Classical limit of irregular blocks and Mathieu functions

The Nekrasov-Shatashvili limit of the N=2 SU(2) pure gauge (Omega-deformed) super Yang-Mills theory encodes the information about the spectrum of the Mathieu operator. On the other hand, the Mathieu equation emerges entirely within the frame of two-dimensional conformal field theory (2d CFT) as the classical limit of the null vector decoupling equation for some degenerate irregular block. Therefore, it seems to be possible to investigate the spectrum of the Mathieu operator employing the techniques of 2d CFT. To exploit this strategy, a full correspondence between the Mathieu equation and its realization within 2d CFT has to be established. In our previous paper [1], we have found that the expression of the Mathieu eigenvalue given in terms of the classical irregular block exactly coincides with the well known weak coupling expansion of this eigenvalue in the case in which the auxiliary parameter is the noninteger Floquet exponent. In the present work we verify that the formula for the corresponding eigenfunction obtained from the irregular block reproduces the so-called Mathieu exponent from which the noninteger order elliptic cosine and sine functions may be constructed. The derivation of the Mathieu equation within the formalism of 2d CFT is based on conjectures concerning the asymptotic behaviour of irregular blocks in the classical limit. A proof of these hypotheses is sketched. Finally, we speculate on how it could be possible to use the methods of 2d CFT in order to get from the irregular block the eigenvalues of the Mathieu operator in other regions of the coupling constant.


Introduction
In a last few years much attention was paid to the study of the connections among two-dimensional conformal field theory (2d CFT), N " 2 supersymmetric gauge theories and integrable systems, cf. e.g. [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. 3 This kind of research was inspired by the discovery of certain dualities, in particular, the AGT [24] and Bethe/gauge [25,26,27] correspondences. 4 The AGT correspondence states that the Liouville field theory (LFT) correlators on the Riemann surface C g,n with genus g and n punctures can be identified with the partition functions of a class T g,n of four-dimensional N " 2 supersymmetric SU(2) quiver gauge theories: (1.1) of the Liouville theory in which, for each pant decomposition σ, the integrand is built out of the holomorphic and the anti-holomorphic Virasoro conformal blocks F pσq c,∆ p r∆ i spZq andF pσq c,∆ p r∆ i spZq multiplied by the DOZZ 3-point functions [31,32]. The Virasoro conformal block F pσq c,∆ p r∆ i spZq on C g,n depends on the following quantities: the cross ratios of the vertex operators locations denoted symbolically by Z, the external conformal weights t∆ i u i"1,...,n , the intermediate conformal weights t∆ p u p"1,...,3g´3`n and the central charge c.
On the other hand, the partition function Z pσq T g,n can be written as the integral over the holomorphic times the anti-holomorphic Nekrasov partition functions [33,34]: where rdas is some appropriate measure. The Nekrasov partition function can be written as a product of three factors Z Nekrasov " Z class Z 1´loop Z inst . The first two factors Z class Z 1´loop ": Z pert describe the contribution coming from perturbative calculations. Supersymmetry implies that there are contributions to Z pert only at the tree-(Z class ) and 1-loop (Z 1´loop ) levels. Z inst is the instanton contribution. The Nekrasov partition function Z Nekrasov pq,ã,m, 1 , 2 q depends on the set of parameters:q,ã,m, 1 , 2 . The components ofq " texp 2πτ 1 , . . . , exp 2πτ 3g´3`n u are the gluing parameters associated with the pant decomposition of C g,n , where the τ p " θ p 2π`4 πi g 2 p are the complexified gauge couplings. The multiplet m " tm 1 , . . . , m n u contains the mass parameters. Moreover,ã " ta 1 , . . . , a 3g´3`n u, where the a's are the vacuum expectation values of the scalar fields in the vector multiplets. Finally, 1 , 2 represent the complex Ω-background parameters.
Comparing the integral representations of both sides of eq. (1.1) it is possible, thanks to AGT hypothesis, to identify separately in the holomorphic and anti-holomorphic sectors the Virasoro conformal blocks F c,∆ p r∆ i spZq on C g,n and the instanton sectors Z inst of the Nekrasov partition functions for the super Yang-Mills theories T g,n .
The AGT correspondence works at the level of the quantum Liouville field theory. It is intriguing to ask, however, what happens if we proceed to the semiclassical limit of the Liouville correlation functions. This is the limit in which the central charge c, the external ∆ i and intermediate ∆ p conformal weights tend to infinity in such a way that their ratios are fixed ∆ p {c " ∆ i {c " const., cf. [32]. For the standard parametrization of the central charge c " 1`6Q 2 , where Q " b`1 b and for heavy weights p∆ p , ∆ i q " 1 b 2 pδ p , δ i q with δ p , δ i " Opb 0 q, the classical limit corresponds to b Ñ 0. It is commonly believed that in the classical limit the conformal blocks behave exponentially with respect to Z: The function f is known as the classical conformal block.
The AGT correspondence dictionary says that b " a 2 { 1 . Therefore, the semiclassical limit b Ñ 0 of the conformal blocks corresponds to the so-called Nekrasov-Shatashvili limit 2 Ñ 0 ( 1 being kept Figure 1: The triple correspondence in the case of the Virasoro classical conformal blocks links the latter to the SU(2) instanton twisted superpotentials which describe the spectra of some quantum-mechanical systems. The Bethe/gauge correspondence on the r.h.s. connects the SU(N) N " 2 SYM theories with the N-particle quantum integrable systems. An extension of the above triple relation to the case N ą 2 needs to consider on the l.h.s. the classical limit of the W N symmetry conformal blocks according to the known extension [35] of the AGT conjecture. finite) of the Nekrasov partition functions. In [25] it was observed that in the limit 2 Ñ 0 the Nekrasov partition functions have the following asymptotical behavior: where Wp¨, 1 q " W pert p¨, 1 q`W inst p¨, 1 q is the effective twisted superpotential of the corresponding two-dimensional gauge theories restricted to the two-dimensional Ω-background.
