]>NUPHB14508S0550-3213(18)30337-710.1016/j.nuclphysb.2018.11.021High Energy Physics – TheorySolving Heun's equation using conformal blocksMarcinPia̧tekac⁎piatek@fermi.fiz.univ.szczecin.plradekpiatek@gmail.comArtur R.Pietrykowskibcpietrie@theor.jinr.ruaInstitute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, PolandInstitute of PhysicsUniversity of Szczecin, Wielkopolska 15Szczecin70-451PolandbInstitute of Theoretical Physics, University of Wrocław, pl. M. Borna, 950-204 Wrocław, PolandInstitute of Theoretical PhysicsUniversity of Wrocławpl. M. BornaWrocław950-204PolandcBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, RussiaBogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubna141980Russia⁎Corresponding author at: Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland.Institute of PhysicsUniversity of Szczecin, Wielkopolska 15Szczecin70-451PolandEditor: Leonardo RastelliAbstractIt is known that the classical limit of the second order BPZ null vector decoupling equation for the simplest two 5-point degenerate spherical conformal blocks yields: (i) the normal form of the Heun equation with the complex accessory parameter determined by the 4-point classical block on the sphere, and (ii) a pair of the Floquet type linearly independent solutions. A key point in a derivation of the above result is the classical asymptotic of the 5-point degenerate blocks in which the so-called heavy and light contributions decouple. In the present work the semi-classical heavy–light factorization of the 5-point degenerate conformal blocks is studied. In particular, a mechanism responsible for the decoupling of the heavy and light contributions is identified. Moreover, it is shown that the factorization property yields a practical method of computation of the Floquet type Heun's solutions. Finally, it should be stressed that tools analyzed in this work have a broad spectrum of applications, in particular, in the studies of spectral problems with the Heun class of potentials, sphere–torus correspondence in 2d CFT, the KdV theory, the connection problem for the Heun equation and black hole physics. These applications are main motivations for the present work.1Introduction1.1The Heun equationTo begin with let us consider the Heun equation, i.e., the Fuchsian equation with four singularities. The most familiar form of the Heun equation reads as follows [1]11Eq. (1.1) is called now the Heun equation in honor of its first investigator — German mathematician Karl Heun (1859–1929).(1.1)d2Φdz2+[γz+ωz−1+ϵz−x]dΦdz+αβz−Qz(z−1)(z−x)Φ=0.Eq. (1.1) has four regular singular points at 0,1,x and ∞. In (1.1) it is assumed the condition ϵ=α+β−γ−ω+1 needed to ensure regularity of the point at ∞. The complex number Q is called the accessory parameter. Another representation of the Heun equation known as the normal form looks as follows [2](1.2)d2Ψdz2−[Az+Bz−1+Cz−x+Dz2+E(z−1)2+F(z−x)2]Ψ=0. Eqs. (1.1) and (1.2) are linked by the substitution Ψ(z)=zγ/2(z−1)ω/2(z−x)ϵ/2Φ(z), whereA=−γω2−γϵ2x+Qx,B=γω2−ωϵ2(x−1)−Q−αβx−1,C=γϵ2x+ωϵ2(x−1)−xαβ−Qx(x−1) andD=12γ(12γ−1),E=12ω(12ω−1),F=12ϵ(12ϵ−1).In the present paper we study a realization of the Heun equation in a two-dimensional conformal field theory (CFT2) and discuss some of its consequences. In CFT2, the Heun equation occurs in the normal form (1.2) usually written as(1.3)d2Ψdz2+[δ1z2+δ2(z−x)2+δ3(1−z)2+δ1+δ2+δ3−δ4z(1−z)+x(1−x)c2(x)z(z−x)(1−z)]Ψ=0. From (1.2) and (1.3) one has(1.4)−A=δ1+δ2+δ3−δ4+c2(x)(x−1),−B=δ4−δ1−δ2−δ3−c2(x)x,−C=c2(x),−D=δ1,−E=δ3,−F=δ2. Here we just only briefly announce that eq. (1.3) emerges in Liouville field theory (LFT) and in a model independent chiral CFT2. These two contexts are related to computations of certain conformal blocks. For this reason, before we express our concrete goals and motivations of the present work, we would like to spell out basic information about the quantum and classical conformal blocks. In particular, we will list some interesting current research topics related to them.1.2Quantum and classical conformal blocksLet Cg,n be the Riemann surface with genus g and n punctures. The basic objects of any two-dimensional conformal field theory living on Cg [3,4] are the n-point correlation functions of primary physical vertex operators defined on Cg,n. Any correlation function can be factorized according to the pattern given by the pant decomposition of Cg,n and written as a sum (or an integral for theories with a continuous spectrum) which includes the terms consisting of the holomorphic and anti-holomorphic conformal blocks times the 3-point functions of the model for each pair of pants. The Virasoro conformal block Fc,Δp[Δi](Z) on Cg,n depends on the cross ratios of the vertex operators locations denoted symbolically by Z and on the 3g−3+n intermediate conformal weights Δp. Moreover, it depends on the n external conformal weights Δi and on the central charge c.In the operator formalism [5,6] the conformal blocks on the Riemann sphere are defined as matrix elements (vacuum expectation values) of radially ordered compositions of the primary chiral vertex operators (CVO's) acting between Verma modules — the highest weight representations of the Virasoro algebra. Conformal blocks on the torus are traced cylinder matrix elements of 2d Euclidean ‘space–time’ translation operators and CVO's insertions, i.e., ‘chiral partition functions’. Conformal blocks are fully determined by the underlying conformal symmetry. These functions possess an interesting, although not yet completely understood analytic structure. In general, they can be expressed only as a formal power series and no closed formula is known for its coefficients.Among the issues concerning conformal blocks which are still not fully understood there is the problem of their classical limit [7]. This is the limit in which all parameters of the conformal blocks tend to infinity in such a way that their ratios are fixed:Δi,Δp,c⟶∞,Δic=Δpc=const. For the standard parametrization of the central charge c=1+6Q2, where Q=b+1b and for ‘heavy’ weights (Δp,Δi)=1b2(δp,δi) with δp,δi=O(b0) the classical limit corresponds to b→0. There exist many convincing arguments that in the classical limit the conformal blocks behave exponentially with respect to Z:Fc,Δp[Δi](Z)∼b→0e1b2fδp[δi](Z). The functions fδp[δi](Z) are known as the classical conformal blocks [7,8].Conformal blocks may also include the ‘light’ conformal weights Δlight which appear as contributions from the ‘light’ chiral vertex operators VΔlight(y). The light conformal weights are defined by the property limb→0b2Δlight=0. It is known (but not proven in general) that light insertions have no influence to the classical limit, i.e., do not contribute to the classical blocks:(1.5)〈∏i=1nVΔheavy(zi)VΔlight(y)〉∼b→0Ψ(y)e1b2fδp[δi](Z).Recently, a considerable progress in the theory of conformal blocks and their applications has been achieved. This is mainly due to the discovery of the so-called AGT correspondence [9]. The AGT conjecture states that the Liouville field theory correlators on the Riemann surface Cg,n with genus g and n punctures can be identified with the partition functions of a class Tg,n of four-dimensional N=2 supersymmetric SU(2) quiver gauge theories. A significant part of the AGT conjecture is an exact correspondence between the Virasoro blocks on Cg,n and the instanton sectors of the Nekrasov partition functions of the gauge theories Tg,n. Soon after its discovery, the AGT hypothesis has been extended, in particular, (i) to the correspondence between conformal Toda correlators