Accessory parameters in confluent Heun equations and classical irregular conformal blocks
AAccessory parameters in confluent Heun equationsand classical irregular conformal blocks
O. Lisovyy a , A. Naidiuk a a Institut Denis-Poisson, Université de Tours, CNRS, Parc de Grandmont, 37200 Tours, France
Abstract
Classical Virasoro conformal blocks are believed to be directly related to accessory parameters of Floquettype in the Heun equation and some of its confluent versions. We extend this relation to another class ofaccessory parameter functions that are defined by inverting all-order Bohr-Sommerfeld periods for confluentand biconfluent Heun equation. The relevant conformal blocks involve Nagoya irregular vertex operators ofrank 1 and 2 and conjecturally correspond to partition functions of a 4D N = N f = The Heun’s differential equation (HE), written in normal form, is given by ψ (cid:48)(cid:48) ( z ) = V ( z ) ψ ( z ), V ( z ) = θ − z + θ − ( z − + θ t − ( z − t ) + θ ∞ − θ − θ − θ t + z ( z − + (1 − t ) E z ( z − z − t ) . (1.1)Every 2nd order ODE with at most 4 regular singular points on the Riemann sphere can be reduced to (1.1) bya Möbius transformation mapping these points to 0, t ,1, ∞ . The exponents of local monodromy are encodedinto the Riemann scheme 0 1 t ∞ ± θ ± θ ± θ t ± θ ∞ The main new feature of (1.1), as compared to the case of 3 regular singularities described by the Gauss hyper-geometric equation, is the presence of an accessory parameter E , which does not influence the local behavior ofsolutions. This makes the global analysis of Heun’s equation considerably more difficult, the central questionbeing the dependence of E on monodromy.Indeed, the Riemann-Hilbert correspondence assigns to every choice of E and t a point in the space ofmonodromy data which consists of conjugacy classes of 4-tuples of SL(2, (cid:67) )-matrices with fixed spectrum thatmultiply to identity: M = (cid:169) M , M , M t , M ∞ ∈ SL(2, (cid:67) ) : M ∞ M M t M = ,Tr M k = − πθ k for k = t , ∞ (cid:170)(cid:177) ∼ . (1.2)The space M is generically 2-dimensional and ( E , t ) can therefore be regarded as a pair of local coordinates onit. Any other convenient coordinate ξ on M may be viewed as a function of ( E , t ). Inverting this relation, to anychoice of ξ we may assign an accessory parameter function E [ ξ ] ( t | ξ ), where the upper index is introduced toemphasize that E [ ξ ] depends not only on the value of ξ but also on how this coordinate was chosen. A standardchoice is to take as ξ one of the three Floquet exponents σ kt defined byTr M k M t = − πσ kt , k = ∞ . (1.3)We thereby obtain three different accessory parameter functions — however, since different singular points of(1.1) can be exchanged by Möbius transformations, these functions turn out to be related by a permutation ofparameters (cid:126) θ = ( θ , θ , θ t , θ ∞ ) combined with an appropriate transformation of t . [email protected] [email protected] a r X i v : . [ m a t h - ph ] J a n he direct monodromy problem for Heun’s equation consists essentially in finding σ kt ’s for given ( E , t ); itcan be formally solved using Hill determinants. Remarkably, the inverse monodromy problem of reconstruc-tion of Heun’s equation with prescribed monodromy exponent σ kt can be solved more efficiently. The