Multi-point passage probabilities and Green's functions for SLE{}_{8/3}
aa r X i v : . [ m a t h - ph ] F e b Multi-point passage probabilities and Green’sfunctions for SLE / Abstract
We consider a loop representation of the O ( n ) model at the criticalpoint. When n = 0 the model represents ensembles of self-avoidingloops (i.e., it corresponds to SLE with κ = 8 / c = 0. We focus on the correlation functions in the upper-half planecontaining the twist operators in the bulk, and a pair of the boundaryone-leg operators. By using a Coulomb gas representation for thecorrelation functions, we obtain explicit results for probabilities of theSLE / trace to wind in various ways about N ≥ Schramm-Loewner evolution (SLE) provides a conventional framework tostudy fractal curves or sets growing into simply connected planar domains
D ∈ C [1]. This approach focuses on constructing measures on randomcurves that occur in such systems. In the simplest setting of SLE from x to x (such that x , x ∈ ∂ D ) the measure is generated dynamically by evolvingthe curve starting from one end point. Conventionally, the domain is takento be the upper-half plane H = { z ∈ C : Im z > } . Then, the curve γ t evolving up to time t (or rather its hull K t ) is characterized by the conformalmapping, g t : H \ K t → H , normalized so that g t ( z ) ∼ z + 2 t/z + O ( z − ) as z → ∞ . This function satisfies Loewner equation: dg t ( z ) dt = 2 g t ( z ) − √ κB t , (1)where B t is a standard Brownian motion, and the real parameter κ has abig influence on the geometric properties of SLE κ . The random curves aresimple paths provided that κ ≤
4; when 4 < κ ≤ κ ≥ P κ ( z ), that the curvepasses to the left of a given point z ∈ H [3]: P κ ( z ) = 12 + Γ(4 /κ ) √ π Γ( − κ κ ) xy F (cid:18) , κ ,
32 ; − x y (cid:19) , (2)where F ( a, b, c ; x ) is a hypergeometric function, and z = x + iy . When κ = 8 / P / ( z ) = + x | z | . Later, an analogousformula for the two-point function was predicted by Simmons and Cardy byusing conformal field theory (CFT) techniques provided that κ = 8 / P / ( z, w ), that the SLE / curve passes to theleft of both points, z = x + iy and w = u + iv , has the form: P / ( z, w ) = P / ( z ) P / ( w ) (cid:18) yx + | z | vu + | w | G ( σ ) (cid:19) , (3)with G ( σ ) = 1 − σ F (cid:18) ,
43 ; 53 ; 1 − σ (cid:19) . (4)Here σ = | z − w | / | z − ¯ w | is a cross-ration of the points { z, w, ¯ z, ¯ w } , andbar stands for complex conjugation.Let us briefly discuss the result of Simmons and Cardy [4]. Their approachuses an intimate relation between SLE κ and CFT, which allows one to studycritical curves using CFT methods [5]. It is well known that many two-dimensional statistical systems, e.g., O ( n ) model and percolation, can bemapped to an equivalent loop representation [6]. Various loop ensemblescan be conventionally described in terms of SLE [1, 2]. Alternatively, theloop model can be mapped to a height model via Coulomb gas. In thecontinuum limit the latter model is described by CFT. Hence, it becomespossible to study loop ensembles in the CFT framework. We briefly describethis relation in Section 2.An essential part of the Simmons-Cardy construction is the identificationof the twist operators with the 0-weight Schramm’s operator provided that κ = 8 /
3. Twist operators at the points z i ∈ H modify statistical weights ofthe loops which wind in various ways about these points. Roughly speaking,the correlation function containing a single twist operator at the point z ∈ H counts an expected number of loops which separate z from the boundary.In Section 3.1 we show, that the correlation function is closely connected toSchramm’s formula (2). In a similar way, the correlation function containingtwo twist operators in H can be used to count an expected number ofloops which separate both point from the boundary. In Ref. [4] Cardy andSimmons showed that this function is closely connected to the two-pointprobability (3). 2ne purpose of this article is to generalize Simmons-Cardy result to thecase of N ≥ H . In this case the system of PDEs which governsthe corresponding probabilities is very difficult to solve directly. We takeadvantage of the CFT technique, namely, the Coulomb gas formalism [7, 8],which provides a tractable approach to constructing explicit solutions. Asthe result, we obtain explicit expressions for probabilities of the SLE / trace to wind in various ways about N ≥ H . Remarkably, thisresult can be used to study SLE multi-point Green’s functions. Indeed, theprobability that the SLE trace passes between the points z , z ∈ H becomesthe one-point SLE Green’s function as the points collapse to one. Similarly,one expects that the 2 N -point passage probability becomes the N -pointGreen’s function as the points z , z , . . . , z N collapse pairwise.The structure of the paper is straightforward. In Section 2 we brieflyreview the O ( n ) model, which serves as a connection between SLE and CFT.We also introduce the twist and legs operators, and describe their