and SU(N) gauge theories partition functions [10]; (ii) to the relation (cf. [11]) between irregular conformal blocks [12–14] and Nekrasov's instanton partition functions for ‘non-conformal’ N=2, SU(2) super Yang–Mills theories.The irregular conformal blocks discovered by Gaiotto were introduced in his work [11] as products of some new states belonging to the Hilbert space of CFT2. These novel irregular Gaiotto states are kind of coherent vectors for some Virasoro generators. It is also known that irregular blocks can be obtained from standard (regular) conformal blocks in properly defined decoupling limits of the external conformal weights, cf. [12,13]. Furthermore, the Gaiotto vectors can be understood as a result of suitable defined collision limit of locations of vertex operators in their operator product expansion (OPE), cf. [14].Interestingly, the classical limit exists also for irregular blocks and consistently defines the classical irregular blocks. This claim first time has appeared in [15] as a result of non-conformal AGT relations and yet another new duality, the so-called Bethe/gauge correspondence [16].Let us recall that the AGT correspondence works at the level of the quantum Liouville field theory. At this point it arises the question as to what happens if we proceed to the classical limit of the Liouville theory. It turns out that the semi-classical limit of the LFT correlation functions [7] corresponds to the Nekrasov–Shatashvili (NS) limit of the Nekrasov partition functions [16]. In particular, a consequence of that correspondence is that the classical conformal blocks can be identified with the instanton sectors of the effective twisted superpotentials [17,18].22On the relation between the quantization of the Hitchin system, SYM theories, CFT2 and the geometric Langlands program, see also [19]. The latter quantities determine the low energy effective dynamics of the two-dimensional gauge theories restricted to the so-called Ω-background. The twisted superpotentials play also a pivotal role in another duality, the aforementioned Bethe/gauge correspondence [16] that maps supersymmetric vacua of the N=2 theories to Bethe states of quantum integrable systems (QIS). A result of that duality is that the twisted superpotentials are identified with the Yang–Yang functions which describe spectra of the corresponding quantum integrable systems. Hence, combining the classical–NS limit of the AGT duality and the Bethe/gauge correspondence one thus gets the triple correspondence:CFT22dN=2 SU(2) SYM2-particle QIS

Classical Virasoro blockstwisted superpotentialsspectra of Schrödinger operators

which links the classical Virasoro blocks to the SU(2) twisted superpotentials and then to spectra of some Schrödinger operators. Indeed, let us note that 2-particle QIS are nothing but quantum-mechanical systems. The above correspondence can be extended to the following:CFT22dN=2 SU(N) SYMN-particle QIS

Classical Toda blockstwisted superpotentialsYang–Yang functions

if we take the classical–NS limit of the generalized AGT conjecture [10].Concluding, the interest in the classical conformal blocks and their uses has recently dramatically increased. Almost every day one may find in a literature more new, additional to those described above, fascinating contexts, in which these functions emerge. Some of these topics are: spectral problems for some Schrödinger operators [15,20–22]; the Painlevé VI equation [23]; the KdV equation [24]; the isomonodromic deformation problem [25]; S-duality in N=2 super Yang–Mills theories [26]; entanglement entropy in CFT2 and CFT2/AdS3 holography [27,28]; quantum chaos [29]; the connection problem for the Heun equation and scattering in black hole backgrounds [30–32]; black holes, holography and information paradox [33]; semi-classical spectra and restrictions on holographic two-dimensional CFT's [28,34]; holographic interpretation of classical blocks and classical bootstrap [34–41].1.3Objectives, motivations, outlineAs has been already mentioned, in two-dimensional conformal field theory the Heun equation emerges in two circumstances.First, eq. (1.3) one gets in the classical limit from the BPZ null vector decoupling equation [3] for the Liouville 5-point function of primary fields with one light degenerate field VΔ2,1(z,z¯), where Δ2,1=−12−34b2. The accessory parameter c2(x) in eq. (1.3) is determined in this framework by the classical Liouville action on the 4-punctured sphere.On the other hand, the degenerate 5-point blocks in the Liouville correlation function fulfill the same null vector decoupling equation as the physical correlator itself. Applying to this equation the asymptotic (1.5) with n=5 and Δlight=Δ2,1 one gets the normal form Heun equation with the two linearly independent solutions (Ψ+,Ψ−) known as the path-multiplicative or Floquet type solutions, cf. [23,42]. Here, the Heun accessory parameter c2(x) is a complex-valued function determined by the classical 4-point block on the sphere. In fact, one obtains the following claim:Proposition 1.1The accessory parameter c2(x) in the Heun equation (1.3) is given by the derivative of the classical 4-point block on the Riemann sphere w.r.t. the modular parameter x,(1.6)c2(x)=∂∂xfδ[δ4δ1δ3δ2](x),δ:=14(1−λ2) iff the two linearly independent solutions (Ψ+(z),Ψ−(z)) of the equation (1.3) are of the Floquet type and have diagonal monodromy along a curve γ(0,x) encircling both 0 and x, and the corresponding monodromy matrix Mγ(0,x) obeys TrMγ(0,x)=−2cos(πλ).Recall that the path-multiplicative (Floquet) solutions of the Heun equation are related to two singularities, for example, z=0 and z=1. Let γ be a smoothly parametrized closed contour encircling these points: γ:z=φ(s) for s∈[0,1], φ(0)=φ(1)=z0. A Floquet solution y(z) is defined as a solution satisfying y(φ(1))=e2πiσy(φ(0)). So, it is a solution which is multiplied by a constant factor e2πiσ if we pass around a simple closed contour in the z-plane that encircles two of the three singularities 0,1,x. The value σ is called a Floquet exponent or path exponent or characteristic exponent. It depends on the parameters of the Heun equation [42]. Within the CFT realization the characteristic exponent σ is determined by the intermediate classical conformal weight δ.To get the claim above one can use perturbative methods (see [43,44], where the implication ⇐ was shown) and a standard conformal field theory machinery. It should be stressed that the CFT tools, such as the OPE, allows to find monodromy properties of the solutions (Ψ+(z),Ψ−(z)) without their explicit computation. However, it would be interesting to know if these solutions can be calculated within CFT2. For this reason in the present paper we study a mechanism of the semi-classical ‘heavy–light’ factorization (1.5) in the case of the degenerate 5-point blocks. Our aim here is to answer the question whether the formula (1.5) for n=5 and Δlight=Δ2,1 really allows to explicitly compute the Floquet type Heun's solutions.Indeed, little is known about a concrete form of the Floquet type solutions (cf. the end of subsection 3.2.1 in [42]) and an elaboration of any method useful for practical numerical calculation is of great importance. In particular, in [42] one can read that the path-multiplicative Heun's functions ‘theoretically can be constructed as series’(1.7)∑−∞+∞cnfσ+n(z) with conjectured form of the coefficients fσ+n(z) given by the hypergeometric function fσ+n(z)=zσ2F1(⋅;⋅;⋅;z), where dots denote dependence on the combinations of parameters of the Heun