Geometrical four-point functions in the two-dimensional critical Q -state Potts model: The interchiral conformal bootstrap
GGeometrical four-point functions in the two-dimensional critical Q -state Potts model: The interchiral conformal bootstrap Yifei He , Jesper Lykke Jacobsen , , , Hubert Saleur , Universit´e Paris-Saclay, CNRS, CEA, Institut de Physique Th´eorique, 91191, Gif-sur-Yvette, France Laboratoire de Physique de l’Ecole Normale Sup´erieure, ENS, Universit´e PSL, CNRS,Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France Sorbonne Universit´e, ´Ecole Normale Sup´erieure, CNRS, Laboratoire de Physique (LPENS), 75005 Paris, France Department of Physics, University of Southern California, Los Angeles, CA 90089, USA
Abstract
Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to de-termine four-point correlation functions of the geometrical connectivities in the Q -state Potts model.Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degen-eracy of fields with conformal weight h r, , with r ∈ N ∗ , and are related to the underlying presence ofthe “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized”recursions, replacing those that follow from the degeneracy of the field Φ D in Liouville theory, andobtain the first few such recursions in closed form. This hints at the possibility of the full analyticaldetermination of correlation functions in this model. Contents A aaaa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.2 A abab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.3 A aabb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Singularities and exact amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4.1 Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4.2 Non-diagonal Liouville theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 “Renormalized” Liouville recursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 a r X i v : . [ h e p - t h ] M a y Conclusions 44A α , β , γ from [3] 46B More details on the numerical bootstrap 47 B.1 Basic checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 C A L and the Liouville recursions from [4] 50 Introduction and summary
Recent years have witnessed the power of the modern bootstrap approach to conformal field theories (CFT)starting with the seminal work of [5]. Since then many rigorous results on critical phenomena in dimension d > d > d = 2 CFT still remainto be answered. Prime among those is the issue of geometrical critical phenomena, where the definition ofcorrelation functions involves non-local aspects. One typical example is the Q -state Potts model which, inthe Q → c <
1. The next natural, yet highly non-trivial step, is to extendthis development to geometrical four-point functions [17]. We stress here that we are exclusively interestedin the bulk geometry, which presents fundamental difficulties not present in the boundary case [18].An interesting strategy towards the determination of the geometrical four-point functions was proposedin a recent work [19] using the conformal bootstrap philosophy. The idea can be stated simply: To obtainthe amplitudes of the primary fields entering a given correlation function, one solves the crossing equationnumerically with a proposed spectrum for the conformal weights of the participating primaries. Thishas led to a simple conjecture for the Potts spectrum in [19] with apparent agreement with Monte-Carlosimulations [19, 20]. It was however shown in [1] that, unfortunately, the simple spectrum of [19] does not correctly describethe geometrical correlations in the Potts model, although to the precision of Monte-Carlo simulations itappears as a rather convincing approximation. Moreover, [1] made a more involved proposal for the spec-trum, based on the representation theory of the affine Temperley-Lieb algebra, and verified its correctnessthrough analytical checks in a number of solvable cases, by analytically arguing that the extra states in thecorrected spectrum are actually necessary to avoid certain singularities which would otherwise be present,and finally by carrying out high-precision numerical verifications using a transfer matrix approach which iscapable of targeting the amplitudes of the added parts of the spectrum.To summarize, the spectrum of [1] is now understood to provide the correct description of geometricalcorrelations in the Potts model. Meanwhile, the correlation functions associated with the simpler spectrumof [19] were solved analytically in [4] and understood later to provide a certain analytic continuation ofcorrelation functions in type-D minimal models, or a non-diagonal generalization of the Liouville theory.Although the spectrum used in [19] is thus not correct for describing the geometrical correlation functionsof interest, the other main idea of that work—namely, to study numerically the bootstrap equations—iscertainly valid and worth further exploitation. The obvious suggestion is thus to revisit this idea, but in thecontext of the corrected Potts spectrum obtained in [1]. This investigation is the focus of our work here.To guide the readers through the bulk of this paper, we draw in fig. 1 a chart which highlights the logicalrelations between the parts of this work, while locating the “landmarks” of our findings.We consider geometrical four-point functions which involve one or two FK clusters, i.e., the probabilitiesof the four points belonging to one or two distinct clusters. They are denoted as P aaaa , P aabb , P abab , P abba , which can be seen as variants of CFT four-point functions of the spin operator Φ , : (cid:104) Φ , Φ , Φ , Φ , (cid:105) . See [21,22] for related studies on the torus. See also [23] for a recent study of the four-spin correlations using the CoulombGas approach. eometrical four-point functions determinespectra amplitudes ATL modulesuniversal amplitude ratiosrecursionsinterchiral conformal blocksinterchiral bootstrap equations exact amplitudesrenormalized Liouville recursion latticedegeneracydeduceconstruct interchiral bootstrap c on s tr a i n t s Potts from Liouville
Figure 1: Chart of the contents of this paper.In [1], the s -channel spectra of the probabilities were obtained using a combination of algebraic and numer-ical methods. They are encoded by the affine Temperley-Lieb (ATL) modules W j, z whose continuum limit gives rise to the conformal fields. The t - and u -channel spectra follow from geometricconsiderations. See eqs. (2.19)–(2.21).Further studies on the lattice model, carried out in [3], reveal interesting features regarding how thesefields contribute to the geometrical four-point functions: There exist universal ratios which relate eitherthe amplitudes of the fields in different Potts probabilities, or the Potts amplitudes themselves, with theamplitudes appearing in the non-diagonal Liouville theory of [19]. We have already established in [3], thatthe CFT four-point functions in the Liouville theory have a geometric interpretation in terms of clustersvery similar to that of the Potts model, however with different weights assigned to the topologically non-trivial clusters. The amplitude ratios are universal in the sense that they depend only on the ATL moduleto which a given field belongs, and on the parameter Q . We briefly review these results in section 3.1 and3.2 and redefine the universal amplitude ratios R β , R α , R ¯ α for further convenience. See eqs. (3.7)–(3.12) for definitions and their explicit expressions up to a certainlevel, as obtained from the lattice computations in [3].The existence of the universal amplitude ratios strongly hints at the objects we call the “interchiralconformal blocks” F j, z , which organize the fields in the spectra according to the ATL modules they belong to (this “organization”corresponds, in the continuum limit, to the action of an interchiral algebra [2], whence the name). Our4rucial observation is the following: since the universal amplitude ratios depend only on the ATL modules,it is the same interchiral conformal blocks that enter various Potts probabilities as well as the non-diagonalLiouville theory of [19]. Only the global amplitudes associated with entire ATL modules are modified bythe change in the cluster weights implied [3] by the passage from the true Potts correlators to non-diagonalLiouville correlators. By contrast, the relations among the fields within the same ATL module remain thesame, i.e., the structure of the interchiral blocks F j, z is rigid. We discuss this in details in section 3.3.There, using the interchiral block expansion of the four-point function in the non-diagonal Liouville theoryand comparing with the Potts probabilities, we see that the bootstrap problem originally considered in [19]has non-unique solutions. Furthermore, using the amplitude ratios R , one can in fact extract some of thePotts amplitudes A from the known amplitudes A L of the non-diagonal Liouville theory as obtained in [4]: A ← A L . The results are given in eq. (3.19).With the existence of the interchiral conformal blocks established, the determination of the geometricalfour-point functions reduces to solving for the global amplitudes A aaaa ( W j, z ) , A aabb ( W j, z ) , A abab ( W j, z ) , A abba ( W j, z )of the entire ATL modules. The bootstrap idea proposed in [19] then comes into play. We proceed byfully exploiting this idea using the interchiral block expansions of all four probabilities as related throughcrossing, and writing down the interchiral bootstrap equations (3.32). This is a linear system of the globalamplitudes A ( W ) in (3.31) whose relations are further constrained through the amplitude ratios R α , R ¯ α .In addition, the Potts amplitudes A extracted from the Liouville amplitudes A L further constrain thebootstrap problem. The precise ingredients of the bootstrap we carry out are indicated in blue on the chart(see fig. 1).To implement the bootstrap, we need to construct the interchiral blocks F j, z . For this, we first observethe degeneracy in the Potts spectra: the ATL module W , q consists of Kac modules in the CFT, i.e.,they are degenerate representations of the Virasoro algebra. This includes in particular the field Φ D , .Degeneracy of this field (as well as Φ D , ) are known to appear in the diagonal and non-diagonal Liouvilletheories [4, 24–26] which lead to recursions in the amplitudes when the Kac indices ( r, s ) are shifted by 2units, eventually providing the full analytic solutions to those theories. Here, in the Potts model, with thesole degeneracy of Φ D , (but not Φ D , ), by focusing on four-point functions of the spin operator, we obtaininstead recursions R D , R N of amplitudes where the Kac index r is shifted by 1 unit. The explicit expressions are given in eqs. (4.14)and (4.5) and the D, N here label the diagonal and non-diagonal fields in the spectra. Such recursionsexactly relate the amplitudes of the fields within the same ATL module, and we use them to re-sum theordinary Virasoro conformal blocks F into the interchiral conformal blocks F . This construction is givenin section 4.1 where the explicit interchiral blocks are given after eq. (4.16) and illustrated with fig. 2. Wepresent the detailed derivation of the recursions from the degeneracy in section 4.1.1.The results of the numerical bootstrap are given in section 4.2 for A ( W j, z ), with the ATL index j ≤ < Q < Q at certain rational values of the centralcharge. The following section 4.3 is then devoted to analyzing these poles in details from a combination ofperspectives: the requirement of smoothness as a function of Q for the geometrical four-point functions, theamplitude ratios R which relate the amplitudes in the different geometries, and the corresponding differencein their respective spectrum. Through this analysis, we obtain certain exact amplitudes at special valuesof Q which interpolate smoothly between the numerical bootstrap results as displayed in figs. 16–19. Thebootstrap results are subsequently compared with lattice computations and the approximate descriptiongiven by the non-diagonal Liouville theory of [19] in section 4.4.One interesting observation from the bootstrap on the Potts amplitudes is the following: While thedegeneracy of the field Φ D , in the (non-diagonal) Liouville theory, and therefore the resulting recursion for5hifting the Kac s -index, are absent in the case of the Potts model, there exists a “renormalized” version ofthe Liouville recursion, with the renormalization factors given by ratios of polynomials in Q . On the onehand, this is obtained from the extraction of the Potts amplitudes A from the Liouville amplitudes A L . Onthe other hand, we also obtain the remaining renormalized Liouville recursion up to level j = 4 from theaccurate numerical bootstrap results. They are given in eqs. (4.110) and (4.111). It is natural to speculatethat by fully understanding these renormalized Liouville recursions, combined with our interchiral blockconstructions, it would be possible to solve the Potts geometrical four-point functions analytically, whichwe will leave for future work.We have in the above provided a complete guide and summary of the results contained in this paper.For the readers’ convenience, we also give background reviews and supplementary materials. In section 2,we give a review of the critical Potts model and the conformal bootstrap with emphasis on the applicationof the bootstrap approach to the Potts geometrical four-point functions. While the numerical aspects areimportant in the determination of the Potts amplitudes we present in section 4.2, we leave the technicaldetails to appendix B. In addition, we recall in appendix A the original amplitude ratios obtained in [3]and in appendix C the relevant analytic results of the non-diagonal Liouville theory, as they are used invarious places in the paper. We recall in this section the ingredients necessary to set up the conformal bootstrap for the geometricalcorrelation functions in the Potts model.