result isgiven by a perturbative series such as E [ σ t ] ( t | σ ) = ∞ (cid:88) n = E [ σ t ] n ( σ ) t n , (1.4)whose coefficients E [ σ t ] k can be found recursively using continued fractions. Both Hill determinant and con-tinued fraction method are well-known since the end of 19th century in the context of Mathieu equation,see e.g. [WW, Chapter 19], [AS, Chapter 20], [Hei], [DLMF, Eq.28.15.E1]. The Mathieu counterparts of func-tions σ t ( E , t ) and E [ σ t ] ( t | σ ) are implemented in Mathematica as MathieuCharacteristicExponent and
MathieuCharacteristicA . Although extension of continued fractions to Heun is straightforward, it does notseem to be widely known and is included to Appendix A.The second class of special functions studied in this work are Virasoro conformal blocks (CBs). The regular4-point spherical s -channel CB, F ( t | c ,{ ∆ k }) = t ∆ σ − ∆ − ∆ t (cid:183) + ∞ (cid:88) n = F n ( c ,{ ∆ k }) t n (cid:184) . (1.5)depends on the anharmonic ratio t and six complex parameters: the Virasoro central charge c and five confor-mal dimensions ∆ t , ∞ , σ . The coefficients F n are rational in c ,{ ∆ k } and are fixed by the commutation relationsof the Virasoro algebra [BPZ]. Their explicit form can also be obtained using the AGT correspondence [AGT]identifying F ( t | c ,{ ∆ k }) with Nekrasov partition function [Nek1, NO] of N = N f = :• Conjecture A (exponentiation) . It states that in the quasiclassical limit c , ∆ t , ∞ , σ → ∞ , 6 ∆ k c → δ k , k = t , ∞ , σ (1.6)conformal block (1.5) has WKB type asymptotics F (cid:161) t (cid:175)(cid:175) { ∆ k } (cid:162) ∼ exp c W (cid:161) t (cid:175)(cid:175) { δ k } (cid:162) . (1.7)The series W (cid:161) t (cid:175)(cid:175) { δ k } (cid:162) = ( δ σ − δ − δ t )ln t + ∞ (cid:88) n = W n ({ δ k }) t n (1.8)appearing in this asymptotics is called classical conformal block .• Conjecture B (Heun/classical CB correspondence) . Classical CB and Heun accessory parameter functionare related by E [ σ t ] (cid:161) t | σ , (cid:126) θ (cid:162) = t ∂∂ t W (cid:161) t (cid:175)(cid:175) { δ k } (cid:162) . (1.9)where the rescaled conformal dimensions are identified with Heun monodromy exponents via δ σ = − σ , δ k = − θ k , k = t , ∞ . (1.10)Conjecture A is supported by semiclassical arguments in Liouville QFT and on the gauge theory side of the AGTcorrespondence where W ( t ) is interpreted as Nekrasov-Shatashvili effective twisted superpotential [NS]. Thestrongest evidence comes from the explicit computation of large-order CB coefficients. We also mention a re-cent paper [BDK] which claims to prove the exponentiation of the 4-point CB (1.5). Conjecture B follows froma refined version of the exponentiation (the so-called heavy-light factorization) for conformal blocks with de-generate fields combined with BPZ decoupling equations [Tes, LLNZ] (see also [PP3] for details). It can also betested directly by comparing the classical CB series (1.8) order by order with the accessory parameter expansioncomputed from continued fractions or by other approaches [NC, LN, HK, Men]. [Zam] further refers to [BPZ]; however, the discussion of the quasiclassical limit in [BPZ] is limited to a footnote on p. 357 which doesnot contain a neat formulation of any of the two hypotheses. .2 Confluent Heun equations Our aim