Large-c conformal (n \leq 6)-point blocks with superlight weights and holographic Steiner trees
LLarge- c conformal ( n ≤ -point blocks with superlightweights and holographic Steiner trees Mikhail Pavlov a a I.E. Tamm Department of Theoretical Physics,P.N. Lebedev Physical Institute,Leninsky ave. 53, 119991 Moscow, Russia
E-mail: [email protected]
Abstract:
In this note we study CFT Virasoro conformal blocks with heavy operatorsin the large- c limit in the context of AdS /CFT correspondence. We compute the lengthsof the holographic Steiner trees dual to the 5-point and 6-point conformal blocks using thesuperlight approximation when one or more dimensions are much less than the others. Theseresults are generalized for N -point holographic Steiner trees dual to ( N + 1)-point conformalblocks with superlight weights. a r X i v : . [ h e p - t h ] F e b ontents N = 4 and N = 5 non-ideal trees in the superlight approximation 8 c conformal blocks 11 c conformal blocks and monodromy method 123.2 Examples of conformal blocks 13 The study of the AdS/CFT correspondence [1, 2] provides many new ideas and fruitful ob-servations related to computations in QFT. In the case of AdS /CFT correspondence it isessential to consider the large- c limit which corresponds to the weak gravitational couplingin the bulk according to Brown-Henneaux formula [3]. One of the most elaborated issues isthe correspondence between Virasoro conformal blocks with heavy operators in the large- c limit and probe particles propagating in the AdS background with conical defects originallyobtained for lower-point blocks [4–10]. The large- c n -point conformal blocks were studied in[17, 19–21]. However, exact expressions for large- c conformal blocks are still unknown.In this work, we continue to study large- c conformal blocks as holographic Steiner treeson the Poincare disk [22]. We consider holographic Steiner trees with N = 4 and N = 5endpoints in the superlight approximation where one of more weights are much less thanthe others. Their lengths are calculated by making use of the hyperbolic trigonometry re-lations. On the boundary, such Steiner trees are dual to the large- c conformal blocks withsuperlight operators [8]. Also, we find the lengths of (2 M + 1)-point holographic Steiner treesin the superlight approximation corresponding to the (2 M + 2) large- c conformal blocks withsuperlight operators.The paper is organized as follows. In Section we study the Steiner tree problem onthe Poincare disk and calculate the lengths of the holographic Steiner trees with N = 4 and N = 5 endpoints in the superlight approximation. Section applies the monodromy method Other recent related research focuses on p -adic AdS/CFT correspondence [11–13], entanglement entropy[14, 15] and OTOC computations [16–18]. – 1 –o calculate the large- c conformal blocks in the heavy-light approximation extended furtherby the superlight approximation. Here, we show the holographic correspondence relationbetween large- c conformal blocks and the lengths of the Steiner trees obtained in Section .Concluding Section summarizes our results. In the context of AdS /CFT correspondence the Poincare disk with an angle deficit arises asa constant-time slice of AdS space [6, 8]. In this section we focus on the Steiner tree problemon the Poincare disk for the special class of trees called holographic [22]. We use hyperbolictrigonometry to calculate particular holographic Steiner trees with N = 3 , , N -point Steiner trees. Let D α denote the Poincare disk with the angle deficit which isparametrized by α ∈ (0 , z, ¯ z ) it is defined as D α = {| z | < , arg( z ) ∈ [0 , πα ) } and the boundary is a part of the circle ∂ D α = {| z | = 1 , arg( z ) ∈ [0 , πα ) } . After reparameterization arg( z ) → α arg( z ) we obtain the Poincare disk model D .In what follows we do all calculations on the Poincare disk and then recover parameter α .The length of a geodesic segment between two points z and z is given by L D ( z , z ) = log 1 + u − u , u = | z − z || − ¯ z z | . (2.1)The regularized length (see e.g. appendix A in [22] for details) of the geodesic connecting twoboundary endpoints z i = exp[ iw i ] and z j = exp[ iw j ] takes the form L ε D ( w i , w j ) = a ij − ε , a ij ≡ log (cid:2) w ij (cid:3) , w ij ≡ w i − w j , (2.2)where the regulator ε → z = r exp[ iϕ ] and the boundary point z i = exp[ iw i ] is given by L ε D ( w i , r, ϕ ) = b − log ε , b ≡ log 2 (cid:0) r − r cos( ϕ − w i ) + 1 (cid:1) − r . (2.3)We denote by L D the finite part of the regularized length on the Poincare disk which isobtained by discarding the ε -dependent terms in (2.2) and (2.3). Steiner trees.
