A Coefficients F^{(r,s)}

## Abstract

The Nekrasov–Shatashvili limit of the SU(2) pure gauge (-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 ( 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 CFT. To exploit this strategy, a full correspondence between the Mathieu equation and its realization within 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 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 CFT in order to get from the irregular block the eigenvalues of the Mathieu operator in other regions of the coupling constant.

Classical limit of irregular blocks and Mathieu functions

Marcin Piatek1          Artur R. Pietrykowski2

Institute of Physics and CASA*, University of Szczecin

ul. Wielkopolska 15, 70-451 Szczecin, Poland

Institute of Theoretical Physics

University of Wrocław

pl. M. Borna 9, 50-204 Wrocław, Poland

Bogoliubov Laboratory of Theoretical Physics,

Joint Institute for Nuclear Research, 141980 Dubna, Russia

## 1 Introduction

In a last few years much attention was paid to the study of the connections among two-dimensional conformal field theory ( CFT), 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 with genus and punctures can be identified with the partition functions of a class of four-dimensional supersymmetric SU(2) quiver gauge theories:

 ⟨n∏i=1VΔi⟩LFTCg,n=Z(σ)Tg,n. (1.1)

Let us recall that for a given pant decomposition of the Riemann surface , both sides of the equation above have an integral representation. Indeed, LFT correlators can be factorized according to the pattern given by the pant decomposition of and written as an integral over a continuous spectrum of the Liouville theory in which, for each pant decomposition , the integrand is built out of the holomorphic and the anti-holomorphic Virasoro conformal blocks and multiplied by the DOZZ 3-point functions [31, 32]. The Virasoro conformal block on depends on the following quantities: the cross ratios of the vertex operators locations denoted symbolically by , the external conformal weights , the intermediate conformal weights and the central charge .

On the other hand, the partition function can be written as the integral over the holomorphic times the anti-holomorphic Nekrasov partition functions [33, 34]:

 Z(σ)Tg,n=∫[da]Z(σ)Nekrasov¯Z(σ)Nekrasov,

where is some appropriate measure. The Nekrasov partition function can be written as a product of three factors . The first two factors describe the contribution coming from perturbative calculations. Supersymmetry implies that there are contributions to only at the tree- () and 1-loop () levels. is the instanton contribution. The Nekrasov partition function depends on the set of parameters: , , , , . The components of are the gluing parameters associated with the pant decomposition of , where the are the complexified gauge couplings. The multiplet contains the mass parameters. Moreover, , where the ’s are the vacuum expectation values of the scalar fields in the vector multiplets. Finally, , represent the complex -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 on and the instanton sectors of the Nekrasov partition functions for the super Yang–Mills theories .

Soon after its discovery, the AGT conjecture has been extended to the conformal Toda/ SU(N) gauge theories correspondence [35, 36], and to the so-called ‘nonconformal’ cases [37, 38, 39] (see also [22, 40, 41, 42]), which will be of main interest in the present work.

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 , the external and intermediate conformal weights tend to infinity in such a way that their ratios are fixed , cf. [32]. For the standard parametrization of the central charge , where and for heavy weights with , the classical limit corresponds to . It is commonly believed that in the classical limit the conformal blocks behave exponentially with respect to :

 F\lx@stackrelb→0∼e1b2f.

The function is known as the classical conformal block.

The AGT correspondence dictionary says that . Therefore, the semiclassical limit of the conformal blocks corresponds to the so-called Nekrasov–Shatashvili limit ( being kept finite) of the Nekrasov partition functions. In [25] it was observed that in the limit the Nekrasov partition functions have the following asymptotical behavior:

 ZNekrasov(⋅,ϵ1,ϵ2)\lx@stackrelϵ2→0∼exp{1ϵ2W(⋅,ϵ1)}, (1.2)

where is the effective twisted superpotential of the corresponding two-dimensional gauge theories restricted to the two-dimensional -background.