is to extend Conjectures A and B to the irregular setting. Indeed, there is a number of confluent ver-sions of the Heun’s equation [SL] which we organize into a geometric degeneration diagram: H VI ∞ t V H III (cid:48) H III H III H III H IV H II (cid:48) H II H I Each entry of this diagram represents an ODE of the form ψ (cid:48)(cid:48) ( z ) = V ( z ) ψ ( z ) on (cid:67)(cid:80) . The form of the poten-tial V ( z ) can be read off from the corresponding Riemann surface. Every hole corresponds to a pole of thequadratic differential V ( z ) d z whose order is equal to the number of cusps plus 2, so that non-cusped holecorresponds to a regular singular point. Thus H VI is the usual HE, H V has two regular singular points and anirregular one, etc. We record the potentials in the following table:notation equation name potential V ( z )H V confluent HE θ − z + θ t − ( z − t ) + + θ ∗ z − E z ( z − t ) H IV biconfluent HE θ − z − E z + θ • + ( z + t ) H III doubly confluent HE t z + t θ (cid:63) z − E z + θ ∗ z + H III reduced doubly confluent HE tz − E z + θ ∗ z + H III doubly reduced doubly confluent HE tz − E z + z H II triconfluent HE (cid:161) z + t (cid:162) + θ ◦ z + E H I reduced triconfluent HE 4 z + t z + E H III (cid:48) reduced confluent HE θ − z + θ t − ( z − t ) + z − E z ( z − t ) H II (cid:48) reduced biconfluent HE θ − z + E z + t + z Table 1: Notation for confluent Heun equations3he parameters such as θ , θ t , θ ∗ , θ • , θ (cid:63) , θ ◦ are exponents of local monodromy around regular singularpoints or formal monodromy around the irregular ones; E denotes the accessory parameter. The “time” (or“coupling”) t parameterizes the exponential behavior of ψ ( z ) at irregular singularities, except in the case of H V and H III (cid:48) , where it can be equivalently chosen as the position of one of the 2 regular singular points.A Laplace transform of H III and H II (applied to their canonical forms) transforms them into H III (cid:48) and H II (cid:48) .The above scheme thus contains seven inequivalent confluent HEs. It is not a surprise that the geometricconfluence diagram exactly reproduces the one suggested in [CM, CMR] for Lax pairs of Painlevé equations.Indeed, there exists a connection between Heun and Painlevé functions [FIKN, Section 2.1.3 and Chapter 15],[Nov] going back to classical papers of Fuchs [Fuchs] and Garnier [Gar] which was also the subject of one of thelast works of Boris Dubrovin [DK].Confluent HEs appear in a wide range of physical applications. To mention a few examples, H I and H II areSchroedinger equations for cubic and quartic oscillator; H III is equivalent to the Mathieu equation (quantumpendulum) through the change of variables f ( x ) = e − ix ψ (cid:161) (cid:112) t e ix (cid:162) ; H V arises in the black hole scattering [BPM,Lea, STU, Fiz, CCN] and Rabi model of quantum optics [MPS, ZXBL, CCR], etc. There already exist a few irregular extensions of Conjectures A and B. Immediately after the AGT discoveryof 4D/2D duality it was realized that Nekrasov partition functions of N = N f = confluent CBs ofthe 1st kind . The algebraic construction of these CBs [Gai, BMT, GT] involves rank 1 Whittaker modules forthe Virasoro algebra. They are given by weak coupling ( t →