Given N points (outer vertices) belonging to D or ∂ D we consider a con-nected tree G N with N outer edges attached to outer vertices and N − N − G N reads L N D = (cid:88) { outer edges } (cid:15) i L i + (cid:88) { inner edges } ˜ (cid:15) j ˜ L j , (2.4) The Euclidean Steiner tree problem in context of QFT is considered in [23]. – 2 –here (cid:15) i and ˜ (cid:15) j are weights of outer and inner edges, respectively. The Steiner problem is tofind positions of inner vertices for given tree and weights such that the weighted length (2.4)is minimal. In this case the inner vertices are called Fermat–Torricelli (FT) points and G N iscalled the Steiner tree (see Fig. ). Also, for further purposes one can consider a hyperbolic N -gon with corners at the outer vertices of the Steiner tree (outer polygon). c ab γ ac γ bc γ ab Figure 1 . N = 6 Steiner tree. FT points are indicated by black points, different colors correspond todifferent weights, the angles are given by formula (2.5). The outer hexagon is shown in dashed lines. One can show that the angles between edges with weights (cid:15) a , (cid:15) b , (cid:15) c intersecting at FTpoint are given bycos γ ac = − (cid:15) c − (cid:15) b + (cid:15) a (cid:15) a (cid:15) c , cos γ bc = − (cid:15) c + (cid:15) b − (cid:15) a (cid:15) c (cid:15) b , cos γ ab = (cid:15) c − (cid:15) b − (cid:15) a (cid:15) a (cid:15) b , (2.5)supplemented by the triangle inequalities (cid:15) a + (cid:15) b ≥ (cid:15) c , (cid:15) a + (cid:15) c ≥ (cid:15) b , (cid:15) b + (cid:15) c ≥ (cid:15) a . (2.6)The relations (2.5) and (2.6) follow from the requirement that the Steiner tree has a minimallength and fix the positions of the FT points.In what follows we focus on two types of Steiner trees [22]: 1) N boundary endpoints,2) N − D . We will refer to them asideal and non-ideal holographic Steiner trees, respectively. Superlight approximation.
Suppose now that one of the three weights in (2.5) is muchless than the other two, which are assumed to be equal, (cid:15) c (cid:28) (cid:15) a = (cid:15) b : γ ab = π , γ ac = γ bc = π/ . (2.7)We see that two edges of the vertex merge into a single geodesic segment while the thirdedge stretches in a perpendicular direction. Then, the case of three arbitrary weights can beregarded as a perturbation of this configuration in the small parameter (cid:15) c . For more detailed analysis, see [24–26]. – 3 – yperbolic trigonometry.
The lengths of the edges of the Steiner tree in (2.4) are deter-mined by the coordinates of the FT points. For N = 3 Steiner trees the coordinates of theFT point can be calculated explicitly but for case N ≥ N = 3 Steiner trees cut the outer triangle into three trianglesand the edges of the trees can be considered as the sides of the hyperbolic triangles. Herewe provide the hyperbolic trigonometry relations that will be useful in calculating the edgelengths of holographic Steiner trees.Given a hyperbolic triangle with sides A, B, C and interior angles α, β, γ opposite to
A, B, C the first and second cosine theorem, and the sine theorem read ascosh A = cosh B cosh C − sinh B sinh C cos α , cosh C sin α sin β = cos γ + cos α cos β , sinh A sin α = sinh B sin β = sinh C sin γ . (2.8)When one of the vertices is on the boundary ( β = 0), the first cosine law can be cast into theform exp[ A ] = exp[ C ](cosh B − sinh B cos α ) + O ( ε ) , (2.9)where A and C denote the regularized lengths of sides connected to the vertex. For twovertices on the boundary ( β = γ = 0) the regularized lengths A, B, C are related as A = B + C + 2 log sin α O ( ε ) . (2.10) In this section, the lengths of N = 3 ideal and non-ideal Steiner trees and N = 4 idealSteiner tree are found for arbitrary weights. On the other hand, N = 4 , N andconsider a particular example of the N -point non-ideal Steiner tree. N=3 trees.