The Q -state Potts model [27] is defined on a lattice where at each site resides a spin variable taking Q possible values σ i = 1 , ..., Q and the nearest neighbors have interaction energy − Kδ σ i ,σ j . The partitionfunction is given by Z = (cid:88) { σ } (cid:89) { ij } e Kδ σi,σj , (2.1)where { ij } indicate the edges on the lattice and the sum is over all spin configurations { σ } . It is easy torecognize that the familiar Ising model corresponds to the case of Q = 2.While the original definition (2.1) is restricted to integer values of Q , a more general definition is givenby the Fortuin-Kasteleyn (FK) clusters [13] where, by setting v = e K −
1, the partition function (2.1)becomes Z = (cid:88) D v |D| Q κ ( D ) . (2.2)In this formulation, the partition function in given by configurations of bonds formed between neighboringlattice sites when they share the same spin value, with a probability v/ (1 + v ). The sum in (2.2) is over alldiagrams D , where |D| is the number of bonds and κ ( D ) denotes the number of connected components—theso-called FK clusters—within a diagram. We henceforth focus on the two-dimensional square lattice. Atthe critical value [28], v c = (cid:112) Q , (2.3)for 0 ≤ Q ≤
4, the system goes through a second-order phase transition and is described by a conformalfield theory (CFT) [27, 28]. Notice that in the FK-cluster description, the number of states Q from theoriginal definition enters the partition funciton (2.2) as a parameter and therefore the model is analyticallycontinued to real values of Q .Another equivalent formulation of the Potts model is through loops [29]. Taking the midpoint of eachedge to form another lattice, the loops are formed by connecting the nearest-neighboring sites such that6hey bounce on the FK clusters and internal cycles. As a result, two distinct Potts clusters are separatedby an even number of loops. The partition function in this case becomes Z = Q | V | / (cid:88) D (cid:16) vn (cid:17) |D| n (cid:96) ( D ) . (2.4)The (cid:96) ( D ) is the number of loops in a certain configuration and the loop fugacity is n = (cid:112) Q = q + q − , (2.5)where q is a quantum-group related parameter. At criticality (2.3), the partition function (2.4) only dependson the number of loops. In particular, all contractible and non-contractible loops get weight n .On the lattice, one naturally considers the correlation functions G a ,a ,...,a N = (cid:104)O a ( σ i ) O a ( σ i ) · · · O a N ( σ i N ) (cid:105) , (2.6)where the spin operator (the order parameter) is defined by O a ( σ i ) ≡ Qδ σ i ,a − . (2.7)More generic correlations are defined as a probability: P P = 1 Z (cid:88) v |D| Q κ ( D ) I P ( i , i , . . . , i N ) , (2.8)usually labelled by N ordered symbols in P . The indicator function I P is defined such that two sites i j , i k belong to the same FK cluster if and only if the corresponding ordered symbols in P are the same. Theseprobabilities, which are well-defined for arbitrary real value of Q , are the geometrical correlations that wewill study. In particular, we focus on the four-point geometrical correlations involving one or two clusters: P aaaa , P aabb , P abba and P abab . On the lattice, they can be formally related to the lattice spin correlationfunctions by studying the combinatorics [17]: G aaaa = ( Q − Q − Q + 3) P aaaa + ( Q − ( P aabb + P abba + P abab ) , (2.9a) G aabb = (2 Q − P aaaa + ( Q − P aabb + P abba + P abab , (2.9b) G abba = (2 Q − P aaaa + P aabb + ( Q − P abba + P abab , (2.9c) G abab = (2 Q − P aaaa + P aabb + P abba + ( Q − P abab . (2.9d)Notice that the left-hand side is only well-defined for integer values of Q .In the continuum limit at criticality, the Potts model can be parameterized as [30] (cid:112) Q = 2 cos (cid:18) πx + 1 (cid:19) , x ∈ [1 , ∞ ] , (2.10)where the parameter x is related to the central charge of the CFT by c = 1 − x ( x + 1) , (2.11)and the quantum group related parameter q = e iπx +1 . One can adopt a Kac parameterization for theconformal dimensions of the primary fields as: h r,s = [( x + 1) r − xs ] − x ( x + 1) (2.12) For more details of the loop formulations, see section 2.1 of [3]. h − r, − s = h r,s , although in this case the x is not restricted to be an integer as in the minimal modelsand furthermore, the Kac indices ( r, s ) can be fractions. In particular, the order parameter—i.e., the spinoperator—is known to be given by the Kac indices ( r, s ) = ( ,
0) [31, 32]. For convenience, we will also useanother parameterization β for the central charge: β = xx + 1 , ≤ β ≤ , (2.13)which is closely related to c < r, s )whose left and right conformal dimensions are given by( h, ¯ h ) = (cid:40) ( h r,s , h r,s ) , diagonal , ( h r,s , h r, − s ) , non-diagonal . (2.14)We shall often label such a primary by the superscript D for diagonal, or N for non-diagonal. Its totalconformal dimension is h + ¯ h (2.15)and the conformal spin is ¯ h − h = (cid:40) , diagonal ,rs, non-diagonal . (2.16)The four lattice sites in the continuum limit become ( i , i , i , i ) → ( z , z , z , z ) and the s, t, u -channelsare defined as s -channel : z → z , t -channel : z → z , u -channel : z → z . (2.17)From the relations with the lattice spin correlations (2.9), it is natural to consider the four-point geometricalcorrelations P aaaa , P aabb , P abba and P abab as four-point functions of the spin operator Φ , of the type: (cid:104) Φ , ( z , ¯ z )Φ , ( z , ¯ z )Φ , ( z , ¯ z )Φ , ( z , ¯ z ) (cid:105) (2.18)and they are therefore given by the conformal blocks of the spectra in the fusion channels. Accounting forthe geometry, the spectra of each channel in the geometrical correlations are related through crossing:probability s -channel t -channel u -channel P aaaa S S S P aabb S S S P abab S S S P abba S S S (2.19)In particular, under s ↔ t which we will focus on below, the spectra of P aaaa and P abab are symmetric while P aabb and P abba get interchanged.The spectra (2.19) were determined in [1] by focusing on the s -channel, using a combination of algebraicand numerical methods. They are given in terms of the affine Temperley-Lieb (ATL) modules W j, z :spectrum ATL modules Parities S W , − ∪ W j,e iπp/M j ∈ N ∗ , jp/M even S W , − ∪ W , q ∪ W j,e iπp/M j ∈ N ∗ , jp/M even S W j,e iπp/M j ∈ N ∗ , jp/M integer (2.20) The spin operator Φ / , , as determined by the representation theory of the symmetric group S Q , has a non-zero N -pointfunction as long as each point belongs to the same FK cluster as at least one other point. The four-point function thusprovides non-zero contributions to precisely the probabilities P aaaa , P aabb , P abba and P abab , with an overall multiplicity thatcan be computed from the representation theory [34–36]. Whenever we wish to single out one of these contributions, we musttherefore specify the corresponding labels. p, M are coprime integers with in particular p = 0 allowed. The ATL modules each contains a towerof primary fields and their descendants with the following Kac indices: W j,e iπp/M : ( r, s ) = ( Z + pM , j ) N , non-diagonal (2.21a) W , q : ( r, s ) = ( N ∗ , D , diagonal (2.21b)where we recall that N, D stand for non-diagonal and diagonal respectively, with the conformal dimensionsgiven by (2.14). In particular, the Virasoro modules in W , q are given by Kac modules, where the nulldescendant of the primaries with ( r, s ) ∈ N ∗ at level rs is removed.Notice that jp/M indicates the conformal spin rs of the leading primary in a module W j,e iπp/M and inthe following we will often need to refer to the modules with even and odd spins separately. We will thususe the notation W + j,e iπp/M , jp/M even , (2.22a) W − j,e iπp/M , jp/M odd , (2.22b)for the rest of the paper. Consider a generic four-point function of identical operators: (cid:104) Φ( z , ¯ z )Φ( z , ¯ z )Φ( z , ¯ z )Φ( z , ¯ z ) (cid:105) . (2.23)After mapping the four points ( z , z , z , z ) to ( z, , ∞ ,
1) through a global conformal transformation, the s, t, u channels (2.17) become s -channel : z → , t -channel : z → , u -channel : z → ∞ , (2.24)and the four-point function (2.23) can be written in terms of the conformal block expansions: G ( z, ¯ z ) = (cid:88) ( h, ¯ h ) ∈S ( c ) A ( c ) ( h, ¯ h ) F ( c ) h ( z ) ¯ F ( c )¯ h (¯ z ) , with c = s, t, u. (2.25)The constant coefficient A ( c ) ( h, ¯ h ) here—which we will henceforth refer to as the amplitude for field ( h, ¯ h )—arises from the structure constant in the fusionΦ × Φ ( c ) −→ ( h, ¯ h ) (2.26)as A ( c ) ( h, ¯ h ) = C (Φ , Φ , ( h, ¯ h )) C (( h, ¯ h ) , Φ , Φ) , (2.27)where we have chosen the normalization of the two-point functions of identical primaries to be 1 besidesposition-dependent factor. Note that our discussions below are independent of this normalization which wetake merely for convenience and notation simplicity. The structure constant is symmetric under permutationof the three fields and in the following we will also use the notation C ( r i ,s i )( r j ,s j )( r k ,s k ) , (2.28)where the indices ( r, s ) = ( r, s ) D,N represent diagonal or non-diagonal fields, as specified by the superscript.Notice that the fusion with the identity operator Φ D (1 , gives:Φ D , × ( h, ¯ h ) → ( h, ¯ h ) , (2.29) As explained in footnote 3, the correlation function decomposes into probabilities P aaaa , P aabb , P abba and P abab , dependingon the chosen geometry. The amplitudes similarly depend on the geometry, but in the general reasoning presented here weshall keep that dependence implicit and only specify it when needed. C (1 , D ( r,s )( r,s ) = 1 , (2.30)since this reduces to the normalized two-point function of ( r, s ).The essence of the conformal bootstrap approach lies in the crossing equation for the four-point functions,which states the equivalence of the conformal-block expansions in different fusion channels [37]. In the caseof the four-point function (2.23), this is to say that the conformal-block expansions (2.25) in differentchannels c = s, t, u give the same four-point functions, as a result of the associativity of the fusion algebra(2.26). Such equivalence puts strong constraints on the spectra S ( c ) and the amplitudes A ( c ) ( h, ¯ h ), andin certain cases can uniquely define the theory of interest. Combining with positivity constraints fromunitarity and powerful numerical implementations, the conformal bootstrap approach has recently led tomany rigorous results in d > P ( z, ¯ z ) = P aaaa , P abab have the same spectrum in the s - and t -channel and aregiven by: P ( z, ¯ z ) = (cid:88) ( h, ¯ h ) ∈S A ( h, ¯ h ) F ( s ) h ( z ) F ( s )¯ h (¯ z ) = (cid:88) ( h, ¯ h ) ∈S A ( h, ¯ h ) F ( t ) h ( z ) F ( t )¯ h (¯ z ) , (2.31)where S = S ( s ) = S ( t ) . For P aaaa one has S = S and A ( h, ¯ h ) = A aaaa ( h, ¯ h ) while for P abab , S = S and A ( h, ¯ h ) = A abab ( h, ¯ h ). Rewritten as (cid:88) ( h, ¯ h ) ∈S A ( h, ¯ h ) (cid:0) F ( s ) h ( z ) F ( s )¯ h (¯ z ) − F ( t ) h ( z ) F ( t )¯ h (¯ z ) (cid:1) = 0 , (2.32)this is a linear system for the amplitudes A ( h, ¯ h ). Using the method proposed in [19], one can numericallysolve this linear system by sampling the points z i . Doing this multiple times provides statistics on theamplitudes which were used in [19] as a measurement of crossing symmetry.Here we are going to take this approach one step further since we have the spectra (2.20) for all fourprobabilities. While the crossing-symmetric probabilities, P aaaa and P abab , can be expanded using (2.31),the other two probabilities, P abba and P aabb , get interchanged under s ↔ t and thus have the followingconformal block expansions: P aabb = (cid:88) ( h, ¯ h ) ∈S A aabb ( h, ¯ h ) F ( s ) h ( z ) F ( s )¯ h (¯ z ) = (cid:88) ( h, ¯ h ) ∈S A abba ( h, ¯ h ) F ( t ) h ( z ) F ( t )¯ h (¯ z ) , (2.33) P abba = (cid:88) ( h, ¯ h ) ∈S A abba ( h, ¯ h ) F ( s ) h ( z ) F ( s )¯ h (¯ z ) = (cid:88) ( h, ¯ h ) ∈S A aabb ( h, ¯ h ) F ( t ) h ( z ) F ( t )¯ h (¯ z ) . (2.34)As discussed in [1], the fields ( r, s ) with even and odd spins have the following amplitude relations: A abab = A abba , rs even , (2.35a) A abab = − A abba , rs odd , (2.35b)and therefore the symmetric and anti-symmetric combinations, P abab + P abba and P abab − P abba , only involvefields with even and odd conformal spin, respectively.Eq. (2.32) for P aaaa , P abab and eqs. (2.33)–(2.34) for P aabb , P abba together define our problem of solvingthe Potts geometrical correlations. In unitary CFTs, the amplitudes (2.27) are positive, as the squares of the structure constants or the matrix constructedfrom pairwise products of the structure constants are positive-definite. .2.1 Conformal blocks One main ingredient in the conformal bootstrap approach to the four-point functions is the computationof conformal blocks. For practical implementations, we use the Zamolodchikov recursive formula [38] tocompute the Virasoro conformal blocks of the primary fields appearing in (2.20). In particular, in the caseof the four-spin correlations (2.18) with external dimensions h , , the s -channel conformal block for aninternal field with dimension h is given by: F ( s ) h ( z ) = (16 q ) h − c − ( z (1 − z )) − c − − β θ ( q ) − c − − β H h ( q ) , (2.36)and in the t -channel we have: F ( t ) h ( z ) = F ( s ) h (1 − z ) . (2.37)In the above expressions, the elliptic nome q and the Jacobi theta function θ ( q ) are given by: q ( z ) = e iπτ , τ = i K (1 − z ) K ( z ) , θ ( q ) = ∞ (cid:88) n = −∞ q n , (2.38)where K ( z ) is the complete elliptical integral of the first kind. The H h ( q ) is given by the recursive relation H h ( q ) = 1 + ∞ (cid:88) m,n =1 (16 q ) mn h − h m,n R m,n H h m, − n ( q ) , (2.39)where the R m,n in the case of four-spin conformal blocks are given explicitly by [19]:R m,n = (cid:40) , n odd, − − mn λ mn (cid:81) mm (cid:48) =1 − m (cid:81) nn (cid:48) =1 − n λ ( − n (cid:48) +1 m (cid:48) ,n (cid:48) , n even, (2.40)with λ m (cid:48) ,n (cid:48) = − m (cid:48) β + n (cid:48) β and the products exclude ( m (cid:48) , n (cid:48) ) = (0 , r, s ∈ N ∗ ,appearing in the modules W j, . This poses problems for the computation of the conformal blocks forthese primaries which have poles for r, s ∈ N ∗ , as is obvious in (2.39). Of course, the appearance of suchpole terms in the Zamolodchikov recursive formula is associated with the fact that in minimal models thedegenerate representations have the null descendants decoupling from the spectrum, while in the case ofthe Potts model, we do not expect such decoupling to occur. This means that the theory is genericallylogarithmic [39]. The fields with r, s ∈ N ∗ in W j, are expected to have logarithmic partners, and thepresence of Jordan