remembering that for ϕ and ϕ the integrals over the real axis run in theopposite directions, we obtain finally:18 πi Z Σ R (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z + 18 πi Z Σ R (cid:0) ∂ϕ ¯ ∂ϕ + 4 λe ϕ (cid:1) d z (231)+ 12 π Z s R ϕ dθ + log R + 12 π Z s R ϕ dθ + log R = − n p n − n p n − " n − p n − − n + p n − +2 iπN + 2 iπN − log λ − . Now we turn to the calculation of the integrals living on the defect. The sumof the last two terms in (103), according to the equations of motion (33) and (34)is − πi Z ∞−∞ d τ ( ∂ − ¯ ∂ )( ϕ − ϕ ) = (232)1 π Z ∞−∞ d τ (cid:18) n s τ + 2 m τ + v − n s τ + 2 m τ + v (cid:19) = n p n − − n p n − . We see that (232) cancels the first two terms in the third line of (231).39ow let us compute the second term on the defect:18 πi Z ∞−∞ d τ ˜Λ( ∂ + ¯ ∂ )( ϕ − ϕ ) =1 πi Z ∞−∞ d τ log [( τ − z )( τ − z )] (cid:18) s τ + m s τ + 2 m τ + v − s τ + m s τ + 2 m τ + v (cid:19) (233)= log " − m + i p n − − z s m + i p n − z s − log " − m + i p n − − z s m + i p n − z s . To simplify this expression one can show, introducing an angle e iξ = z − z | z − z | ,that − m j + i q n j − − z s j = i (cid:16) ( − ) j ie − ν j e iξ − n j + q n j − (cid:17) , (234) m j + i q n j − z s j = i (cid:16) ( − ) j +1 ie ν j e iξ + n j + q n j − (cid:17) , j = 1 , . We can also prove (cid:16) ( − ) j ie − ν j e iξ − n j + q n j − (cid:17) (cid:16) ( − ) j +1 ie ν j e − iξ − n j + q n j − (cid:17) , (235)= ( − ) j (cid:16) − n j + q n j − (cid:17) (cid:0) ie − ν j e iξ − ie ν j e − iξ − ( − ) j n j (cid:1) , and writing z = x + iy and z = x + iy , one obtains that (cid:0) ie − ν j e iξ − ie ν j e − iξ − − ) j n j (cid:1) = − i sinh ν j | z − z | ( x − x + i ( y + y )) . (236)And finally we need (cid:16) ( − ) j +1 ie ν j e − iξ − n j + q n j − (cid:17) (cid:16) ( − ) j +1 ie ν j e iξ + n j + q n j − (cid:17) (237)= − e ν j (cid:16) cosh ν j + ( − ) j i cos ξ q n j − − ) j n j sin ξ (cid:17) . Using all these identities, and noting that the terms in the r.h.s. of (236), inde-pendent on j , get canceled, we obtain18 πi Z ∞−∞ d τ ˜Λ( ∂ + ¯ ∂ )( ϕ − ϕ ) = (238)log (cid:16) cosh ν + i cos ξ p n − n sin ξ (cid:17) sinh ν (cid:16) − n + p n − (cid:17)(cid:16) cosh ν − i cos ξ p n − − n sin ξ (cid:17) sinh ν (cid:16) n − p n − (cid:17) + ν − ν . The third multipliers in the numerator and in the denominator of the argumentof the logarithm in (238) together cancel the third term in the third line of (231).40t is easy to see that the remaining logarithmic term after this cancellation is apure argument since the remaining numerator and denominator have the samemodulus:1sinh ν j h (cosh ν j + ( − ) j n j sin ξ ) + ( n j −
1) cos ξ i = ( x − x ) + ( y + y ) | z − z | . (239)And finally the first integral on the defect is − πi Z ∞−∞ d τ ( ϕ ∂ τ ϕ − ϕ ∂ τ ϕ ) = (240) − πi Z ∞−∞ d τ h log( s τ + 2 m τ + v ) s τ + m s τ + 2 m τ + v − log( s τ + 2 m τ + v ) s τ + m s τ + 2 m τ + v i =log " − s m + s m + i ( s p n − s p n − s m − s m + i ( s p n − s p n − . Obviously this is also a pure argument. After cumbersome but straightforwardcalculation one can show that:( − s m + s m + i ( s p n − s p n − s m − s m + i ( s p n − s p n − ν + i cos ξ p n − n sin ξ ) sinh ν (cosh ν − i cos ξ p n − − n sin ξ ) sinh ν = 1(241)and therefore (240) cancels the remaining logarithmic terms in (238). Collectingall we obtain: b S = 2 iπ ( N + N ) − log λ − ν − ν . (242) D Defect two-point function