The twisted superpotentials play a pivotal role in the already mentioned Bethe/gauge correspondence [25, 26, 27] which maps supersymmetric vacua of the 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 SU(N) (pure gauge) and the 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 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 and the classical 1–point block on the torus determine the spectra of the Mathieu and Lamé operators in the same way as their gauge theory counterparts, i.e.:  and . 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 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 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 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

### 2.1 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 called chiral or vertex algebra;

• the Hilbert space of states of the theory is a direct sum of irreducible representations of the algebra :

 H=N⨁i=1Ui⊗Ui. (2.1)

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 . Therefore, each chiral algebra contains as a subalgebra the Virasoro algebra ,

 [Ln,Lm]=(n−m)Ln+m+c12(n3−n)δn+m,0 . (2.2)

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 and define CVO’s as operators acting between Verma modules.

Let be the free vector space generated by all vectors of the form

 |νnΔ,I⟩=L−I|νΔ⟩=L−k1…L−kj−1L−kj|νΔ⟩ (2.3)

where is an ordered () sequence of positive integers of the length , and is the highest weight vector:

 L0|νΔ⟩=Δ|νΔ⟩,Ln|νΔ⟩=0∀n>0. (2.4)

The -graded representation of the Virasoro algebra determined on the space:

 Vc,Δ=∞⨁n=0Vnc,Δ

by the relations (2.2) and (2.4) is called the Verma module of the central charge and the highest weight . The dimension of the subspace of all homogeneous elements of degree is given by the number of partitions of (with the convention ). It is an eigenspace of with the eigenvalue .

On there exists the symmetric bilinear form uniquely defined by the relations

 ⟨νΔ|νΔ⟩=1and(Ln)†=L−n.

The Gram matrix of the form is block-diagonal in the basis with blocks

 [Gnc,Δ]IJ=⟨νnΔ,I|νnΔ,J⟩=⟨νΔ|(L−I)†L−J|νΔ⟩.

In particular, one finds

• : ,

 Gn=1c,Δ=⟨L−1νΔ|L−1νΔ⟩=⟨νΔ|L1L−1νΔ⟩=2Δ,
• : ,

 Gn=2c,Δ=(⟨L−2νΔ|L−2νΔ⟩⟨L2−1νΔ|L−2νΔ⟩⟨L−2νΔ|L2−1νΔ⟩⟨L2−1νΔ|L2−1νΔ⟩)=(c2+4Δ6Δ6Δ4Δ(2Δ+1)),
• : ,

 Gn=3c,Δ=⎛⎜⎝2c+6Δ10Δ24Δ10ΔΔ(c+8Δ+8)12Δ(3Δ+1)24Δ12Δ(3Δ+1)24Δ(Δ+1)(2Δ+1)⎞⎟⎠.

The Verma module is irreducible if and only if the form is non-degenerate. The criterion for irreducibility is vanishing of the determinant of the Gram matrix, known as the Kac determinant, given by the formula [51, 52, 53, 54, 55, 56]:

 detGnc,Δ=Cn∏r,s∈N,s≤r1≤rs≤nΦrs(c,Δ)p(n−rs). (2.5)

In the equation above is a constant and

 Φrs(c,Δ)=(Δ+r2−124(c−13)+rs−12)(Δ+s2−124(c−13)+rs−12)+(r2−s2)216.

The Kac determinant vanishes for

 Δrs(c) = (13−c)(r2+s2)+√(c−25)(c−1)(r2−s2)−24rs−2+2c48, r,s∈Z,r≥1,s≥1,1≤rs≤n

or

 crs(Δ) = 13−6(Trs(Δ)+1Trs(Δ)), Trs(Δ) = rs−1+2Δ+√(r−s)2+4(rs−1)Δ+4Δ2r2−1, r,s∈Z,r≥2,s≥1,1≤rs≤n.

For these values of and the representations or are reducible.

The set of the degenerate conformal weights can be parametrized as follows

 Δrs(c)=Δ0+β2rs4,βrs=rβ++sβ−, (2.6)

where

 β±(c)=√1−c±√25−c2√6,Δ0=−14(β++β−)2=c−124.

Sometimes, it is also convenient to use the alternative parametrization:7

 Missing or unrecognized delimiter for \right (2.7)

for which the central charge is given by with .

The non-zero element of degree is called a null vector if , and , . Hence, is the highest weight state which generates its own Verma module , which is a submodule of . One can prove that each submodule of the Verma module is generated by a null vector. Then, the module is irreducible if and only if it does not contain null vectors with positive degree.