The twisted superpotentials play a pivotal role in the already mentioned Bethe/gauge correspondence [25,26,27] which maps supersymmetric vacua of the N " 2 theories to Bethe states of quantum integrable systems (QIS's). A result of that duality is that the twisted superpotentials are identified with the Yang-Yang (YY) functions [43] which describe the spectra of some QIS's. Therefore, combining both the classical/Nekrasov-Shatashvili limit of the AGT duality and the Bethe/gauge correspondence one thus gets a triple correspondence which connects the classical blocks with the twisted superpotentials and then with the Yang-Yang functions (cf. Fig.1).
For example, the twisted superpotentials for the N " 2 SU(N) N f " 0 (pure gauge) and the N " 24 SU(N) SYM theories determine respectively the spectra of the N-particle periodic Toda (pToda) and the elliptic Calogero-Moser (eCM) models [25]. In the case of the SU(2) gauge group these QIS's are simply quantum-mechanical systems whose dynamics is described by some Schrödinger equations. Concretely, for the 2-particle pToda and eCM models these Schrödinger equations correspond to the celebrated Mathieu and Lamé equations with energy eigenvalues expressed in terms of the twisted superpotentials. This correspondence can be used to investigate nonperturbative effects in the Mathieu and Lamé quantum-mechanical systems, cf. [44]. On the other hand, the Mathieu and Lamé equations emerge entirely within the framework of 2d CFT as the classical limit of the null vector decoupling (NVD) equations for the 3-point degenerate irregular block and for the 2-point block (projected 2-point function) on the torus with one degenerate light operator [1,15,45]. It turns out that the classical irregular block f irr and the classical 1-point block on the torus f torus determine the spectra of the Mathieu and Lamé operators in the same way as their gauge theory counterparts, i.e.: W SUp2q,N f "0 inst and W SUp2q,N"2i nst . Therefore, it seems that there is a way to study the spectrum of the Mathieu and Lamé operators using two-dimensional conformal field theory methods. 5 However, in order to exploit this possibility it is necessary to establish a full correspondence between the Mathieu and Lamé equations and their realizations within 2d CFT. The missing element is to understand how the solutions of the equations obtained in the classical limit from the NVD equations are connected to the eigenfunctions of the Mathieu and Lamé operators. It is also important to know what kind of solutions are possible to be obtained. An answer to these questions in the case of the Mathieu equation is our main goal in the present paper.
The organization of the paper is as follows. In section 2 the necessary tools of 2d CFT are introduced. In section 3 the simplest irregular blocks are defined and some of their properties are described. In particular, an exponentiation of the pure gauge irregular block within the classical limit is proved at the leading order. After that, the NVD equations for certain degenerate irregular blocks are derived. Section 4 is devoted to the derivation of the Mathieu equation within the formalism of 2d CFT. The calculation presented there provides formulas for the Mathieu eigenvalue and the related eigenfunction in terms of the classical limit of irregular blocks. It is shown that these formulas reproduce the well known noninteger order weak coupling expansion of the Mathieu eigenvalue and the corresponding Mathieu function. In subsection 4.2 a factorization property of the degenerate irregular block with the light operator and its representation in the classical limit as a product of light and heavy parts is proved at the leading order. This factorization property is crucial for deriving the Mathieu equation. Section 5 contains our conclusions. In particular, the problems that are still open and the possible extensions of the present work are discussed.
2 Conformal blocks in the operator formalism

Chiral vertex operators
Starting from the Belavin-Polyakov-Zamolodchikov axioms [46], Moore and Seiberg [47] have constructed formalism of the so-called rational conformal field theories (RCFT's), 6 where -the operator algebra of local fields contains purely holomorphic subalgebra A called chiral or vertex algebra; -the Hilbert space of states of the theory is a direct sum of irreducible representations of the algebra A ' A: In RCFT's the summation in (2.1) is over a discrete finite set. However, one can generalize and successfully apply the Moore-Seiberg formalism to the case of two-dimensional conformal field theories with continuous spectrum, cf. e.g. [49,50]. In such a case the direct sum in eq. (2.1) becomes a direct integral. In any 2d CFT there exist at least two chiral fields, i.e., the identity operator and its descendant -the holomorphic component of the energy-momentum tensor T pzq " ř nPZ z´n´2L n . Therefore, each chiral algebra A contains as a subalgebra the Virasoro algebra Vir " In the Moore-Seiberg formalism the 'physical' fields of [46] are built out of more fundamental objects -the so-called chiral vertex operators (CVO's). These are intertwining operators acting between representations of the vertex algebra. In the present paper we confine ourselves to the simplest case when A " Vir and define CVO's as operators acting between Verma modules.
Let V n c,∆ be the free vector space generated by all vectors of the form where I " pk 1 , . . . , k j´1 , k j q is an ordered (k 1 ě . . . ě k j ě 1) sequence of positive integers of the length |I| " k 1`. . .`k j " n, and | ν ∆ y is the highest weight vector: The Z-graded representation of the Virasoro algebra determined on the space: by the relations (2.2) and (2.4) is called the Verma module of the central charge c and the highest weight ∆. The dimension of the subspace V n c,∆ of all homogeneous elements of degree n is given by the number ppnq of partitions of n (with the convention pp0q " 1). It is an eigenspace of L 0 with the eigenvalue ∆`n.
On V n c,∆ there exists the symmetric bilinear form x¨|¨y uniquely defined by the relations x ν ∆ | ν ∆ y " 1 and pL n q : " L´n.
The Gram matrix G c,∆ of the form x¨|¨y is block-diagonal in the basis t| ν ∆,I yu with blocks In particular, one finds n " 1: tL´1| ν ∆ yu, The Verma module V c,∆ is irreducible if and only if the form x¨|¨y is non-degenerate. The criterion for irreducibility is vanishing of the determinant det G n c,∆ of the Gram matrix, known as the Kac determinant, given by the formula [51,52,53,54,55,56]: (2.5) In the equation above C n is a constant and Φ rs pc, ∆q "´∆`r .