0) expansions similar to (1.5) whose coefficientsare nothing but suitable limits of F n ( c ,{ ∆ k }). The same applies to quasiclassical limit where confluent CBsof the 1st kind are expected to exponentiate [PP1, RZ1, RZ2, PP2] and produce Nekrasov-Shatashvili twistedsuperpotentials of the relevant gauge theories.On the other hand, for confluent equations H V , H III , H III , H III there exists a natural counterpart of theaccessory parameter function E [ σ t ] ( t | σ ). Recall that the latter is fixed by the choice of monodromy parame-ter σ t , which in the confluent cases should be replaced by the Floquet exponent of (non-formal!) monodromyaround irregular singular point at ∞ . The relevant cycles are shown in red color in the degeneration diagram.Slightly abusing the notation, we will denote the corresponding confluent accessory parameter function by E [F] ( t | σ ) and call it Floquet characteristic . Its small t expansion can be computed using continued fractionsexactly as in the Heun case (see Appendix B). We also have E [F]V ( t | σ ) = lim Λ →∞ E [F]VI (cid:161) t Λ | σ (cid:162)(cid:175)(cid:175)(cid:175) θ = Λ + θ ∗ , θ ∞ = Λ − θ ∗ , (1.11a) E [F]III ( t | σ ) = lim Λ →∞ (cid:104) E [F]V (cid:161) t Λ | σ (cid:162) − θ − θ t + (cid:105) θ = Λ − θ (cid:63) , θ t = Λ + θ (cid:63) , (1.11b) E [F]III ( t | σ ) = lim θ (cid:63) →∞ E [F]III ( t / θ (cid:63) | σ ), (1.11c) E [F]III ( t | σ ) = lim θ ∗ →∞ E [F]III ( t / θ ∗ | σ ). (1.11d)A relation between the Floquet characteristic E [F]III ( t | σ ) (i.e. accessory parameter in the Mathieu equation)with Nekrasov-Shatashvili twisted superpotential W N f = ( t ) of the pure gauge theory has appeared in [NS] asthe simplest example of Bethe/gauge correspondence. Using the AGT relation, this correspondence was refor-mulated in terms of classical irregular CBs in [PP1]. Further extensions to N f = N f = III , H III and H V . In all four cases, theanalog of Conjecture B has the same form as (1.9): E [F] ( t | σ ) = t ∂∂ t W N f ( t ). (1.12)These developments are briefly reviewed in Subsection 2.2 and Appendix B for convenience of the readers withlittle prior exposition to 4D/2D vocabulary, and also to provide general statements in a uniform notation . For N f =
2, Refs. [PP2, RZ1] deal (on the differential equations side) with the Whittaker-Hill equation which is a special case of H
III .Ref. [CCC] uses gauge theory quantities to define the analog of the right side of (1.12) for N f = strong coupling ( t → ∞ ) regime. In this setting, it is much less clear what one should expect at bothsides of the putative analog of the relation (1.9). First, one needs to introduce a new basis in the space ofirregular CBs to describe the confluent limit of the regular CBs in the u -channel. Several candidates for these confluent CBs of the 2nd kind were suggested in the work of Gaiotto and Teschner [GT] which defines a collisionlimit of the regular u -channel CBs leading to finite answers. An algebraic definition of such CBs was proposedby Nagoya in [Na1, Na2]; its main new feature is an irregular vertex operator acting between two Virasoro-Whittaker modules.Note that the series for confluent CBs of the 2nd kind cannot be expected to converge. Instead, they shouldbe considered as formal series in t which represent asymptotic expansions of an actual function inside