equation and the characteristic exponent σ. However, further one can find in [42] that ‘in the general case, the form of the dependence of the path exponents σ on the parameters of the Heun equation is lacking, and hence expansions (1.7) cannot be used for practical numerical calculations.’The answer to the above question allows to make contact with the perturbative techniques used to solve the monodromy problem for the Heun equation determining the classical 4-point block, cf. [43,44]. Both, the CFT and complementary perturbative methods can be capitalized in concrete applications. In particular, it seems very promising to use the CFT methods in the study of scalar perturbations of certain black hole (BH) backgrounds described by the Heun equation. This particular application is our main motivation for the present work. We spell out it in more details below.It is known that the Klein–Gordon equation in the Kerr–AdS5 background [45] can be reduced to the two (angular and radial) Heun equations by a separation of variables [46].33In fact, a bit more general situation can be considered, than those in [46], namely, for the metric of the 5-dimensional charged AdS BH, cf. e.g. [47]. Such BH solution contains the electric charge Q in addition and for Q=0 reduces to the metric discussed in [45,46]. Having in mind two contexts in which the Heun equation appears in 2-dimensional CFT one may consider an idea of the formal correspondence between the dynamics of a scalar field in Kerr–AdS5 and the Liouville theory/chiral CFT2. Such concept has recently occurred in [48].44See also [30–32]. First, it would be nice to explore this relation at the formal level, i.e. to complete a dictionary and to describe formal implications, as it has been partially done in [48]. Interestingly, this identification has many common features with what is known as the Kerr/CFT correspondence [49]. It would be very interesting to examine whether it is an accidental similarity or indeed something deeper.Then, one may ask about its possible application in the computation of the so-called BH quasi-normal modes (QNMs), cf. [50]. These quantities are certain complex frequency modes that resonate when the BH is weakly perturbed, e.g. by the weak scalar field. The quasi-normal modes are important for studying a stability of the black holes. They are also useful in the study of a plasma through the finite temperature AdS/CFT correspondence. For the Kerr–AdS5 BH a preliminary analysis shows that the parameter in the radial and angular Heun equations, which can be interpreted as the frequency of the BH oscillations, is determined by the complex accessory parameter, and so, by the calculable complex-valued classical 4-point block fδ[δ4δ1δ3δ2](x). Using such an identification one can express the frequency as a function of the other parameters in the radial/angular Heun equation and the characteristic exponent related to δ. However, to interpret the frequency parameter as the BH frequency QNMs the scalar field, and hence, its radial and angular parts have to fulfill physically appropriate boundary conditions at the horizon (purely ingoing waves) and at the spatial infinity (purely outgoing waves).55Exactly, the presence of the horizon implies that the boundary value problem, which must be solved to determine BH QNMs, is non-hermitian and associated eigenvalues (= QNMs) are complex. Therefore, one of the key points that needs to be clarified here is to answer the question whether the solutions available within CFT2 obey these boundary conditions.The above ideas should work also for other BH backgrounds. For instance, the radial equation for the scalar perturbation of the Kerr–NUT–(A)dS BH in four dimensions is reducible to the Heun equation. Next, confluent Heun equations determine the scalar perturbations of the Kerr metric in four dimensions.66These equations are known in the literature as the Teukolsky equations (cf. [42] and refs. therein). It is an interesting question of how to get the latter, i.e., the confluent Heun equation together with its certain solutions entirely within the formalism of CFT2. Perhaps one needs to analyze the collision and classical limits of the BPZ equation for the 5-point blocks. In such a way the quantum and classical irregular conformal blocks should come to the game, cf. [31].77Moreover, it is reasonable to expect that there exists a monodromy problem for the confluent Heun equation whose solution matches the CFT results.Closely related to the scattering problems in the black hole backgrounds is the connection problem for the Heun and confluent Heun equations, cf. e.g. [42,51]. It is a question of how to express any of the pairs of linearly independent solutions of eq. (1.1) around some singular point 0 or 1 or x or ∞ as a linear combination of any one of the other three such pairs. Recently some progress in the solution of the connection problem has been achieved mainly due to the application of the so-called ‘isomonodromic approach’ and results relating regular and irregular conformal blocks to Painlevé VI and Painlevé V τ-functions, cf. [30–32].The structure of the paper is as follows. In subsection 2.1 a well-known fact is reminded, namely, it is shown how the CFT machinery determines a monodromy of the Heun's solutions referred to in the Proposition 1.1. In subsection 2.2 the Heun equation is studied within purely mathematical representation theoretic formalism of the conformal blocks. The formula for the linearly independent solutions of the normal form Heun equation in terms of the classical limit of the conformal blocks is derived. In subsection 3.1 the heavy–light factorization property of the degenerate 5-point blocks is studied. This analysis leads to a computation of the limit defining the Floquet type Heun's functions and in fact yields a new method of the calculation of the latter. A concrete example of the path-multiplicative Heun function computed in this way is presented there and in the appendix B. In subsection 3.2 the x→0 limit of the solution extracted from the conformal blocks is computed. This calculation confirms that the ‘CFT solution’ of the Heun equation has for x→0 the expected form in terms of a hypergeometric function. In subsection 3.3 a comparison with the perturbative techniques is discussed. In section 4 conclusions of the present work and open problems for further research are collected.2The Heun equation in CFT22.1Classical limit of BPZ equation for the degenerate five-point functionLet us consider the projected 5-point function on the sphere in the diagonal theory (Δi=Δ¯i):GΔ(z,x):=〈V4(∞,∞)V3(1,1)V−b2(z,z¯)PΔ,ΔV2(x,x¯)V1(0,0)〉, where Vα=−b2 is the degenerate field with the conformal weightΔ−b2=Δα=−b2=α(Q−α)=−12−34b2,Q=b+1b and Vi's are the four heavy primary operators (Δi=b−2δi, δi=O(1)). The function GΔ(z,x) satisfies the following null vector decoupling (NVD) equation [3]:(2.8)[∂2∂z2−b2(1z−11−z)∂∂z]GΔ(z,x)=−b2[Δ1z2+Δ2(z−x)2+Δ3(1−z)2+Δ1+Δ2+Δ3+Δ−b2−Δ4z(1−z)+x(1−x)z(z−x)(1−z)∂∂x]GΔ(z,x). Our aim now is to consider the classical limit of the equation above. The key point here is an observation that in the limit b→0 only the operator with weight Δ−b2 remains light (Δ−b2=O(1)) and its presence in the correlation function has no influence on the classical dynamics. Then, for b→0(2.9)GΔ(z,x)∼Ψ(z)e−1b2(Scl(δ4,δ3,δ)+Scl(δ,δ2,δ1)−fδ[δ4δ1δ3δ2](x)−f¯δ[δ4δ1δ3δ2](x¯)). Indeed, assuming that the light field does not contribute to the classical limit we are left with the projected 4-point function of the heavy