Let us consider N = 3 ideal Steiner tree with three boundary endpoints w i and outer edges of lengths X i , i = 1 , , ).– 4 – w w w w Figure 2 . (a) N = 3 ideal tree, (b) N = 3 non-ideal tree. The outer triangles are depicted indashed lines, different colours correspond to different weights. Since the Steiner tree splits the outer triangle into three triangles with two vertices onthe boundary we apply (2.10) to each of them and find a = X + X + 2 log sin γ ,a = X + X + 2 log sin γ ,a = X + X + 2 log sin γ , (2.11)where a ij and γ ij are given by (2.2) and (2.5). Solving this system of linear equations we findthe weighted length defined by (2.4) as L (3) D ( w i | (cid:15) i ) = ( (cid:15) + (cid:15) − (cid:15) ) log sin w + ( (cid:15) + (cid:15) − (cid:15) ) log sin w + ( (cid:15) + (cid:15) − (cid:15) ) log sin w + C , (2.12)where C = 2 (cid:16) log sin γ γ γ (cid:17) . (2.13)A similar analysis in the case of non-ideal N = 3 tree is a bit more complicated (see (b)Fig. ). Let Y and Z be the lengths of outer edges of weights (cid:15) , and X be the length of theradial line of weight (cid:15) . The outer triangle has a vertex in the center of D and two boundaryvertices w , . In this case, the outer triangle is cut by the Steiner tree into two triangles withone boundary vertex and one triangle with two boundary vertices. Again, using (2.9) and(2.10) we find 2 = exp[ Y ](cosh X − sinh X cos γ ) , Z ](cosh X − sinh X cos γ ) ,a = Y + Z + 2 log sin γ . (2.14)– 5 –he weighted length of the N = 3 non-ideal tree is found to be L (3) D ( w | (cid:15) i ) = (cid:15) (cid:34) Arcth (cid:34) cos w (cid:112) − β sin w (cid:35) + γ log sin w (cid:35) − (cid:15) β (cid:18) β cos w + (cid:113) − β sin w (cid:19) + ˜ C , (2.15)where γ = (cid:15) + (cid:15) (cid:15) , β = (cid:15) − (cid:15) (cid:15) , (2.16)and ˜ C is given by˜ C = (cid:15) (cid:18) log γ − γ + 1)(1 − β ) + γ log γ − β ( γ − − β ) + β log γ + β (1 − β )( γ − β ) (cid:19) . (2.17) Ideal N = 4 tree. Here we consider an N = 4 ideal tree with two FT points (see (a) Fig. ). This Steiner tree has four outer edges with weights (cid:15) i , i = 1 , ..., (cid:15) connecting two FT points. The minimum length condition here is encoded by sixangles α k (three at each of the FT points) given by (2.5). w w w w w w w w Figure 3 . (a) N = 4 ideal tree with five independent weights depicted in different colors and outertetragon in dashed lines. (b) The auxiliary triangle (in black lines) dissecting the outer tetragon. Let
A, B, C, D denote the regularized lengths of outer edges and R be the length of theinner edge. Consider an auxiliary triangle whose vertices are two boundary endpoints w and w and the FT point (see (b) Fig. ). Here, K and K are the regularized lengths of thesides attached to the boundary points w and w and λ and λ (cid:48) are the angles between R and Originally, this length was obtained in the context of the wordline approach [6]. For the analysis in thecontext of Steiner trees see [22]. The particular case of the tree with weights (cid:15) = (cid:15) and (cid:15) = (cid:15) was studied in [27]. – 6 – and K , respectively. Using the relations (2.9) and (2.10) one findsexp[ K ] = (cosh R − sinh R cos α ) exp[ A ] , exp[ K ] = (cosh R − sinh R cos α ) exp[ B ] ,A + B + 2 log sin α a . (2.18)Since K , together with the edges C, D cut the outer tetragon into three triangles, one hasexp[ K D ](1 − cos( α − λ )) = a , exp[ K C ](1 − cos( α − λ (cid:48) )) = a , exp[ A ] = exp[ K ](cosh R − sinh R cos λ ) , exp[ B ] = exp[ K ](cosh R − sinh R cos λ (cid:48) ) ,C + D + 2 log sin α a . (2.19)Eliminating K , and λ, λ (cid:48) from equations (2.18) and (2.19) we obtain R = log (cid:114) γ − γ + 1 (cid:114) γ − γ + 1 (cid:16) U − β β + (cid:112) ( β − β ) + 4 U ( U + 1 − β β ) (cid:17)(cid:112) (1 − β )(1 − β ) , (2.20)where U ≡ exp[ 12 ( a + a − a − a )] = sin w sin w sin w sin w ,γ = (cid:15) + (cid:15) ˜ (cid:15) , γ = (cid:15) + (cid:15) ˜ (cid:15) , β = (cid:15) − (cid:15) ˜ (cid:15) , β = (cid:15) − (cid:15) ˜ (cid:15) . (2.21)The lengths of the outer edges can be found from (2.18) and (2.19) together withexp[ K D ](1 − cos( α + λ (cid:48) )) = a , exp[ K C ](1 − cos( α + λ )) = a . (2.22)Finally, the weighted length (2.4) takes the form L (4) D ( w i | (cid:15) i , ˜ (cid:15) ) = ˜ (cid:15) ( γ log sin w + γ log sin w + R ) +˜ (cid:15) ( β + β )2 log (cid:32) − β − β + 2 U (1 + β β ) − ( β + β ) (cid:112) ( β − β ) + 4 U ( U + 1 − β β )(sin w ) − sin w (1 + U ) (cid:33) +˜ (cid:15) ( β − β )2 log (cid:32) U ( β β − − ( β − β ) + ( β − β ) (cid:112) ( β − β ) + 4 U ( U + 1 − β β )(sin w ) − sin w U (cid:33) , (2.23)where we dropped the weight-dependent constants. In the case β = β = 0, which corre-sponds to equal dimensions (cid:15) = (cid:15) and (cid:15) = (cid:15) , the length is given by L (4) D ( w i | (cid:15) , (cid:15) , ˜ (cid:15) ) = 2 (cid:15) log sin w + 2 (cid:15) log sin w + 2˜ (cid:15) log( √ U + √ U ) . (2.24)– 7 – .3 N = 4 and N = 5 non-ideal trees in the superlight approximation The lengths of N ≥ N = 4non-ideal tree can be considered as a perturbation of the N = 3 non-ideal tree with respectto one of outer weights [8]. In this section we calculate N = 4 and N = 5 non-ideal trees inthe superlight approximation by perturbing N = 3 ideal tree and disconnected N = 4 trees. Non-ideal N = 4 tree from disconnected N = 4 tree. Let us consider a N = 4 non-ideal tree as a perturbation of a disconnected N = 4 tree (see (a) and (b) Fig. ). The resulting N = 4 non-ideal tree has one inner edge with the weight ˜ (cid:15) (cid:28) (cid:15) , and two pairs of outeredges: the first one with weights (cid:15) = (cid:15) is a geodesic connecting w and w according to(2.7), and the second one is a radial line with weight (cid:15) = ˜ (cid:15) . However, the radial length is aweight-dependent constant so that it can be omitted. w w w w w w w w w s w s Figure 4 . Disconnected N = 4 tree (a) and N = 4 non-ideal tree (b). The green line in (b) carriesthe superlight weight ˜ (cid:15) , the non-deformed tree is shown in red lines. (c) shows an auxiliary bridgetree associated with the N = 4 non-ideal tree. The N = 4 non-ideal tree without the radial line can be obtained by cutting an auxiliary N = 4 ideal tree as shown on (c) of Fig. . Such an auxiliary tree has four outer edges withweights (cid:15) and the outer vertices of the tree are located at points ( w , w , w s , w s ), where w s = w + 2 w − w and w s = 2 w + w are identified by reflecting endpoints w and w relative to the radius connecting the center of D and the endpoint w . Using (2.24) we findthat the length of the N = 4 non-ideal tree takes the form L (4) D ( w i | (cid:15) , ˜ (cid:15) ) = 2 (cid:15) log sin w + ˜ (cid:15) log( (cid:112) U + (cid:112) ˜ U ) , ˜ U = sin( w − w ) sin w sin w − w . (2.25) Non-ideal N = 4 tree from ideal N = 3 tree. Another example of a N = 4 non-idealtree is obtained by adding an outer edge with superlight weight ˜ (cid:15) to the N = 3 ideal tree(see (b) Fig. ). According to (2.18) the outer edge (denoted by K ) is the perpendicular tothe third edge of the N = 3 ideal tree. – 8 – w w w w w w w w Figure 5 . (a) N = 3 ideal tree, (b) N = 4 non-ideal tree with the superlight weight ˜ (cid:15) . The greenline represents a perpendicular to the inner edge. (c) An auxiliary Steiner tree shown in black lines,the outer triangle shown in dashed lines. Let us consider an auxiliary triangle with two boundary vertices and a third vertex inthe center of D (see (c) Fig. ). An auxiliary Steiner tree of the triangle consists of edges X , X of the N = 3 ideal tree and the edge A stretched to the center of D . To simplify thecalculations here we assume (cid:15) = (cid:15) . Using the trigonometric relations (2.8) and (2.9) we findexp[ X ](cosh A − sinh A cos( γ + α )) = 2 , exp[ X ](cosh A − sinh A cos( γ − α )) = 2 , sinh K sin α = sinh A , (2.26)where α is the angle between edges X and A . Solving equations (2.26) in the variable K weobtain sinh K = sin 2 w − w − w w − w w − w . (2.27)Then, the length of the non-ideal N = 4 tree takes the form L (4) D ( w i | (cid:15) , (cid:15) , ˜ (cid:15) ) = L (3) D ( w i | (cid:15) , (cid:15) , (cid:15) ) + ˜ (cid:15) Arcsinh sin 2 w − w − w w − w w − w , (2.28)where L (3) D ( w i | (cid:15) , (cid:15) , (cid:15) ) is given by (2.12). Non-ideal N = 5 tree from N = 4 disconnected tree. Here, we consider a N =5 non-ideal tree with two superlight weights ˜ (cid:15) , ˜ (cid:15) , see Fig. . The unperturbed N = 4disconnected tree is given by two geodesics with weights (cid:15) , (cid:15) connecting pairs w , w and w , w , respectively. – 9 – w w w Figure 6 . N = 5 non-ideal tree. The unperturbed N = 4 tree is shown in red. The green linesrepresent the inner edge of the tree and the radial line with superlight weights ˜ (cid:15) and ˜ (cid:15) , respectively. In the superlight approximation the length of the tree is given by the sum