cells for L , ¯ L should lead to finite confomal blocks regularizing the naive divergencesin Zamolodchikov’s formula, with, in particular, a ln( z ¯ z ) dependency. This will be studied in detail in aforthcoming paper. For the time being, we content ourselves with a “naive” regularization procedure fornumerical implementations. As can be seen from (2.36) and (2.39), the residue of the pole at h = h m,n isgiven by R m,n F ( s ) h m, − n , (2.41)where F ( s ) h m, − n is the conformal block of the descendant with conformal dimension h = h m, − n = h m,n + mn .We therefore subtract this pole term from the (left and right) block of F ( s ) h m,n and include the term F ( s ) h m, − n ( z ) F ( s ) h m, − n (¯ z ) (2.42)in the four-point function with a free coefficient. This takes into account certain contributions of thedescendants in the four-point functions. However, it is worth stressing that this serves as an approximationin the numerical bootstrap, since in general the coefficient involved should have logarithmic dependence in We thank S. Ribault for various discussions on the topic of conformal blocks. This is similar to the regularization procedure in [19], however we do not assume a specific z -dependence. and the modification here could change the higher-level structure of the blocks. While this may introduceinstabilities into the numerics, we will discuss below extra constraints to impose on the numerical bootstrapin order to stabilize the solution.Note that in the module W , q of (2.20), the fields also have degenerate indices ( r, W , q . The conformal blocks in this caseare thus exact. In [19], the authors conjectured a simple spectrum for some of the geometrical correlations in the Pottsmodel which, using the bootstrap approach, was checked to satisfy the crossing equation (2.32). While itprovides a numerical description of the Potts probabilities that appeared to be in accord with Monte-Carlosimulations [20], the proposed spectrum was finally shown in [1] to be only a subset of the true Pottsspectrum (2.20). Later it was understood [4] that the spectrum of [19] was in fact valid for a generalizationof type-D minimal models, when the β in (2.13) was taken to irrational values.In [3], we studied the CFT four-point functions given by the spectrum of [19] from the lattice point of viewand revealed its connection with the Potts probabilities: the four-point functions of the operators of interestinvolve the same types of diagrammatic expansion in terms of clusters/loops as the Potts probabilities weconsider here, however with different weights assigned to the topologically non-trivial loops. We referred tothe geometrical correlations thus obtained as the “pseudo-probabilities” (see eq. (3.3) below for a precisedefinition). In the work [3] we have also studied the Potts probabilities in a lattice regularization—i.e.,on semi-infinite cylinders of finite circumference L —and observed, to arbitrarily high numerical precision,several striking facts regarding the contribution of the fields to the Potts probabilities and to the pseudo-probabilities. Crucially, these facts were observed to be independent of L , and can hence be presumed tocarry over to the continuum limit as well. In this section, we briefly summarize these results and explain howthey can be used to extract information about the Potts model from minimal models (i.e., the generalizationto generic central charges) and also as input for the bootstrap of the Potts model itself. In [19], the authors found a crossing-symmetric spectrum S r,s = S Z + , Z at generic values of 0 < Q < for the s ↔ t crossing-symmetric four-point function conjectured to describe the Potts probabilities (cid:104) V D , V N , V D , V N , (cid:105) ∝∼ P aaaa + 2 Q − P abab , (3.1)which is approximately true for generic Q and becomes exact for Q = 0 , ,
4. The fields V D , and V N , haveconformal dimensions ( h , , h , ), i.e., same as the spin operator, and have their origin in the diagonaland non-diagonal sectors, respectively, of the type-D minimal models (here and below D and N standfor diagonal and non-diagonal). Initially proposed as the spectrum for the Potts probabilities, it is nowunderstood [4] that this spectrum arises from a certain limit of minimal models when the β in (2.13) istaken to irrational values, although numerically it gives a reasonable approximation of some of the Pottsprobabilities [20]. The structure constants appearing in the four-point function were later obtained analy-tically in [4] and the corresponding CFT at generic central charges is in fact a non-diagonal generalizationof Liouville theory [26]. From now on, we will refer to the analytically-known amplitudes (the square of In [19], the spectrum was found to be crossing symmetric for complex values of Q . Here we focus on real 0 ≤ Q ≤ The factor Q − was fixed later in [20] and the claim in [19] that (3.1) was exact was modified to an approximation in [20].We show here the approximate nature of the proportionality by the symbol ∝∼ . The exact relation between this non-diagonal Liouville theory and the well-known diagonal Liouville theory is howeverunclear. See [40] for a recent study on this. A L , where L stands for Liouville. See appendixC for explicit expressions of A L that are relevant in this paper.Regarding this intriguing relation between the Potts model and minimal models, we studied in [3] thecluster interpretation of the minimal-model four-point functions and its irrational limit and thus provideda geometric picture of (3.1). We have seen there that the four-point function in question is, in fact, givenby the cluster expansion on the lattice of the type (cid:104) V D , V N , V D , V N , (cid:105) ∝ P aaaa + ˜ P abab , (3.2)where we have defined the pseudo-probability˜ P abab = 1 Z Potts (cid:88)
D∈D abab W Potts ( D ) M ( k ( D )) , (3.3)with the sum over all diagrams of the type D abab , i.e., points 1 , , M ( k ( D )) is defined as the weight of a diagram D with respect to the Potts weightwhen the two marked clusters—i.e., those labelled a and b —are separated by k (necessarily even) non-contractible loops. This difference in weighing a certain diagram is ultimately due to the different weightsassigned to the non-contractible loops in the Potts model and minimal models. In contradistinction to(3.3), the true Potts probability is P abab = 1 Z Potts (cid:88)
D∈D abab W Potts ( D ) . (3.4)Note that the two quantities (3.3) and (3.4) are expanded by the same set of diagrams D ∈ D abab , with thedifference in the weight summarized into the multiplicity M ( k ( D )). The explicit expression of M is givenby [3] M ( k = 2 l ) = 2 Q l l (cid:88) m = − l (cid:18) ll + m (cid:19) q m + q − m , (3.5)and can be written in terms of ratios of polynomials in QM ( k = 2) = 2 Q − , (3.6a) M ( k = 4) = 2(3 Q − Q − Q − Q + 2) , (3.6b)...Similar expressions hold for the geometries D abba , D aabb giving rise to the relations between probabilities P abba , P aabb and pseudo-probabilities ˜ P abba , ˜ P aabb . Through numerical studies on the lattice, we have further investigated in [3] the relation between the geom-etry of the lattice models and the contribution of the spectrum to the geometrical correlation functions. Wefound facts about how the fields in (2.20) contribute through their amplitudes to various Potts probabilities,and to their counterparts—the pseudo-probabilities—where the geometric content is modified. These factsstate the existence of universal amplitude ratios of eigenvalues of the lattice transfer matrix and, amazingly,such ratios do not depend on the lattice size. It is therefore natural to assume that the same ratios hold Non-contractible on the four-time punctured sphere at the marked points. In the case of the Potts probabilities, the non-contractible loops each gets the weight √ Q as in (2.5) while in the pseudo-probability (3.3), one sums over the algebra of the type-D Dynkin diagram for the non-contractible loop weight. See [3] formore details.
13n the continuum limit in the corresponding CFT and is translated into ratios of amplitudes of the fields. Here we restate these facts directly in the CFT language and they are of the following two types:1. When the same field contributes to both a Potts probability and to the corresponding pseudo-probability, the ratio of the two corresponding amplitudes depends only on the ATL module thatthe field belongs to, and on Q .2. When the same field contributes to two different Potts probabilities, the ratio of the correspondingamplitudes depends only on the ATL module that the field belongs to, and on Q .The facts of the first type give the ratios between the amplitudes A in the true Potts probabilities ( P abab , P abba and P aabb ) and ˜ A in the pseudo-probabilities ( ˜ P abab , ˜ P abba and ˜ P aabb ). Note that the probability P aaaa does not involve any non-contractible loops and therefore there is no corresponding pseudo-probability.We now define the following ratios for a certain ATL module related to the β given in appendix A: R β ( W j, z ) ≡ ˜ A abab A abab ( W j, z ) = (cid:80) jk =2 even β ( k ) j, z M ( k ) (cid:80) jk =2 even β ( k ) j, z . (3.7)Using the explicit expressions of β ( k ) j, z given in appendix A, we have the following: R β ( W , − ) = 2 Q − , (3.8a) R β ( W , − ) = − Q − Q − Q − Q + 2) . (3.8b)Notice that for W , − , the denominator in the last expression of (3.7) actually vanishes at Q = 1 and Q = 4, indicating that the module decouples from P abab at these values of Q . This is partially taken careof by the factor of Q − Q = 4 the module disappears from ˜ P abab as well, since M ( k ) = 1 at Q = 4. One can similarly define R γ as R γ ( W j, z ) ≡ ˜ A aabb A aabb ( W j, z ) , (3.9)which is related to the γ in appendix A. We shall however not use its explicit expression in this paper.In addition, we also have the ratios relating amplitudes in different Potts probabilities from the secondtype of facts: R ¯ α ( W + j, z ) ≡ ¯ α j, z A abab A aaaa ( W + j, z ) = A abba A aaaa ( W + j, z ) , (3.10)and R α ( W + j, z ) ≡ α j, z = A aabb A aaaa ( W + j, z ) , (3.11)where we have used the definitions of α and ¯ α in (A.1) and that A abab ( W + j, z ) = A abba ( W + j, z ) due to (2.35a)and (2.22a). Note that R ¯ α and R α are not defined for W − j, z , i.e., jpM odd, since A aaaa ( W − j, z ) = 0. Using It is crucial for this translation between lattice quantities and the continuum limit that the affine Temperley-Lieb mod-ules W j, z —the centerpiece of our algebraic understanding of the lattice model—have well-defined continuum limits, and inparticular their labels j and z can be cleanly interpreted in both contexts [1]. This is because any loop surrounding the four points can be contracted at “infinity” on the sphere. See [3] for more details. α and ¯ α in appendix A, we have the following expressions for R α and R ¯ α : R α ( W , − ) = − , (3.12a) R α ( W , ) = 11 − Q , (3.12b) R ¯ α ( W , ) = 2 − Q , (3.12c) R α ( W , − ) = 2 − Q , (3.12d) R ¯ α ( W , − ) = ( Q − Q − , (3.12e) R α ( W , ) = − Q − Q + 15 Q − Q + 4 Q − Q − Q + 1) , (3.12f) R ¯ α ( W , ) = − ( Q − Q + 2)( Q − Q − . (3.12g)As discussed in [3], the ratios (3.8) and (3.12) were obtained as a numerical lattice observation whosefirst-principle derivation is still unknown. These thus comprise all the ratios of the two types for ATLmodules up to j = 4. In the following, we will first use the ratios (3.8) to analytically extract certainPotts amplitudes from the well-known Liouville amplitudes A L . We will then use this together with theratios (3.12) to bootstrap the Potts probabilities. The key observation now from the results we summarized above is that, while the geometric feature ischanged from the Potts probability (3.4) to the pseudo-probability (3.3) through the multiplicity M ( k ),only the global amplitudes A ( W j, z ) associated with entire ATL modules are modified. The relationsbetween the amplitudes of fields belonging to the same ATL module remain the same and this relationpermeates into the continuum—as manifested in the existence of the universal amplitude ratios. Thisallows us to define the “interchiral conformal blocks” F j, z , which group the Virasoro conformal blocksaccording to the ATL modules they belong to. From (3.7), (3.10) and (3.11) we see that it is the sameinterchiral conformal blocks F j, z that enter various probabilities and pseudo-probabilities. The existence ofthese blocks is ultimately due to the degeneracy of the field Φ D , in W , q and can be constructed explicitlyas an infinite sum of products of left and right conformal blocks, which we shall discuss in details in section4.1. As suggested in [2], the underlying algebra can be considered as an extension of the product of leftand right Virasoro algebras via fusion with Φ D , , leading to the object dubbed the interchiral algebra inthat reference. This algebra, in turn, can be obtained as the continuum limit of the affine Temperley-Liebalgebra.Consider now the combination P aaaa + ˜ P abab . (3.13)The corresponding CFT correlation function (3.2) is well-known to be given by the non-diagonal Liouvilletheory of [19]. Note that its spectrum S Z + , Z belongs to the ATL modules W j, − . We can then expand itin terms of the interchiral conformal blocks as P aaaa + ˜ P abab = ∞ (cid:88) j =0 even A L ( W j, − ) F j, − . (3.14)Meanwhile, from the lattice study in [3], we have seen that the modules W , − and W j, z from the Pottsspectrum (2.20) all appear, where the amplitudes for W j, z are modified from their corresponding values in As mentioned in appendix D of [3], it is numerically impossible to obtain the complete set of ratios for j = 6 using thecurrent lattice-computation approach, although we have provided in that reference some partial results. R β ( W j, z ) defined in (3.7). The combination (3.13) can thus be writtenas P aaaa + ˜ P abab = A aaaa ( W , − ) F , − + (cid:88) j odd ˜ A abab ( W − j, − ) F j, − + (cid:88) j even (cid:0) A aaaa ( W + j, − ) + ˜ A abab ( W + j, − ) (cid:1) F j, − + (cid:88) jpM even (cid:0) A aaaa ( W + j, z ) + ˜ A abab ( W + j, z ) (cid:1) F j, z + (cid:88) jpM odd ˜ A abab ( W − j, z ) F j, z = A aaaa ( W , − ) F , − + (cid:88) j odd R β A abab ( W − j, − ) F j, − + (cid:88) j even A aaaa ( W + j, − ) (cid:0) R β R ¯ α (cid:1) F j, − + (cid:88) jpM even A aaaa ( W + j, z ) (cid:0) R β R ¯ α (cid:1) F j, z + (cid:88) jpM odd R β A abab ( W − j, z ) F j, z . (3.15)In the case the the true Potts probabilities, one has instead the combination: P aaaa + P abab = A aaaa ( W , − ) F , − + (cid:88) j odd A abab ( W − j, − ) F j, − + (cid:88) j even (cid:0) A aaaa ( W + j, − ) + A abab ( W + j, − ) (cid:1) F j, − + (cid:88) jpM even (cid:0) A aaaa ( W + j, z ) + A abab ( W + j, z ) (cid:1) F j, z + (cid:88) jpM odd A abab ( W − j, z ) F j, z = A aaaa ( W , − ) F , − + (cid:88) j odd A abab ( W − j, − ) F j, − + (cid:88) j even A aaaa ( W + j, − ) (cid:0) R ¯ α (cid:1) F j, − + (cid:88) jpM even A aaaa ( W + j, z ) (cid:0) R ¯ α (cid:1) F j, z + (cid:88) jpM odd A abab ( W − j, z ) F j, z . (3.16)Note that F in (3.15) and (3.16) represent s - or t -channel blocks since in both cases we have combinationsthat are crossing-symmetric under s ↔ t .One remark on the bootstrap problem follows immediately. By comparing eqs. (3.15)–(3.16), it is obviousthat the crossing-symmetric spectrum proposed in [19], which gives a complete description of eq. (3.15),is also a solution to the conformal block expansion of the true Potts probabilities (3.16) using the fullspectrum (2.20), with of course the amplitudes for the whole ATL modules given by ˜ A abab instead of A abab as in (3.15). This means that within the full spectrum (2.20) of the Potts model, the states which do notappear in the spectrum of [19] have A aaaa + ˜ A abab = 0, i.e.,1 + R β ( W + j, z ) R ¯ α ( W + j, z ) = 0 , (3.17a) R β ( W − j, z ) = 0 , (3.17b)for pM (cid:54) = , as we have checked explicitly in [3] for all the ATL modules up to j = 4. This suggests that thesolution to the original bootstrap problem considered in [19], i.e., the bootstrap of the probabilities P aaaa + P abab , is not unique: The spectrum and amplitudes in [19] is one solution, while the true Potts spectrumwith its amplitudes provides another one, and possibly there exists (infinitely many?) further solutions.Geometrically this can be understood as the freedom of assigning weights to cluster/loop configurations, a16echanism which we have seen explicitly at play above, where it involves two different ways of assigningweights to the non-contractible loops. This complexity of the solution to the bootstrap problem is perhapsrooted in the irrationality of the theory in general, and the simple spectrum given by [19]—being itself ageneralization of minimal models—stands out as special, or “minimal” in a(nother) sense, as it sees a largenumber of fields decouple from the spectrum, cf. (3.17).Now focus on eqs. (3.15)–(3.16). It is fascinating to see that we can, in fact, extract the true Pottsamplitudes from the known Liouville amplitudes A L for the modules that contribute to both combinations: A aaaa ( W , − ) = A L ( W , − ) , (3.18a) R β A abab ( W − j, − ) = A L ( W − j, − ) , j , (3.18b) A aaaa ( W + j, − )(1 + R β R ¯ α ) = A L ( W + j, − ) , j . (3.18c)Using eqs. (3.8) and (3.12), we have the following expressions of the Potts amplitudes in terms of theLiouville amplitudes: A aaaa ( W , − ) = A L ( W , − ) , (3.19a) A abab ( W , − ) = Q − A L ( W , − ) , (3.19b) A aaaa ( W , − ) = ( Q − Q − Q + 2) Q ( Q − A L ( W , − ) . (3.19c)Writing this out explicitly, the Potts probabilities are given by the A L and the interchiral blocks as P aaaa = A L ( W , − ) F , − + ( Q − Q − Q + 2) Q ( Q − A L ( W , − ) F , − + . . . , (3.20a) P abab = Q − A L ( W , − ) F , − + ( Q − Q − Q − Q − Q + 2)4 Q ( Q − A L ( W , − ) F , − + . . . , (3.20b)where we have used (3.12e) in writing the second term of (3.20b), and the left-out terms ( . . . ) are to bedetermined by the bootstrap computations.Let us now turn to another combination of probabilities P aaaa + ˜ P aabb . (3.21)In the s -channel, as studied in [3], we have P aaaa + ˜ P aabb = (cid:0) A aaaa ( W , − ) + ˜ A aabb ( W , − ) (cid:1) F ( s )0 , − + (cid:88) jpM even (cid:0) A aaaa ( W + j, z ) + ˜ A aabb ( W + j, z ) (cid:1) F ( s ) j, z + (cid:88) a ˜ A aabb ( W , q a ) F ( s )0 , q a = (cid:0) A aaaa ( W , − ) + R γ A aabb ( W , − ) (cid:1) F ( s )0 , − + (cid:88) jpM even (cid:0) A aaaa ( W + j, z ) + R γ A aabb ( W + j, z ) (cid:1) F ( s ) j, z + (cid:88) a ˜ A aabb ( W , q a ) F ( s )0 , q a , (3.22)where the last term involves diagonal Verma modules with the conformal dimensions( h e + a x +1 , , h e + a x +1 , ) , e ∈ Z . (3.23)It was argued in [3], by comparison with the CFT results [41], that the s -channel here involves purelydiagonal fields so the first two terms disappear from (3.22) with the amplitude ratios R γ , R α as we have17hecked explicitly in [3]. In the limit of irrational β , the sum in the last term becomes an integral over acompact set, (cid:90) π d θ , (3.24)where we have introduced the variable θ = a πx +1 , and (3.23) becomes( h r, , h r, ) , r ∈ R , (3.25)i.e., a continuous diagonal spectrum. Geometrically speaking this indicates that the non-contractable loopweights are integrated over the fugacities n z = z + z − , with z = e iθ . (3.26)In contrast with this, we have in the Potts model: P aaaa + P aabb = (cid:0) A aaaa ( W , − ) + A aabb ( W , − ) (cid:1) F ( s )0 , − + (cid:88) jpM even (cid:0) A aaaa ( W j, z ) + A aabb ( W j, z ) (cid:1) F ( s ) j, z + A aabb ( W , q ) F ( s )0 , q = A aaaa ( W , − ) (cid:0) R α ( W , − ) (cid:1) F ( s )0 , − + (cid:88) jpM even A aaaa ( W j, z ) (cid:0) R α ( W j, z ) (cid:1) F ( s ) j, z + A aabb ( W , q ) F ( s )0 , q . (3.27)First notice that for the first two terms, we do not expect in general a cancellation and thus, the s -channelspectrum here in the true Potts probabilities involves non-diagonal modules W j, z . Furthermore, the lastterm, as argued in [1] already, comes from the requirement that in the Potts model, all loops—contractibleor non-contractible—carry the weight √ Q . Therefore one fixes [1]:z = q ± , (3.28)and obtains the module W , q . As mentioned before, this includes the Kac modules of diagonal primarieswith conformal dimensions ( h r, , h r, ) , r ∈ N ∗ , (3.29)a discrete spectrum. By comparing (3.22) with (3.27), we see that the diagonal spectrum in the s -channel ofthe geometry D aabb , where the two FK clusters are separated by a large number of non-contractible loops,encodes important geometric information on the non-contractible loop weight in the lattice model. We will consider the bootstrap problem of the following probabilities as related by crossing: P ( s ) aaaa = P ( t ) aaaa , (3.30a) P ( s ) abab = P ( t ) abab , (3.30b) P ( s ) aabb = P ( t ) abba , (3.30c) P ( s ) abba = P ( t ) aabb . (3.30d)Eqs. (3.30) are simply a shorthand rewriting of eqs. (2.32), (2.33) and (2.34) which should be interpreted asfollows: the subscripts indicate the spectrum for the interchiral block expansions and thus the correspondingamplitudes, while the superscripts indicate which channel of the blocks to use. The basic idea is now to writethe Potts probabilities in terms of the interchiral block expansions with the amplitudes A ( W ) associatedwith the whole ATL modules. Eqs. (3.30) are then a coupled linear system for these amplitudes and willbe used for the “interchiral conformal bootstrap”. 18he amplitudes involved here are: A aaaa ( W , − ) , A aaaa ( W + j, z ) ,A abab ( W + j, z ) , A abab ( W − j, z ) ,A abba ( W + j, z ) , A abba ( W − j, z ) ,A aabb ( W , − ) , A aabb ( W + j, z ) , A aabb ( W , q ) , (3.31)and we can then write eq. (3.30) as the interchiral bootstrap equations : A aaaa F ( s )0 , − + (cid:88) {W + j, z2 } A aaaa F ( s ) j, z = A aaaa F ( t )0 , − + (cid:88) {W + j, z2 } A aaaa F ( t ) j, z , (3.32a) (cid:88) {W + j, z2 } A abab F ( s ) j, z + (cid:88) {W − j, z2 } A abab F ( s ) j, z = (cid:88) {W + j, z2 } A abab F ( t ) j, z + (cid:88) {W − j, z2 } A abab F ( t ) j, z , (3.32b) A aabb F ( s )0 , − + A aabb F ( s )0 , q + (cid:88) {W + j, z2 } A aabb F ( s ) j, z = (cid:88) {W + j, z2 } A abba F ( t ) j, z + (cid:88) {W − j, z2 } A abba F ( t ) j, z , (3.32c) (cid:88) {W + j, z2 } A abba F ( s ) j, z + (cid:88) {W − j, z2 } A abba F ( s ) j, z = A aabb F ( t )0 , − + A aabb F ( t )0 , q + (cid:88) {W + j, z2 } A aabb F ( t ) j, z , (3.32d)where we have omitted the arguments for the amplitudes for notation simplicity. Notice that we can furtherimpose the constraints R α and R ¯ α from (3.12). Recall also that W + j, z have the same amplitudes A abab and A abba , while W − j, z have opposite amplitudes due to (2.35). This gives us: A aaaa R α = A aabb , for W , − , W + j, z (3.33a) A aaaa R ¯ α = A abab , for W + j, z (3.33b) A abab = A abba , for W + j, z (3.33c) A abab = − A abba , for W − j, z . (3.33d)In addition, we have obtained analytically (3.19) which, with proper normalizations, can be imposed asextra constraints into the bootstrap. With the setup given by eqs. (3.32) and (3.33), we are now ready to bootstrap the amplitudes (3.31). In thissection, we start by constructing the interchiral conformal blocks F j, z explicitly and show how they arisefrom the degeneracy of the field Φ D , . We then present and study the bootstrap results on the amplitudes.The numerical details of the bootstrap will be discussed in appendix B. In the conformal bootstrap approach to the diagonal Liouville theory [24,25], the degeneracy of the diagonalfields Φ r,s = Φ D , and Φ D , are used to obtain the recursions when the structure constants, and thus theamplitudes, are related through shifting the Kac indices by 2 units: s ± r ±
1, which eventuallyleads to a full solution of the theory. The key idea is to consider the four-point functions involving thesedegenerate fields which, in the conformal block expansions, truncate to only two terms. One can then writethe relations of the structure constants using the fusing matrix—the linear transformation between theconformal blocks in the two channels as constructed from the solutions of BPZ equations. This techniquewas further generalized to the non-diagonal case in [4, 26] which gives the analytic amplitudes A L of thenon-diagonal Liouville theory in [19]. 19n the case of the Potts model, the degeneracy of Φ D , is absent and therefore this technique does notapply directly. However, we see in the spectrum (2.20) that the field Φ D , ∈ W , q is degenerate and oneexpects the recursions in the (diagonal and non-diagonal) Liouville theory for shifting the first Kac index, r ±
1, to hold in this case. In fact, for the Potts geometrical correlations P aaaa , P abab , P aabb , P abba whichcan be considered (up to the remarks in footnote 3) as four-point functions of the spin operator (2.18): (cid:104) Φ , Φ , Φ , Φ , (cid:105) , (4.1)the degeneracy of Φ D , indicates a recursion of the amplitudes with r shifted by 1 which we will giveexplicitly below, deferring the derivation to section 4.1.1. Such recursion exactly relates the amplitudesof the primaries belonging to the same ATL module and organizes the corresponding Virasoro conformalblocks into the interchiral conformal blocks.Recall from the spectrum (2.20) that the non-diagonal primary fields in the modules W j,e iπp/M havethe conformal dimensions ( h pM + e,j , h pM + e, − j ) , e ∈ Z . (4.2)The module W j,e iπp/M therefore contains fields related by e → e +1, with the leading amplitude A ( h pM ,j , h pM , − j ).We will therefore take the overall amplitude associated with the interchiral blocks to be: A ( W j,e iπp/M ) = (cid:40) A ( h ,j , h , − j ) , for p = 0 , A ( h pM ,j , h pM , − j ) , otherwise , (4.3)where the factor of 2 in the second line accounts for the identification of the amplitudes: A ( h r,s , h r, − s ) = A ( h r, − s , h r,s ) = A ( h − r,s , h r,s ) , (4.4)since the two non-diagonal fields ( r, s ) and ( r, − s ) have the same total conformal dimension (2.15) and spin(2.16) (up to a sign). The amplitudes of the other fields within the module are related by the recursion R Ne,j = A ( h e +1 ,j , h e +1 , − j ) A ( h e,j , h e, − j )= Γ( − j − e β )Γ(1 + j − e β )Γ( − j + e β )Γ( j + e β )Γ( − j + e β )Γ( j + e β )Γ( j + e β )Γ(1 − j + e β )Γ( j − e β )Γ( − j − e β )Γ( j − e β )Γ( − j − e β ) , (4.5)which has the properties 1 R N − e − , − j = R Ne,j , (4.6a)1 R N − e − ,j = R Ne,j , (4.6b)where (4.6a) is explicit while (4.6b) is expected due to (4.4) and can be easily checked using that j ∈ Z .Notice that in the special case of e = 0, the expression (4.5) includes the divergent factor Γ( − j )Γ(1 − j ) . In thiscase one can use the property (4.6b) to obtain instead: R N ,j = Γ( j )Γ(1 − j − β )Γ( − j + β )Γ( j + β )Γ(1 + j )Γ( − j + β )Γ( − j − β )Γ( j − β ) (4.7)which is divergence-free.Note that due to the definition (4.3) and the identification (4.4), we have also A ( W j,e iπp/M ) = 2 A ( h pM ,j , h pM , − j ) = 2 A ( h − pM ,j , h − pM , − j ) = 1 R N − pM ,j A ( W j,e iπ (1 − p/M ) ) , (4.8)20.e., the recursion (4.5) also relates the global amplitudes for the modules W j,e iπp/M ↔ W j,e iπ (1 − p/M ) , (4.9)which corresponds to z ↔ z − . (4.10)This reduces the independent non-diagonal amplitudes to A ( W j,e iπp/M ) with0 ≤ pM ≤ . (4.11)The module W , q involves diagonal primaries with conformal dimensions( h e, , h e, ) , e ∈ N . (4.12)Again the corresponding Kac modules are related by e → e + 1 with the leading ( h , , h , ) and thereforewe take A ( W , q ) = A ( h , , h , ) . (4.13)The amplitudes of diagonal fields have the recursion: R De,s = A ( h e +1 ,s , h e +1 ,s ) A ( h e,s , h e,s )= Γ( s − e β )Γ(1 + s − e β )Γ( − s + e β ) Γ( − s + e β ) Γ( − s + e β )Γ(1 − s + e β )Γ( s − e β ) Γ( s − e β ) , (4.14)which has the explicit property 1 R D − e − , − s = R De,s (4.15)as expected since h r,s = h − r, − s .The interchiral conformal blocks involved in the bootstrap equations (3.32) are the following: F ( c ) j, − , F ( c ) j, z = e iπp/M , F ( c )0 , q , (4.16)where here and below the channels are denoted with c = s, t . Writing explicitly, we define: F ( c ) j, − = 12 (cid:88) e ∈ N R e + ,j (cid:18) F ( c ) h e + 12 ,j ( z ) F ( c ) h e + 12 , − j (¯ z ) + F ( c ) h e + 12 , − j ( z ) F ( c ) h e + 12 ,j (¯ z ) (cid:19) , = (cid:88) e ∈ N R e + ,j Re (cid:20) F ( c ) h e + 12 ,j ( z ) F ( c ) h e + 12 , − j (¯ z ) (cid:21) (4.17)where R e + ,j ≡ A ( h e + ,j , h e + , − j ) A ( h ,j , h , − j ) = (cid:40) , for e = 0 , (cid:81) e − i =0 R Ni + ,j , for e (cid:54) = 0 , (4.18)and we have used (4.4) to write the block of ( h − e − ,j , h e + ,j ) as F ( c ) h e + 12 , − j ( z ) F ( c ) h e + 12 ,j (¯ z ) (4.19) Recall that z is related with the phase acquired by the non-contractible lines as they wind around the axis of the cylinder[42]. Switching z and z − amounts to switching clockwise and counterclockwise. In the spectrum (2.20) we have only s = 1, but here we give the most general recursion as the result of the degenerateΦ D , . Note that compared to (4.5) we have changed the notation j → s since the module W , q corresponds to j = 0 but theconformal weights can be written with s = 1 using (2.12). See eq. (5.21) in [1].