somesectors (or even only along some directions) at ∞ . Different sectors/directions can in principle lead to differentbases of confluent CBs of the 2nd kind and it may well turn out that the constructions of [GT, Na1, Na2] do notcover all of them. In any case, it is straightforward to test the quasiclassical asymptotics for confluent CBs ofthe 2nd kind that are already available. We address this question in Subsection 2.3 by checking exponentiation(Conjecture A) in two cases: (a) 3-point CB with 2 regular punctures and 1 irregular puncture of rank 1 (b)2-point CB with 1 regular and 1 irregular puncture of rank 2.Trying to define the accessory parameter side of the confluent Conjecture B for large t , one faces a con-ceptual problem of “good” choice of the monodromy parameter defining the accessory parameter function, aswell as more technical issue of its efficient computation. We propose to replace the Floquet exponent σ by theall-order Bohr-Sommerfeld (BS) period ν = π i (cid:73) γ S odd ( z ) d z , S odd ( z ) = ∞ (cid:88) n = S n − ( z ), (1.13)for a suitably chosen closed contour γ , with S − ( z ) = (cid:112) V ( z ) and S (cid:48) n + (cid:80) n + k =− S k S n − k =
0. The computation of BSperiods is a standard tool of N = (cid:178) (the other one being 0 in the NS limit). In our setup, however, the relevant WKB parameter isthe inverse coupling t .We focus on two examples: (a) confluent and (b) biconfluent Heun equations H V and H IV . Inverting therelation ν = ν ( E , t ), it is fairly easy to obtain accessory parameter E = E [BS] ( t | ν ) in the form of a formal asymp-totic series in t . These series are then identified with classical confluent CBs via Conjectures 3.1 and 3.2. Wefind that, similarly to (1.9) and (1.12), E [BS]V ( t | ν ) = t ∂∂ t U N f = ( t ), E [BS]IV ( t | ν ) = ∂∂ t ˜ U ( t ), (1.14)where U N f = ( t ) and ˜ U ( t ) are classical counterparts of CBs introduced in [GT, Na1]. This observation is con-sistent with a conjectural relation between conformal blocks of the 2nd kind and partition functions of stronglycoupled gauge/Argyres-Douglas theories [GT, BLMST, NU], and may be viewed as a quasiclassical variant ofthe AGT or Bethe/gauge correspondence. Regular CBs are matrix elements of compositions of vertex operators between states in the highest weight rep-resentations of the Virasoro algebra[ L m , L n ] = ( m − n ) L m + n + c n (cid:161) n − (cid:162) δ m + n ,0 . (2.1)A highest weight representation V ∆ and its dual V ∗ ∆ are generated from the states | ∆ 〉 , 〈 ∆ | defined by L | ∆ 〉 = ∆ | ∆ 〉 , L n > | ∆ = 〈 ∆ | L = ∆ 〈 ∆ | , 〈 ∆ | L n < =
0. (2.2)5here is a canonical bilinear pairing 〈|〉 : V ∗ ∆ × V ∆ → (cid:67) satisfying 〈 u | L n · | v 〉 = 〈 u | · L n | v 〉 for all 〈 u | ∈ V ∗ ∆ , | v 〉 ∈ V ∆ and n ∈ (cid:90) . It will be normalized by 〈 ∆ | ∆ 〉 =