of the lengthof the N = 4 ideal tree and the length of the radial line. The length of the radial line givenby the first term in (2.15) under the condition ˜ (cid:15) (cid:28) (cid:15) , is equal to L ( r ) D ( w ) = ˜ (cid:15) log cot w . (2.29)The length of the bridge line with weight ˜ (cid:15) (cid:28) (cid:15) , stretched between the geodesics is givenby the last terms in formula (2.24) as L ( b ) D ( w i ) = ˜ (cid:15) log( √ U + √ U ) U = sin w sin w sin w sin w . (2.30)In this case, the lengths (2.29) and (2.30) are determined only by coordinates w i and do notdepend on the structure of the unperturbed tree, i.e.weights (cid:15) , . Finally, the weighted lengthof the N = 5 non-ideal tree takes the form L (5) D ( w i | (cid:15) , (cid:15) , ˜ (cid:15) , ˜ (cid:15) ) = 2 (cid:15) log sin w + 2 (cid:15) log sin w + ˜ (cid:15) log( √ U + √ U ) + ˜ (cid:15) log cot w . (2.31) Multi-point trees.
The superlight approximation allows one to calculate the length of amulti-point non-ideal Steiner tree with N = 2 M +1 , M = 3 , , , ... outer vertices. The tree isa perturbation of a disconnected N = 2 M Steiner tree consisting of M geodesics with weights (cid:15) i , i = 1 , ..., N which connect the points w i − , w i . The inner bridge lines with superlightweights ˜ (cid:15) j , j = 1 , ..., N − D carries the weight (cid:15) r (see Fig. ).– 10 – w w w w w . . . w M − w M Figure 7 . N = 2 M + 1 non-ideal Steiner tree in the superlight approximation. Red lines correspondto connecting outer vertices w i − and w i . The inner edges with weights ˜ (cid:15) j and the radial line withthe weight (cid:15) r are shown in green. Since the lengths of the radial and bridge lines for the tree are given by (2.29) and (2.30),then the weighted length of the N = 2 M + 1 non-ideal tree takes the form L (2 M +1) D ( w i | (cid:15), ˜ (cid:15) ) = L (2 M ) D ( w i | (cid:15) ) + 2 M − (cid:88) i =1 ˜ (cid:15) j log (cid:16)(cid:112) U i +1 + 1 + (cid:112) U i +1 (cid:17) + (cid:15) r log cot w M − , M , (2.32)where L (2 M ) D ( w i | (cid:15) ) = 2 M (cid:88) i =1 (cid:15) i log sin w i − , i , U i − = sin w i +1 , i sin w i +2 , i − sin w i, i − sin w i +2 , i +1 . (2.33)Note that this analysis can be generalized to other cases of non-ideal Steiner trees in thesuperlight approximation. For example, one can consider a disconnected M = 3 N treeconsisting of N ideal Steiner trees with three boundary endpoints as an unperturbed tree.However, the example is more complicated from a computational point of view. c conformal blocks Here, we discuss the n -point large- c conformal blocks with heavy operators in the heavy-lightapproximation. To this end, we use the monodromy method [4, 5, 9, 19, 21, 28] to demonstratethe holographic correspondence relation (3.7) for particular examples of 5-point and 6-pointblocks with superlight operators. This analysis is generalized to the (2 M + 2)-point conformalblock with superlight operators. – 11 – .1 Large- c conformal blocks and monodromy method Consider primary operators O i ( z i , ¯ z i ) , i = 1 , ..., n at fixed points ( z, ¯ z ) = { ( z , ¯ z ) , ..., ( z n , ¯ z n ) } .Let F n ( z | ∆ i , ˜∆ p , c ) be the corresponding holomorphic conformal block which depends on con-formal dimensions ∆ i and exchange dimensions ˜∆ p , p = 1 , ..., n − c [29]. Assuming that in the limit c → ∞ dimensions ∆ and ˜∆ are proportional to the centralcharge one can check perturbatively up to a sufficiently high order that the conformal blocktakes the exponential form [30] F n ( z | ˜∆ p , ∆ i , c ) = exp (cid:104) c f n ( z | (cid:15) i , ˜ (cid:15) p ) (cid:105) + O (cid:18) c (cid:19) , (cid:15) i ≡ i c , ˜ (cid:15) p ≡ p c , (3.1)where f ( z | (cid:15) i , ˜ (cid:15) p ) is a large- c block, (cid:15) i , ˜ (cid:15) p are classical dimensions which are finite in the large- c limit. z , (cid:15) z , (cid:15) z n − , (cid:15) n − · · · · · · z n , (cid:15) n z n − , (cid:15) h ˜ (cid:15) ˜ (cid:15) n − ˜ (cid:15) n − · · · · · · Figure 8 . The n -point perturbative conformal block with two background operators depicted by boldblack lines. In what follows, we work within the heavy-light approximation [5–7, 21, 22] when twoexternal operators with (cid:15) n = (cid:15) n − = (cid:15) h are assumed to be heavier than the other externaland exchange operators (see Fig. ) (cid:15) h (cid:28) (cid:15) i , ˜ (cid:15) p , i = 1 , ..., n − , p = 1 , ..., n − . (3.2) Monodromy method and heavy-light approximation.