operators:〈V4(∞,∞)V3(1,1)PΔ,ΔV2(x,x¯)V1(0,0)〉=C(Δ4,Δ3,Δ)C(Δ,Δ2,Δ1)F1+6Q2,Δ[Δ4Δ1Δ3Δ2](x)F¯1+6Q2,Δ[Δ4Δ1Δ3Δ2](x¯)∼b→0e−1b2(Scl(δ4,δ3,δ)+Scl(δ,δ2,δ1)−fδ[δ4δ1δ3δ2](x)−f¯δ[δ4δ1δ3δ2](x¯)). The quantities Scl(δ3,δ2,δ1), known as the classical 3-point actions, are the classical limits of the structure constants C(Δi,Δj,Δk). The functions(2.10) and(2.11)fδ[δ4δ1δ3δ2](x)=limb→0b2logF1+6Q2,Δ[Δ4Δ1Δ3Δ2](x) are quantum, and classical 4-point blocks on the sphere respectively. The substitution of eq. (2.9) into the NVD eq. (2.8) leads, within the limit b→0, to the normal form of the Heun equation (1.3) with the accessory parameter c2(x) determined by the classical 4-point block, according to (1.6), cf. [23].Alternatively, the parameter c2(x) can be found once the solution of certain Bethe-like saddle point equation defining the twisted superpotential of the SU(2) Nf=4 N=2 SYM theory is known, cf. [52].Let us stress that the holomorphic accessory parameter c2(x) is related to that which emerges in the uniformization theory of the 4-punctured sphere. The real-valued accessory parameter for C0,4 can be found by taking the classical limit of the NVD equations obeyed by the physical Liouville 5-point function on the sphere with Vα=−b2, cf. [7,8,52,53]. The latter is nothing but the projected correlation function discussed above integrated over the Liouville field theory spectrum Δ=Δα, α=12Q+iR+ with the structure constant C(Δ3,Δ2,Δ1) identified as the Liouville 3-point function [7,54].Monodromy properties of the solutionsLet us consider monodromy properties of the independent solutions Ψ± of eq. (1.3) with the accessory parameter given by (1.6). It turns out that the substitution (2.9) automatically fixes the solutions Ψ± to be of the Floquet type with the characteristic exponent σ determined by the intermediate classical conformal weight δ=14(1−λ2) [23]. First, note that the functions on both sides of eq. (2.9) should have the same monodromy properties along the contour encircling the points 0 and x. Secondly, from a completeness of the intermediate states follows that the monodromy properties w.r.t. z of the 5-point function GΔ(z,x) along a curve encircling both 0 and x are the same as the monodromy of the 4-point correlator:(2.12)〈V4(∞,∞)V3(1,1)VΔ−b2(z,z¯)VΔα(0,0)〉 for a curve encircling 0. The z dependence (i.e., an overall prefactor) of the correlator (2.12) can be read off from (holomorphic part of) the OPE:V−b2(z)Vα(0)=zΔ+−Δ−b2−ΔαC(Δ+,Δ−b/2,Δα)[Vα+(0)+descendants]+zΔ−−Δ−b2−ΔαC(Δ−,Δ−b/2,Δα)[Vα−(0)+descendants], where Δ±≡Δα±, α±=α±b2, and explicitly,Δ+=Δα−αb+b24+12,Δ−=Δα+αb−3b24−12. Assuming that the intermediate weight Δα is heavy, i.e.,α=12b(1−λ)⇔limb→0b2Δα=14(1−λ2)=δ then in the limit b→0 one gets Δ±−Δ−b2−Δα→12(1±λ). Therefore, in the space of solutions of eq. (1.3) there exist basis solutions Ψ±(z)∝z12(1±λ) which analytically continued in z along the path encircling the points 0 and x satisfy the condition:(2.13)Ψ±(e2πiz)=−e±iπλΨ±(z). This corresponds to the monodromy matrix with trace equal −2cos(πλ).As a final remark in this subsection let us notice that knowing monodromies (2.13) of the solutions of eq. (1.3) one can determine the classical 4-point block solving the Riemann–Hilbert problem formulated as follows: adjust c2(x) in such a way that the eq. (1.3) admits solutions with the monodromy around 0 and x given by (2.13). The latter was Zamolodchikov's idea which allowed him to find (i) a large classical intermediate weight behavior of the classical 4-point block, (ii) a large quantum intermediate weight behavior of the quantum 4-point block and its nome q(x) expansion [55].2.2Classical limit of BPZ equation for the degenerate five-point blocksLet Vc,Δ(n) denote the vector space generated by all vectors of the form:(2.14)|ΔIn〉=L−I|Δ〉≡L−k1…L−kℓ(I)|Δ〉,n=k1+…+kℓ(I)=:|I|, where I=(k1≥…≥kℓ(I)≥1) is a partition of n,88We will use the notation I⊢n. Ln's are the Virasoro generators obeying(2.15)[Ln,Lm]=(n−m)Ln+m+c12(n3−n)δn+m,0, and |Δ〉 is the highest weight state with the following property:(2.16)L0|Δ〉=Δ|Δ〉,Ln|Δ〉=0,∀n>0. The representation of the Virasoro algebra on the space:Vc,Δ=⨁n=0∞Vc,Δ(n),Vc,Δ(0)=R|Δ〉 defined by the relations (2.15), (2.16) is called the Verma module with the central charge c and the highest weight Δ. It is clear that dimVc,Δ(n)=p(n), where p(n) is the number of partitions of n (with the convention p(0)=1). On Vc,Δ(n) exists symmetric bilinear form 〈⋅|⋅〉 uniquely defined by the relations 〈Δ|Δ〉=1 and (Ln)†=L−n.Let VΔ be the Verma module with the highest weight state |νΔ〉. We define the chiral vertex operator (CVO) as the linear mapV∞Δ3zΔ20Δ1:VΔ2⊗VΔ1⟶VΔ3 such that for all |ξ2〉∈VΔ2 the operatorV(ξ2|z)≡V∞Δ3Δ2z0Δ1(|ξ2〉⊗⋅):VΔ1⟶VΔ3 satisfies the following conditions:(2.17)[Ln,V(ν2|z)]=zn(z∂∂z+(n+1)Δ2)V(ν2|z),n∈Z(2.18)V(L−1ξ2|z)=∂∂zV(ξ2|z),(2.19)V(Lnξ2|z)=∑k=0n+1(kn+1)(−z)k[Ln−k,V(ξ2|z)],n>−1,(2.20)V(L−nξ2|z)=∑k=0∞(n−2n−2+k)zkL−n−kV(ξ2|z)+(−1)n∑k=0∞(n−2n−2+k)z−n+1−kV(ξ2|z)Lk−1,n>1 and(2.21)〈νΔ3|V(νΔ2|z)|νΔ1〉=zΔ3−Δ2−Δ1. The commutation relation (2.17) defines the primary vertex operator corresponding to the highest weight state |ν2〉∈VΔ2. Eqs. (2.18)–(2.20) characterize the descendant CVO's.Let |0〉 denote the vacuum state, i.e., the highest weight state in the vacuum module with the highest weight Δ=0. Conformal blocks on the Riemann sphere are defined as the matrix elements(2.22)〈0|VΔn(zn)…VΔ1(z1)|0〉 of compositions of the primary chiral vertex operators99We adopt here three equivalent notations for primary CVO's.:VΔj(z)≡V(νj|z)≡Vαk,αiαj(z):VΔi⟶VΔk,Δl=Δαl=αl(Q−αl), cf. [5,6]. By inserting projection operators1010PΔ˜p are identity operators in Vc,Δ˜p built out of the basis vectors (2.14) and their duals (cf. calculations below). PΔ˜p on the intermediate conformal weights Δ˜p, p=1,…,n−3 into the internal channels of the spherical blocks (2.22) one gets the latter in terms of the formal power series. Coefficients of these series consist of three-point blocks (forms ρ defined below) and inverses (Gc,Δ(n))IJ of the Gram matrix(2.23)(Gc,Δ(n))IJ=〈ΔIn|ΔJn〉.For a given triple Δ1,Δ2,Δ3 of conformal weights we define the trilinear mapρ∞Δ3zΔ20Δ1:VΔ3⊗VΔ2⊗VΔ1⟶C induced by the matrix element of a single chiral vertex operatorρ∞Δ3zΔ20Δ1(ξ3,ξ2,ξ1)=〈ξ3|V(ξ2|z)|ξ1〉,∀|ξi〉∈VΔi,i=1,2,3. The form ρ∞Δ3zΔ20Δ1 (three-point block) is uniquely determined by the conditions (2.17)–(2.20). In particular,1.for L0-eingenstates1111Note that for the basis vectors {|νi,I〉} one has Δi(νi,I)=Δi+|I|. L0|ξi〉=Δi(ξi)|ξi〉,i=1,2,3 one gets(2.24)ρ∞Δ3zΔ20Δ1(ξ3,ξ2,ξ1)=zΔ3(ξ3)−Δ2(ξ2)−Δ1(ξ1)ρ∞Δ31Δ20Δ1(ξ3,ξ2,ξ1);2.for basis vectors νi,I≡|νΔi,I〉∈VΔi, i=1,2,3 one finds(2.25)ρ∞Δ31Δ20Δ1(ν3,I,ν2,ν1)=γΔ3[Δ1Δ2]I,ρ∞Δ31Δ20Δ1(ν3,ν2,ν1,I)=γΔ1[Δ3Δ2]I,ρ∞Δ31Δ20Δ1(ν3,ν2,I,ν1)=(−1)|I|γΔ2[Δ3Δ1]I, where for a given partition I=(k1,…,kℓ(I)),ki≥kj≥1,i<j,(2.26)γΔ[Δ1Δ2]I≡∏i=1ℓ(I)(Δ+kiΔ2−Δ1+∑i<jℓ(I)kj).Let us stress once again that using the three-point blocks and the inverse (Gc,Δ(n))IJ of the Gram matrix (2.23) one can construct the matrix elements of various compositions of the primary chiral vertex operators.1212It should be stressed that the compositions are well defined when the weights in the internal channels are non-degenerate. For instance, the four-point block in the s channel reads as follows(2.27)Precisely, eqs. (2.25) and (2.27) yield a function introduced in (2.10),Curious class of CVO's form the degenerate operators [3] VΔrs(z) having conformal weights:Δr,s=Q24−14(rb+sb)2,Q=b+1b,r,s∈N being zeros of the Kac determinant [56–60]:det(Gc,Δ(n))IJ=det〈ΔIn|ΔJn〉=const.