1. The primary regular vertex operators V ∆ ∆ , ∆ ( t ) : V ∆ → V ∆ aredefined by the commutation relation (cid:104) L n , V ∆ ∆ , ∆ ( t ) (cid:105) = t n (cid:179) t ∂∂ t + ( n + ∆ (cid:180) V ∆ ∆ , ∆ ( t ). (2.3)It determines V ∆ ∆ , ∆ ( t ) uniquely up to normalization (multiplicative constant independent of t ), which wefurther fix by setting 〈 ∆ | V ∆ ∆ , ∆ ( t ) | ∆ 〉 = t ∆ − ∆ − ∆ . Spherical 4-point conformal block (1.5) is defined as F ( t | c ,{ ∆ k }) : = (cid:173) ∆ ∞ (cid:175)(cid:175) V ∆ ∆ ∞ , ∆ σ (1) V ∆ t ∆ σ , ∆ ( t ) (cid:175)(cid:175) ∆ (cid:174) = ∞ t ∆ ∆ t ∆ ∞ ∆ σ ∆ (2.4)The low-order expansion coefficients in F ( t | c ,{ ∆ k }) = t ∆ σ − ∆ − ∆ t (cid:163) + (cid:80) ∞ n = F n ( c ,{ ∆ k }) t n (cid:164) can be computedby inserting the resolution of the identity operator in a suitable basis of states in V ∆ σ between the two vertexoperators in (2.4). For example, one has F ( c ,{ ∆ k }) = ( ∆ σ − ∆ + ∆ t )( ∆ σ − ∆ ∞ + ∆ )2 ∆ σ , (2.5a) F ( c ,{ ∆ k }) = ( ∆ σ − ∆ + ∆ t )( ∆ σ − ∆ + ∆ t + ∆ σ − ∆ ∞ + ∆ )( ∆ σ − ∆ ∞ + ∆ + ∆ σ (1 + ∆ σ ) + (2.5b) + (1 + ∆ σ ) (cid:179) ∆ + ∆ t + ∆ σ ( ∆ σ − − ∆ − ∆ t ) + ∆ σ (cid:180)(cid:179) ∆ ∞ + ∆ + ∆ σ ( ∆ σ − − ∆ ∞ − ∆ ) + ∆ σ (cid:180) − ∆ σ ) + c − + ∆ σ ) .The form of F n ’s becomes quite involved with the growth of n but all coefficients can still be explicitly com-puted for arbitrary n thanks to the AGT relation. For concrete formulas, see e.g. [AFLT] (or [LNR, Eqs. (1.2)–(1.4)], whose notation is close to the present paper).The quasiclassical limit described by (1.6)–(1.8) more precisely means that one should consider c ln F ( t | c ,{ ∆ k }) as a formal series in t , and every coefficient of this series has a finite limit (Conjecture A).E.g. from (2.5a)–(2.5b) we find W ({ δ k }) = ( δ σ − δ + δ t )( δ σ − δ ∞ + δ )2 δ σ , (2.6a) W ({ δ k }) = ( δ σ − δ + δ t ) ( δ σ − δ ∞ + δ ) δ σ (cid:183) δ σ − δ + δ t + δ σ − δ ∞ + δ − δ σ (cid:184) + (2.6b) + (cid:161) δ σ + δ σ ( δ + δ t ) − δ − δ t ) (cid:162)(cid:161) δ σ + δ σ ( δ ∞ + δ ) − δ ∞ − δ ) (cid:162) δ σ (4 δ σ +
3) .Already at the order O (cid:161) t (cid:162) , the existence of the limit is not immediately obvious and involves a cancellation ofseparately divergent contributions from F and F .In the discussion of confluent limits, it will be occasionally convenient to use the Liouville CFT parameter-ization of the Virasoro central charge and conformal dimensions, c = + (cid:161) b + b − (cid:162) , ∆ = c − + P . (2.7)The classical limit of the regular CB corresponds to setting P k = i b − θ k ( k = t , ∞ , σ ) and sending b to 0,so that c ∼ b − and ∆ k ∼ b − δ k with δ k = − θ k . When it will be useful to indicate the dependence of CBssuch as F ( t | c ,{ ∆ k }) and W ( t | { δ k }) on various conformal dimensions more explicitly, we will accordingly use anotation such as F (cid:181) P P t P σ P ∞ P ; t (cid:182) and W (cid:179) θ θ t σθ ∞ θ ; t (cid:180) . 6 .2 Confluent conformal blocks of the 1st kind In the confluent case, in addition to the highest weight representations for the Virasoro algebra, one also needsto consider Whittaker modules. A Whittaker module V W (cid:126) λ of rank r is generated from a joint eigenstate (cid:175)(cid:175) (cid:126) λ (cid:174) =| λ r ,..., λ r 〉 of Virasoro generators L r ,..., L r [Gai, BMT, GT]: L n | λ r ,..., λ r 〉 = (cid:40) λ n | λ r ,..., λ r 〉 , r ≤ n ≤ r ,0, n > r , (2.8)which is called a Whittaker vector. Note that e sL n (cid:175)(cid:175) (cid:126) λ (cid:174) with n = r − s ∈ (cid:67) is again a Whittaker vector.For example, t L | λ r ,..., λ r 〉 ∼ (cid:175)(cid:175) t r λ r ,..., t r λ r (cid:174) , (2.9)This property of e sL n (cid:175)(cid:175) (cid:126) λ (cid:174) will be used later to fix some of the eigenvalues in the definitions of confluent CBs.The