For these values of ∆ and c the representations V c,∆ rs pcq or V c rs p∆q,∆ are reducible. The set t∆ rs pcqu of the degenerate conformal weights can be parametrized as follows where β˘pcq " ? 1´c˘?25´c Sometimes, it is also convenient to use the alternative parametrization: 7 ∆ rs pcq " for which the central charge is given by c " 1`6Q 2 with Q " b`b´1.
The non-zero element | χ rs y P V c,∆ rs pcq of degree n " rs is called a null vector if L 0 | χ rs y " p∆ rs`r sq | χ rs y, and L k | χ rs y " 0, @ k ą 0. Hence, | χ rs y is the highest weight state which generates its own Verma module V c,∆ rs pcq`rs , which is a submodule of V c,∆ rs pcq . One can prove that each submodule of the Verma module V c,∆ rs pcq is generated by a null vector. Then, the module V c,∆ rs pcq is irreducible if and only if it does not contain null vectors with positive degree. For non-degenerate values of ∆, i.e. for ∆ ∆ rs pcq, there exists in V n c,∆ the 'dual' basis t| ν t,n ∆,I yu whose elements are defined by the relation x ν t,n ∆,I | ν n ∆,J y " δ I J for all | ν n ∆,J y P t| ν n ∆,J yu. The dual basis vectors | ν t,n ∆,I y have the following representation in the standard basis Let V ∆ be the Verma module with the highest weight state | ν ∆ y. The chiral vertex operator is the linear map such that for all | ξ 2 y P V ∆ 2 the operator satisfies the following conditions rL n , Vpν 2 |zqs " z nˆz B Bz`p n`1q∆ 2˙V pν 2 |zq , n P Z (2.8)
3 Quantum and classical zero flavor irregular blocks

Definition and basic properties
To begin with, let us consider the following (coherent) vector in the Verma module V c,∆ discovered by D. Gaiotto in [37] and constructed by A. Marshakov, A. Mironov and A. Morozov in [38]: 10 The summation in eq. (3.1) runs over all partitions or equivalently over their pictorial representations -Young diagrams. The symbol p1 |I| q in eq. (3.1) denotes a single-row Young diagram, where the total number of boxes |I| " n equals the number of columns pIq, i.e. pIq " |I| " n.
In [38] it was shown that the vector (3.1) obeys the Gaiotto defining conditions: The zero flavor N f " 0 qunatum irregular block is defined as the inner product of the Gaiotto state [37,38]: In fact, there are much more Gaiotto's states and therefore irregular blocks. 11 In the present paper we confine ourselves to study irregular blocks which are built out of (3.1). Possible extensions of the present work taking into account the existence of the other Gaiotto states will be discussed soon in a forthcoming publication. 12 Let C g,n denotes a Riemann surface with genus g and n punctures. Let x be the modular parameter of the 4-punctured Riemann sphere C 0,4 . Then, the s-channel conformal block on C 0,4 is defined as the following formal x-expansion: 10 With some abuse of nomenclature, we will call 'zero flavor' both the Gaiotto state and the irregular block. The reason for that is that the irregular block corresponds to the Nekrasov instanton function of the N " 2 (Ω-deformed) pure gauge (zero flavor N f " 0) super Yang-Mills theory, in accordance with the 'non-conformal' extension of the AGT conjecture, see below. 11 See for instance [58] and refs. therein. 12 Cf. conclusions. where Let q " e 2πiτ be the elliptic variable on the torus with modular parameter τ, then the conformal block on C 1,1 is given by the following formal q-series: The irregular block (3.3) can be recovered from the conformal blocks on the torus and on the sphere in a properly defined decoupling limit of the external conformal weights [38,39]. Indeed, employing the AGT inspired parametrization of the external weights∆, ∆ i and the central charge c, i.e.: and introducing the dimensionless expansion parameter Λ "Λ{p´ 1 2 q 1 2 it is possible to prove the following limits [38,39]: Due to the 'non-conformal' AGT relation, the N f " 0 irregular block can be expressed through the SU(2) pure gauge Nekrasov instanton partition function [37,40,22,42]: The identity (3.9), which in particular is understood as term by term equality between the coefficients of the expansions of both sides, holds for In [25] it was observed that in the limit 2 Ñ 0 the Nekrasov partition functions Z Nekrasov " Z pert Z inst behave exponentially. In particular, for the instantonic sector we have Therefore, taking into account the AGT relation (3.9), the fact that b " p 2 1 q 1 2 and the Nekrasov-Shatashvili limit (3.12) of the instanton function, one can expect that the irregular block has the following exponential behavior in the limit b Ñ 0: The semiclassical asymptotical behavior (3.13) is a very nontrivial statement concerning the quantum N f " 0 irregular block. 13 First, the existence of the classical zero flavor irregular block f 0 δ`Λ { 1˘c an be checked by direct calculation. Indeed, from the power expansion of the quantum irregular block (3.3) and eq. (3.13) one finds where the coefficients f 0,n δ up to n " 6 take the form: 17884 δ 6´9 6187 δ 5´1 56432 δ 4`3 88737 δ 3´7 317 δ 2´1 38348 δ`34020 1280 δ 9 pδ`2qpδ`6qp4δ`3q 3 p4δ`15q , As a further consistency check of our approach, let us observe that combining (3.9)-(3.11) and (3.13) it is possible to identify the classical irregular block with the SU(2) N f " 0 effective twisted superpotential: . We stress that the classical conformal weight δ in eq. (3.16) above is expressed in terms of the gauge theory parameters a, 1 . Indeed, it is easy to see that . 13 Cf. considerations in subsection 3.2 and conclusions of the present paper.