(n)×∏1≤rs≤n(Δ−Δr,s)p(n−rs). Let us consider the degenerate chiral vertex operators of the form(2.28)VΔ2,1(±)(z)≡Vβ±,β−b/2(z),Δ2,1=Δ−b/2=−12−34b2. If β±=β±b2 then1. the CVO's VΔ2,1(±)(z) satisfy the differential equation(2.29)1b2d2dz2VΔ2,1(±)(z)+:TVΔ2,1(±)(z):=0, where(2.30):TVΔ(z):≡∑n⩽−2z−n−2LnVΔ(z)+∑n⩾−1VΔ(z)Lnz−n−2;2. conformal blocks with VΔ2,1(±)(z) obey certain partial differential equations (PDE's), in particular, the 5-point degenerate conformal blocks1313Here |αi〉≡|Δαi〉 and the operator-state correspondence is assumed.: obey the following PDE1414Eqs. (2.31) are nothing but eq. (2.8) rewritten for conformal blocks.:(2.31)[∂2∂z2−b2(1z−11−z)∂∂z+b2(Δ1z2+Δ2(z−x)2+Δ3(1−z)2+Δ1+Δ2+Δ3+Δ−b2−Δ4z(1−z))+b2x(1−x)z(z−x)(1−z)∂∂x]F±(z,x)=0.In what follows we will compute the classical limit of eqs. (2.31). Let us assume that all ‘momenta’ αi, β are heavy, i.e.1515Here β is the same as α in the previous subsection and 2η=1−λ.:(2.32)αi=ηib,β=ηb then, in the limit b→0 one gets(2.33)F±(z,x)∼b→0Ψ±(∞,1,z,x,0)e1b2fδ[δ4δ1δ3δ2](x), where fδ[δ4δ1δ3δ2](x) is the classical spherical 4-point block (2.11) with δ=limb→0b2Δβ=η(1−η), δi=limb→0b2Δi=ηi(1−ηi) and(2.34)Ψ±(∞,1,z,x,0)=limb→0〈α4|Vα4,β±α3(1)V−b/2β±,β(z)Vβ,α1α2(x)|α1〉〈α4|Vα4,βα3(1)Vβ,α1α2(x)|α1〉.To see that the semi-classical asymptotic of F±(z,x) has the factorized form (2.33) let us define for Z=(z4,z3,z) the ratio (cf. appendix C.2 in [61])(2.35)Ψ±,n(Z)≡〈0|V0,α4α4(z4)Vα4,β±α3(z3)Vβ±,β−b/2(z)|βIn〉〈0|V0,α4α4(z4)Vα4,βα3(z3)|βIn〉, where |βIn〉≡|Δβ,In〉 are the basis vectors of the form (2.14). Let us note that the numerator in (2.35) is a four-point block of two generic primary CVO's, a degenerate primary CVO and a general descendant CVO, respectively. On the other hand, the denominator in (2.35) is a three-point block built out of two generic primary CVO's and the generic descendant.One can see that the functions Ψ±,n(Z) are light in the limit b→0, i.e., Ψ±,n(Z)∼O(b0). Let us employ the notation X:=V0,α4α4(z4)Vα4,β±α3(z3) and consider first the case n=0,(2.36)Ψ±,0(Z)=〈0|XV−b/2β±,β(z)|Δβ〉〈0|X|Δβ〉. Note that the denominator of Ψ±,0(Z) for Z=(∞,1,z) equals one, cf. (2.21). Then, the functions Ψ±,0(∞,1,z) are nothing but the degenerate four-point blocks, which are well known to be given by the hypergeometric function F12, e.g.:Ψ+,0(∞,1,z)=zΔβ+−Δ2,1−Δβ(1−z)Δ(α3−b2)−Δ3−Δ2,1×2F1(b(α3−α4+β¯−b2),b(α3+α4−β−b2);b(2β¯−b);z), where α¯i=Q−αi and Δ(α−b2)−Δα−Δ2,1=bα. In the classical limit αi's and β scale like (2.32) therefore the functions Ψ±,0(∞,1,z) are really light in this limit.1616Let's remind that a relation between the four-point block in arbitrary locations of the vertex operators and the block in (z4,z3,z2,z1)=(∞,1,z,0) is of the form〈0|VΔ4(z4)VΔ3(z3)VΔ(−b2)(z)VΔβ(z1)|0〉=(z4−z1)Δ3+Δ−b2−Δ4−Δβ(z4−z)−2Δ−b2(z4−z3)Δ−b2+Δβ−Δ4−Δ3×(z3−z1)Δ4−Δβ−Δ−b2−Δ3〈Δ4|VΔ3(1)VΔ(−b2)(z)|Δβ〉 as it follows from the SL(2,C) invariance of the vacuum state. Next, let us choose a basis state of the form L−k|Δβ〉. Then,(2.37)Ψ±,k(Z)=〈0|XV−b/2β±,β(z)L−k|Δβ〉〈0|XL−k|Δβ〉. Using (2.36) and (2.17) one can compute〈0|XVβ±,β−b/2(z)L−k|Δβ〉=∑i=4,3,z[(k−1)Δizik−1zik−1∂i]〈0|XVβ±,β−b/2(z)|Δβ〉=∑i=4,3,z[(k−1)Δizik−1zik−1∂i]Ψ±,0(Z)〈0|X|Δβ〉=Ψ±,0(Z)〈0|XL−k|Δβ〉+z−k(k−1)Δ−b2Ψ±,0(Z)〈0|X|Δβ〉−∑i=4,3,z(zi1−k∂iΨ±,0(Z))〈0|X|Δβ〉, where ∂i:=∂zi, zz:=z, Δz:=Δ−b2. Dividing both sides of this equation by 〈0|XL−k|Δβ〉 and taking into account the definition (2.37) one gets1717Note that for z4=∞ we have 〈0|X|Δβ〉=z3Δ4−Δ3−Δβ and 〈0|XL−k|Δβ〉=z3Δ4−Δ3−Δβ−k (cf. (2.21)).Ψ±,k(Z)=Ψ±,0(Z)+z−k(k−1)Δ−b2Ψ±,0(Z)〈0|X|Δβ〉〈0|XL−k|Δβ〉−∑i=4,3,z(zi1−k∂iΨ±,0(Z))〈0|X|Δβ〉〈0|XL−k|Δβ〉. The above calculation can be simply generalized to any basis state (2.14).So, one sees that the functions Ψ±,n(Z) are light in the limit b→0. Second property of Ψ±,n(Z) (already seen in the calculation above) is that the ‘lightness property’ of the ratios of the type (2.35) does not depend on the level n. Indeed, using (2.35) one may find〈0|XVβ±,β−b/2(z)L−m|βIn〉=∑i=4,3,z[(m−1)Δizim−1zim−1∂i]〈0|XVβ±,β−b/2(z)|βIn〉=∑i=4,3,z[(m−1)Δizim−1zim−1∂i]Ψ±,n(Z)〈0|X|βIn〉=Ψ±,n(Z)〈0|XL−m|βnI〉+z−m(m−1)Δ−b2Ψ±,n(Z)〈0|X|βIn〉−∑i=4,3,z(zi1−m∂iΨ±,n(Z))〈0|X|βIn〉. As before dividing both sides of this equation by 〈0|XL−m|βIn〉 we see that the ‘shifted’ ratio(2.38)Ψ±,n+m(Z):=〈0|XV−b/2β±,β(z)L−m|βIn〉〈0|XL−m|βIn〉=Ψ±,n(Z)+z−m(m−1)Δ−b2Ψ±,n(Z)〈0|X|βIn〉〈0|XL−m|βIn〉−∑i=4,3,z(zi1−m∂iΨ±,n(Z))〈0|X|βIn〉〈0|XL−m|βIn〉is also light.Using the fact that the ‘lightness property’ of the ratios (2.35) does not depend on the level n and taking into account that the generic four-point block with heavy weights exponentiates in the classical limit,Fc,Δβ[Δ4Δ1Δ3Δ2](x)≡〈α4|Vα4,βα3(1)Vβ,α1α2(x)|α1〉∼b→0exp{1b2fδ[δ4δ1δ3δ2](x)}, one can computeF±(z,x)=〈α4|Vα4,β±α3(1)Vβ±,β−b/2(z)PβVβ,α1α2(x)|α1〉=∑n≥0∑I,J⊢n(Gβ(n))IJ〈α4|Vα4,β±α3(1)Vβ±,β−b/2(z)|βIn〉〈βJn|Vβ,α1α2(x)|α1〉=∑n≥0∑I,J⊢n(Gβ(n))IJΨ±,n(Z)〈α4|Vα4,βα3(1)|βIn〉〈βJn|Vβ,α1α2(x)|α1〉∼b→0Ψ±(∞,1,z,x,0)exp{1b2fδ[δ4δ1δ3δ2](x)}. In eqs. above (Gβ(n))IJ is the inverse of the Gram matrix (Gβ(n))IJ=〈Δ(β)In|Δ(β)Jn〉.1818Here and below we use the equivalent notation Δβ≡Δ(β). The above calculation defines the functions Ψ±(Z), so one can write(2.39)Ψ±(∞,1,z,x,0)=limb→0F±(z,x)Fc,Δβ[Δ4Δ1Δ3Δ2](x).Turning to the problem of the classical limit of eqs. (2.31) it is easy to see that the limit b→0 taken from eqs. (2.31) after the substitution (2.33) yields the normal form of the Heun ordinary differential equation (1.3) with a pair of Floquet type linearly independent solutions given by eq. (2.39).1919For consistency of our calculations one can check that limb→0b2∂zΨ±=0 and limb→0b2∂xΨ±=0.In the next section we will look once again into depths of eq. (2.33) in order to explicitly compute the limit (2.39).3Heavy–light factorization and Floquet type Heun's solutions3.1Path-multiplicative solutions from conformal blocksIn this subsection we analyze the way the degenerate five-point blocks factorize in the classical limit. Without loss of generality, let us focus on F+. The analysis of the case F− is analogous. Its expansion in terms of intermediate states reads(3.40)F+(z,x)=〈Δ4|VΔ3(1)PΔβ+VΔ2,1(z)PΔβVΔ2(x)|Δ1〉=xΔβ−Δ2−Δ1zΔβ+−Δ2,1−Δβ∑m,n≥0xnzm−n〈m;Δβ+|VΔ2,1(1)|n;Δβ〉︸=:Am,n, where we used the following notation(3.41)VΔa(x)|Δb〉=xΔ−Δa−Δb∑n≥0xn|n;Δ〉,|n;Δ〉:=∑I⊢nβΔ[ΔaΔb]IL−I|Δ〉,βΔ[ΔaΔb]I:=∑J⊢n(GΔ(n))IJγΔ[ΔaΔb]J. Formulas for bra vectors can be inferred from those above. The formula (3.40) can be given yet another form that is more useful for further study, namely(3.42a)F+(z,x)=zΔβ+−Δ2,1−ΔβFΔβ[Δ3Δ2Δ4Δ1](x)∑m∈Zzmχm(x), where(3.42b)χm>0(x):=∑k≥0xkAm+k,k∑k≥0xkF(k),χm<0(x):=∑k≥0xk−mAk−m,k∑k≥0xkF(k),χm=0(x):=∑k≥0xkAk,k∑k≥0xkF(k). The prefactor of eq. (3.42a) is the four-point conformal block defined as(3.43a)FΔβ[Δ3Δ2Δ4Δ1](x)=xΔβ−Δ2−Δ1∑n≥0xnF(n), and(3.43b)F(n):=FΔβ(n)[Δ3Δ2Δ4Δ1]:=∑I⊢nγΔβ[Δ3Δ4]IβΔβ[Δ2Δ1]I. Let us consider for definiteness χm>0. The matrix element Am+n,n has the following structure(3.44)Am+n,n=∑M⊢n〈n+m;Δβ+|VΔ2,1(z)L−M|Δβ〉βΔβ[Δ2Δ1]M|z→1. Developing the matrix element under the sum yields(3.45)〈m+n;Δβ+|VΔ2,1(z)L−M|Δβ〉=∑s=0ℓ(M)∑1≤i1<…<is≤ℓ(M)(−1)s〈m+n;Δβ+|L−Ms∁ad−MsVΔ2,1(z)|Δβ〉, where Ms∁ is a complement of a partition Ms such that Ms∁∪Ms=M for any s. Explicitly, for 1≤i1<…<is≤ℓ(M) they readMs∁=(k1,…,ki1−1,ki1+1,…,kis−1,kis+1,…kℓ(M)),Ms=(ki1,…,kis). In particular M0∁=Mℓ(M)=M, and M0=Mℓ(M)∁={0}. We also denotedad−MsVΔ2,1(z):=[L−kis,…[L−ki1,VΔ2,1(z)]]. In order to get further insight in to the formula (3.45) we notice that operating the Virasoro algebra element L−m on the state 〈Δ;n| yields〈Δ;n|L−m=(Δ+mΔb−Δa+n−m)〈Δ;n−m|,m≤n. The generalization of this formula to L−M:=L−k1⋯L−kℓ(M), |M|=m≤n reads(3.46)〈Δ;n|L−M=γΔ+n−m[ΔbΔa]M〈Δ;n−m|. Using the above result to the matrix element in eq. (3.45) and noticing that |Ms∁|+|Ms|=n we obtain the general form of the term that contributes to the expansion (3.45)(3.47)(−1)s〈m+n;Δβ+|L−Ms∁ad−MsVΔ2,1(z)|Δβ〉|z→1=FΔβ+(m+|Ms|)[Δ3Δ2,1Δ4Δβ]γΔβ++m+|Ms|[Δ3Δ4]Ms∁γΔβ[Δ2,1Δβ++m+|Ms|]Ms. Observe that the first factor in the above equation is m+|Ms|-th coefficient of the four-point conformal block with a degenerate weight Δ2,1. As it has been already mentioned, the latter is well known to be related to hypergeometric function F12, namelyFΔβ+[Δ3Δ2,1Δ4Δβ](z)=zΔβ+−Δ2,1−Δβ∑n≥0znFΔβ+(n)[Δ3Δ2,1Δ4Δβ]=zΔβ+−Δ2,1−Δβ(1−z)Δ(α3−b2)−Δ3−Δ2,1×2F1(b(α3−α4+β¯−b2),b(α3+α4−β−b2);b(2β¯−b);z), where α¯i=Q−αi. Thus, the coefficient of the conformal block can be read off from the above relation, which yields (Δ(α−b2)−Δα−Δ2,1=bα)(3.48)FΔβ+(m+|Ms|)[Δ3Δ2,1Δ4Δβ]=∑k=0m+|Ms|(−1)m+|Ms|−k×(bα3)m+|Ms|−k(b(α3−α4+β¯−b2))k(b(α3+α4−β−b2))k(m+|Ms|−k)!k!