conformalproperties. In Section 3 we use CFT technique to calculating probabilitiesof SLE / trace to wind in various ways about 1 , , . . . , N marked point inthe upper-half plane. In particular, in Section 3.4 we obtain Coulomb gasrepresentation for the N -point passage probabilities of SLE curves. Section 4is devoted to multi-point Green’s functions of SLE / curves in the upper-half plane. We obtain explicit expressions for the Green’s functions in termsof the correlation functions of 1/3-weight operators in the bulk, and 1-legoperators on the boundary in c = 0 logarithmic CFT in H . Finally, we drawour conclusion. O ( n ) model, CFT and SLE Let us start with a standard loop representation of the O ( n ) model with n -component spins s ( r i ), such that s ( r i ) = 1, on the lattice. The partitionfunction of the O ( n ) model has the form: Z = Tr Y h ij i (1 + x s ( r i ) · s ( r j )) , (5)where x is a parameter of the model, and the product in (5) is over pairsof nearest neighbors. One can expand the product into a sum of 2 K terms(where K is the number of nearest neighbors), so that each term is associatedto a graph on the lattice in what follows: the bond between r i and r j isincluded in the graph if the factor x s ( r i ) · s ( r j ) appears in the expansion.Note, that only the graphs composed of closed loops contribute to the sum.The partition function takes a particularly simple form if the model isconsidered on a honeycomb lattice, where the loops can visit each site amaximum of one time. Because Tr s a ( r i ) s b ( r j ) = δ ab , each loop contributesa total weight n to the partition function. Besides, each occupied bond3ontributes by the factor x . Hence, the partition function is equivalent to Z = X Λ n N x L , (6)where the sum is taken over all closed non-intersecting loop configurationsΛ on the honeycomb lattice, N is the number of loops, and L is the totallength of loops in each configuration.The long loops are suppressed for small value of x , so that the modelflows to vacuum under the renormalization group flow. For large valuesof x the system flows to a fixed point of densely packed loops. At theboundary between these two regimes there exists a critical point at x = x c with x c = (2+ √ − n ) − / , for which the mean loop length diverges, and thesystem flows to the dilute fixed point. At this point the model is supposedto be conformally invariant. Hence, it becomes possible to study the O ( n )model in the CFT framework.The loop model can be mapped to the Coulomb gas by replacing asum over closed loops in (6) by a sum over configurations with orientedloops. This can be achieved by inserting the factors e iπχ ( e − iπχ ) at eachvertex where the curve turns to the right (left), and sum over two possibleorientations of each loop. As the result, each closed loop on the honeycomblattice contributes the factor e πiχ + e − πiχ to the partition function. Becausethe contribution of each loop should be n , one concludes n = 2 cos 6 πχ. (7)The model of oriented loops can be mapped further into a height model.The directed loops can be treated as the level lines of a height variable, h ( r ),on the dual of the lattice, provided that the height variable changes by π ( − π ) whenever one crosses a loop pointing to the right (left). There existsa one-to-one correspondence between a given configuration of heights and aunique graph of oriented loops. One can argue that under the renormalizationgroup flow the height model flows into a free field theory with the action S [ h ( r )] = ( g ( n ) / π ) R ( ∂h ( r )) d r , where g ( n ) is a constant determined by n . Hence, it becomes possible to use field theoretical methods in order tomake precise calculations in the continuum limit of the model.As explained, the O ( n ) model describes ensembles of closed paths onthe lattice. One can argue, that in the continuum limit the measure on thecurves is given by SLE κ , and (see, e.g., Ref [9]) n = 2 cos ( κ − πκ , (8)where 2 < κ < < κ for a dense phase. Thementioned correspondence between the O ( n ) model and CFT implies that4he loop models can be described by rational CFTs with central charge andconformal weights given by c = (6 − κ )(3 κ − κ , h r,s = ( κr − s ) − ( κ − κ . (9)Below, we consider only the dilute regime with 2 < κ < κ = 8 / c = 0. LCFTs are characterized by presence oflogarithmic structure in the operator product expansion explained by indecomposablerepresentations that occur in fusion of primary operators [10, 11]. In otherwords, there exist primary operators with degenerate scaling dimensionconstituting a Jordan block structure.The so-called twist operators introduced in Ref. [12] play a crucial rolein Simmons-Cardy construction of the probabilities for the SLE trace towind in various ways about marked points in H . A pair of twist operatorschanges the weights of all loops that separate them. Because the weightsof the loops separating the twist operators is − n , the partition function forthe loop model in presence of twist operators