By comparison of the expansion (3.14)-(3.15) with that of the twisted superpotential obtained independently from the instanton partition function, the identity (3.16) may be confirmed up to desired order.

Towards a proof of the classical limit
The equations (3.15) constitute a direct premise for the existence of the classical irregular conformal block. The rigorous proof of this statement, however, has not yet been performed, although there are many convincing arguments in favor of its validity. 14 In what follows we discuss the leading order of the coefficients of the quantum irregular block and extend the discussion beyond the leading order to provide yet more arguments for the existence of the classical irregular block.

Classical irregular block at the leading order
In order to find the leading contribution to the classical irregular block we examine the coefficients of the expansion of the quantum irregular block. Since these are functions of the matrix elements of the Virasoro algebra, we analyze their dependence on c and ∆ to find out how they scale with respect to b within the classical limit.
The quantum irregular block can be rewritten explicitly as where M ppnq,ppnq p∆, cq is the greatest principal minor of the Kac determinant at the level n p|∆y " |ν ∆ yq 15 As it was mentioned earlier the matrix elements x∆|L I u L´I v |∆y are polynomials in ∆, and c. Although in the classical limit both parameters are important we restrict our attention to ∆ dependence. The reason is that, due to Virasoro algebra (2.2), ∆ appears as a factor in a matrix element x∆|L I u L´I v |∆y either additively accompanied by c or alone, which takes place when I u and I v have part one in common with nonzero multiplicity. In order to find the greatest power of ∆ in general matrix element at level n we take advantage of the argument used by Kac and Raina in ref. [56]. Let us first consider the diagonal matrix element. Making use of the following notation: u :" pI u q, as well as Virasoro algebra (2.2) we obtain for the arbitrary diagonal matrix element 14 See the discussion in the conclusions on this point. 15 We use conventions as in eq. (2.3). I $ n means that I is a partition of n.
which, in accord with the formula pa`kiq, pa; 1q n " paq n :" Γpa`nq L Γpaq, (3.20) amounts to The symbol ' deg " ' indicates that a polynomial on the left hand side and the one on the right hand side are equal up to the term with the greatest power in variable ∆ which, by definition, determines its degree. Let us now consider arbitrary off-diagonal term x∆|L I u L´I v |∆y. Let us assume that both partitions have, say, N parts ti s u N s"1 in common with nonzero multiplicities m i s pI u q and m i s pI v q. Then, repeating the above reasoning, one finds that the general off-diagonal element takes the form Using the generalized formula in eq. (3.20) for m ď n we get where ϑpxq " xθpxq and θpxq is Heaviside's theta function. These general results in eqs. (3.21) and (3.23) allows us to draw the following conclusions for the matrix elements as a polynomials in ∆ and c. For any I u , Since the degree of a matrix element as a polynomial in ∆ depends on the length of partition those of greatest degree yield the leading contribution to both M ppnq,ppnq and det G pnq ∆,c within the classical limit. In the following discussion the explicit form of particular matrix elements prove useful x∆|L n 1 L ń 1 |∆y " n!p2∆q n , (3.24) The above results enable to conclude that (L ń 1 " L´p 1 n q " L´I ppnq ) Moreover, let us note that the minor produced by crossing out the i th diagonal element when treated as a polynomial in ∆ and c has the same coefficient in the highest degree term as the product of diagonal elements of the Gram matrix, namely With these results at hand we can proceed to estimate the contribution to the classical conformal block at the leading order within the limit b Ñ 0. In order to do this we expand the Kac determinant along the Let us observe that in view of our analysis concerning matrix elements as polynomials in ∆ the leading contribution of the latter to M u,ppnq can be found as follows. By means of the elementary operations on columns we obtain where c u " tx∆|L I j L´I u |∆yu " b´2 n n!p2δq n , (3.31) Hence, the formula in eq. (3.30) within the classical limit amounts to Since n " max then it is seen that the first term dominates over the rest for b Ñ 0. Therefore, the coefficient of the irregular block in eq. (3.17) within this limit readś G pnq c,∆¯p This, in accord with eq. (3.14), yields the first coefficient f 0,1 δ of the classical irregular block expansion given in eq. (3.15), namely Classical irregular block beyond the leading order The above computations show that in the estimations of the quantum irregular block coefficients based on the leading order contribution all but the first term in the classical irregular block expansion are neglected. Therefore a more accurate analysis is required. In general the sought expression takes the form where for the sake of brevity we have introduced notation F pnq "`G pnq˘p 1 n q p1 n q . The logarithm in eq. (3.33) has the following expansion In order to find the limit of the above coefficient ofΛ{ 1 the knowledge of F piq is necessary. Unfortunately the exact form of M ppnq,ppnq as a polynomial in ∆ and c is not known and it is necessary to compute it term by term which is the major obstacle in finding the limit. In order for the limit in eq. (3.33) to exist each coefficient should be proportional to b 2 . Thus the complete rigorous proof of the mentioned limit is still an open problem which in order to be solved must be attacked in fact by another methods. 16

The null vector decoupling equations
In this subsection we shall derive the partial differential equations obeyed by the N f " 0 degenerate irregular blocks, cf. [15]. We define the latters as matrix elements of the degenerate chiral vertex operators V˘pzq " Vp|ν ∆˘y |zq between the states (3.1): In the above equation: Moreover, in order to apply the null vector decoupling theorem we will assume that the weights ∆ 1 "∆ and ∆ 3 " ∆ 1 of the in and out states are related by the fusion rule: 17 Let us consider the descendant chiral vertex operator which corresponds to the null vector | χ`y " χ`p0q| 0 y "ˆL´2`1 b 2 L 2 1˙| ν ∆`y appearing at the second level of the Verma module V ∆`. Then, by the NVD theorem, we have that In order to convert eq. (3.38) to the PDE obeyed by the degenerate irregular block Ψ 0 pΛ, zq, one needs to employ the following Ward identity: Using the formula [46]: 17 In the parameterization ∆ β i " 1 24 pc´1q`1 4 β i (see also (2.6)) used in the NVD theorem the fusion rule (3.36) reads as follows: ∆ β 3 " ∆ β 1´β`. In another commonly used parametrization, in which ∆pα i q " α i pQ´α i q, we have ∆pα 3 q " ∆pα 1`b 2 q.