(b(2β¯−b))k. Combining eqs. (3.47), (3.45) and (3.44) we arrive at the explicit form of coefficient Am+n,n, and hence, the nominator of χm(x) that, when written in a suggestive form, reads∑n≥0xnAm+n,n=∑n≥0∑M⊢n∑s=0ℓ(M)∑1≤i1<…<is≤ℓ(M)x|Ms∁|γΔβ++m+|Ms|[Δ3Δ4]Ms∁βΔβ[Δ2Δ1]M×x|Ms|FΔβ+(m+|Ms|)[Δ3Δ2,1Δ4Δβ]γΔβ[Δ2,1Δβ++m+|Ms|]Ms. Instead of summing over levels n we can rearrange the sum so that it runs over |Ms∁| and |Ms| separately. This rearrangement has an effect in the appearance of the symmetry factor multiplying βΔβ[Δ2Δ1]M that comes from different ways one subpartition is immersed into the other. For instance, keeping summation index |Ms∁| and partition Ms∁ fixed while summing over |Ms| we find that there are many terms with the same shape of total partition M=Ms∁∪Ms that differ only by the distribution of parts of the partition Ms∁ between parts of Ms which occurs provided both have equal parts. As a result of this reshuffling of terms in the series we get (|Ms∁|≡|I|=r, |Ms|≡|J|=u)∑n≥0xnAm+n,n=∑u≥0xu∑J⊢u(∑r≥0xr∑I⊢r(I∪JI)γΔβ++m+u[Δ3Δ4]IβΔβ[Δ2Δ1]I∪J)×FΔβ+(m+u)[Δ3Δ2,1Δ4Δβ]γΔβ[Δ2,1Δβ++m+u]J, where (I∪JI) is the number of ways the partition I is immersed into the partition I∪J. As it may be inferred it equals(3.49)(I∪JI):=∏i≥1(mi(I∪J)mi(I)), where mi(I∪J) is a multiplicity of a part i in a partition I∪J, whereas mi(I) denotes a multiplicity of i in a partition I. The term in the parentheses can be rewritten in yet another form(3.50)∑n≥0xnAm+n,n=∑u≥0xuFΔβ+(m+u)[Δ3Δ2,1Δ4Δβ]×∑J⊢u(∑r≥0xr∑I⊢rγΔβ++m+u[Δ3Δ4]IβΔβ[Δ2Δ1]I(I∪JI)βΔβ[Δ2Δ1]I∪JβΔβ[Δ2Δ1]IβΔβ[Δ2Δ1]J)×γΔβ[Δ2,1Δβ++m+u]JβΔβ[Δ2Δ1]J. Observe, that the expression in the parentheses resembles the contribution to the conformal block (see eq. (3.43b)) with the intermediate weight Δβ shifted as Δβ+=Δβ+12+b24−bβ, multiplied by ratio of components of β forms. In the classical limit in which weights scale asΔi∼δib−2,fori=1,…,4,Δβ:=β(b+b−1−β)∼δb−2,β∼ηb−1,c∼6b−2, the term exponentiates to the classical conformal block provided the mentioned ratio decouples. This occurs if and only if the quotient of components of β forms has the following asymptotic behavior(3.51)As we have checked in a few cases (cf. appendix A), this indeed takes place. It means that the expression in parentheses in eq. (3.50) in the classical limit looses the dependence on J and therefore decouples from the series over u in the classical limit. The decoupled series exponentiates to the classical block, so that we are finally left with the function∑n≥0xnAm+n,n∼b→0e1b2fδ[δ3δ2δ4δ1](x)hm(x), wherehm(x):=∑s≥0hm,s(δ1,…,δ4,δ)xs=limb→0χm(x), andhm,n(δ1,…,δ4,δ)=FΔβ+(m+n)[Δ3Δ2,1Δ4Δβ]FΔβ(n)[Δ2,1Δ2Δβ++m+nΔ1]|b=0+… becomes the coefficient of the Heun's function in vicinity of x=0. The Heun's function obtained as a limit b→0 of (3.42a) reads2020Note that 1−η=12(1+λ) ⇒ Ψ+ has correct monodromy, cf. eq. (2.13).(3.52)Ψ+(z,x):=limb→0F+(z,x)FΔβ[Δ3Δ2Δ4Δ1](x)=z1−η∑m∈Zzmhm(x), where the coefficients for m≥0 are given in the appendix B. As for the part of the Heun's function with m<0 we get∑k≥0Ak−m,kxk−m=x|m|∑k≥0Ak+|m|,kxk, so that(3.53)hm<0(x)=x−m∑s≥0h−m,sxs, and the coefficients are the same as for m≥0 and are presented in eqs. (2.65a–2.65c).3.2Hypergeometric limitSolutions in the limit x→0Let us consider the ratio of the matrix element (3.42a) and the four-point conformal block which we denote as(3.54)H+(z,x):=F+(z,x)FΔβ[Δ3Δ2Δ4Δ1](x)=zbβ¯∑m∈Zχm(x)zm, in vicinity of the point x=0. As it follows from eq. (3.42b) at this point the only contribution to H comes from χm with m≥0. Expansion of this coefficient about x=0 yieldsχm≥0(x)=∑k≥0xkAm+k,k1+∑k>0xkF(n)=Am,0+(Am+1,1−F(1)Am,0)x+O(x2). Hence, at this point it amounts to(3.55)χm>0(0)=Am,0=〈m;Δβ+|VΔ2,1(1)|Δβ〉=FΔβ+(m)[Δ3Δ2,1Δ4Δβ], so that, taking into account the relationship of the four-point conformal block with one degenerate weight with the hypergeometric function, H+ from eq. (3.54) amounts to(3.56)H+(z,0)=FΔβ+[Δ3Δ2,1Δ4Δβ](z)=zbβ¯(1−z)bα32F1(b(α3−α4+β¯−b2),b(α3+α4−β+b2);b(2β¯−b);z). Since at the classical limit parameters αi and β scale as αi∼ηi/b,β∼η/b, we can immediately take the limit b→0 and obtain explicit form of the Heun's function at x=0 that reads(3.57)limb→0H+(z,0)=z1−η(1−z)η32F1(η3−η4−η+1,η3+η4−η;2(1−η);z).The hypergeometric equation vs. the Heun equationSince the accessory parameter (1.6) takes the explicit form(3.58)c2(x)=(δ−δ1−δ2)1x+∑n>0nxn−1f(n), the content of the square bracket in eq. (1.3) amounts in the limit x→0 to(3.59)δ1z2+δ2(z−x)2+δ3(1−z)2+δ1+δ2+δ3−δ4z(1−z)+x(1−x)c2(x)z(z−x)(1−z)→x→0zδ3+(z−1)(zδ4−δ)(z−1)2z2, so that we finally arrive at the following equation(3.60)AΨ:=[d2dz2+zδ3+(z−1)zδ4−(z−1)δ(z−1)2z2]Ψ=0. This equation can be transformed into the form of the hypergeometric equation. Namely, the transformation UAU−1UΨ=A˜Ψ˜, where U=z−a(1−z)−b yieldsA˜=d2dz2+2(a−(a+b)z)z(1−z)ddz+(1−z)z[(a+b)(1−(a+b))−δ4]−(1−z)(a(1−a)−δ)−z(b(1−b)−δ3)(z−1)2z2. The operator A˜ can be brought to the canonical hypergeometric formH:=z(1−z)d2dz2+(γ−(α+β+1)z)ddz−αβ, by multiplying by z(1−z), redefining δ=η(1−η),δ3=η3(1−η3) and setting(3.61)a(1−a)−η(1−η)=0,b(1−b)−η3(1−η3)=0. The solution of the above equations reads(3.62)a=η∨a=1−ηandb=η3∨b=1−η3 such that we obtainA¯=z(1−z)d2dz2+2(a−(a+b)z)ddz+((a+b)(1−(a+b))−δ4). Depending on the choice of the solution (3.62) of eqs. (3.61) we get the relationship between the classical block parameters η,η3,η4 and those from the hypergeometric operator H, i.e., α,β,γ. For a=1−η and b=η3 we haveα=η3−η4−η+1,β=η3+η4−η,γ=2(1−η). Hence, the solution for a=1−η and b=η3 canonical at z=0 is the hypergeometric function F12 with the following parametersΨ˜+=2F1(η3−η4−η+1,η3+η4−η;2(1−η);z). Thus, the original function being a solution to eq. (3.60) takes the form(3.63)Ψ+=U−1Ψ˜+Ψ+=z1−η(1−z)η32F1(η3−η4−η+1,η3+η4−η;2(1−η);z)fora=1−η,b=η3. The above solution has been labeled with Ψ+ to match the one in eq. (3.57) that stems from the five-point degenerate block in the classical limit with a given choice of fusion rule. The second solution of the equation (3.60) readsΨ−=U−1Ψ˜−=zη(1−z)η32F1(η+η3−η4,η+η3+η4−1;2η;z)fora=η,b=η3.3.3Comparison with other methodsThe path-multiplicative solutions of the Heun equation can be computed by making use of the perturbative methods. Such technics have been developed lately by Menotti in [43] to solve the monodromy problem determining the complex-valued accessory parameter in the Heun equation. An elegant and useful generalization of the perturbative approach has been recently developed in [62] and used to calculate the monodromy representation of the (generalized) Heun's opers.In [43] (see also [44]) the author study the monodromy problem for the eq.