21o restrict the summation to e ≥ F ( c ) j, z = e iπp/M with pM (cid:54) = we have F ( c ) j,e iπp/M = 12 (cid:88) e ∈ N R e + pM ,j (cid:18) F ( c ) h e + pM ,j ( z ) F ( c ) h e + pM , − j (¯ z ) + F ( c ) h e + pM , − j ( z ) F ( c ) h e + pM ,j (¯ z ) (cid:19) , = (cid:88) e ∈ N R e + pM ,j Re (cid:20) F ( c ) h e + pM ,j ( z ) F ( c ) h e + pM , − j (¯ z ) (cid:21) (4.20)with R e + pM ,j ≡ A ( h e + pM ,j , h e + pM , − j ) A ( h pM ,j , h pM , − j ) = (cid:40) , for e = 0 , (cid:81) e − i =0 R Ni + pM ,j , for e (cid:54) = 0 . (4.21)Notice that we have used (4.4) to group the blocks in the module W j,e iπ (1 − p/M ) with e < W j,e iπp/M with e ≥ F j,e iπp/M , and vice versa. See figure 2 for an illustration ofthis for the case of F ,i and F , − i . Also keep in mind that due to (4.8), the blocks F j,e iπp/M and F j,e iπ (1 − p/M ) can be further grouped into F j,e iπp/M + R N − pM ,j F j,e iπ (1 − p/M ) . (4.22)In the special case of p = 0, we have F ( c ) j, = 12 (cid:88) e ≥ R e,j (cid:18) F ( c ) h e,j ( z ) F ( c ) h e, − j (¯ z ) + F ( c ) h e, − j ( z ) F ( c ) h e,j (¯ z ) (cid:19) , = (cid:88) e ≥ R e,j Re (cid:104) F ( c ) h e,j ( z ) F ( c ) h e, − j (¯ z ) (cid:105) (4.23)with R e,j ≡ (cid:40) , for e = 0 , A ( h e,j ,h e, − j ) A ( h ,j ,h , − j ) = 2 (cid:81) e − i =0 R Ni,j , for e (cid:54) = 0 , (4.24)where the difference in the definition of R e (cid:54) =0 ,j and R ,j takes into account the special choice of (4.3) for p = 0.Finally, the block F , q is given by: F ( c )0 , q = (cid:88) e ∈ N ∗ R e, F ( c ) h e, ( z ) F ( c ) h e, (¯ z ) , (4.25)where R e, ≡ A ( h e, , h e, ) A ( h , , h , ) = (cid:40) , for e = 1 , (cid:81) e − i =1 R Di, , for e (cid:54) = 1 . (4.26)In figure 2, we give explicit examples of the construction of various interchiral blocks. In this subsection, we derive the recursions (4.5) and (4.14) using the degeneracy of Φ D , . The key isto study the four-point functions involving the degenerate field Φ D , , as done in the conformal bootstrapapproach to the diagonal and non-diagonal Liouville theory in [4, 24–26]. We first briefly summarize thegeneral formalism and then explore its consequences on the geometrical correlation of the type (4.1).Consider a generic four-point function with the degenerate field Φ D , : (cid:104) Φ D , Φ r ,s Φ r ,s Φ r ,s (cid:105) , (4.27)22igure 2: Examples of the construction of interchiral conformal blocks F j, z and F , q . The dots indicate thefields ( r, s ) = ( e + pM , j ) N , ( e + 1 , D whose amplitudes satisfy the recursion R D,N in eqs. (4.14) and (4.5)under e → e + 1 as illustrated by the arrows. The leading primary in each ATL module whose amplitudeis taken as A ( W ) in (4.3) and (4.13) is labeled red. We give explicit examples of the coefficients R for sub-leading fields entering the blocks indicated with blue. The fields in modules W j,e iπp/M and W j,e iπ (1 − p/M ) are regrouped into the blocks F j,e iπp/M and F j,e − iπp/M as illustrated in the magenta boxes.where Φ r i ,s i represent either diagonal or non-diagonal fields. Due to the degeneracy of Φ D , , the fusioninvolves only two terms Φ D , × Φ Dr,s → Φ Dr +1 ,s + Φ Dr − ,s , (4.28a)Φ D , × Φ Nr,s → Φ Nr +1 ,s + Φ Nr − ,s . (4.28b)and the s - and t -channels of the four-point function (4.27) are illustrated in figure 3. The conformal blockexpansions of (4.27) in the s - and t -channel therefore truncate to two terms: (cid:104) Φ D , Φ r ,s Φ r ,s Φ r ,s (cid:105) = (cid:2) F + F − (cid:3) ( s ) a ( s ) (cid:20) ¯ F + ¯ F − (cid:21) ( s ) = (cid:2) F + F − (cid:3) ( t ) a ( t ) (cid:20) ¯ F + ¯ F − (cid:21) ( t ) , (4.29)where ± represents ( r i ± , s i ) and we have omitted the dependence on the external fields. Note that23igure 3: The s - and t -channel of the four-point function (cid:104) Φ D , Φ r ,s Φ r ,s Φ r ,s (cid:105) .depending on the external fields, the s - and t -channels can independently involve either diagonal or non-diagonal fields. We will in the following study four-point functions of various types and use the labels: DD, N N, DN, N D (4.30)where the first letter indicates the type of field in the s -channel, and the second letter similarly specifiesthe type for the t -channel. (According to the fusion (4.28), these are just the labels for the fields Φ r ,s andΦ r ,s .) The amplitude matrix a ( c ) in (4.29) is given by a ( c ) = (cid:34) a ( c )+ a ( c ) − (cid:35) , diagonal , (cid:34) a ( c )+ a ( c ) − (cid:35) , non-diagonal , for c = s, t. (4.31)The amplitudes a ( c ) ± come from the structure constants: a ( c ) ± = (cid:40) C (2 , D ( r ,s )( r ± ,s ) C ( r ± ,s )( r ,s )( r ,s ) , for c = s,C (2 , D ( r ,s )( r ± ,s ) C ( r ± ,s )( r ,s )( r ,s ) , for c = t, (4.32)where ( r, s ) represent either diagonal or non-diagonal fields obeying the fusion (4.28). The s - and t -channelconformal blocks are related through the fusing matrix: (cid:20) F + F − (cid:21) ( s ) = (cid:20) F ++ F + − F − + F −− (cid:21) (cid:20) F + F − (cid:21) ( t ) , (4.33)and similarly (cid:20) ¯ F + ¯ F − (cid:21) ( s ) = (cid:20) ¯ F ++ ¯ F + − ¯ F − + ¯ F −− (cid:21) (cid:20) ¯ F + ¯ F − (cid:21) ( t ) , (4.34)where F ±± is given by F st = Γ(1 − s β λ r ,s )Γ( t β λ r ,s ) (cid:81) + , − Γ( ± β λ r ,s − s β λ r ,s + t β λ r ,s ) , with s , t = ± (4.35)24nd ¯ F is obtained by replacing λ with ¯ λ , defined as: λ r i ,s i = − r i β + s i β , (4.36a)¯ λ r i ,s i = (cid:40) λ r i ,s i , diagonal ,λ − r i ,s i , non-diagonal . (4.36b)Plugging (4.33) and (4.34) into (4.29), we obtain (cid:20) F ++ F + − F − + F −− (cid:21) T a ( s ) (cid:20) ¯ F ++ ¯ F + − ¯ F − + ¯ F −− (cid:21) = a ( t ) , (4.37)which gives the relations among a ( s ) ± and a ( t ) ± . Keep in mind that the explicit relations depend on theproperties (4.30) and therefore the explicit form of a ( c ) as in (4.31).In the conformal bootstrap approach to solve the diagonal [24, 25] and non-diagonal [4, 26] Liouvilletheory, the ratio ρ = a ( s )+ a ( s ) − (4.38)has been exploited in various four-point functions of the type (4.27) to obtain the recursion for shifting theamplitudes with r ± a ( t ) ± and a ( s ) ± . From (4.37), this relation can be extracted for differenttypes of the four-point function as labeled with (4.30). Defining the ratios χ st = a ( t ) t a ( s ) s , with s , t = ± , (4.39)we will need the following explicit expressions from (4.37): χ DN − + = F − + ¯ F −− + ρ DN F ++ ¯ F + − ,χ ND + − = F + − ¯ F −− + 1 ρ ND F −− ¯ F + − ,χ DD − + = F − + ¯ F − + + ρ DD F ++ ¯ F ++ ,χ DD + − = F + − ¯ F + − + 1 ρ DD F −− ¯ F −− ,χ DD −− = F −− ¯ F −− + ρ DD F + − ¯ F + − , (4.40)where ρ DN = − F − + ¯ F − + F ++ ¯ F ++ , ρ ND = − F − + ¯ F + − F ++ ¯ F −− , ρ DD = − F − + ¯ F −− F ++ ¯ F + − . (4.41)The superscript should be interpreted as in (4.30): for example, χ DN − + corresponds to the amplitude ratioof the non-diagonal t -channel field ( r + 1 , s ) N with the diagonal s -channel field ( r − , s ) D .We are now ready to derive the recursions (4.5) and (4.14) for non-diagonal and diagonal fields in thefour-point function (4.1).For non-diagonal fields, consider the following four-point function of the type (4.1): (cid:104) Φ N , Φ D , Φ N , Φ D , (cid:105) , (4.42)where the fusion gives the non-diagonal fields in the Potts spectrum (2.20):Φ N , × Φ D , → ( h e,j , h e, − j ) (4.43) This corresponds to the Liouville momentum P s,r used in [4]. A ( h e,j , h e, − j ) = C ( , N ( , D ( e,j ) N C ( e,j ) N ( , D ( , N . (4.44)The desired recursion (4.5) is then written as R Ne,j = A ( h e +1 ,j , h e +1 , − j ) A ( h e,j , h e, − j ) = C ( , N ( , D ( e +1 ,j ) N C ( e +1 ,j ) N ( , D ( , N C ( , N ( , D ( e,j ) N C ( e,j ) N ( , D ( , N . (4.45)Now consider the four-point function G DN = (cid:104) Φ D , Φ D , Φ N , Φ Ne,j (cid:105) , (4.46)whose crossing equation and the relevant fusion channels are illustrated in figure 4. The correspondingamplitudes in the two channels come from the structure constants and give the following ratio: χ DN − + , = C (2 , D ( e,j ) N ( e +1 ,j ) N C ( e +1 ,j ) N ( , D ( , N C (2 , D ( , D ( − , D C ( − , D ( , N ( e,j ) N . (4.47)Keep in mind the identification C ( − , D ( , N ( e,j ) N = C ( , D ( , N ( e,j ) N , (4.48)since ( ,
0) and ( − ,
0) represent the same spin field.Figure 4: The s - and t -channels of the four-point function (cid:104) Φ D , Φ D , Φ N , Φ Ne,j (cid:105) . Notice that since ( ,
0) and( − ,
0) both represent the same spin field, we obtain the structure constants C ( − , D ( , N ( e,j ) N in the s -channel and C ( e +1 ,j ) N ( , D ( , N in the t -channel which relates two fields with e → e + 1.We then turn to the four-point function: G ND = (cid:104) Φ D , Φ Ne,j Φ Ne,j Φ D , (cid:105) (4.49)as illustrated in figure 5. Notice that in this case we have a ( t ) − = C (2 , D (2 , D (1 , D C (1 , D ( e,j ) N ( e,j ) N = 1 , (4.50)where (1 , D represents the identity field and the structure constants in (4.50) are given by (2.30), due tothe normalization of the two-point function. Therefore, one has χ ND + − , = 1 C (2 , D ( e,j ) N ( e +1 ,j ) N C ( e +1 ,j ) N ( e,j ) N (2 , D . (4.51) We have, for convenience, chosen (2.30) which means the constant in the two-point functions are normalized to 1. With adifferent normalization, the derivation here still holds, since all the normalization factors cancel in the final expression (4.55)below. Therefore, the recursions we obtain here are independent of the normalization. s - and t -channels of the four-point function (cid:104) Φ D , Φ Ne,j Φ Ne,j Φ D , (cid:105) . Notice the appearance of theidentity field Φ D , in the t -channel and the corresponding amplitude a ( t ) − = 1, due to the normalization ofthe two-point functions.Finally, consider the four-point function G DD = (cid:104) Φ D , Φ D , Φ D , Φ D , (cid:105) (4.52)as illustrated in figure 6. Similar to the previous case, one has a ( t ) − = C (2 , D (2 , D (1 , D C (1 , D ( , D ( , D = 1 , (4.53)and therefore χ DD −− , = 1 C (2 , D ( , D ( − , D C ( − , D ( , D (2 , D . (4.54)Figure 6: The s - and t -channels of the four-point function (cid:104) Φ D , Φ D , Φ D , Φ D , (cid:105) .It is now easy to see that the recursion (4.45) can be expressed as R Ne,j = χ DN − + , χ ND + − , χ DN − + , χ DD −− , , (4.55)where we have used the permutation symmetry of the three-point structure constants. After plugging inthe explicit expressions of (4.40)–(4.41), eq. (4.55) becomes (4.5).27n the diagonal case, the derivation of (4.14) is completely analogous. We then consider the followingfour-point function of the type (4.1): (cid:104) Φ D , Φ D , Φ D , Φ D , (cid:105) , (4.56)where the diagonal fields arise from the fusionΦ D , × Φ D , → ( h e,s , h e,s ) (4.57)with the amplitudes A ( h e,s , h e,s ) = C ( , D ( , D ( e,s ) D C ( e,s ) D ( , D ( , D . (4.58)The recursion (4.14) is then given by R De,s = A ( h e +1 ,s , h e +1 ,s ) A ( h e,s , h e,s ) = C ( , D ( , D ( e +1 ,s ) D C ( e +1 ,s ) D ( , D ( , D C ( , D ( , D ( e,s ) D C ( e,s ) D ( , D ( , D . (4.59)Going through the same procedure as in the non-diagonal case, but replacing (4.46), (4.49) with G DD = (cid:104) Φ D , Φ D , Φ D , Φ De,s (cid:105) , (4.60a) G DD = (cid:104) Φ D , Φ De,s Φ De,s Φ D , (cid:105) , (4.60b)one arrives at the expression for the recursion (4.59) given by R De,s = χ DD − + , χ DD + − , χ DD − + , χ DD −− , . (4.61)Plugging in (4.40)–(4.41), we obtain (4.14).Note that to obtain (4.55) and (4.61), it is important that we are studying a four-point function of thespin operator where the amplitudes are given by three-point structure constants as in (4.44) and (4.58),since in this case the four-point function of figure 4 involving Φ D , gives rise to both C ( , , e,j ) and C ( , , e +1 ,j ) in their s - and t -channels. In this section, we give the bootstrap results on the amplitudes (3.31) associated with the ATL modulesup to j = 4 and leave the numerical details to appendix B. As discussed in section 3.4, this involvessolving numerically the truncated interchiral bootstrap equations (3.32) combined with the constraints ofthe amplitude ratios (3.33) and the analytic results (3.19). For this last constraint, we have in fact imposedthe ratios of A abab ( W , − ) A aaaa ( W , − ) , A aaaa ( W , − ) A aaaa ( W , − ) (4.62)as obtained from (3.19) without fixing the overall normalization. It is worth pointing out that (4.62) canin fact be (partially) bootstrapped as a consistency check. See figure 29 and the related discussions inappendix B.1. Notice that in the Potts spectrum (2.20), the leading primary in the module W , q has theconformal dimension ( h , , ¯ h , ) = (0 , P aabb , it is natural to use thenormalization A aabb ( W , q ) = 1 (4.63)for the bootstrap equations (3.32).From the discussions in [1], it is expected that some amplitudes should display singularities at rationalvalues of β , the effect of which is to cancel the overall singularities and thus lead to smooth geometricalcorrelations. We will study this in more details in the next section, while here we simply point out thelocations of the singularities. 28 .2.1 A aaaa Up to j = 4, the following amplitudes appear in the interchiral block expansion of P aaaa : A aaaa ( W , − ) , A aaaa ( W , ) ,A aaaa ( W , − ) , A aaaa ( W , ) . (4.64)All the other amplitudes of the primaries can be obtained using the recursions which have been incorporatedinto the interchiral blocks for the numerical bootstrap. With the normalization (4.63), we obtained theamplitude A aaaa ( W , − ) given in figure 7, where we also plot the analytic amplitude (3.19a). The explicitexpression of the latter is given in (C.4) of appendix C, as obtained originally in [4] and reproducedin [20], where it was also found to agree with Monte-Carlo simulations. It was pointed out in [20] that thisspecific normalization for the amplitude A ( h , , h , ) (i.e., our A aaaa ( W , − ) here) underlies the three-pointstructure constants describing the probability P aaa of three points belonging to the same FK cluster [14].Here we can clearly see that the agreement with bootstrap result is perfect. Q A aaaa � � - � Figure 7: The amplitude A aaaa ( W , − ). Red dots are the numerical bootstrap result and the black curveis the analytic expression (3.19a). They agree perfectly (the black curve being only visible behind the redpoints close to Q = 4).In figure 8, we show on the left the amplitude A aaaa ( W , ) and on the right A aaaa ( W , − ) as givenin (3.19c). In both cases, the amplitudes have simple poles at Q = 2 and no other poles in the range0 < Q < (a) (b) Figure 8: The bootstrapped A aaaa ( W , ) on the left and the analytic A aaaa ( W , − ) on the right.29he amplitude A aaaa ( W , ) is shown in figure 9. It has simple poles at: Q = 4 cos (cid:18) π (cid:19) = 0 . . . . , (4.65a) Q = 4 cos (cid:16) π (cid:17) = 3 . . . . , (4.65b)of which we also plot the details in the zoomed-in regions of 0 < Q < < Q < A aaaa ( W , ) and its detailed pole structure in the regions0 < Q < < Q < A abab In P abab , up to j = 4, we have the following amplitudes A abab ( W , ) , A abab ( W , − ) ,A abab ( W , ) , A abab ( W , − ) ,A abab ( W ,i ) , A abab ( W , − i ) . (4.66)The second amplitude A abab ( W , − ) was obtained analytically in (3.19b), and for the modules W , , W , , W , − the corresponding amplitudes are related to the A aaaa through R ¯ α in (3.12). This was in fact usedas input in the bootstrap for the final results we present here. However for completeness we plot all theseamplitudes below.The amplitudes A abab ( W , ) and A abab ( W , − ) are shown in figure 10. They are smooth with nosingularities in the whole range 0 < Q <