it is now possible to compute the matrix element x ∆ 1 , Λ 2 | p L´2pzqV`pzq |∆, Λ 2 y with the help of eq. (3.39). Applying Cauchy's integral formula one finds that Finally, taking into account that the matrix element of the descendant operator p L 2 1 pzqV`pzq yields B 2 z Ψ 0 pΛ, zq, we get from (3.37), (3.38) (3.40) the desired partial differential equation, determining Ψ 0 pΛ, zq: Replacing ∆`with ∆´and repeating all the steps described above one can get an analogous equation for Ψ 0 pΛ, zq. In the next section we will consider the limit b Ñ 0 of eq.
where κ " ∆ 1´∆`´∆ . Let us observe that Φ 0 pΛ, zq can be split into two parts, i.e. when r " s and r s: Then, one can write Ψ 0 pΛ, zq " z κ exp log`Φ 0 r"s pΛq`Φ 0 r s pΛ, zq˘( where the following notation has been introduced (4.6) Note that the 'diagonal' part Φ 0 r"s of the degenerate irregular block and thus Y 0 do not depend on z. The substitution of (4.2) into eq. (3.41) yields Our aim now is to find the limit b Ñ 0 of eq. (4.7). To this purpose it is convenient to replace the parameter σ in∆ and ∆ 1 (cf. (3.36)) with ξ " bσ and Λ with the new parameterΛ " Λ 1 b. After this rescaling, we have (4.10) Note that ∆`b Ñ0 " Opb 0 q. The next step needed to complete our task is to determine the behavior of the normalized degenerate irregular block Φ 0 " z´κΨ 0 when b Ñ 0. For Λ "Λ{p 1 bq and ∆ 1 ,∆ bÑ0 " 1 b 2 δ, it is reasonable to expect that Moreover, comparing the r.h.s. of eq. (4.11) with eqs. (4.5)-(4.6) one arrives at the following results: The meaning of eq. (4.11) is that the light (∆`b Ñ0 " Opb 0 q) degenerate chiral vertex operator does not contribute to the classical limit. In other words, its presence in the matrix element does not affect the 'classical dynamics' (i.e. the 'classical action'). Let us note that eq. (4.11) is a 'chiral version' of Zamolodchikovs' conjecture [32] (see also [62]) concerning the semiclassical behavior of the Liouville correlators with heavy and light vertices on the sphere. Let us stress that there are only a few explicitly known tests verifying Zamolodchikovs' hypothesis. For instance, the derivation of the large intermediate conformal weight limit ∆ Ñ 8 of the 4-point block on the sphere as well as its expansion in powers of the so-called elliptic variable is based on that assumption in the case of the semiclassical behavior of the 5-point function with the light degenerate vertex operator [59,63]. The calculation performed in this section is a new test of the semiclassical behavior of the form (4.11) (see also [64]). Regardless of the attempts to prove (cf. subsection 4.2), eq. (4.11) can be well confirmed, first, by direct calculations, secondly, by its consequences. Indeed, one can check order by order that the limits (4.12) and (4.13) exist. Moreover, the latter limit reproduces the classical zero flavor irregular block.
Therefore, from (4.7) and (4.11) for b Ñ 0 one gets The nontrivial point here is the observation that lim bÑ0 b 2Λ BΛϕ 0 " 0. This result has been checked up to high orders of the expansion of (4.12). At this point, we define the new function ψ 0 pΛ{ 1 , zq related to the old one by The analogue of eq. (4.14) in the case of ψ 0 pΛ{ 1 , zq is Since for z " e w the derivatives transform as`z 2 B 2 z`z B z˘ψ 0 pzq " B 2 w ψ 0 pe w q, it turns out that eq. (4.16) becomes Finally, the substitution w "´2ix, x P R in (4.17) yields In conclusion, what we have obtained is the following claim: 1. For the coupling constant h " 2Λ{ 1 and the Floquet exponent ν " 2ξ the eigenvalue λ of the Mathieu operator: "´d 2 dx 2`2 h 2 cos 2x is given by the following formula (4.20) where δ " 1 4´ξ 2 .

The classical asymptotic of the degenerate irregular block with the light insertion
In order to understand the factorization phenomenon into 'heavy' and 'light' factors of the degenerate irregular block within the classical limit it suffices to examine the behavior of the two factors in eq. (4.5) as b Ñ 0. According to eq. (4.2) the main ingredients of the degenerate irregular block expansion are the inverse of the Gram matrix and the three form rho. Their dependence on ∆ is crucial for study of the classical limit. As we will see in what follows it is enough to confine oneself to the leading order in b. In section 3.2 we found the leading behavior of the ppnqˆppnq component of the inverse Gram matrix in the classical limit. However, from that analysis one can infer also the leading behavior for all the elements of ppnq th column of the inverse Gram matrix. Indeed, from eq. (3.29) we find that (4.23) As for the rho form its analysis is much more involved. By definition, for any two partitions I $ r, J $ s it takes the form ρ ∆ 1 ∆`∆ 8 1 0`ν ∆ 1 ,I , ν ∆`, ν∆ ,J˘" x∆ 1 |L I V ∆`p zqL´J|∆yˇˇz "1 . (4.24) Making use of the Virasoro algebra (2.2) this can be developed into the form x∆ 1 |L I V ∆`p zqL´J|∆y " x∆ 1 |V ∆`p zqad I pL´Jq |∆y`x∆ 1 |ad : I p1q`V∆`p zq˘ad 9 I p1q pL´Jq |∆ỳ x∆ 1 |ad : I p2q`V∆`p zq˘ad 9 I p2q pL´Jq |∆y`. . .`x∆ 1 |ad I`V∆`p zq˘L´J|∆y, (4.25) where for the sake of brevity we have used the following notation ad I pL´Jq :" rL k pIq pIq , . . . , rL k 1 pIq , L´Js . . .s, @ mPt1,..., pIqu I " 9 I pmq Y : I pmq , 9 I pmq :"`k 1 pIq, . . . , k pIq´m pIq˘.