takes the form: Z = X Λ ( − N s n N x L , (10)where N s is the number of loops separating the twist operators. Hence, thetwist operators can be used to count loops with weights − n rather than n .The scaling dimension of twist operators can be calculated explicitly [12].Remarkably, the twist operators correspond to CFT primary fields , Φ , ,and, therefore, their Kaˇc weights read: h twist = h , = 3 κ − , (11)The twist operators are spinless, so that the antiholomorphic dimensioncoincides with the holomorphic one, ¯ h twist = h twist .Another set of operators considered in the SLE/CFT correspondenceare the so-called boundary K -leg operators anchoring SLE traces to theboundary of the domain. In the Coulomb gas framework these operatorschange the boundary conditions by K steps within ǫ -neighbor of their insertion,and can be identified with the boundary primary operators, Φ ,K +1 , withthe weights: h K − leg = h ,K +1 = K (4 + 2 K − κ )2 κ . (12)We will use a pair of the 1-leg boundary operators at the points x , x ∈ R to encode the SLE process in H from x to x . The two-point correlationfunction of such operators is fixed by scale invariance, H ( x , x ) = h Φ , ( x )Φ , ( x ) i H = ( x − x ) − h , , (13) Here and below, we adapt the notation from Ref. [4] h· · · i H denotes the correlation function in H , and we set the normalizationconstant to 1 by choosing an appropriate normalization of the fields. / trace In Ref. [4] it is shown, that the correlation function containing a pair of the1-leg boundary operators, Φ , , and the bulk twist operator, Φ , , determinesprobabilities of the SLE / trace to wind in various ways about the pointin H . This result can be easily generalized to the case of N ≥ / trace. It is closely connected tothe correlation function of the boundary 1-leg operators in presence of thetwist defect at the point z ∈ H [4]: H ( z, ¯ z ; x , x ) = h Φ , ( z, ¯ z )Φ , ( x )Φ , ( x ) i H , (14)where bar stands for complex conjugation, and x , x ∈ R . As per usual CFTapproach, the correlation function H ( z, ¯ z, x , x ) in H can be representedas the correlation function in C [13], H ( z, z ∗ ; x , x ) = h Φ , ( z )Φ , ( z ∗ )Φ , ( x )Φ , ( x ) i , (15)subjected to certain constraints on R specified below. By h· · · i in (15) wedenoted the correlation function in the complex plane C , and the points z, z ∗ are treated as the independent variables (one sets z ∗ = ¯ z at the end of thecomputation).CFT methods allow one to derive a set of second orders PDEs satisfiedby the correlation functions containing null state operators, e.g., Φ , andΦ , [5]. In particular, one can show that the correlation function (15)satisfies the following equations: (cid:20) ∂ z h , ) − h , ( z ∗ − z ) + ∂ z ∗ z ∗ − z − h , ( x − z ) + ∂ x x − z − h , ( x − z ) + ∂ x x − z (cid:21) H = 0 , (cid:20) ∂ x h , ) − h , ( z ∗ − x ) + ∂ z ∗ z ∗ − x − h , ( z − x ) + ∂ z z − x − h , ( x − x ) − ∂ x x − x (cid:21) H = 0 . (16)These equations have the common solution, H ( z, z ∗ ; x , x ) = ( z − z ∗ ) − h , ( x − x ) − h , G ( η ) , (17)6here G ( η ) is the function of the cross ratio η : G ( η ) = 2 − η √ − η , η = ( z − z ∗ )( x − x )( z − x )( x − z ∗ ) . (18)The function G ( η ) has a branch cut from 1 to ∞ . By noting that 1 − η =[( z ∗ − x )( x − z )] / [( z − x )( x − z ∗ )] and setting { z, z ∗ , x , x } → { z, ¯ z, , ∞} ,we conclude, that the choice of the branch of the square root is determinedby the argument of z . We obtain H ( z, ¯ z ; 0 , ∞ ) H (0 , ∞ ) = (2 Im z ) − κ/ Re z | z | , (19)where H is the two-point function of the boundary 1-leg operators (13).Remarkably, one can obtain exact solution to the system of equations (16)via Coulomb gas formalism introduced by Dotsenko and Fateev [7, 8]. In thisapproach one uses a representation of the conformal fields in terms of thevertex operators build from a free boson with specific boundary conditions.In particular, there exists a one-to-one correspondence between the primaryfields Φ r,s ( z ) with the conformal weights (9) and the vertex operators V r,s ( z ) ≡ V α r,s = e i √ α r,s ϕ ( z ) , (20)where ϕ ( z ) is a free boson specified by the two-point function h ϕ ( z ) ϕ ( w ) i = − ln( z − w ), and α r,s is the so-called charge of the vertex operator: α r,s = 12 (1 − r ) α − + 12 (1 − s ) α + . (21)Here α ± are determined by the central charge c of CFT as follows: c = 1 − α , α + + α − = 2 α , α + α − = − . (22)Below, we will use the parametrization relevant for SLE/CFT correspondence: α + = 2 √ κ , α − = − √ κ . (23)We refer to α in (22) as to the background charge, because Coulombgas formalism implies an existence of the chiral operator with the charge − α at infinity. The background charge specifies the conformal dimensionof the vertex operator, V α ( z ), as follows: h α = α ( α − α ) . (24)Note, that the conformal dimension (24) is invariant under α → α − α ,so that the vertex operators V α and V α − α have the same dimensions.Therefore, the conformal field Φ r,s can be associated to two different