This method is discussed indetails in [5, 6] for 4-point conformal blocks and generalized to n -point blocks in [4, 8, 19, 21].Below we summarize the main steps.Let Ψ( y | z ) be an auxiliary n +1-point conformal block with one degenerate operator V (1 , inserted in the point ( y, ¯ y ) and n primary operators O i . In the large- c limit the auxiliaryblock is factored into a product of the formΨ( y | z ) (cid:12)(cid:12)(cid:12) c →∞ = ψ ( y | z ) exp (cid:104) c f n ( z | (cid:15) i , ˜ (cid:15) p ) (cid:105) , (3.3) Conformal blocks beyond these limits limit are considered in [31, 32]. For recent study of the blockexponentiation see [33]. The case of three or more heavy operators is considered in [27, 34]. – 12 –here ψ ( y | z ) is a semiclassical contribution of the operator V (1 , . On the other hand, theauxiliary block satisfies the BPZ equation which is reduced to the Fuchsian-type equationwith n singular regular points (cid:20) d dy + T ( y | z ) (cid:21) ψ ( y | z ) = 0 ,T ( y | z ) = n (cid:88) j =1 (cid:15) j ( y − z j ) + c j y − z j , c j = ∂f n ( z | (cid:15) i , ˜ (cid:15) p ) ∂z j , (3.4)where gradients c j are accessory parameters. In the first order in the heavy-light approxima-tion it leads to the monodromy equations [21] I ( n | k )+ − I ( n | k ) − + + (cid:16) I ( n | k )++ (cid:17) = − π ˜ (cid:15) k , k = 1 , ... , n − , (3.5)where I ( n | k )+ − = 2 πiα (cid:16) α(cid:15) + n − (cid:88) j =2 X j − k +1 (cid:88) j =2 (1 − z j ) α ( X j − (cid:15) j α ) , α = √ − (cid:15) h ,I ( n | k ) − + = I ( n | k )+ − (cid:12)(cid:12) α →− α , I ( n | k )++ = 2 πiα n − (cid:88) j = k +2 X j , X j = c j (1 − z j ) − (cid:15) j . (3.6)These are n − Holographic correspondence relation.
The duality between large- c conformal blocksand Steiner trees on the Poincare disk is given by the holographic correspondence relation f n ( z k | (cid:15) k , ˜ (cid:15) p ) = − L ( n − D ( αw k | (cid:15) k , ˜ (cid:15) p ) + i n − (cid:88) k =1 (cid:15) k w k , w k = i log(1 − z k ) , (3.7)where L ( n − D ( αw k | (cid:15) k , ˜ (cid:15) p ) is the weighted length of the Steiner tree corresponding to the n -point block with weights (cid:15) k , ˜ (cid:15) p that are equal to the classical dimensions of the block. Notethat the length L ( n − D ( αw k | (cid:15) k , ˜ (cid:15) p ) depends on the rescaled coordinates αw k due to the factthat α is the angle deficit of the Poincare disk (see Section ). In this section we calculate 5-point and 6-point large- c conformal blocks dual to the lengthsof Steiner trees computed in Section . By virtue of (3.7) the relations (2.6) define thefusion rules for such blocks. We use the following variables P j = (1 − z j ) α , j = 2 , ..., n − , (3.8)and set w = 0 in the lengths of Steiner trees due to the condition P = 1.– 13 – -point non-identity blocks with superlight operators. Here we suppose that one ofthe exchange operator dimensions ˜ (cid:15) or ˜ (cid:15) is superlight: ˜ (cid:15) , (cid:28) (cid:15) , , . The first examplecorresponds to ˜ (cid:15) (cid:28) (cid:15) , , . The weighted length of the dual Steiner tree is given by (2.25)and from the fusion rules (2.6) we get (cid:15) = (cid:15) and ˜ (cid:15) = (cid:15) . Then, using the holographiccorrespondence relation (3.7) we find f ( z | (cid:15) , (cid:15) , ˜ (cid:15) ) = (cid:15) ( − α ) log P /α − ( (cid:15) + α ˜ (cid:15) ) log P /α − (2 (cid:15) + ˜ (cid:15) ) log[1 − P ] + ˜ (cid:15) log[ P − P − (cid:112) (1 − P )( P − P )] . (3.9)The accessory parameters corresponding to the