Let us examine a matrix element that contributes to the sum in eq. (4.25). It takes the form x∆ 1 |ad : I pmq`V∆`p zq˘ad 9 I pmq pL´Jq |∆y " x∆ 1 |rL k pIq pIq , . . . , rL k pIq´m`1 pIq , V ∆`p zqs . . .srL k pIq´m pIq , . . . , rL k 1 pIq , L´Js . . .s|∆y. (4.26) It is nonzero provided t " | 9 I pmq | " ř k i p 9 I pmq q ď |J| " s. For definiteness let us assume that t ă s. From the commutator formula between the Virasoro generator and vertex operator (2.8) we obtain ad : Hence, the matrix element (4.26) assumes the form x∆ 1 |ad : I pmq`V∆`p zq˘ad 9 I pmq pL´Jq |∆y " pIq ź i" pIq´m`1 z k i pIq´z B z``ki pIq`1˘∆`¯x∆ 1 |V ∆`p zqad 9 I pmq pL´Jq |∆y. (4.27) The nested commutator encoded in ad 9 I pmq pL´Jq on the right hand side of the above formula provides possible factors containing∆ and c. These factors typically appear if one or more parts of 9 I pmq coincides with those in the partition J. Let us rewrite the matrix element on the right hand side of eq. (4.27) in terms of multiplicities x∆ 1 |V ∆`p zqad 9 I pmq pL´Jq |∆y " x∆ 1 |V ∆`p zqrL k pIq´m pIq , . . . , rL k 1 pIq , L´Js . . .s|∆y and similarly for parts j u . Let us assume for definiteness that j u " i u for u P t1, . . . , Nu, N ď l. Then, according to the formula in eq. (3.22) an overall factor that appears in front of the resulting matrix element, up to leading term in∆, reads x∆ 1 |V ∆`p zqad 9 I pmq pL´Jq |∆y deg " Poly 9 I pmq , 9 J p∆, cqx∆ 1 |V ∆`p zqL´: J |∆y, (4.29a) where 9 I pmq , 9 J $ t and : J $ s´t.
where ϑpxq " xθpxq and θpxq is Heaviside's theta function. Poly 9 I pmq , 9 J p∆, cq is a polynomial in∆ and c and ' deg " ' has the same meaning as in section 3.2. Its degree in∆, in general, varies between 0 ď deg∆ Poly I i ,I j p∆, cq ď t, I i , I j $ t, i, j " t1, . . . , pptqu.
Since`2∆`c 12 pi 2´1 q; i˘n deg " 2 n∆n as a polynomial in∆ then from eq. (4.29b) we get deg∆ Poly 9 The above formula enables one to estimate the contribution of the corresponding term in the sum (4.25) within the classical limit. Since for b Ñ 0 conformal weights scale as ∆ 1 ,∆ " δ{b 2 and ∆`"´1{2 we conclude that the two products in the second and third line of eq. (4.32) reduce to the ξ dependent numerical factors and the only factor that determines the classical behavior of the matrix element is the polynomial Poly 9 I pmq , 9 J p∆, cq. The degree of the latter depends on the number of factors. Therefore, the term with the greatest number of factors will dominate entire sum within the classical limit. The term in question is the first one in eq. (4.25). This analysis allows us to conclude that x∆ 1 |L I V ∆`p 1qL´J|∆y " x∆ 1 |V ∆`p zqad I pL´Jq |∆yˇˇz "1`. . . The argument concerning the maximal degree of the polynomial entails that the dominant contribution to the sum over all partitions with fixed |I| " r, |J| " s comes from the term with the maximal possible multiplicity, i.e., if r ă s then I " p1 r q and J " : J Y p1 r q. In this case ρ ∆ 1 ∆`∆ 8 1 0`ν ∆ 1 ,p1 r q , ν ∆`, ν∆ ,J˘" r!p2∆q r p´1q p :  We are at the point where we have all necessary ingredients to prove the factorization phenomenon for the three point irregular conformal block stated in eq. (4.11). Let us consider the case where r " s. Then from eqs. (3.32) and (4.34) we get that | : J| " 0 as well as C p´q : J pξq " 1 which, when applied to eq. (4.3), yields Thus the function Φ 0 r s pΛ, zq splits into two parts with positive and negative power of variable z Φ 0 r s pΛ, zq " φ r s 1 pΛ, zq`φ r s 2 pΛ, zq, (4.38b) Let us consider the second one and compute the classical limit of F pr,sq . From eqs. (4.23) and (4.34) and recalling the notation J " : J`p1 r q " b 2pr`sq 1 b 2r p2δq r r! The last line of the above formula is noting but the exponent of the first coefficient of the irregular classical block as in eq. (4.35) and the second one is the leading order approximation to the Mathieu function. The same argument applies to φ r s 1 , such that we can conclude with the following formula n´ζ p´q n pξqz´n`ζ p`q n pξqz n¯. (4.40) Having found the classical limit of Φ 0 r"s pΛq in eq. (4.35) and Φ 0 r s pΛ, zq in eq. (4.40) we can combine them to find the limit that defines the factor in eq. (4.12) deriving from the light field. As a result we find n´ζ p´q n pξqz´n`ζ p`q n pξqz n¯. (4.41) The above formula does not depend on b and is a finite expression which shows in the leading order approximation that the factorization of the light operator insertion in the three point irregular conformal block indeed takes place.