First let us briefly explain how to derive the Cardy-Lewellen cluster condition fordefects in rational theories without multiplicities [7, 9]. Suppose we have a two-dimensional rational conformal field theory with primary fields Φ i . The vacuumstate is attributed i = 0. A topological defect X is a sum of projectors X = X i D i P i (243)where P i = X N, ¯ N ( | i, N i ⊗ | i, ¯ N i )( h i, N | ⊗ h i, ¯ N | ) (244)41ere | i, N i and | i, ¯ N i are vectors of orthonormal bases of left and right copies ofthe highest weight representations R i respectively. Two-point functions with adefect X insertion can be written as h Φ i ( z , ¯ z ) X Φ i ( z , ¯ z ) i = D i ( z − z ) i (¯ z − ¯ z ) i , (245)where D i = D i C ii (246)and C ii is a two-point function.The fields Φ i via the operator product expansion (OPE) form an algebra withstructure constant C kij [49, 50]:Φ i ( z , ¯ z )Φ j ( z , ¯ z ) = X k C kij ( z − z ) ∆ i +∆ j − ∆ k (¯ z − ¯ z ) ∆ i +∆ j − ∆ k Φ k ( z , ¯ z )+descendants . (247)We need also to introduce the fusion number N kij . This is the number of occur-rence of the field Φ k in the operator product expansion of Φ i and Φ j . Here weassume that N kij takes two values: 0 and 1. Consider the following four-pointcorrelation function with the defects insertions on a torus: h Φ j ( z , ¯ z )Φ i ( z , ¯ z ) X Φ i ( z , ¯ z )Φ j ( z , ¯ z ) X i . (248)Using (247) and (245) one can compute (248) in two pictures. In the first pic-ture at the beginning we use OPE (247) for the pairs Φ j ( z , ¯ z )Φ i ( z , ¯ z ) andΦ i ( z , ¯ z )Φ j ( z , ¯ z ) and then (245) for the fields produced in this process. Thisresults in X k D k D C kij F k " i ij j , (249)where F k " i ij j is the so called conformal block [49,50] giving the contributionof the descendant fields in the OPE (247). It appears squared since it is separatelyproduced by the left and right modes.In the second picture we move the field Φ j ( z , ¯ z ) to the rightmost position: h Φ i ( z , ¯ z ) X Φ i ( z , ¯ z )Φ j ( z , ¯ z ) X Φ j ( z , ¯ z ) i (250)42nd then use twice (245) resulting in D i D j F " i ji j + · · · . (251)Using the fusing matrix: F k " i ij j = X m F km " j ji i F m " i ji j , (252)we obtain X k D D k C kij F k " j ji i = D i D j . (253)This is the Cardy-Lewellen cluster condition for defects.Using that for rational conformal field theory the structure constants and thefusion matrix satisfy the relation [51] C pij F p, " j ji i = ξ i ξ j ξ ξ p , (254)where ξ i = p C ii F i , (255)and F i ≡ F , " i ii i , (256)the Cardy-Lewellen condition for defects (253) simplifies to X k D D k N kij (cid:18) ξ i ξ j ξ ξ k (cid:19) = D i D j . (257)Define Ψ k as D k D = Ψ k (cid:18) ξ k ξ (cid:19) . (258)Eq. (257) becomes the following equation for Ψ k X k Ψ k N kij = Ψ i Ψ j . (259)And finally to find the coefficient D i of the defects expansion to projectors weshould, according to (246), divide D i by the two-point function.43et us now apply this machinery to the Liouville theory. Liouville theory isa non-rational theory, but we can overcome the difficulties caused by the infinitenumber of primaries. First of all it is shown in [8] that the relation (254) worksalso in