(3.64)y″(z)+Q(z)y(z)=0, whereQ(z)=δ1z2+δ2(z−x)2+δ3(1−z)2+δ1+δ2+δ3−δ4z(1−z)+C(x)z(z−x)(1−z) and δi=14(1−λi2). The ‘potential’ Q(z) in eq. (3.64) is almost the same as in eq. (1.3). The only difference is in the definition of the accessory parameter, namely, C(x)=x(1−x)c2(x). Further, it is assumed that C(0)=δ−δ1−δ2. This assumption is precisely met by x(1−x)c2(x), according to the formula (3.58). The monodromy problem considered in [43] is the same as stated in the last paragraph of subsection 2.1, i.e., the accessory parameter C(x) must be adjusted so that the monodromy of the fundamental solutions along a contour encircling both 0 and x has the trace −2cos(πλ).To reconstruct the accessory parameter C(x) as a power series in x, a specific contour including both the origin 0 and x is chosen in [43] for the monodromy calculation. Then, the appropriate fundamental solutions (y(1)(z),y(2)(z)) are computed in the form y(i)(z)=y0(i)(z)+y1(i)(z)x+… by perturbing the Heun equation in small x. More precisely, the method relies on an expanding of Q(z) in small x, Q=Q0+xQ1+x2Q2+O(x3), where in particular one getsQ0=δz2+δ3(z−1)2+δ4−δ3−δz(z−1),Q1=2δ2−C′(0)z2(z−1)+2δ2−C(0)z3(z−1),Q2=−C″(0)2z2(z−1)+3δ2−C′(0)z3(z−1)−3δ2+C(0)z4(z−1). The calculation starts with the solutions (y0(1)(z),y(2)0(z)) to the zeroth-order equation y″(z)+Q0(z)y(z)=0 and then subsequent corrections in powers of x are computed. The accessory parameter C(x) thus obtained exactly matches the formula (3.58).In order to compare the method quoted above with the calculations made in previous subsections let us point out that the zeroth-order equation is nothing but the eq. (3.60) whose solutions are given in terms of the hypergeometric functions. In refs. [43,44] these zeroth-order solutions are built out of the hypergeometric functions canonical at z=1. An argument that appears in [43] for such a choice is a reasonable observation that ‘working near z=0 is difficult due to the singular nature of the kernel’ Q(z). On the other hand, we have obtained that in the limit x→0 the solutions extracted from the conformal blocks yield the zeroth-order solutions built out of the hypergeometric functions canonical at z=0. In the present work we leave this discrepancy without an explanation. It requires a better understanding of a mechanism of an analytic continuation of the solutions parallelly within perturbative and CFT approaches.4Concluding remarksIn the present paper we have studied the mechanism of the heavy–light factorization which occurs in the classical limit of conformal blocks with the heavy and light contributions. We have examined the factorization property in the case of the simplest 5-point degenerate spherical conformal blocks F±. Our goal was to answer the question whether this semi-classical asymptotical behavior of F± determines the linearly independent Floquet type solutions of the normal form Heun equation. Recall that the Heun equation in its normal form can be obtained in the classical limit from the null vector decoupling equations obeyed by F±. Analyzing the semi-classical factorization in the case under consideration we have identified the mechanism responsible for the decoupling of the heavy and light contributions. In particular, a crucial observation made in the present work is the limit (3.51) which we have checked in many cases. This leads to an interesting novel way of computing the path-multiplicative Heun's functions (see the result written down in appendix B). Indeed, if the observation (3.51) is true then an analysis performed in subsection 3.1 yields a practical method of computation of the Floquet type Heun's solutions which is suitable for numerical calculations. In subsection 3.2 we have computed the x→0 limit of the Heun's solution within CFT framework and have confirmed that this has an expected form in terms of the hypergeometric function. A demonstration of a complete compatibility of our approach with the perturbative methods requires further study of the mechanism of the analytic continuation of the solutions. We aim to analyze this problem soon.Finally, let us stress that methods developed in this work have immediate applications in black hole physics problems listed in the introduction. We plan to examine these contexts very soon. Moreover, we plan to continue an exploration of the correspondence between the Heun equation, the classical limit of the null vector decoupling equation for the 5-point function/blocks and the BC1 Inozemtsev integrable model. The latter is nothing but the Schrödinger spectral problem for a class of elliptic so-called Treibich–Verdier potentials built out of the Weierstrass elliptic ℘-function. This spectral problem is yet one more incarnation of the Heun equation and on the other hand it can be obtained in the classical limit from the null vector decoupling equation written in elliptic variables living on the torus (cf. [63] and refs. therein). It turns out that this identification has some unspoken so far intriguing consequences for: (i) the theory of elliptic solitons, i.e., the solutions of the Korteweg–de Vries equation given by the above-mentioned Treibich–Verdier potentials; (ii) the study of modular properties of N=2 gauge theories; (iii) the correspondence between the sphere and torus correlation functions in the Liouville theory as well as between the sphere and torus conformal blocks.Appendix ARatio of beta forms in the classical limitIn what follows we present the results of computations of asymptotic of the ratio of coefficients of β in the limit b→0 given in eq. (3.51) which we denote here asRI∪J:=βΔβ[Δ2Δ1]I∪JβΔβ[Δ2Δ1]IβΔβ[Δ2Δ1]J.•|I∪J|=2, |I|=1, |J|=1R(1)∪(1)=12+−2(3δ2+(5δ−3)δ2)δ1+(5δ−3)δ12+(δ−δ2)(δ(δ+3)+(3−5δ)δ2)4δ(4δ+3)(δ−δ1+δ2)2b2+O(b4). The value of ((1)∪(1)(1))=2.•|I∪J|=3, |I|=2, |J|=1►I=(2),J=(1)R(2)∪(1)=1+2(δ1−δ2)(−2(5δ2+(7δ−6)δ2)δ1+(7δ−6)δ12+(δ−δ2)(3δ(δ+2)+(6−7δ)δ2))3δ(δ+2)(δ−δ1+δ2)(−3δ12+2(δ+3δ2)δ1+(δ−δ2)(δ+3δ2))b2+O(b4). The value of ((2)∪(1)(2))=1.►I=(1,1),J=(1)R(1,1)∪(1)=13+(−2(3δ2+(5δ−3)δ2)δ1+(5δ−3)δ12+(δ−δ2)(δ(δ+3)+(3−5δ)δ2))3δ(4δ+3)(δ−δ1+δ2)2b2+O(b4).The value of ((1,1)∪(1)(1,1))=3.•|I∪J|=4, |I|=3, |J|=1►I=(3),J=(1)R(3)∪(1)=1+O(b2),((3)∪(1)(3))=1.►I=(2,1),J=(1)R(2,1)∪(1)=12+O(b2),((2,1)∪(1)(2,1))=2.►I=(1,1,1),J=(1)R(1,1,1)∪(1)=14+O(b2),((1,1,1)∪(1)(1,1,1))=4. In the last item for |I∪J|=4, because of their sizable forms, we have omitted the terms of order O(b2).Appendix BCoefficients of the Heun's functionIn this appendix we present the explicit form of numerical derivation of the coefficients of the Heun's function given asΨ+(z,x)=z1−η∑m∈Zhm(x)zm,hm(x)=∑s≥0hm,s(δ1,…,δ4,η)xs, where particular values read(65a)h0(x)=1−(δ1−δ2+(η−1)η)(η((η−1)η−δ3+δ4−1)+2δ3−2δ4+1)4(η−1)2ηx+…,(65b)h1(x)=δ3−δ4+(1−η)η2(η−1)+(δ1−δ2+(η−1)η)8(η−1)2η(2η−3)((−2δ3+2δ4+1)η3+(8δ3−6δ4+4)η2+(δ32−2(δ4+5)δ3+(δ4−1)(δ4+3))η−3(δ3−δ4)2+3(δ3+δ4)+η4(η−3))x+…(65c)h2(x)=(5−2δ3+2δ4)η2+(6δ3−4δ4−2)η+δ32+(δ4+2)(δ4−2δ3)+η3(η−4)4(2η2−5η+3)−(δ1−δ2+(η−1)η)48(η−2)(η−1)2η(2η−3)((−3δ3+3δ4+16)η5+(24δ3−18δ4−7)η4+(3δ32−3(2δ4+25)δ3+3δ4(δ4+11)−23)η3+(−21δ32+6(6δ4+19)δ3−3δ4(5δ4+4)+32)η2+(−δ33+(3δ4+49)δ32−3(δ4(δ4+22)+28)δ3+(δ4−1)δ4(δ4+18)−12)η+12(2δ3+δ4)+2(δ3−δ4)(2δ32−(4δ4+17)δ3+2δ42+δ4)+η7−7η6)x+…References[1]A.RoveauxP.ArscottHeun's Differential Equations1995Oxford University Press[2]J. Olver, A.B.O. Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark et al., NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.13 of 2016-09-16.