vertex7perators, V r,s and V − r, − s , implying that the correlation functions of conformalfields may be evaluated in several different but equivalent ways.Because the two-point function of the free boson ϕ ( z ) has a simple form,the correlation function of vertex operators can be written as h V α ( z ) V α ( z ) · · · V α n ( z n ) i = Y i 0, and theexpression on the right-hand side of (35) goes to H ( x , x ). Therefore, thecoefficient in front of Π in equals 1. Next, to find coefficient of Π out we send z, ¯ z → x ∈ R \ [ x , x ]. In this limit we have Π in → 0, Π out → 1, while theexpression on the right-hand side goes to − 1, thus justifying the coefficient − out .The system of equations (35) determines probabilities for the SLE / trace to wind in various ways about the point z ∈ H . In particular, the These coefficients depend on κ . However, in the case κ = 8 / ± z from the interval [0 , ∞ ],i.e., the left-crossing probability, P L ( z ), reads P L ( z ) = Π out Π out + Π in = 12 − H ( z, ¯ z ; x , x )2 H ( x , x ) = 12 + cos(arg( z ))2 . (37)In the last equality we set x = 0 and x = ∞ , and took into account (33).As expected, we obtain Schramm’s formula (2) for κ = 8 / / passage probabilities We already noted that the SLE / left-crossing probability is determined bythe correlation function containing the twist operator, Φ , , and a pair ofthe boundary 1-leg operators, Φ , . Similarly, in presence of two markedpoints in the bulk, z , z ∈ H , the passage probabilities are determined bythe correlation functions of two twist operators at the points z , z , and apair of the 1-leg boundary operators at the points x , x ∈ R : H ( z , ¯ z , z , ¯ z ; x , x ) = h Φ , ( z , ¯ z )Φ , ( z , ¯ z )Φ , ( x )Φ , ( x ) i H . (38)The correlation function (38) is specified by the boundary conditions, whichdetermine an appropriate linear combination of two conformal blocks thatcontribute to the correlation function.Before defining conformal blocks, it is convenient to recast the correlationfunction in H into the correlation function in C . By replacing the antiholomorphiccoordinates, ¯ z , ¯ z ∈ H , by the holomorphic coordinates, z ∗ , z ∗ ∈ C , weconsider the 6-point correlation function H ( z , z ∗ , z , z ∗ ; x , x ) = h Y i =1 Φ , ( z i )Φ , ( z ∗ i )Φ , ( x )Φ , ( x ) i . (39)The conformal symmetry implies that it can be written in the form: H ( z , z ∗ , z , z ∗ ; x , x ) = G ( η , η , η )( x − x ) h , ( z − z ∗ ) h , ( z − z ∗ ) h , . (40)Here G ( η , η , η ) is a function of the cross-ratios η = η ( z ∗ ), η = η ( z ),and η = η ( z ∗ ), where η ( s ) = ( z − s )( x − x )( z − x )( s − x ) . (41)In particular, η ( s ) = 1 − s/z when x = 0, and x → ∞ . One can alsoconsider the correlation function H (0 , η , η , η ; 1 , ∞ ) which can be writtenin the form H (0 , η , η , η ; 1 , ∞ ) = G ( η , η , η ) η h , ( η − η ) h , . (42)11y eliminating the function G from eqs. (38) and (42) we obtain thefollowing relation for the correlation function H : H ( z , z ∗ , z , z ∗ ; x , x ) = η h , ( η − η ) h , H (0 , η , η , η ; 1 , ∞ )( x − x ) h , ( z − z ∗ ) h , ( z − z ∗ ) h , . (43)The null state conditions for the operators Φ , and Φ , lead to six PDEsfor G ( η , η , η ). The required solution to these equations must satisfy thefollowing limiting condition:lim z − z →∞ H ( z , ¯ z , z , ¯ z , x , x ) H ( x , x ) = H ( z , ¯ z , x , x ) H ( z , ¯ z , x , x ) . (44)In Ref. [4] Simmons and Cardy proposed a unique solution to the system ofthe equations, which satisfies the limiting condition (44). However, in thecase of N ≥ V , , V , and V − , − : F (0 , η , η , η ; 1 , ∞ ; γ , γ ) == h V , (0) Y i =1 V , ( η i ) V , (1) V − , − ( ∞ )( Q − ) i , (45)where we inserted two screening charges Q − inside the correlation functionof vertex operators in order to satisfy the neutrality condition (26).We will refer to the correlation function (45) as to the conformal block .It depends on the integration contours, γ and γ , which determine thescreening charges (27). The correlation function of primary fields (39) isgiven by an appropriate linear combination of these blocks: H (0 , η , η , η ; 1 , ∞ ) = X i,j N ( γ i , γ j ) F (0 , η , η , η ; 1 , ∞ ; γ i , γ j ) , (46)where the coefficients N ( γ i , γ j ) depends on the boundary conditions. Inthe case of the 6-points function (45) there exist 10 natural couples ofthe contours ( γ i , γ j ). However, one can argue that only one choice of thecontours is relevant, namely, γ and γ are simple paths connecting 0 , η and η , η respectively. Below, we present simple reasoning in support ofthe statement. Note, however, that our conclusion is justified by explicitcalculations [4].One can show, that the bulk-boundary fusion, Φ , ( z )Φ , ( z ∗ ) as z , z ∗ → x ∈ R , can be realized via the identity channel only [4] (we briefly discuss the12igure 2: The conformal block F ( z , z ∗ , z , z ∗ ; x , x ) is shown with respectto the bulk-boundary fusion. The boundary is shown by the solid line. Thedashed lines represent the integration paths