diagonal non-rational theories. In particular it is shown in [8] that in theLiouville theory ξ ( α ) = p W (0) W ( Q ) W ( α ) . (260)and (254) takes the form: C α α ,α F α , " α α α α = W (0) W ( α ) W ( α ) W ( α ) , (261)where W ( α ) is ZZ function (77). The second problem is that in the Liouvilletheory the OPE of primary fields with generic α , and α contains infinite numberof intermediate primary states, which makes the use of the equation (259) ratherproblematic. This difficulty can be resolved via Teschner’s trick [52]. Teschner’stricks relies on the existence of degenerate fields in the Liouville field theory. Thefields V α with α belonging to the set α m,n = 1 − m b + 1 − n b , m, n ∈ N (262)produce in the OPE with other fields just a finite number of the fields. Teschner’strick suggests to take as Φ j one of the fields V α m,n . This choice will yield onlyfinite number of terms in the l.h.s. of (259). The simplest of the fields (262) is V − b/ . With a generic field V α it has the OPE: V α V − b/ ∼ C α − b/ − b/ ,α V α − b/ + C α + b/ − b/ ,α V α + b/ . (263)With j = − b , i = α , and k = α ± b/
2, the equations (259) and (258) take theform: Ψ( α )Ψ( − b/
2) = Ψ( α − b/
2) + Ψ( α + b/ , (264)and D ( α ) D (0) = Ψ( α ) (cid:18) W (0) W ( α ) (cid:19) (265)The solution of the equation (264) isΨ m,n ( α ) = sin( πmb − (2 α − Q )) sin( πnb (2 α − Q ))sin( πmb − Q ) sin( πnbQ ) , (266)Using (265) we obtain for the defect two-point function44 m,n ( α ) = − √ πmb − (2 α − Q )) sin( πnb (2 α − Q )) W ( α ) . (267)And finally dividing on S ( α ) (75) we get D m,n ( α ) = sin( πmb − (2 α − Q )) sin( πnb (2 α − Q ))sin πb − (2 α − Q ) sin πb (2 α − Q ) . (268)Note that the defect given by ( m, n ) = (1 ,
1) is the identity defect.But this is not the end of the story. Let us now explain how to obtain two-point function for the continuous family of defects. We will use the strategydeveloped in [23, 53] in the context of the Liouville and Toda theories with aboundary. Assume that we have a family of defects parameterized by κ . In thiscase D ( − b/ V − b/ in thepresence of defect, will be a function of κ and b . Denote the ratio D ( − b/ /D (0)by A ( κ, b ) and define D ( α ) = ˜Ψ( α ) W ( α ) . (269)Substituting A ( κ, b ) and ˜Ψ( α ) in (257) again for j = − b , i = α , and k = α ± b/ α ): (cid:18) W ( − b/ W (0) (cid:19) A ˜Ψ( α ) = ˜Ψ( α − b/
2) + ˜Ψ( α + b/ . (270)The solution of (270) is indeed a one-parametric family,˜Ψ s ( α ) = − / cosh(2 πs (2 α − Q )) , (271)with a parameter s related to A by2 cosh 2 πbs = A (cid:18) W ( − b/ W (0) (cid:19) . (272)Substituting (271) in (269) we obtain for D s ( α ) and D s ( α ) respectively D s ( α ) = − / cosh(2 πs (2 α − Q )) W ( α ) . (273) D s ( α ) = cosh(2 πs (2 α − Q ))2 sin πb − (2 α − Q ) sin πb (2 α − Q ) . (274)We would like to finish by a remark on the world-volume of the defects (268) and(274). Recall the notion of the defect world-volume [54].45he values of the Liouville fields φ and φ on a point τ of the defect lineform a point ( φ ( τ ) , φ ( τ )) in the plane R . The set of all such points may berestricted to belong to a submanifold Q of the plane R , depending on the defectcondition. The submanifold Q is called the world-volume of the defect. 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