4. 30 a) (b)
Figure 10: The bootstrapped A abab ( W , ) on the left and the analytic A abab ( W , − ) on the right.The numerical A abab ( W , ) and the analytic A abab ( W , − ) are plotted in figure 11, where the latter areobtained using (3.19c) and (3.12e). Notice that the amplitude A abab ( W , ) is smooth for 0 < Q <
4, due tothe cancellation of the zeros of R ¯ α ( W , ) in (3.12g) with the poles in A aaaa ( W , ), which further confirmsthat the singularities at (4.65a) and (4.65b) in A aaaa ( W , ) appear as simple poles. (a) (b) Figure 11: The bootstrapped A abab ( W , ) on the left and the analytic A abab ( W , − ) on the right.The modules W ,i and W , − i only appear in P abab (and P abba with the same amplitude but the oppositesign) and hence were obtained purely through the numerical bootstrap. Recall that they are in fact relatedby (4.8). The amplitudes display poles at Q = 4 cos (cid:18) π (cid:19) = 0 . . . . , (4.67a) Q = 4 cos (cid:16) π (cid:17) = 3 . . . . . (4.67b)These poles were in fact already observed in [1] for A ( h , , h , − ) (i.e., A abab ( W ,i ) in the present notation)which we will analyze in more details in the next section. The results are plotted in figures 12 and 13together with their detailed pole structures in the bottom parts of those figures.31igure 12: The bootstrap result of the amplitude A abab ( W ,i ) and its detailed pole structures in the regions0 < Q < < Q < A abab ( W , − i ) and its detailed pole structures in the regions0 < Q < < Q <
4. 32 .2.3 A aabb In the probability P aabb , we have the following amplitudes A aabb ( W , q ) , A aabb ( W , − ) ,A aabb ( W , ) ,A aabb ( W , − ) , A aabb ( W , ) . (4.68)While the first amplitude provides the normalization (4.63), we plot last three in figures 14 and 15. (Notehere that A aabb ( W , − ) is trivially related to A aaaa ( W , − ) in figure 7 by a minus sign as in (3.12a).)The analytic structures of these amplitudes can be seen from that of A aaaa and R α from (3.12b), (3.12d)and (3.12f). In particular, R α ( W , ) and R α ( W , ) indicate new poles in A aabb ( W , ) at Q = 1 and in A aabb ( W , ) at Q = 4 cos (cid:18) π (cid:19) = 0 . . . . , (4.69a) Q = 4 cos (cid:16) π (cid:17) = 2 . . . . . (4.69b) (a) (b) Figure 14: The bootstrapped A aabb ( W , ) on the left and the analytic A aabb ( W , − ) on the right. As pointed out in [1], the proposal of [19] cannot be the accurate description of the Potts geometricalcorrelations due to the appearance of divergences in Q in their correlation functions, whereas the Pottsprobabilities are expected to be smooth functions in Q . The spectrum of (2.20), on the other hand, hasthe effect of canceling such unwanted singularities, as already studied in [1] through an example. Wenow proceed further along this line to analyze in full detail the bootstrapped amplitudes that we havepresented in the previous section. We will see that combining with the analytic amplitudes that we gavein section 3.3 and with the recursions that we established in section 4.1, this gives us exact amplitudes atspecial values of Q corresponding to rational β given by (2.13). Such rational values of β are currentlynot directly accessible to the numerical bootstrap. In the meantime, we will see the intricate interplaybetween the spectra involved in various Potts probabilities and the analytic structures in the amplitudes.This provides a CFT interpretation of some of the amplitude ratios in (3.12), which were originally obtainedas an observation in the lattice-model computations. The Zamolodchikov recursive formula for computing conformal blocks is singular at rational β and therefore we do notbootstrap directly at the corresponding values of Q . A aabb ( W , ) and its detailed pole structures in the regions0 < Q < < Q < A abab ( W ,i )In [1], it was argued that the leading field ( h , , h , − ) is necessary in addition to the field ( h , , h , − ) ofthe spectrum of [19] in order for P abab to be a smooth function of Q . As our first case, we now make thisanalysis more precise and explain the poles in the amplitudes A abab ( W ,i ) at (4.67).At Q = 4 cos (cid:0) π (cid:1) , one finds a coincidence of conformal dimensions: h , = ¯ h , , ¯ h , = ¯ h , , h , = h , . (4.70)The contribution of ( h , , h , − ) in P abab therefore has a divergent term A abab ( W , − ) R , Re (cid:104) F h , ( z ) F h , − (¯ z ) (cid:105) = A abab ( W , − ) R , R , h , − h , Re (cid:104) F h − , ( z ) F h , − (¯ z ) (cid:105) + . . . , (4.71)where we have used (4.3), (4.17), (4.18) and (2.41). The divergence is necessarily canceled by A abab ( W ,i )Re (cid:104) F h , ( z ) F h , − (¯ z ) (cid:105) , (4.72)which requires A abab ( W ,i ) = − A abab ( W , − ) R , R , h , − h , + O (1) . (4.73)Extracting the residue, we obtain at Q = 4 cos (cid:0) π (cid:1) :Res [ A abab ( W ,i )] | Q =4 cos π = − A L ( W , − ) 174607744311 π Γ( − )Γ( − )Γ( )Γ( )Γ( )Γ( )5777653760000 √ − )Γ( − )Γ( − )Γ( )Γ( )Γ( ) , (4.74)34here we have used the explicit expression (3.19b) of R , , and the expression of A L ( W , − ) is given in(C.4) in appendix C.Similarly, at Q = 4 cos (cid:0) π (cid:1) , one finds¯ h , = ¯ h , , h , = ¯ h , , h , = h , . (4.75)A completely parallel calculation to the above leads to the exact result A abab ( W ,i ) = − A abab ( W , − ) R , R , h , − h , + O (1) (4.76)and explicitly:Res [ A abab ( W ,i )] | Q =4 cos π = A L ( W , − ) 1932805 π Γ( − )Γ( )Γ( ) Γ( )Γ( )501377302265856 √ − )Γ( − ) Γ( )Γ( )Γ( ) . (4.77)In figure 16, we plot (4.74) and (4.77) together with the bootstrap results in the respective regions of Q . As can be seen, the exact results interpolate smoothly between the numerical bootstrap results.Figure 16: The residues of the amplitude A abab ( W ,i ) at Q = 4 cos (cid:0) π (cid:1) (left) and Q = 4 cos (cid:0) π (cid:1) (right)given by the exact expressions (4.74) and (4.77) are indicated with black dots. The slightly smaller bluedots are the bootstrap results in the nearby region.While the above analysis focuses on a single probability P abab , in the following we consider the compa-rison of analytic structures of the amplitudes in different probabilities which are explicitly related by (3.12).We will focus on how such differences are related to the corresponding differences of the spectra in (2.20).This will give an analytic explanation of some of the ratios R , as well as exact results on the amplitudes. R α ( W , ) and W , q Consider now the amplitudes A aabb ( W , ) in figure 14a. Compared to A aaaa ( W , ), it has an extra poleat Q = 1, as can be seen explicitly from the ratio R α ( W , ) in (3.12b). One naturally suspects that suchdifference in the analytic structure is directly related to the difference in the spectra of the two probabilitiesinvolved, the module W , q in this case. Indeed, at Q = 1, one finds a collision of the conformal dimensions h , = ¯ h , = h , . (4.78)This means that the left and right conformal blocks for the identity field include the following divergentterm: F h , ( z ) = ˜ F h , ( z ) + R , h , − h , F h , − ( z ) , (4.79a) F ¯ h , (¯ z ) = ˜ F h , (¯ z ) + R , ¯ h , − h , F h , − (¯ z ) . (4.79b) We have in this section omitted the superscript of the conformal blocks indicating the channels. The arguments here applyto either one of the s - and t -channels whose blocks are related by (2.37). F still have divergences due to the coincidence of ( h , , ¯ h , ) with other fields, with howeverdifferent z, ¯ z -dependence. With the normalization A aabb ( W , q ) = 1 from (4.63), the identity field entersthe s -channel of P aabb as F h , ( z ) F ¯ h , (¯ z ) = R , ( h , − h , ) F h , − ( z ) F h , − (¯ z ) + 2R , h , − h , Re (cid:104) ˜ F h , ( z ) F h , − (¯ z ) (cid:105) + . . . . (4.80)First, notice that the double pole in the first term is canceled exactly within the block of W , q . Dueto the coincident dimensions h , = ¯ h , = h , − , (4.81)the block F , q from (4.25) includes the term R , F h , ( z ) F h , (¯ z ) = R , F h , − ( z ) F h , − (¯ z ) , (4.82)where R , has a double pole at Q = 1 whose residue cancels the residue of R , ( h , − h , ) exactly, as can beeasily checked.Now, in order to cancel the simple pole in the second term of (4.80), it is necessary for the amplitudeof ( h , , h , − ) at Q = 1 to be of the form A aabb ( h , , h , − ) = − R , h , − h , + O (1) , (4.83)where we recall the identification (4.4) to account for the factor of 2. Notice that the blocks in the secondterm of (4.80) are precisely the regular part of the blocks for the field ( h , , h , − ) after removing the pole,as described in (2.41)–(2.42). We then deduce A aabb ( W , ) = − R , ( h , − h , ) R N , + O (1) , (4.84)where R N , is given by the recursion (4.7). Now, using (3.12b), we obtain the exact amplitude A aaaa ( W , )at Q = 1: A aaaa ( W , ) | Q =1 = 5 π Γ( − )Γ( )144 √ )Γ( ) . (4.85)In figure 17a, we plot the value of (4.85) together with the bootstrapped amplitude A aaaa ( W , ) in theregion around Q = 1.We have seen above that from the CFT point of view, the amplitude ratio R α ( W , ) is necessary tointroduce the pole at Q = 1 in the amplitude A aabb ( W , ), in order to cancel the simple pole generated bythe conformal blocks of W , q appearing in the s -channel of P aabb . This picture is quite generic as we shallnow see in another example. R ¯ α ( W , ) and W , − From figures 8a and 10a, one can see that the amplitudes A aaaa ( W , ) has a pole at Q = 2 which is canceledby R ¯ α ( W , ) of (3.12c) in A abab ( W , ). This difference could easily be understood from the participationof the module W , − in P aaaa . At Q = 2, one finds h , = ¯ h , = h , = h , (4.86) Even though our treatment of conformal blocks for fields with degenerate indices is not exact due to the logarithmicstructure, we believe the regular part is accurate. P aaaa : F h , ( z ) F ¯ h , (¯ z ) = R , ( h , − h , ) F h , − ( z ) F h , − (¯ z ) + R , ( h , − h , ) F h , − ( z ) F h , − (¯ z )+ 2R , h , − h , Re (cid:104) ˜ F h , ( z ) F h , − (¯ z ) (cid:105) + 2R , h , − h , Re (cid:104) ˜ F h , ( z ) F h , − (¯ z ) (cid:105) + . . . (4.87)with an overall amplitude A aaaa ( W , − ). The two double poles are again canceled exactly within the block F , − by the terms R , F h , ( z ) F h , (¯ z ) + R , F h , ( z ) F h , (¯ z ) (4.88)due to the coincident dimensions h , = ¯ h , = h , − , h , = ¯ h , = h , − , (4.89)as can be easily checked. To cancel the simple poles one needs the amplitudes for ( h , , h , − ) and( h , , h , − ) to be − A aaaa ( W , − ) R , h , − h , , − A aaaa ( W , − ) R , h , − h , . (4.90)This on one hand requires the recursion R N , = A ( h , , h , − ) A ( h , , h , − ) = − R , R , , (4.91)which can indeed be shown to be true. On the other hand, one finds that at Q = 2: A aaaa ( W , ) = − A aaaa ( W , − ) R , ( h , − h , ) R N , + O (1) . (4.92)Using (3.12c), this gives the exact value of A abab ( W , ) at Q = 2: A abab ( W , ) | Q =2 = A L ( W , − ) 21 π Γ( − )Γ( )2048 √ ) , Γ( ) , (4.93)where the expression of A L ( W , − ) is given in (C.4). In figure 17b, we show this exact amplitude at Q = 2together with the bootstrapped amplitudes in the region around Q = 2. (a) (b) Figure 17: The amplitudes A aaaa ( W , ) at Q = 1 (left) and A abab ( W , ) at Q = 2 (right). The red andblue dots are the bootstrap results and the slightly bigger black dots are the exact expressions (4.85) and(4.93) obtained from the requirement of singularity cancellations.37e have seen in the above how singularities in the amplitudes cancel the divergences in the conformalblocks at special values of Q . In the last part of this section, we shall see another type of divergences whicharises from the R in the construction of the interchiral conformal blocks in section 4.1 (which ultimatelycomes from the recursions) and how it leads to singularities in the amplitudes, and also provides exactresults. Canceling divergences from R As we have seen in figure 15, the ratio R α ( W , ) in (3.12f) introduces poles in A aabb ( W , ) at Q − Q +1 = 0,viz., those given in (4.69). This is naturally due to the module W , q . At Q = 4 cos (cid:0) π (cid:1) , one finds thecoincidence of dimensions between the leading field in W , with the diagonal field ( h , , h , ) in W , q : h , = ¯ h , = h , = ¯ h , . (4.94)Meanwhile recall that the contribution of ( h , , h , ) is given by