Therefore, the eigenfunction ψ 0 ν pxq given by our formulae (4.21), (4.51) is nothing but the Floquet solution me ν px, h 2 q, ν Z known as the Mathieu exponent (cf. Example 17.1 in [65]). 23 The second solution is of the form ψ 0 ν p´xq " me ν p´x, h 2 q and it is the Floquet solution for´ν and the same eigenvalue (4.22). It is known that me ν px, h 2 q and me ν p´x, h 2 q obey (http://dlmf.nist.gov/28.12.iii): Functions ce ν and se ν constitute another fundamental system of solutions. So far only the solutions of the noninteger order (ν Z) have been discussed. Hence, the question arises at this point of how to get from the classical limit of the irregular block the Mathieu eigenvalues and eigenfunctions corresponding to the integer values of the Floquet exponent. Recall, such solutions are periodic. 24 In particular, one can construct the solutions of periods π or 2π (q " h 2 ): -the cosine-elliptic ce m px; qq, m " 0, 1, 2, . . ., that corresponds at q " 0 with cos mx, for instance: ce 1 px; qq " cos x´1 8 q cos 3x`1 64 q 2ˆ1 3 cos 5x´cos 3x1 512 q 3ˆ1 3 cos 3x´4 9 cos 5x`1 18 cos 7x˙`. . . ; -the sine-elliptic se m px; qq, m " 1, 2, . . ., that corresponds at q " 0 with sin mx, for example: However, the expression (4.52) does not satisfy (4.53). 24 Cf. appendix B.
The functions ce m and se m have period π if m is even and period 2π if m is odd. The corresponding eigenvalues λ denoted by a m pqq for ce m and b m pqq for se m are called characteristic numbers, e.g.: For any q ą 0 characteristic numbers form the band/gap structure: a 0 ă b 1 ă a 1 ă b 2 ă a 2 . . .. For large m the leading terms of the a m and b m are (http://dlmf.nist.gov/28.6.E14): 32 pm 2´4 q pm 2´1 q 3``9 m 4`5 8m 2`2 9˘q 6 64 pm 2´9 q pm 2´4 q pm 2´1 q 5`. . . .
Notice that for m " ν the above expression matches (C.5) and therefore can be recovered from the classical irregular block with δ " 1 4 p1´m 2 q (cf. (4.20), (4.22)) or from the gauge theory counterpart of (4.20), i.e.: 1 where a " 1 2 m 1 , cf. [1,44]. To conclude, regardless of the coincidence described above, work is in progress in order to find a mechanism which allows to derive from the conformal blocks the eigenvalues in the case of the finite integer values of m and the corresponding integer order solutions.

Concluding remarks and open problems
In the present paper we have shown that the N f " 0 classical irregular block solves the eigenvalue problem for the Mathieu operator. The statement that the Mathieu eigenvalue for small h 2 " 4Λ 2 { 2 1 (weakly-coupled region) and noninteger characteristic exponent ν Z is determined by the classical zero flavor irregular block has been already stated in our previous work [1]. The new result of the present work is the derivation of an expression of the corresponding eigenfunction. Moreover, it has been shown that the formula (4.21) reproduces the known solution of the Mathieu equation with the eigenvalue (4.22). Therefore, we have established a link between the Mathieu equation and its realization within two-dimensional CFT. This result paves the way for a new interesting line of research. Concretely, one can try to study other regions of the spectrum of the Mathieu operator by means of 2d CFT tools. Indeed, two interesting questions arise at this point: (i) How is it possible to derive from the irregular block the solutions with integer values of the Floquet parameter? (ii) How within 2d CFT one can get the solutions in the other regions of the spectrum? The answer to the first question needs more studies. It seems that also the second question is reasonable. Let us remember that the quantum irregular block can be obtained from the four-point block on the sphere in the so-called decoupling limit of the external conformal weights (cf. eq. (3.8)). However, to our knowledge, such limit has been discussed only for the s-channel four-point block [38]. Therefore, it arises at this point the question of what happens if we take the decoupling limit from the four-point blocks in the other channels, i.e. t or u. The latter case appears to be especially interesting since in the u-channel four-point block the invariant ratio is 1 x . Hence, the question is whether we can obtain the irregular block with the expansion parameter 1 Λ " pΛ 1 b q´1. Secondly, if this is possible, do we get in the classical limit the classical irregular block determining the strongly-coupled region of the Mathieu spectrum? In addition, let us note that the s and u-channel fourpoint blocks are related by the braiding relation which, in general, has the form of an integral transform with a complicated kernel -the so-called braiding matrix (cf. e.g. [68]). However, when one of the four external conformal weights becomes degenerate, then the integral transform reduces to the known formula for the analytic continuation of the hypergeometric function from the vicinity of a point x to that of the point 1 x . It seems to be technically possible to take combinations of the decoupling and classical limits on both sides of the braiding relation (at least) in the degenerate case and to obtain as a result a 'duality relation' for the classical irregular blocks. As one can expect, in this way it becomes feasible to establish a formula that continues the Mathieu eigenvalue from the weakly-coupled region to that of strong coupling. Work is in progress in order to verify this hypothesis.