[3]A.BelavinA.M.PolyakovA.ZamolodchikovInfinite conformal symmetry in two-dimensional quantum field theoryNucl. Phys. B2411984333380[4]T.EguchiH.OoguriConformal and current algebras on general Riemann surfaceNucl. Phys. B2821987308328[5]G.W.MooreN.SeibergPolynomial equations for rational conformal field theoriesPhys. Lett. B2121988451[6]G.FelderJ.FröhlichG.KellerOn the structure of unitary conformal field theory II: representation theoretic approachCommun. Math. Phys.130May 1990149[7]A.B.ZamolodchikovA.B.ZamolodchikovStructure constants and conformal bootstrap in Liouville field theoryNucl. Phys. B4771996577605arXiv:hep-th/9506136[8]L.HadaszZ.JaskolskiM.PiatekClassical geometry from the quantum Liouville theoryNucl. Phys. B7242005529554arXiv:hep-th/0504204[9]L.F.AldayD.GaiottoY.TachikawaLiouville correlation functions from four-dimensional gauge theoriesLett. Math. Phys.912010167197arXiv:0906.3219[10]N.WyllardAN−1 conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theoriesJHEP09112009002arXiv:0907.2189[11]D.GaiottoAsymptotically free N=2 theories and irregular conformal blocksarXiv:0908.0307[12]A.MarshakovA.MironovA.MorozovOn non-conformal limit of the AGT relationsPhys. Lett. B6822009125129arXiv:0909.2052[13]V.AlbaA.MorozovNon-conformal limit of AGT relation from the 1-point torus conformal blockJETP Lett.902009708712arXiv:0911.0363[14]D.GaiottoJ.TeschnerIrregular singularities in Liouville theory and Argyres–Douglas type gauge theories, IJHEP12122012050arXiv:1203.1052[15]M.PiatekA.R.PietrykowskiClassical irregular block, N = 2 pure gauge theory and Mathieu equationJ. High Energy Phys.122014032arXiv:1407.0305[16]N.A.NekrasovS.L.ShatashviliQuantization of Integrable Systems and Four Dimensional Gauge Theories2009[17]N.NekrasovA.RoslyS.ShatashviliDarboux coordinates, Yang–Yang functional, and gauge theoryNucl. Phys. Proc. Suppl.21620116993arXiv:1103.3919[18]J.TeschnerQuantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence IAdv. Theor. Math. Phys.152011471564arXiv:1005.2846[19]J.TeschnerQuantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondencearXiv:1707.07873[20]M.PiatekA.R.PietrykowskiClassical limit of irregular blocks and Mathieu functionsJHEP012016115arXiv:1509.08164[21]M.PiatekA.R.PietrykowskiClassical irregular blocks, Hill's equation and PT-symmetric periodic complex potentialsJHEP072016131arXiv:1604.03574[22]G.BasarG.V.DunneResurgence and the Nekrasov–Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systemsJHEP022015160arXiv:1501.05671[23]A.LitvinovS.LukyanovN.NekrasovA.ZamolodchikovClassical conformal blocks and Painleve VIJ. High Energy Phys.072014144arXiv:1309.4700[24]W.HeQuasimodular instanton partition function and the elliptic solution of Korteweg–de Vries equationsAnn. Phys.3532015150162arXiv:1401.4135[25]J.TeschnerClassical conformal blocks and isomonodromic deformationsarXiv:1707.07968[26]A.-K.Kashani-PoorJ.TroostQuantum geometry from the toroidal blockJ. High Energy Phys.82014117arXiv:1404.7378[27]T.HartmanEntanglement entropy at large central chargearXiv:1303.6955[28]C.T.AsplundA.BernamontiF.GalliT.HartmanHolographic Entanglement Entropy from 2d CFT: heavy states and local quenchesJ. High Energy Phys.22015171arXiv:1410.1392[29]D.A.RobertsD.StanfordTwo-dimensional conformal field theory and the butterfly effectPhys. Rev. Lett.1152015131603arXiv:1412.5123[30]B.Carneiro da CunhaF.NovaesKerr–de Sitter greybody factors via isomonodromyPhys. Rev. D932016024045arXiv:1508.04046[31]B.Carneiro da CunhaF.NovaesKerr scattering coefficients via isomonodromyJ. High Energy Phys.112015144arXiv:1506.06588[32]F.NovaesB.Carneiro da CunhaIsomonodromy, Painlevé transcendents and scattering off of black holesJ. High Energy Phys.72014132arXiv:1404.5188[33]T.AnousT.HartmanA.RovaiJ.SonnerBlack hole collapse in the 1/c expansionJ. High Energy Phys.072016123arXiv:1603.04856[34]C.-M.ChangY.-H.LinBootstrapping 2D CFTs in the semiclassical limitJ. High Energy Phys.082016056arXiv:1510.02464[35]K.B.AlkalaevMany-point classical conformal blocks and geodesic networks on the hyperbolic planeJ. High Energy Phys.122016070arXiv:1610.06717[36]K.B.AlkalaevV.A.BelavinHolographic interpretation of 1-point toroidal block in the semiclassical limitJ. High Energy Phys.062016183arXiv:1603.08440[37]K.B.AlkalaevV.A.BelavinFrom global to heavy–light: 5-point conformal blocksJ. High Energy Phys.032016184arXiv:1512.07627[38]K.B.AlkalaevV.A.BelavinMonodromic vs geodesic computation of Virasoro classical conformal blocksNucl. Phys. B9042016367385arXiv:1510.06685[39]K.B.AlkalaevV.A.BelavinClassical conformal blocks via AdS/CFT correspondenceJ. High Energy Phys.082015049arXiv:1504.05943[40]C.-M.ChangY.-H.LinBootstrap, universality and horizonsJ. High Energy Phys.102016068arXiv:1604.01774[41]P.BanerjeeS.DattaR.SinhaHigher-point conformal blocks and entanglement entropy in heavy statesJ. High Energy Phys.052016127arXiv:1601.06794 [hep-th][42]S.Y.SlavyanovW.LaySpecial Functions, A Unified Theory Based on SingularitiesOxford Mathematical Monographs2000[43]P.MenottiOn the monodromy problem for the four-punctured sphereJ. Phys. A47412014415201arXiv:1401.2409 [hep-th][44]P.MenottiClassical conformal blocksMod. Phys. Lett. A312720161650159arXiv:1601.04457 [hep-th][45]S.W.HawkingC.J.HunterM.TaylorRotation and the AdS/CFT correspondencePhys. Rev. D591999064005arXiv:hep-th/9811056[46]H.K.KunduriJ.LuciettiIntegrability and the Kerr–(A)dS black hole in five dimensionsPhys. Rev. D712005104021arXiv:hep-th/0502124[47]A.N.AlievO.DeliceSuperradiant instability of five-dimensional rotating charged AdS black holesPhys. Rev. D792009024013arXiv:0808.0280[48]J.B.AmadoB.Carneiro da CunhaE.PallanteOn the Kerr–AdS/CFT correspondenceJ. High Energy Phys.82017094arXiv:1702.01016[49]M.GuicaT.HartmanW.SongA.StromingerThe Kerr/CFT correspondencePhys. Rev. D802009124008arXiv:0809.4266[50]E.BertiV.CardosoA.O.StarinetsQuasinormal modes of black holes and black branesClass. Quantum Gravity262009163001arXiv:0905.2975 [gr-qc][51]A.CastroJ.M.LapanA.MaloneyM.J.RodriguezBlack hole scattering from monodromyClass. Quantum Gravity302013165005arXiv:1304.3781[52]F.FerrariM.PiatekLiouville theory, N=2 gauge theories and accessory parametersJ. High Energy Phys.12052012025arXiv:1202.2149[53]L.HadaszZ.JaskolskiLiouville theory and uniformization of four-punctured sphereJ. Math. Phys.472006082304arXiv:hep-th/0604187[54]H.DornH.J.OttoTwo and three point functions in Liouville theoryNucl. Phys. B4291994375388arXiv:hep-th/9403141[55]A.ZamolodchikovConformal symmetry in two-dimensional space: recursion representation of conformal blockTheor. Math. Phys.7319871088[56]V.G.KacContravariant form for infinite dimensional lie algebras and superalgebrasW.BeiglbockA.BohmE.TakasugiGroup Theoretical Methods in Physics1978Springer-Verlag441445[57]B.L.FeiginD.B.FuksInvariant skew symmetric differential operators on the line and Verma modules over the Virasoro algebraFunct. Anal. Appl.161982114126[58]B.FeiginD.FuchsRepresentations of the Virasoro algebraA.M.VershikD.P.ZhelobenkoRepresentation of Lie Groups and Related Topicsvol. 71990Gordon and BreachLondon465554[59]C.B.ThornComputing the Kac determinant using dual model techniques and more about the no-ghost theoremNucl. Phys. B2481984551[60]V.G.KacA.K.RainaBombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, Vol. 21987World Scientific[61]A.L.FitzpatrickJ.KaplanM.T.WaltersUniversality of long-distance AdS physics from the CFT bootstrapJ. High Energy Phys.082014145arXiv:1403.6829[62]L.HollandsO.KidwaiHigher length-twist coordinates, generalized Heun's opers, and twisted superpotentialsarXiv:1710.04438 [hep-th][63]A.-K.Kashani-PoorJ.TroostTransformations of spherical blocksJ. High Energy Phys.13102013009arXiv:1305.7408 [hep-th]