For non-degenerate values of , i.e. for , there exists in the ‘dual’ basis whose elements are defined by the relation for all . The dual basis vectors have the following representation in the standard basis

 |νt,nΔ,I⟩=∑J,|J|=n[Gnc,Δ]IJ|νnΔ,J⟩,

where is the inverse of the Gram matrix .

Let be the Verma module with the highest weight state . The chiral vertex operator is the linear map

 VΔ3∞Δ2zΔ10:VΔ2⊗VΔ1→VΔ3

such that for all the operator

 V(ξ2|z)≡VΔ3∞Δ2zΔ10(|ξ2⟩⊗⋅):VΔ1→VΔ3

satisfies the following conditions

 [Ln,V(ν2|z)] = zn(z∂∂z+(n+1)Δ2)V(ν2|z),n∈Z (2.8) V(L−1ξ2|z) = ∂∂zV(ξ2|z), (2.9) V(Lnξ2|z) = n+1∑k=0(n+1k)(−z)k[Ln−k,V(ξ2|z)],n>−1, (2.10) V(L−nξ2|z) = ∞∑k=0(n−2+kn−2)zkL−n−kV(ξ2|z) (2.11) + (−1)n∞∑k=0(n−2+kn−2)z−n+1−kV(ξ2|z)Lk−1,n>1

and

 ⟨νΔ3|V(νΔ2|z)|νΔ1⟩=zΔ3−Δ2−Δ1.

The commutation relation (2.8) defines the primary vertex operator corresponding to the highest weight state . Eqs. (2.9)–(2.11) characterize the decendant CVO’s.

### 2.2 The 3-point block

For a given triple of conformal weights we define the trilinear map

 ρΔ3∞Δ2zΔ10:VΔ3⊗VΔ2⊗VΔ1→C

induced by the matrix element of a single chiral vertex operator

 ρΔ3∞Δ2zΔ10(ξ3,ξ2,ξ1)=⟨ξ3|V(ξ2|z)|ξ1⟩,∀|ξi⟩∈VΔi,i=1,2,3.

The form is uniquely determined by the conditions (2.8)-(2.11). In particular,

1. for -eingenstates8 one gets

 ρΔ3∞Δ2zΔ10(ξ3,ξ2,ξ1)=zΔ3(ξ3)−Δ2(ξ2)−Δ1(ξ1)ρΔ3∞Δ21Δ10(ξ3,ξ2,ξ1); (2.12)
2. for basis vectors , one finds

 ρΔ3∞Δ21Δ10(ν3,I,ν2,ν1) = γΔ3[Δ2Δ1]I, ρΔ3∞Δ21Δ10(ν3,ν2,ν1,I) = γΔ1[Δ2Δ3]I, (2.13) ρΔ3∞Δ21Δ10(ν3,ν2,I,ν1) = (−1)|I|γΔ2[Δ1Δ3]I,

where for a given partition ,

 γΔ[Δ2Δ1]I≡ℓ(I)∏i=1⎛⎝Δ+kiΔ2−Δ1+ℓ(I)∑i

In terms of the trilinear form (3-point block) one can spell out an important result known as the null vector decoupling theorem (Feigin–Fuchs [57]):9

Let be chosen such that , , . Let us assume that

• , (cf. parametrization (2.6)) and

• the vector lies in the singular submodule generated by the null vector , i.e.:

Then, if and only if

 Δj=Δβj≡124(c−1)+14β2jandΔk=Δβk≡124(c−1)+14β2k

satisfy the fusion rules , where and .

## 3 Quantum and classical zero flavor irregular blocks

### 3.1 Definition and basic properties

To begin with, let us consider the following (coherent) vector in the Verma module discovered by D. Gaiotto in [37] and constructed by A. Marshakov, A. Mironov and A. Morozov in [38]:10

 |Δ,Λ2⟩ = ∑IΛ2|I|[G|I|c,Δ](1|I|)IL−I|νΔ⟩ (3.1) = Missing or unrecognized delimiter for \Big

The summation in eq. (3.1) runs over all partitions or equivalently over their pictorial representations — Young diagrams. The symbol in eq. (3.1) denotes a single–row Young diagram, where the total number of boxes equals the number of columns , i.e. .