R , F h , ( z ) F h , (¯ z ) , (4.95)and that R , as defined in (4.26) has a simple pole. To cancel this divergence in the s -channel of P aabb (orthe t -channel of P abba ), we need A aabb ( W , ) = −R , + O (1) , (4.96)or using (3.12f): A aaaa ( W , ) = − R , R α ( W , ) . (4.97)This gives the exact amplitude at Q = 4 cos (cid:0) π (cid:1) : A aaaa ( W , ) | Q =4 cos ( π ) = 9 (cid:113) (5 + √ π Γ( − )Γ( − )Γ( ) Γ( ) √ − ) Γ( ) Γ( )Γ( ) . (4.98)Similarly at Q = 4 cos (cid:0) π (cid:1) , one has instead h , = ¯ h , = h , = ¯ h , . (4.99)and therefore A aabb ( W , ) = −R , + O (1) . (4.100)This means that A aaaa ( W , ) = − R , R α ( W , ) , (4.101)which is explicitly given by A aaaa ( W , ) | Q =4 cos ( π ) = (cid:113) (5 − √ π Γ( − )Γ( − )Γ( ) Γ( ) Γ( ) √ − ) Γ( − ) Γ( ) Γ( )Γ( ) . (4.102)One can carry out the same analysis on the poles of A aaaa ( W , ) and A abab ( W , ) at Q − Q + 2 = 0which disappear in A aabb ( W , ) due to the ratio R ¯ α ( W , ) from (3.12g). This can be understood from themodule W , − with the divergences in R , and R , . We do not repeat the details here but give the exactresults from the cancellation of these divergences: A abab ( W , ) | Q =4 cos ( π ) = − A L ( W , − ) 45(2 + √ π Γ( − )Γ( − )Γ( − )Γ( ) Γ( ) − ) Γ( ) Γ( )Γ( )Γ( ) , (4.103a) A abab ( W , ) | Q =4 cos ( π ) = A L ( W , − ) 823543( √ − − )Γ( − )Γ( )Γ( ) Γ( ) Γ( ) √ − ) Γ( − ) Γ( )Γ( )Γ( )Γ( ) , (4.103b)38here again the A L ( W , − ) is given in (C.4).In figures 18 and 19, we plot the analytic expressions (4.102), (4.98), (4.103a) and (4.103b) togetherwith the bootstrap results.Figure 18: The amplitudes A aaaa ( W , ) in the regions around Q = 4 cos (cid:0) π (cid:1) (left) and Q = 4 cos (cid:0) π (cid:1) (right). The red dots are the bootstrap results and the black dots are the exact expressions (4.98) and(4.102).Figure 19: The amplitudes A abab ( W , ) in the regions around Q = 4 cos (cid:0) π (cid:1) (left) and Q = 4 cos (cid:0) π (cid:1) (right). The blue dots are the bootstrap results and the black dots are the exact expressions (4.103a) and(4.103b). The amplitudes that we have obtained from the bootstrap in section 4.2 can be compared with a fewexisting partial results. Note that while such comparisons provide some sanity checks on the bootstrappedamplitudes, there has not been a complete determination of the amplitudes in the Potts probabilities upto this level before our work. In the following we will discuss the comparison of the bootstrap results withnumerical transfer-matrix computations in the lattice model [1], and with the non-diagonal Liouville theoryof [19] which provides an approximate description.
The approach of computing the amplitudes on the lattice is described in details in [1] where a few exampleswere also given. Here we apply this lattice approach to obtain the amplitudes associated with the primary Specifically we are using the scalar product method exposed in section 4.3.2 of [1], for which ample technical detailswere given in appendix A.2 of that paper. We were generally able to obtain finite-size results for cylinders of circumferences < Q <
4. On one hand, this provides a check on thePotts solution to the bootstrap we obtained above, namely the amplitudes A ( W ) associated with entireATL modules, which we show in this section. On the other hand, the lattice results for the sub-leadingprimaries in a ATL module also serve as basic checks on the interchiral conformal blocks as established insection 4.1, by means of a comparison of the lattice results with the recursions (4.5) and (4.14). We leavethis latter issue to appendix B.1.We now consider the lattice comparisons for the bootstrapped amplitudes A aaaa ( W , ) , A aaaa ( W , ) , A abab ( W ,i ) , A abab ( W , − i ) . (4.104)Note that due to the normalization in the lattice computation (see below), we do not have the lowestamplitude in each probability for the lattice results. On the other hand, since the bootstrap has imposedthe amplitude ratios (3.12), which were obtained originally from lattice computations [3], it suffices toconsider the comparisons of the amplitudes (4.104) and ignore the ones related to them through (3.12).In the lattice computation, one needs to choose a normalization for each probability which we havechosen to be the amplitude of the field with the lowest dimension. In particular, as described in [1, 3], forthe probabilities P abab and P abba , we focus on the symmetric and antisymmetric combinations P S = P abab + P abba , P A = P abab − P abba , (4.105)which consist respectively of modules with even and odd spins, due to (3.33c) and (3.33d). This way, weobtain the following results on the lattice: A aaaa ( W , ) A aaaa ( W , − ) , A aaaa ( W , ) A aaaa ( W , − ) , A abab ( W ,i ) A abab ( W , − ) , A abab ( W , − i ) A abab ( W , − ) (4.106)and plot them in figures 20, 21, 22 and 23 together with the corresponding bootstrap results. In eachof these, the bootstrap and the lattice results agree on the analytic structures (the location of poles andzeros), the order of magnitudes (which vary considerably with the amplitude being considered) and thegeneric behavior as a function of Q (sign, monotonicity, local extrema). The difference in the actual valuesis likely largely due to the finite-size effect of the lattice computations. In particular, for each parity of thelattice size L we have only three points at our disposal, which is a rather precarious situation for performingreliable extrapolations. Overall, given these constraints, we find the agreement on the general features ofthe curves highly satisfactory, while the detailed comparison of the actual values ranges from excellent (forthe lowest-lying amplitudes) to acceptable (for the higher-lying ones). L = 5 , , . . . , L max , with a maximal size L max = 11 for the amplitudes corresponding to the lowest-lying eigenvalues in thetransfer matrix spectrum, and L max = 10 for higher-lying cases. Extrapolations to the scaling limit L → ∞ were doneseparately for even and odd sizes, using the tricks given in section 4.3.3 of [1]. Indicative error bars were estimated from thedifference between the extrapolations through even and odd sizes. Concretely, we extrapolated to the limit L → ∞ using a second-order polynomial in 1 / L , which might not always besufficient due to the amount of curvature observed in the data. A aaaa ( W , ) normalized with the leading amplitude A aaaa ( W , − ) in P aaaa .Comparison of the lattice results (indicated with × ) and bootstrap results (indicated with • ).Figure 21: The amplitude A aaaa ( W , ) normalized with the leading amplitude A aaaa ( W , − ) in P aaaa .Comparison of the lattice results (indicated with × ) and bootstrap results (indicated with • ) in the regions0 < Q < < Q < A abab ( W ,i ) normalized with the leading amplitude A abab ( W , − ) in P abab − P abba .Comparison of the lattice results (indicated with × ) and bootstrap results (indicated with • ) in the regions0 < Q < < Q <
4. 41igure 23: The amplitude A abab ( W , − i ) normalized with the leading amplitude A abab ( W , − ) in P abab − P abba . Comparison of the lattice results (indicated with × ) and bootstrap results (indicated with • ) in theregions 0 < Q < < Q < As claimed by the authors of [20], the spectrum S Z + , Z applied to the four-point function (3.1) providesan approximate description that becomes accurate at Q = 0 , ,
4. This means that the difference betweenthe spectrum (2.20) and S Z + , Z vanishes at these values of Q for the combination in (3.1), as can be easilychecked using our results. This involves the modules W , , W , , W ,i and W , − i .For W , , using (3.12c), one has A aaaa ( W , ) + 2 Q − A abab ( W , ) = 0 (4.107)for all values of Q , which explains why [19] gives a reasonable numerical approximation, since this is thenext-to-leading contribution in the combination P aaaa + P abab after the module W , − . On the other hand,as we have studied in [3], the expression (3.1) is accurate up to diagrams involving two non-contractibleloops, which is reflected in the amplitude identities for modules with j = 2, i.e., eqs. (3.19b) above and(4.107) here.For W , , using (3.12g), one obtains A aaaa ( W , ) + 2 Q − A abab ( W , ) = − A aaaa ( W , ) Q ( Q − Q − Q −
2) (4.108)and indeed, the module disappears in this combination exactly at Q = 0 , ,
4. For generic values of Q , thisdoes not vanish but the values are numerically small—except obviously for the regions near the poles at Q = 2, 4 cos (cid:0) π (cid:1) and 4 cos (cid:0) π (cid:1) —as we show in figure 24.42igure 24: The amplitude of W , in the combination P aaaa + Q − P abab (the right-hand side of (3.1)) whichis approximated by the four-point function of [19] (the left-hand side of (3.1)) whose spectrum does notinclude these fields. The values are generically small and vanishes at Q = 0 , , W ,i and W , − i : since they do not appear in P aaaa , the combination (3.1)involves simply the amplitudes 2 Q − A abab ( W ,i ) , Q − A abab ( W , − i ) (4.109)which—as is clearly seen from figures 12 and 13—vanish at Q = 0 , , Q . In this case, they also give rise to poles at Q = 2, 4 cos (cid:0) π (cid:1) and 4 cos (cid:0) π (cid:1) which appear in thecorrelation functions (3.1). As we have mentioned in section 4.1, the field Φ D , in (non-diagonal) Liouville theory is degenerate, and thisfeature leads to the recursions A L ( W j +1 , − ) A L ( W j − , − ) . The explicit expressions were obtained in [4, 26] and we recallthem in appendix C. In the case of the Potts model, the degeneracy of this field is absent and thereforethe usual Liouville recursions for shifting the j -index do not hold any more. We see however in (3.19), thatthis is replaced by a renormalized version in which the Liouville recursion is dressed by a factor consistingof ratios of polynomials in Q : A aaaa ( W , − ) A aaaa ( W , − ) = ( Q − Q − Q + 2) Q ( Q − A L ( W , − ) A L ( W , − ) , (4.110a) A abab ( W , − ) A abab ( W , − ) = ( Q − Q − Q − Q + 2)2 Q ( Q − A L ( W , − ) A L ( W , − ) . (4.110b)Interestingly, using the bootstrap results, we have managed to conjecture another renormalized Liouvillerecursion: A aaaa ( W , ) A aaaa ( W , ) = ( Q − ( Q − ( Q − Q + 2) A L ( W , ) A L ( W , ) . (4.111)It is certainly remarkable that the precision of the numerical bootstrap results is sufficient for such a relationto be established. Notice that despite of the fields in modules W , and W , being absent in the spectrumof (non-diagonal) Liouville theory, the recursion A L ( W , ) A L ( W , ) exists as a result of the degeneracy of Φ D , there.In the case of the Potts-model probabilities considered here, it is renormalized by a Q -dependent factorsimilar to (4.110) which we have established analytically. In figure 25 and 26, we plot the bootstrap resultsof (4.111) and the analytic expression on the right-hand side and they agree perfectly.43igure 25: The renormalized Liouville recursion of A aaaa ( W , ) A aaaa ( W , ) from the bootstrap result (red dots) matchesperfectly with the analytic expression on the right-hand side of (4.111) (black curve).Figure 26: The agreement of the bootstrap result of A aaaa ( W , ) A aaaa ( W , ) with the right-hand side of (4.111) in theregions 0 < Q < < Q < Our results fully confirms the correctness of the spectrum for the Potts-model four-point functions proposedin [1], and, for all practical purposes, solve the bootstrap problem and determine accurately the leadingamplitudes, hence carrying to its term the program initiated in [19]. To be fair, our treatment of conformalblocks arising from fields in the modules W j, with degenerate conformal weights is a bit unsatisfactory, as wedid not take into account the likely presence of logarithmic terms (ln( z ¯ z )). We do not expect this, however,to affect the numerically determined amplitudes significantly, as witnessed by the excellent agreement withdata from lattice calculations. Nonetheless, we hope to revisit this question in our next paper.In the course of this work, we have also unearthed a lot of structure that remains to be understood.The degeneracy of fields with weight h r, , for r ∈ N ∗ , arising in W , q led naturally to the existence ofinterchiral conformal blocks, a structure deeply related with the underlying affine Temperley-Lieb algebra.This begs further study of the continuum limit of this algebra, which is more than the product of left andright Virasoro algebras, and was postulated in [2] to be described by an interchiral algebra . The results of44his paper should make possible the construction of this algebra beyond the case c = − D familiar in Liouvilletheory. This “structure” manifests itself by an infinite series of rational functions of Q (see, e.g., (4.110)and (4.111)), whose origin remains largely mysterious to us. Understanding these functions would likelyrequire a deeper study of the algebraic structure of the models on the lattice, and result in the full analyticdetermination of the correlation functions in the Potts model and in particular, the analytic expressions forthe amplitudes. This question belongs as well to our list of ongoing investigations.In conclusion, it is worth mentioning that a fully similar approach would lead to geometrical correlationfunctions in the O ( n ) model (involving “polymer lines” instead of clusters). Acknowledgements