conformal block (3.9) must satisfy monodromyequations (3.5) (cid:16) I (5 | (cid:17) + I (5 | − I (5 | − + = − π ˜ (cid:15) , I (5 | − I (5 | − + = − π (cid:15) . (3.10)A few comments are in order. Since the block (3.9) is obtained in approximation ˜ (cid:15) (cid:28) (cid:15) , we require the equation (3.10) to be fulfilled in the first order in ˜ (cid:15) . The first equationdoes not contain terms linear in ˜ (cid:15) but after explicit substitution of the accessory parameterscorresponding to the conformal block (3.9) we can see that it is satisfied exactly. The secondequation is valid up to the first order in ˜ (cid:15) .Next we consider the case ˜ (cid:15) (cid:28) (cid:15) , , . The block is dual to the Steiner tree of the length(2.28) so that we assume (cid:15) = (cid:15) . The fusion rules (2.6) require ˜ (cid:15) = (cid:15) . According to (3.7)the confromal block has the form f ( z | (cid:15) , (cid:15) , ˜ (cid:15) ) = (cid:15) ( − α ) (cid:16) log P /α + log P /α (cid:17) − (cid:15) (log[1 − P ] + log[ P − P ]) − (2 (cid:15) − (cid:15) ) log[1 − P ] + ˜ (cid:15) Arcsinh (cid:20) − i ( P − P )(1 − P )( P − P ) (cid:21) , (3.11)and, substituting the accessory parameters associated with the conformal block into themonodromy equations (3.5), we find (cid:16) I (5 | (cid:17) + I (5 | − I (5 | − + = − π (cid:15) , I (5 | − I (5 | − + = − π ˜ (cid:15) . (3.12)As in the previous case the monodromy equations are satisfied in the first order in ˜ (cid:15) .6 -point identity block with light operators. Let us consider the 6-point identity blockwith ˜ (cid:15) = 0 and denote ˜ (cid:15) = ˜ (cid:15) . According to the fusion rules (2.6) it follows that ˜ (cid:15) = (cid:15) .The length of the corresponding Steiner tree is given by (2.23) and according to (3.7) the For the analysis of other approximations used to calculate 5-point large- c block see [35, 36]. – 14 –onformal block takes the form f ( z | (cid:15) i , ˜ (cid:15) ) = (cid:15) ( α −
1) log P /α + (cid:15) ( α −
1) log P /α + (cid:15) ( α −
1) log P /α − ˜ (cid:15) (cid:18) β log P − β log P + β log P − ( β + β ) log P − P − P + ( β − β ) P − P − P (cid:19) − ˜ (cid:15) (cid:16) γ log[ P − P ] + γ log[1 − P ] + log (cid:104) U − β β + (cid:113) β − + 4 U + 4 (1 − β β ) U (cid:105)(cid:17) + ˜ (cid:15)β + (cid:16) − β − β + 2 U (1 + β β ) − β + (cid:113) β − + 4 U ( U + 1 − β β ) (cid:17) U + ˜ (cid:15)β − (cid:18) U (cid:18) U ( β β − − β − + β − (cid:113) β − + 4 U ( U + 1 − β β ) (cid:19)(cid:19) ,U = (1 − P )( P − P )(1 − P )( P − P ) , β ± = β ± β . (3.13)The monodromy equations (3.5) for the conformal block (3.13) take the form (cid:16) I (6 | (cid:17) + I (6 | − I (6 | − + = − π ˜ (cid:15) , (cid:16) I (6 | (cid:17) + I (6 | − I (6 | − + = − π (cid:15) ,I (6 | − I (6 | − + = 0 , (3.14)and one can explicitly show that the corresponding accessory parameters satisfy these equa-tions without using superlight approximation.6 -point non-identity block with superlight operators. Here we discuss the case ofnon-identity 6-point block with (cid:15) = (cid:15) and (cid:15) = (cid:15) . The fusion rules (2.6) constrain thedimensions as ˜ (cid:15) = (cid:15) . There are two superlight exchange operators with dimensions ˜ (cid:15) , (cid:28) (cid:15) , . Using holographic correspondence relation (3.7) and the length of the correspondingSteiner tree (2.31) we find that the conformal block takes the form f ( z | (cid:15) , (cid:15) , ˜ (cid:15) , ˜ (cid:15) ) = ( − α ) (cid:16) (cid:15) log P /α + (cid:15) log P /α + (cid:15) log P /α (cid:17) + ˜ (cid:15) log √ P − √ P √ P + √ P − (cid:15) log[1 − P ] − (cid:15) log[ P − P ] − (cid:15) log (cid:32)(cid:115) (1 − P )( P − P )(1 − P )( P − P ) + (cid:115) (1 − P )( P − P )(1 − P )( P − P ) (cid:33) . (3.15)After substituting the corresponding accessory parameters into the monodromy equations (cid:16) I (6 | (cid:17) + I (6 | − I (5 | − + = − π ˜ (cid:15) , (cid:16) I (6 | (cid:17) + I (6 | − I (6 | − + = − π (cid:15) ,I (6 | − I (6 | − + = − π ˜ (cid:15) , (3.16)– 15 –e find that they are satisfied up to the first order in ˜ (cid:15) , .(2 M +2) -point conformal block with superlight operators. The foregoing analysis canbe generalized to (2 M + 2)-point conformal block dual to the multi-point Steiner tree (2.32)(see Fig. and ). It has M exchange superlight operators with weights ˜ (cid:15) j , j = 1 , ...., M .The fusion rules are (cid:15) i − = (cid:15) i = ˜ (cid:15) i − , i = 1 , ..., M . (3.17) (cid:15) (cid:15) (cid:15) (cid:15) n − · · · · · · · · · · · · · · · (cid:15) h (cid:15) h ˜ (cid:15) (cid:15) ˜ (cid:15) n − (cid:15) n − · · · · · · · · · · · · · · · Figure 9 . (2 M + 2)-point large- c conformal block with M superlight operators depicted by greenlines. According to (3.7) and (2.32) the block function takes the form f M +2 ( z | (cid:15) i , ˜ (cid:15) j ) = M (cid:88) i =1 f ( z | (cid:15) i ) + M − (cid:88) j =1 ˜ f ( z | ˜ (cid:15) j ) + ˜ (cid:15) n − log √ P n − − √ P n − √ P n − + √ P n − , (3.18)where f ( z | (cid:15) i ) = ( − α ) (cid:15) i − (cid:16) log P /α i − + log P /α i (cid:17) − (cid:15) i − log[ P i − − P i ] , ˜ f ( z | ˜ (cid:15) j ) = − (cid:15) j log (cid:32)(cid:115) ( P j − − P j +1 )( P j − P j +2 )( P j − − P j )( P j +1 − P j +2 ) + (cid:115) P j − − P j +1 )( P j − P j +2 )( P j − − P j )( P j +1 − P j +2 ) (cid:33) . (3.19)It can be explicitly shown that this block satisfies the monodromy equations (3.5) up to thefirst order in the dimensions of superlight operators ˜ (cid:15) j . In this paper we explicitly computed the weighted lengths of the N = 4 , , large- c conformalblocks. On the boundary side, the superlight approximation corresponds to superlight oper-ators. Our results along with previously known are shown in the table below. HL and SLdenote the heavy-light and the superlight approximations, respectively.– 16 – Steiner tree with N endpoints N +1-point con-formal block approximation2 Ideal tree, ref.[5] ref.[5] HL3 Ideal tree, ref.[22] ref.[22] HL3 Non-ideal tree, ref.[6, 8, 22] ref.[6, 8, 22] HL4 Simplest ideal tree, ref.[22] ref.[22] HL4 General ideal tree, eq.(2.23) eq.(3.13) HL4 Non-ideal tree, ref.[8, 10, 35] ref.[8, 10, 35] HL+SL4 Non-ideal tree, eq.(2.25) eq.(3.9) HL+SL4 Non-ideal tree, eq.(2.28) eq.(3.11) HL+SL5 Non-ideal tree, eq.(2.31) eq.(3.15) HL+SLN Non-ideal tree, ref.[19] ref.[19] HL+SLN Various disconnected trees, ref.[22] ref.[22] HLN = 2M+1 Non-ideal tree, eq.(2.32) eq.(3.18) HL+SLOne can analyze N -point Steiner trees as deformations of other unperturbed tree config-urations. Also it would be interesting to compute the lengths of Steiner trees in the secondand next orders in the superlight approximation. Acknowledgements.
I would like to thank K.B. Alkalaev for discussions and the organiz-ers of YRISW 2020 school for the hospitality and productive atmosphere during the visit.The work was supported by RFBR grant No 18-02-01024 and by the Foundation for theAdvancement of Theoretical Physics and Mathematics “BASIS”.
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