for two screening charges Q − .algebraic structure of c = 0 LCFT in Section 4). There exist one conformalblock which satisfy this requirement, namely, F ( z , z ∗ , z , z ∗ ; x , x ) withthe integration path connecting z , z ∗ . This can be easily shown by insertingthe product R z ∗ z V , ( z ) V , ( z ∗ ) V − ( u ) du into the conformal block (28). Thechoice of the integration contour (from z to z ∗ ) implies that the pair oftwist operators, V , ( z ) and V , ( z ∗ ), fuse only via the identity channelwhen z , z ∗ → x ∈ R . Indeed, by fusing these operators we obtain thevertex operator V α ( x ) with the charge α = 2 α , = α , . By adding thescreening charge α − , the total charge vanishes, 2 α , + α − = 0, implyingthat the result is the identity operator. However, the screening chargeis pulled in with the vertex operators only if the path γ contracts to apoint in this process. Hence, the contour γ connects z and z ∗ . Similarly,one can prove that γ connects z and z ∗ . We show the conformal block F ( z , z ∗ , z , z ∗ ; x , x ) in Fig. 2. Under M¨obius transformation (41) thesecontours become γ ( z , z ∗ ) → γ (0 , η ) and γ ( z , z ∗ ) → γ ( η , η ) as proposedbelow eq. (46).By using (25) and evaluating the conformal block (45) we obtain theintegral representation of the correlation function : H (0 , η , η , η ; 1 , ∞ ) = N Y i =1 η κ/ i (1 − η i ) / Y i 1, where η ( u ) is specified by eq. (41), and u i ∈ { z , z ∗ , z , z ∗ , . . . , z n , z ∗ n } . Below, wesuppose that κ = 8 / / trace to wind about the points z , z , . . . , z n ∈ H in what follows. Inpresence of n twist operators the partition function of the SLE / curvecan be decomposed into the sum of the weights depending on the windingof the curve about the points. In order to label these weights, we introducethe following notation. Let I n = { , , . . . , n } be the set of n integers, and I n ⊂ I N . We decompose the set I n into two subsets, I + n and I − n , so that I + n ∪ I − n = I n , I + n ∩ I − n = ∅ . Besides, let us introduce the set of points, Z I = { z i | i ∈ I } , labeled by the integers from the set I . By Π I + n : I − n wedenote the weight of the SLE / traces which separates the points Z I − n awayfrom the interval [ x , x ] ⊂ R , while the points Z I + n remain unseparated.Then, we can decompose the partition function in terms of the weights asfollows (c.f., eqs. (49)): X I + N + I − N = I N ( − I − n Π I + N : I − N = H n ( Z I n ; x , x ) , (57)for n = 0 , , . . . , N . In (57) the sum is taken over all decompositions of I N into two subsets I + N and I − N . The coefficients ( − I − n can be obtainedby sending z i , ¯ z i → x ∈ [ x , x ] ( x ∈ R \ [ x , x ]) as explained in thediscussion below Eq. (35). Therefore, we obtain 2 N linear equations forthe 2 N unknown statistical weights Π I + N : I − N .Eqs. (57) allow us to determine statistical weights of SLE / traces from x to x to pass to the right of the points Z I − N and to the left of the points Z I + N (c.f., Eqs. (50)):Π I + N : I − N = 2 − N N X n =0 X I n ( − I − N ∩ I n ) H n ( Z I n ; x , x ) , (58)and the probability that the SLE / trace from x to x separates the points Z I − N from the interval [ x , x ] reads P I + N : I − N = Π I + N : I − N H ( x , x ) . (59) / / Green’s function In this section we discuss the probability that the SLE / trace passes inthe ǫ -neighborhood of the marked point. This probability is closely related16ith the one-point SLE / Green’s function. More specifically, consider theSLE / trace from x to x . Then, the probability, P { z < ǫ ; x , x } , that thetrace passes in the ǫ -neighborhood of the point z ∈ H vanishes as follows:lim ǫ → ǫ − / P { z < ǫ ; x , x } = cG SLE H ( z ; x , x ) + O ( ǫ ) , (60)where c is a constant, and G SLE H ( z ; x , x ) is called the one-point Green’sfunction of the SLE trace.In order to evaluate the one-point Green’s function we use the resultsobtained in the previous section. We consider the probability that theSLE / trace passes between the points z , z ∈ H . There are two possibletrace configurations, which contribute to the probability, namely, Π andΠ (see Fig. 3). Hence, the probability of the event is given by P ( z , z ; x , x ) = Π + Π Π ∅ + Π + Π + Π ∅ :12 == 12 − H ( z , ¯ z , z , ¯ z ; x , x )2 H ( x , x ) . (61)Further, we set z = z + ǫν/ z = z − ǫν/ 2, where z ∈ H , ǫ ≪ | ν | = 1,and consider the series expansion of P ( z , z ; x , x ) in the limit ǫ → 0. Theleading term of the small- ǫ expansion determines the Green’s function of thetrace as follows:lim ǫ → ǫ − / P (cid:16) z − ǫν , z + ǫν x , x (cid:17) = c G H ( z ; x , x ) , (62)where c is a constant.In order to evaluate (62) one needs to study the series expansion of the4-point correlation function (38) as z → z . In the CFT framework therequired expansion can be obtained by using the so-called operator productexpansion (OPE) of the primary fields Φ , ( z )Φ , ( z ) inside the correlationfunction. The form of the OPE can be deduced from the global conformalinvariance including