This work was supported by the ERC Advanced Grant NuQFT. We thank S. Ribault for many useful discus-sions and for detailed comments on the manuscript. We are also grateful to J. Cardy and A. Zamolodchikovfor their kind encouragements throughout this work, and L. Gr¨ans-Samuelsson for discussions and collab-orations on related topics. 45 α , β , γ from [3] In [3], we have stated several facts regarding how the eigenvalues of the lattice transfer matrix (which inthe continuum limit become the fields in the CFT) contribute to various quantities with different geometriccontent. Here we recall the definitions and expressions which are used in the main text for deriving furtherconsequences on the Potts model.One of the main results in [3] is that the ratios of the amplitudes of these eigenvalues in differentprobabilities, different sub-diagrams contributing to the probabilities, and the diagrams with modified loopweights, depend only on the corresponding ATL modules and Q , which allows one to define the followingquantities: α j, z ≡ A aabb A aaaa ( W j, z ) , ¯ α j, z ≡ A abab + A abba A aaaa ( W j, z ) ,β ( k ) j, z ≡ A ( k ) abab A (2) abab ( W j, z ) , γ ( a ) j, z ≡ A ( a ) aabb A aabb ( W j, z ) . (A.1)Here A ( k ) refers to the amplitude of a module in sub-diagrams with a fixed number k (necessarily even) ofnon-contractible loops between the marked FK clusters, and note that a module W j, z has at most k = j .By definition, β (2) j, z = 1. A ( a ) refers to the amplitude of a module when the non-contractible loop is givenweight n a = q a + q − a , (A.2)with a = 1 in the case of the Potts model. In section 3.2, we have used α j, z , ¯ α j, z , β ( k ) j, z to define the ratios R α , R ¯ α , R β . One can similarly define the ratio R γ ( W j, z ) = ˜ A aabb A aabb ( W j, z ) = p − (cid:88) a =1 odd ( − a − γ ( a ) j, z , (A.3)where p is from the minimal models labeled by M ( p, q ) before taking the irrational limit (see [3] for moredetails).We have then [3]: α , − = − , (A.4a) α , = 11 − Q , (A.4b)¯ α , = 2 − Q , (A.4c) α , − = 2 − Q , (A.4d)¯ α , − = ( Q − Q − , (A.4e) α , = − Q − Q + 15 Q − Q + 4 Q − Q − Q + 1) , (A.4f)¯ α , = − ( Q − Q + 2)( Q − Q − . (A.4g)In addition, we have β (4)4 , − = − Q ( Q − Q − . (A.5)The explicit expressions of β (4)4 , , β (4)4 , ± i and γ ( a ) j, z are also given [3], but since they are irrelevant in this paperwe do not repeat them here. 46 More details on the numerical bootstrap
Our numerical bootstrap follows the general philosophy proposed in [19] but adapted to our bootstrapprogram with the interchiral conformal block construction. Namely we solve the interchiral bootstrapequations (3.32) as a linear system for the amplitudes (3.31) with the coefficients given by the blocks F ( s,t ) evaluated at a set of points { z i } in the region where the s - and t -channel conformal blocks convergefast. See appendix A.2 of [43] for a rewriting of the recursive formula (2.36) convenient for numericalimplementation. We then re-sum the conformal blocks into the interchiral conformal blocks using theanalytic recursions (4.5) and (4.14) and truncate the blocks according to the total conformal dimensionof the primaries (see the caption of figure 27). Solving the bootstrap equations a few times with differentsets of points {{ z i } m } gives a set of amplitudes A m ( { z i } ). As pointed out in [19], since the amplitudesare supposed to be constants arising from the three-point structure constants in the fusion channels, theyshould not depend on { z i } m and therefore have a small variation within the numerical errors. After imposingvarious constraints to fix the solution to the Potts model, as described at the beginning of section 4.2, weuse this criteria of small variation as a check for the stability of the solution, up to the chosen truncation.This variation is defined through the quantity δ ( A ) = (cid:113) (cid:80) Mm=1 ( A m − ¯ A ) M − ¯ A , (B.1)where ¯ A is the average among the set of m = 1 , . . . , M results. Meanwhile, δ ( A ) also provides an estimateof the number of significant figures reliable of the average ¯ A .As we have discussed at the end of section 2.2.1, since for the moment we do not know the exact regu-larization procedure for the pole terms in (2.39) for the modules W j, and therefore leave a free amplitudefor the block F h m, − n ( z ) F h m, − n (¯ z ), the latter amplitudes are not expected to be constants and will haveunstable results. In addition, same as for the primary fields, we also impose the amplitude ratios (3.12) forthese free amplitudes associated with the null descendants, for the modules W , and W , in particular.This is a reasonable constraint as in obtaining these ratios on the lattice in [3], it was observed that theyhold for all fields in the same ATL module, regardless of primaries or descendants. Of course, as statedabove, this has to be taken as a numerical approximation since the possible logarithm is not taken intoconsideration. As a result, the instability of these null descendant amplitudes does not influence our finalresults on the amplitudes that we have presented in section 4.2.The spectrum (2.20) is truncated at a certain total conformal dimension h + ¯ h after which the primaryfields are organized into ATL modules and enter the interchiral conformal block constructions. We havefound that in order to obtain a stable bootstrap result on the amplitudes with j ≤ < Q <
4, it is good to truncate out the modules with j ≥ This being said, it is however necessaryto include the conformal block of the diagonal field ( r, s ) D = (3 , − r, s ) = (3 ,
2) whose block comes with a free amplitude, despite that its total dimension is above thedimension of the lowest j = 6 field ( r, s ) = (0 , < Q < A aaaa ( W , − ) is particular important for obtaining stable amplitudes for j = 4. Thisis likely due to a certain instability introduced by the “naive” regularization we implement in the module W , .For the plots we give in section 4.2, we have typically δ ( A ) ≈ − , − except for a few less stablecases (near Q = 0 for example) with δ ( A ) ≈ − , − . We present here the detail of a typical bootstrap The q -expansion in the Zamolodchikov recursive formula is convergent everywhere on the z -plane except at z = 1 , ∞ forthe s -channel and at z = 0 , ∞ for the t -channel. This speeds up the convergence within { z i || z i | < } ∩ { z i || z i − | < } wherewe evaluate the blocks. By including the j = 6 modules, the resulting amplitudes with j ≤ Q h + h Q h + h Figure 27: The truncation of the Potts spectrum (2.20) in the interchiral bootstrap. On the left we plotthe total dimensions h + ¯ h for the primary fields in the spectrum. After organizing them in terms of ATLmodules, we plot on the right the total dimension of the leading fields in the ATL modules whose amplitudesare taken as the overall amplitude A ( W ) in the interchiral conformal block expansions of the geometricalcorrelations and need to be determined by the bootstrap. We truncate at j = 6 and plotted the totaldimension of ( r, s ) = (0 ,
6) in cyan. However we include the diagonal null descendant ( r, s ) D = (3 , − ≤ Q <
A δ ( A ) A aaaa ( W , − ) 1 . . × − A aaaa ( W , ) 0 . . × − A aaaa ( W , ) 1 . × − . × − A abab ( W ,i ) 6 . × − . × − (B.2)at Q = 1 .
56 and the other amplitudes are obtained using the recursions (4.5), (4.14) and the amplituderatios (3.12). As stated in [19] and mentioned above, the δ ( A ) here gives an estimate of the accuracy inthe bootstrap determination of the amplitudes A . This is also reflected in the few comparisons with theanalytic expressions we give in figs. 7, 25 above. We plot the relative error of these comparisons in figure28.Figure 28: The relative errors in the comparisons of the bootstrap results with the analytic expressions of A aaaa ( W , − ) in figure 7 (left) and A aaaa ( W , ) A aaaa ( W , ) in figure 25 (right). These give a measure of the accuracyin the bootstrapped amplitudes. 48 .1 Basic checks In the bootstrap, we have re-summed the Virasoro conformal blocks into the interchiral conformal blocksusing the recursions obtained from the degeneracy of field Φ D , , as discussed in details in section 4.1. Onthe other hand, we have also imposed the relations (4.62) which, as discussed in section 4.5, is essentially arenormalized version of the Liouville recursions. It is actually a fun exercise to check these constraints withlattice computations or “reduced bootstrap” where the constraints are loosened. We provide some resultson such basic checks in this subsection. A abab ( W , − ) A aaaa ( W , − ) and A aaaa ( W , − ) A aaaa ( W , − ) In figure 29, we plot the analytic results of A abab ( W , − ) A aaaa ( W , − ) and A aaaa ( W , − ) A aaaa ( W , − ) obtained in section 3.3 (seealso eq. (4.110)) compared with lattice-computation and “reduced bootstrap” results. In particular, webootstrapped A abab ( W , − ) A aaaa ( W , − ) using eqs. (3.32a) and (3.32b) with constraints (3.33b), whose result is quitestable with δ ( A ) (cid:46) − in general. We do not have the lattice results for this ratio, since it involves theleading amplitudes in P aaaa and P abab − P abba which are used as normalization for the other amplitudesin the same probabilities. For the second ratio A aaaa ( W , − ) A aaaa ( W , − ) , the bootstrap result is much less stable, with δ ( A ) ≈ − at most. This is obtained by bootstrapping eq. (3.32a) alone. In the plot we show at a fewvalues of Q with δ ( A ) ≈ − and also the lattice results. As can be seen, the bootstrap results match theanalytical values more accurately than the lattice results which have finite-size errors.Figure 29: The analytic results of A abab ( W , − ) A aaaa ( W , − ) and A aaaa ( W , − ) A aaaa ( W , − ) compared with lattice (indicated with × )and reduced bootstrap results (indicated with (cid:35) ). Recursions (4.5)In figure 30, we plot the analytic recursions (4.5) compared with the lattice results and find reasonableagreement. In particular, the lattice data for the last two plots are obtained from amplitudes of W , inboth P aaaa and P abab + P abba . The small discrepancies between these two different determinations appearto be a reasonable measure of the accuracy of the lattice computations, and to within roughly this accuracythe lattice computations are consistent with the analytic results.49igure 30: The recursions (4.5) compared with lattice results. The continuous curves are the analyticexpressions. The “ + ” indicate the lattice results from the amplitudes in P aaaa and “ (cid:3) ” indicate the latticeresults from P abab + P abba . C A L and the Liouville recursions from [4] In section 3.3, we have obtained the analytic form of certain Potts amplitudes in terms of the Liouvilleamplitudes A L , where the latter refers to the non-diagonal generalization of the Liouville theory with theexplicit expressions given in [4]. In addition, we have discussed in section 4.5 how the recursions of theamplitudes in the Potts model with the index j shifted turn out to be a renormalized version of the Liouvillerecursion and provided the renormalization factors for all the modules up to j = 4. In this appendix, wegive the expression of A L and the corresponding Liouville recursions as originally obtained in [4] for thereader’s convenience.The non-diagonal Liouville amplitudes A L were solved in [4] for the four-point function (3.1) and theproportionality in this approximate description of Potts probabilities was fixed to be by comparing withMonte-Carlo simulation in [20]. Due to the exact relation (3.2), the combination of (3.13) is therefore P aaaa + ˜ P abab = 2 (cid:68) V D , V N , V D , V N , (cid:69) , (C.1)whose conformal block expansion is given by the spectrum S Z + , Z : (cid:68) V D , V N , V D , V N , (cid:69) = (cid:88) e ∈ N A L (cid:16) h e + , , ¯ h e + , (cid:17) F ( c ) h e + 12 , ( z ) F ( c ) h e + 12 , (¯ z )+ (cid:88) e ∈ N ,s ≥ A L (cid:16) h e + ,s , ¯ h e + ,s (cid:17) (cid:18) F ( c ) h e + 12 ,s ( z ) F ( c ) h e + 12 , − s (¯ z ) + F ( c ) h e + 12 , − s ( z ) F ( c ) h e + 12 ,s (¯ z ) (cid:19) . (C.2)Comparing (C.2) with our construction of the interchiral blocks F j, − in (4.17) and the definition of A L ( W j, − ) in (3.14), we can identify: A L ( W , − ) = 2 A L (cid:16) h , , ¯ h , (cid:17) , (C.3)whose explicit expression is given by [20]:2 A L (cid:16) h , , ¯ h , (cid:17) =8 π β β − β Γ( β )Γ( β )Γ(2 − β )Γ(2 − β ) × Γ β (cid:18) β + 12 β (cid:19) Γ β (cid:18) β − β (cid:19) Γ β (cid:18) β (cid:19) Γ β (cid:18) β (cid:19) Υ β (cid:18) β − β (cid:19) Υ β (cid:18) β β (cid:19) , (C.4)where Γ β and Υ β are the double-Gamma and Upsilon functions. Eq. (C.4) is the expression we have usedfor plotting the analytic curve in figure 7, where we have found perfect agreement with the bootstrap results The A L ( h r,s , ¯ h r,s ) here should be identified with D ( s,r ) in [20]. s ≥
2, i.e, j ≥
2, the identification is A L ( W j, − ) = 4 A L ( h ,j , ¯ h ,j ) . (C.5)The recursion of shifting the index s for ( h r,s , ¯ h r,s ) was given in [4] as: A L ( h r,s +1 , ¯ h r,s +1 ) A L ( h r,s − , ¯ h r,s − ) = − Γ( − r − s β )Γ( r − s β )Γ( − r + (1 − s ) β )Γ(1 − r − (1 + s ) β )Γ( − r + s β )Γ( r + s β )Γ( − r + (1 + s ) β )Γ(1 − r − (1 − s ) β ) × Γ( − r + s β ) Γ( − r + s β ) Γ( − r + s β ) Γ( − r − s β ) Γ( − r − s β ) Γ( − r − s β ) (C.6)which gives us the following Liouville recursions for j ≥ A L ( W , − ) A L ( W , − ) = 2 A L ( h , , ¯ h , ) A L ( h , , ¯ h , ) , (C.7a) A L ( W , − ) A L ( W , − ) = A L ( h , , ¯ h , ) A L ( h , , ¯ h , ) . (C.7b)Notice that the factor of 2 in (C.7a) is due to the special definition of A L with j = 0 in (C.2). In general,the Liouville recursion is given by A L ( W j +1 ,e iπp/M ) A L ( W j − ,e iπp/M ) = A L ( h pM ,j +1 , ¯ h pM ,j +1 ) A L ( h pM ,j − , ¯ h pM ,j − ) , for j (cid:54) = 1 , (C.8)which we have used in (4.111) for extracting another renormalized Liouville recursion from the bootstrap.Note that despite the amplitudes in the numerator and denominator on the right-hand side of (C.8) notbeing given by the analytic results of [4], the recursion exists and coincides with that of [4], as a result ofthe degeneracy of the field Φ D , there. References [1] J. L. Jacobsen and H. Saleur, “Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q -state Potts model: A study of the s -channel spectra,” JHEP , vol. 01, p. 084,2019, 1809.02191.[2] A. Gainutdinov, N. Read, and H. Saleur, “Associative algebraic approach to logarithmic CFT in thebulk: the continuum limit of the gl (1 |
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