The derivation of the Mathieu equation within the formalism of two-dimensional conformal field theory is based on conjectures concerning the asymptotical behavior of irregular blocks in the classical limit, cf. eqs. (3.13) and (4.11). We recall that the coefficient of the irregular block expansion is a ratio of polynomials in ∆ and c. Our idea of proving the existence of the classical irregular block was to estimate the degree of the irregular block coefficient as a polynomial in the parameter b. To accomplish this task we used methods which had previously been used to prove the Kac determinant formula, cf. [56]. However, these techniques have proven to be too weak to give a complete answer. As a result, we have obtained the classical irregular block at the leading order only. It is therefore necessary to use other methods to try to prove eq. (3.13). It seems that there are two available approaches to solve this problem. The first way is to use a representation of the Gaiotto states in terms of Jack polynomials. As S. Yanagida has shown in ref. [67], it is possible to represent the Gaiotto state for the pure gauge theory using the Jack polynomials. The coefficients relating the Jack polynomials to the Gaiotto state are found explicitly in [67]. The inner product, by means of which one computes the norm of the pure gauge Gaiotto state, induces by the bosonization map and the parameter dependent isomorphism, the inner product in the space spanned by the Jack polynomials. However, these Jack polynomials are not orthogonal within this inner product and one has to expand them in a new basis of Jack polynomials depending on a different parameter and orthogonal with respect to the induced inner product. The coefficients relating the two bases of the Jack polynomials are not known explicitly. The lack of the explicit form of the coefficients hampers the effort to find the classical limit of the pure gauge conformal block.
The second and in our opinion more promising way to get eq. (3.13) in particular and more in general analogous results for the regular blocks, is the application of the Fock space free field realization of the conformal blocks. In the case of discrete spectrum this approach is even mathematically rigorous [69,70,71,72]. As a result one gets the Dotsenko-Fateev(-like) integral representations of the conformal blocks and hence one can try to use the methods of matrix models (beta-ensembles) in the computation of the classical limit of these blocks, cf. [74]. However, it should be stressed that the link between the integral and power series representations of conformal blocks is not completely understood cf. e.g. [75]. In our opinion, also the operator realization of conformal blocks requires further work. For instance, the Fock space representation of the chiral vertex operator with the three independent general conformal weights, whose compositions in the matrix element lead to integral formulas, needs to be developed, cf. [49,50].
The main claim of this work -formulas (4.20) and (4.21) -possess fascinating generalizations. The simplest extension is to consider irregular blocks with N f " 1, 2 flavors. 25 As a preview of the results which will be reported in our next papers let us only mention that also in the cases N f " 1, 2 the classical limit of the irregular blocks exists and yields a consistent definition of the classical blocks. In these cases we have also found explicit formulas for the eigenvalues and the eigenfunctions of operators emergent in the classical limit of the null vector decoupling equations. For N f " 2 one gets a Schrödinger operator containing a generalization of the Mathieu potential. These further developments will confirm the validities of 2d CFT technics in the investigation of different regions of the spectra of the investigated operators.
The Floquet theorem states that the Mathieu eq. (C.1) has a nontrivial solution ψpzq such that ψpz`πq " σ ψpzq (C.2) with σ being a root of the eq.: where ψ 1 pzq and ψ 2 pzq are even and odd, respectively, normalized`ψ 1 ψ 1 2´ψ 1 1 ψ 2˘ˇz "0 " 1 linearly independent solutions. Equivalently, the coefficients c 1 and c 2 in the solution ψ " c 1 ψ 1`c2 ψ 2 , which obeys the condition (C.2), are given as an eigenvector of eq.: A solution of the Mathieu eq. in the form given by the Floquet theorem is called a Floquet solution. In order to gain more understanding of the Floquet solution let us define the quantities ν and y such that σ " e iπν , ypzq " e´i νz ψpzq, where ψpzq fulfills (C.2). The definition has the effect that ypz`πq " e´i νpz`πq ψpz`πq " e´i νz ψpzq " ypzq which shows that ypzq is a periodic function of z with period π. Moreover, ψpzq " e iνz ypzq (C. 3) showing that a Floquet solution ψpzq consists of a periodic function of z multiplied by a complex exponential in z. The quantity ν which controls the exponential behavior is known as the characteristic or Floquet exponent of ψ. The Floquet exponent ν is determined by the eq.: cos πν " ψ 1 pπ; a, qq. (C.4) Eq. (C.4) allows to express the eigenvalue a (or λ) in terms of q (or h 2 ) and Floquet parameter ν. However, usefulness of eq. (C.4) is restricted by an ability to calculate the normalized Mathieu function ψ 1 . The eigenfunction ψ 1 can be found as an expansion in terms of other functions, in particular, in terms of trigonometric functions. Indeed, the meaning of eq. (C.4) is that the Floquet exponent is determined by the value at z " π of the solution which is even around z " 0. To the lowest order, i.e. for q " h 2 " 0, the even solution around z " 0 is ψ p0q 1 pzq " cos ? λz. Therefore, from (C.4) for q " h 2 " 0 one gets ν " ? λ, and more in general ν 2 " λ`O`h 2˘. One can derive various terms of this expansion perturbatively. Indeed, for small q " h 2 the eigenvalue λ as a function of ν and h 2 explicitly reads as follows (http://dlmf.nist.gov/28.15 and cf. Example 17.1 in [65]) λ ν ph 2 q " ν 2`h 4 2 pν 2´1 q``5 ν 2`7˘h8 32 pν 2´4 q pν 2´1 q 3``9 ν 4`5 8ν 2`2 9˘h 12 64 pν 2´9 q pν 2´4 q pν 2´1 q 5`. . . . (C.5) The expansion (C.5) holds for noninteger values of ν Z. The corresponding eigenfunction for small q " h 2 and ν Z is of the form me ν pz, h 2 q " e iνz´h 2 4ˆ1 ν`1 e ipν`2qz´1 ν´1 e ipν´2qz˙`. . . .
The Mathieu equation admits periodic solutions. Indeed, the Floquet solution will be periodic for special values of σ. A necessary condition for periodicity is that |σ| " 1. Since the Floquet solution (C.3) contains the factor y that is periodic with period π, ψ will be periodic with period a) π if ν " 0, 2, . . . ô σ " 1, b) 2π if ν " 1, 3, . . . ô σ "´1, c) sπ if ν " 2r{s, where r, s ą 2 are integers with no common divisors. For physical reasons the solutions of most importance are those with periods π or 2π, and these are the ce m and se m introduced in the main text.