the form of the two- and three-point functions and theirsymmetry properties :Φ h i ( z )Φ h j (0) = z h k − h i − h j X k C ki,j Φ h k (0) + X { n } β k, { n } i,j z |{ n }| Φ ( −{ n } ) h k (0) , (63)where the coefficients β k, { n } i,j are fixed by conformal invariance, and Φ ( −{ n } ) h k denotes the contribution of the |{ n }| -level descendant operators:Φ ( −{ n } ) h k = L − n L − n · · · L − n l Φ h k , (64) This form of the OPE is typical for rational CFTs, while in LCFTs the OPE of certainoperators can be modified. L − n are the generators of the Virasoro algebra [5]. The structureconstants C kij are determined by the two- and three-point functions , C kij = lim z →∞ | z | h i h Φ h i ( z, ¯ z )Φ h j (1)Φ k k (0) i , (65)where the normalization h Φ h i ( z, ¯ z )Φ h i (0) i = | z | − h i is assumed. Note,that the structure constants are not fixed by conformal invariance, and theadditional constraints follow from the request of associativity of the operatoralgebra [5]. However, once the structure constants are known all correlationfunctions can be in principle computed.Let us briefly recall the structure of c = 0 LCFT (see Refs. [14, 15] fordetails .). By V r,s we denote the Verma module generated from the state | Φ r,s i . In c = 0 LCFT the vacuum module is indecomposable M , = V , / V , . Furthermore, the physical module corresponding to the primaryfield Φ , is M , = V , / V , . The fusion of this modules with itself reads: M , × M , = M , + M , , (66)where M , was introduced earlier, and M , turns out to be the irreduciblemodule with h , = 1 / 3. The fusion rule (66) implies the following form ofthe OPE of primary fields Φ , ( ǫ ) and Φ , (0):lim ǫ → Φ , ( ǫ )Φ , (0) = Φ , (0) + C , ǫ / Φ , (0) + O ( ǫ ) , (67)where C , is a fixed OPE coefficient. In this case one can say that the OPEis realized via two channels : the first one involves Φ , , while the second oneinvolves Φ , . However, in the boundary CFT the general form of the OPEcan be modified because of the boundary conditions. As we will see below,the case of c = 0 boundary LCFT is even more tricky.Let us consider the correlation function (39), and examine the bulk-boundary fusion Φ , Φ , . The coulomb gas representation (47) allows us toobtain the OPE explicitly. By shrinking the integration contour connecting z , ¯ z , one obtains the small- ǫ expansion of H ( x + ǫ/ , x − ǫ/ , z , ¯ z , x , x ),where x ∈ R . It has the form g + ǫg + ǫ g + . . . , where g n with n ≥ { x, z , ¯ z , x , x } . By comparingthis expansion with OPE (67) we conclude that the bulk-boundary fusionis realized via the identity fusion channel, while the second channel, Φ , , isforbidden. This result can be also justified by computations in Ref. [4].Further, we consider the bulk-bulk fusion of the fields Φ , Φ , inside thecorrelation function as z → z , ¯ z → ¯ z . Explicit calculations shows thatboth channels, Φ , and Φ , , appear in this case [4]. Hence, the small- ǫ In LCFT certain structure constants become functions containing logarithms. Note, that we follow the notation of Ref. [4], so that the Kˆac indices are in reverseorder to those in [14, 15] H ( z , ¯ z , z , ¯ z ; x , x ), are shown with respect to the bulk-bulk fusion, Φ , Φ , . The fusion can be realized via two channels: Φ , and Φ , .By T we denote the stress-energy tensor, and rectangles [ m, n ] correspond tothe fields Φ m,n . Note, that the boundary operators are connected by doublelines.expansion of the correlation function H containing two twist operators canbe obtained from (67). It readslim ǫ → H ( z + ǫν/ , z − ǫw/ , ¯ z + ǫ ¯ ν/ , ¯ z − ǫ ¯ w/ x , x ) = H ( x , x )++ ( C , ) ǫ / h Φ , ( z, ¯ z )Φ , ( x )Φ , ( x ) i H + O ( ǫ ) . (68)By using (68) we recast the probability (61) in the formlim ǫ → ǫ − / P ( z − ǫη, z + ǫη ; x , x ) = ( C , ) G H ( z ; x , x ) , (69)where the Green’s function (compare (69) with (60)) is written in terms ofthe following correlation function: G H ( z ; x , x ) = − h Φ , ( z, ¯ z )Φ , ( x )Φ , ( x ) i H H ( x , x ) . (70)Hence, we proposed an explicit expression of the one-point SLE / Green’sfunction in terms of the correlation function of the field Φ , in the bulk,and two fields Φ , at the boundary in c = 0 LCFT. Recall, that the SLE / Green’s function defined by (60) can be evaluated explicitly [2], G SLE / H ( z ; 0 , ∞ ) = (Im z ) − / sin (arg( z )) . (71)In the next section we will show that G H ( z ; 0 , ∞ ) = G SLE / H ( z ; 0 , ∞ ) byobtaining an explicit Coulomb gas representation for the correlation functionon the right-hand side of (71). In this section we obtain a Coulomb gas representation for the one-pointSLE / Green’s function. The correlation function on the right-hand side19f eq. (70) can be realized via the correlation functions of vertex operatorscontaining a single screening charge Q + : h Φ , ( z )Φ , ( z ∗ )Φ , ( x )Φ , ( x ) i = X { γ } M ( γ ) H ( z, z ∗ ; x , x ; γ ) , (72)where the conformal blocks, H , are given by H ( z, z ∗ ; x , x ; γ ) = h V , ( z ) V − , − ( z ∗ ) V , ( x ) V , ( x ) Q + i , (73)and the coefficients, M ( γ ), depend on the integration contour. By takinginto account (25) we obtain the following integral representation for theconformal block: H ( z, z ∗ ; x , x ; γ ) = R γ du u /κ − (1 − u ) − /κ ( u − η ) ( z − z ∗ ) h , ( x − x ) h , (1 − η ) , (74)where η is the standard cross-ratio introduced earlier (18).One can argue, that in the case κ = 8 / η . Indeed, explicit calculation shows that the bulk-boundary fusionΦ , Φ , is realized via the weight 2 operator, namely, the stress-energytensor [4]. When z, z ∗ → x ∈ R the operators, V , ( z ) and V − , − ( z ∗ ), fuseto V α ( x ) with α = 2 α . By adding the screening charge α + , we determine thetotal charge 2 α + α + , and the conformal dimension of the operator is equalto h α + α + = 2, i.e., the dimension of the stress-energy tensor. However, thescreening charge is pulled in with the fusion only if the contour γ contractsto a point in this process. Hence, γ is a simple path connecting z and z ∗ .Below, we suppose that κ = 8 / 3. After substituting u = ηt in theintegrand of H ( z, z ∗ ; x , x ; γ ), and taking the integration contour to be asimple path connecting 0 and η , we obtain the following representation forcorrelation function (72), namely, h Φ , ( z, ¯ z )Φ , ( x )Φ , ( x ) i H = M η ( z − ¯ z ) h , ( x − x ) h , (1 − η ) , (75)where M is a normalization constant, and we set z ∗ = ¯ z in order to obtainthe correlation function in H . Thus, we find an explicit expression for theGreen’s function (70): G H ( z ; x , x ) = 14(Im z ) / η − η . (76)In the last expression we set M = 2 − / e iπ/ . By setting x = 0, x → ∞ ,so that η = 1 − e − i arg( z ) , we conclude, that the function (70) takes the formof the one-point Green’s function for SLE / in the upper-half plane (71),i.e., G H ( z ; 0 , ∞ ) = G SLE / H ( z ; 0 , ∞ ). 20 .3 The two-point Green’s function In this section we outline a derivation of the two-point SLE / Green’sfunction. We use the results of the previous sections, where the probabilitiesfor the SLE trace to wind in various ways about 4 marked points, z , z , z ,and z , were obtained. In particular, the probability that the curve passesbetween the points z , z and z , z correspondingly is given by the normalizedlinear combination of 4 trace configurations : P ( z , z , z , z ; x , x ) = Π + Π + Π + Π H ( x , x ) == 14 − H ( z , z ; x , x )4 H ( x , x ) − H ( z , z ; x , x )4 H ( x , x ) + H ( z , z , z , z ; x , x )4 H ( x , x ) . (77)Here H n ( z , . . . , z n ; x , x ) is the n -point correlation function in the upper-half plane (53). Let us set z = z + ǫν/ , z = z − ǫν/ ,z = w + δµ/ , z = w − δµ/ , (78)where z, w ∈ H , ǫ, δ ≪ | ν | , | µ | = 1, and consider the series expansion ofthe probability (77) in the limit ǫ, δ → 0. The leading term in the small- ǫ, δ expansion determines the two-point SLE / Green’s functionlim ǫ,δ → ǫ − / δ − / P (cid:16) z − ǫν , z + ǫν , w − ǫµ , w + ǫµ x , x (cid:17) == c G H ( z, w ; x , x ) , (79)where c is a constant.From (77), (79) it follows, that the two-point Green’s function is determinedby series expansions of the 4-point and 6-point correlation functions, H and H , as the points z , z ∈ H and z , z ∈ H collapse pairwise. Theseries expansion of H was determined in the previous section (see Eq. (68)).Therefore, we focus our attention on the 6-point correlation function, H ( z , z , z , z ; x , x ) = h Y i =1 Φ , ( z i , ¯ z i )Φ , ( x )Φ , ( x ) i H . (80)The leading order terms of the small- ǫ, δ expansion of the correlation functionare specified by possible channels of the fusion M , × M , . As discussed, We denote the function H n ( z , ¯ z , . . . , z n , ¯ z n ; x , x ) as H n ( z , . . . , z n ; x , x ) forbrevity. 21n the case of the bulk-bulk fusion we can use the OPE (67) in order toobtain the series expansion of the correlation function: H ( z , z , z , z ; x , x ) = H ( x , x )++ ǫ / ( C , ) h Φ , ( z, ¯ z )Φ , ( x )Φ , ( x ) i H ++ δ / ( C , ) h Φ , ( w, ¯ w )Φ , ( x )Φ , ( x ) i H ++ ǫ / δ / ( C , ) h Φ , ( z, ¯ z )Φ , ( w, ¯ w )Φ , ( x )Φ , ( x ) i H + O ( ǫ ) + O ( δ ) . (81)Upon substituting this expansion in (77) and taking account of (68) wedetermine the two-point SLE / Green’s function (79), G H ( z, w ; x , x ) = h Φ , ( z, ¯ z )Φ , ( w, ¯ w )Φ , ( x )Φ , ( x ) i H H ( x , x ) . (82)Hence, we conclude that the two-point SLE / function can be written interms of the correlation function in c = 0 boundary LCFT.Note, that the probability of the SLE / trace to pass via two points (79)is expected to possess the following property: it should reduce to the one-point function (62) when the points z and w collapse to one. In terms ofthe Green’s function this property can be written as follows:lim ǫ → ǫ / G H (cid:16) z − ǫν , z + ǫν x , x (cid:17) = cG H ( z ; x , x ) , (83)where ǫ ≪ | ν | = 1, and c is a constant. In the next section we will discussthis property in greater detail.We also note, that the result for the two-point Green’s function (82) canbe easily generalized