In [38] it was shown that the vector (3.1) obeys the Gaiotto defining conditions:

 L0|Δ,Λ2⟩=(Δ+Λ2∂∂Λ)|Δ,Λ2⟩,L1|Δ,Λ2⟩=Λ2|Δ,Λ2⟩,Ln|Δ,Λ2⟩=0∀n≥2. (3.2)

The zero flavor qunatum irregular block is defined as the inner product of the Gaiotto state [37, 38]:

 Fc,Δ(Λ) = ⟨Δ,Λ2|Δ,Λ2⟩=∞∑n=0Λ4n[Gnc,Δ](1n)(1n) (3.3) = 1+Λ412Δ+Λ8c+8Δ4Δ(2cΔ+c+2Δ(8Δ−5)) (3.4) + Λ12(11c−26)Δ+c(c+8)+24Δ224Δ((c−7)Δ+c+3Δ2+2)(2(c−5)Δ+c+16Δ2)+…. (3.5)

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 denotes a Riemann surface with genus and punctures. Let be the modular parameter of the 4-punctured Riemann sphere . Then, the -channel conformal block on is defined as the following formal -expansion:

 Fc,Δ[Δ2Δ3Δ1Δ4](x)=xΔ−Δ3−Δ4(1+∞∑n=1xnFnc,Δ[Δ2Δ3Δ1Δ4]), (3.6)

where

 Fnc,Δ[Δ2Δ3Δ1Δ4] = ∑|I|=|J|=nρΔ1∞Δ21Δ0(νΔ1,νΔ2,νΔ,I)[Gnc,Δ]IJρΔ∞Δ31Δ40(νΔ,J,νΔ3,νΔ4) (3.7) = ∑|I|=|J|=nγΔ[Δ2Δ1]I[Gnc,Δ]IJγΔ[Δ3Δ4]J.

Let be the elliptic variable on the torus with modular parameter , then the conformal block on is given by the following formal -series:

 F~Δc,Δ(q)=qΔ−c24(1+∞∑n=1F~Δ,nc,Δqn),

where

 F~Δ,nc,Δ=∑|I|=|J|=nρΔ∞~Δ1Δ0(νΔ,I,ν~Δ,νΔ,J)[Gnc,Δ]IJ.

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 , and the central charge , i.e.:

 ~Δ=M(ϵ−M)ϵ1ϵ2,Δi=αi(ϵ−αi)ϵ1ϵ2, c=1+6ϵ2ϵ1ϵ2,ϵ=ϵ1+ϵ2, α1=12(ϵ+μ1−μ2),α2=12(μ1+μ2), α3=12(μ3+μ4),α4=12(ϵ+μ3−μ4),

and introducing the dimensionless expansion parameter it is possible to prove the following limits [38, 39]:

 qc24−ΔF~Δc,Δ(q) M→∞−−−−−→qM4=^Λ4 Fc,Δ(Λ), xΔ3+Δ4−ΔFc,Δ[Δ2Δ3Δ1Δ4](x) μ1,μ2,μ3,μ4→∞−−−−−−−−−−→xμ1μ2μ3μ4=^Λ4 Fc,Δ(Λ). (3.8)

Due to the ‘non-conformal’ AGT relation, the irregular block can be expressed through the SU(2) pure gauge Nekrasov instanton partition function [37, 40, 22, 42]:

 Fc,Δ(Λ)=ZSU(2),Nf=0inst(^Λ,a,ϵ1,ϵ2). (3.9)

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

 Λ=^Λ√−ϵ1ϵ2,Δ=ϵ2−4a24ϵ1ϵ2,c=1+6ϵ2ϵ1ϵ2≡1+6Q2 (3.10)

where

 Q=b+1b≡√ϵ2ϵ1+√ϵ1ϵ2⇔b=√ϵ2ϵ1. (3.11)

In [25] it was observed that in the limit the Nekrasov partition functions behave exponentially. In particular, for the instantonic sector we have

 Zinst(⋅,ϵ1,ϵ2)\lx@stackrelϵ2→0∼exp{1ϵ2Winst(⋅,ϵ1)}. (3.12)

Therefore, taking into account the AGT relation (3.9), the fact that 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 :

 F1+6Q