to the case of N marked points in H . We suggest thefollowing expression for the multi-point Green’s function: G H ( { z i } Ni =1 ; x , x ) = ( − N h Q Ni =1 Φ , ( z i , ¯ z i )Φ , ( x )Φ , ( x ) i H H ( x , x ) . (84) We end this section by proposing a Coulomb gas representation for the two-point Green’s function in a somewhat heuristic manner. The correlationfunction on the right-hand side of (82) can be written as a linear combinationof the conformal blocks, H ( z, z ∗ , w, w ∗ ; x , x ; γ , γ , γ , γ ) == h V , ( z ) V , ( z ∗ ) V , ( w ) V , ( w ∗ ) V , ( x ) V − , − ( x )( Q − ) i . (85)22igure 5: Two possible choices of the integration contours for the conformalblock (85). The dashed lines, γ i , i = 1 , , , 4, represent the integrationcontours, while the solid line denotes the boundary.The conformal blocks depend on the contours, γ , γ , γ , and γ , whichdetermine the screening charges (27).Let us discuss possible choices of integration contours for (85). Recall,that the contours are in one-to-one correspondence with the conformal blocks.The structure of c = 0 boundary LCFT imposes strong constraints on theconformal blocks that contribute to the correlation function. In Ref. [4] itwas argued that the theory must contain two logarithmic partners of thestress-energy tensor: Φ , and Φ , . However, both fields (with differentlogarithmic couplings) can not appear in the theory simultaneously, becausesome quantities, e.g., h Φ , Φ , i , are undefinable. Simmons and Cardysuggested that both fields can coexist provided that Φ , appears in thebulk, while Φ , — on the boundary only. This conclusion imposes strongconstraints on the bulk-boundary fusion. Namely, the bulk operators fuseto the boundary through the identity and the stress-energy tensor only.This result suggests the following choice of the integration contours. Twocontours, γ and γ , should connect the operators V , ( x ) and V − , − ( x )(see Fig. 5). By fusing these operators as x → x , and shrinking theintegration contours to a point in this process we obtain the screening charge V α ( x ) with α = 2 α + 2 α − , so that h α +2 α − = 2. This is the conformaldimension of the stress-energy tensor. Note, that this fusion agrees withprevious results. Indeed, recall the conformal block (73) representing theone-point Green’s function. Since the integration contour connects z and z ∗ , the fusion of the vertex operators, V , ( x ) V , ( x ) as x → x , resultsin the operator V α with α = 2 α , . Its conformal dimension, h α , = 2,also equals the conformal dimension of the stress-energy tensor (see alsoFig. 4( b )).Let us consider the other contours, γ and γ , in the conformal block (85).By requiring these contours to be symmetric with respect to the points z, z ∗ , w , and w ∗ , we consider two possibilities shown in Fig. 5: ( a ) the It can be generalized to the case of N points. z, w ), and ( z ∗ , w ∗ ), and ( b ) the contours connect( z, z ∗ ), and ( w, w ∗ ). As discussed, the contours determine possible fusionchannels, which contribute to the OPE of the field Φ , and Φ , . Therefore,it is instructive to recall the fusion of the module M , with itself. Itreads [14] M , × M , = M , + I , , (86)where I , is a staggered module, structurally described by the exact sequence0 → M , → I , → M , → 0. Note, that I , is not itself a highest weightmodule. It is generated by the state | Φ , i with h , = 2, and the field Φ , is a Jordan partner of the stress-energy tensor, L | Φ , i = 2 | Φ , i + L − | i ,and L | Φ , i = − (5 / | i . Remarkably, the staggered module structureleads to the apeearance of logarithms in the correlation functions, e.g., h Φ , ( z )Φ , (0) i = (5 / 4) log( z ) /z .Now, by taking account of the fusion rules (86) we discuss two blockswhich can contribute to the correlation function (85) (see Fig. 5). In thecase ( b ) the bulk-boundary fusion, V , ( z ) V , ( z ∗ ) as z, z ∗ → x ∈ R , resultsin the vertex operator V α ( x ) with the conformal dimension h α , + α − = 1 / , ( x ). However, we already noted atthe begging of this section, that the bulk operators fuse to the boundarythrough the identity and the stress-energy tensor only. Therefore, theconformal block shown in Fig. 5( b ) is forbidden. In the case ( a ) the bulk-bulk fusion V , ( z ) V , ( w ) as w → z results in the operator V α , + α − ( z ) with h α , + α − = 1 / 3. It corresponds to the bulk field, Φ , ( z ), which generatesthe module M , on the right-hand side of (86). Besides, this fusion channelagrees with the limiting property of the two-point Green’s function (85).The outlined reasoning suggests the following Coulomb gas representationof the two-point Green’s function: it is determined by the conformal blockshown in Fig. 5( a ). By computing the correlation function of vertex operators,we arrive with the following expression for the two-point Green’s function: G H ( z, w ; x , x ) = X η / ( η − η ) / ( z − ¯ z ) / ( z − ¯ z ) / ×× Y i =1 η / i − η i Y j
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