present subsection is concerned only with Whittaker modules of rank 1. In this case, there is a canonicalbilinear pairing 〈|〉 : V W , ∗ ( λ , λ ) × V ∆ → (cid:67) which is uniquely determined by the condition 〈 w | · L n | v 〉 = 〈 w | L n · | v 〉 forall 〈 w | ∈ V W , ∗ ( λ , λ ) , | v 〉 ∈ V ∆ and n ∈ (cid:90) , and normalization 〈 λ , λ | ∆ 〉 = N f = N f = F N f = (cid:161) t (cid:175)(cid:175) c ; P , P t , P ∗ , P σ (cid:162) : = (cid:68) P ∗ , (cid:175)(cid:175) V ∆ t ∆ σ , ∆ ( t ) (cid:175)(cid:175) ∆ (cid:69) = ∞ t P ∗ , ) ∆ t ∆ σ ∆ (2.10)Its small t expansion may be found by expanding V ∆ t ∆ σ , ∆ ( t ) (cid:175)(cid:175) ∆ (cid:174) in a suitable basis of V ∆ σ and computing thepairing with (cid:173) P ∗ , (cid:175)(cid:175) . The first few coefficients are explicitly given by F N f = (cid:161) t (cid:175)(cid:175) c ;{ P k } (cid:162) = t ∆ σ − ∆ − ∆ t (cid:183) + ( ∆ σ − ∆ + ∆ t ) P ∗ ∆ σ t + (2.11) + (cid:181) ( ∆ σ − ∆ + ∆ t )( ∆ σ − ∆ + ∆ t + P ∗ ∆ σ (1 + ∆ σ ) + (cid:179) ∆ + ∆ t + ∆ σ ( ∆ σ − − ∆ − ∆ t ) + ∆ σ (cid:180)(cid:161) + ∆ σ − P ∗ (cid:162) ∆ σ − ∆ σ + c (1 + ∆ σ ) (cid:182) t + O (cid:161) t (cid:162)(cid:184) .This series can also be obtained by taking termwise confluent limit of the expansion (1.5) where 2 of the 5conformal dimensions become infinite in a correlated way: F N f = (cid:161) t (cid:175)(cid:175) c ;{ P k } (cid:162) = lim Λ →∞ Λ ∆ σ − ∆ − ∆ t F (cid:195) Λ + P ∗ P t P σ Λ − P ∗ P ; t Λ (cid:33) . (2.12)The exponentiation conjecture for the quasiclassical asymptotics of the confluent CB (2.10) is similar to theregular case but involves in addition a rescaling of t . More specifically, F N f = (cid:179) itb (cid:175)(cid:175) c ; i θ b , i θ t b , i θ ∗ b , i σ b (cid:180) b → ∼ const · exp (cid:110) b − W N f = (cid:161) t (cid:175)(cid:175) θ , θ t , θ ∗ , σ (cid:162)(cid:111) , (2.13)where the constant (independent of t ) prefactor does not necessarily have a finite limit as b →
0. Using (2.11),one finds W N f = (cid:161) t (cid:175)(cid:175) θ , θ t , θ ∗ , σ (cid:162) = ( δ σ − δ − δ t )ln t − ( δ σ − δ + δ t ) θ ∗ δ σ t ++ (cid:34) (cid:161) δ σ − ( δ − δ t ) (cid:162) θ ∗ δ σ − (cid:161) θ ∗ + δ σ (cid:162)(cid:161) δ σ + δ σ ( δ + δ t ) − δ − δ t ) (cid:162) δ σ (3 + δ σ ) (cid:35) t + O (cid:161) t (cid:162) , (2.14)where δ = − θ , δ t = − θ , δ σ = − σ as above. 7 .2.3 N f = Π ∆ denote the identity operator on V ∆ , considered as an element of V ∆ ⊗ V ∗ ∆ . For instance, writing thecontributions of descendant states up to level 2, we have Π ∆ = | ∆ 〉〈 ∆ | + L − | ∆ 〉〈 ∆ | L ∆ + (cid:161) L − | ∆ 〉 L − | ∆ 〉 (cid:162)(cid:181) ∆ (2 ∆ +
1) 6 ∆ ∆ ∆ + c (cid:182) − (cid:181) 〈 ∆ | L 〈 ∆ | L (cid:182) + ... (2.15)Confluent CBs of the 1st kind corresponding to N f = F N f = (cid:161) t (cid:175)(cid:175) c ; P (cid:63) , P ∗ , P σ (cid:162) : = (cid:173) P ∗ , (cid:175)(cid:175) t L Π ∆ σ (cid:175)(cid:175) P (cid:63) , (cid:174) , (2.16a) F N f = (cid:161) t (cid:175)(cid:175) c ; P ∗ , P σ (cid:162) : = (cid:173) P ∗ , (cid:175)(cid:175) t L Π ∆ σ (cid:175)(cid:175) (cid:174) , (2.16b) F N f = (cid:161) t (cid:175)(cid:175) c ; P σ (cid:162) : = (cid:173) (cid:175)(cid:175) t L Π ∆ σ (cid:175)(cid:175) (cid:174) . (2.16c)Small t expansions of these CBs can be computed using (2.15): F N f = (cid:161) t (cid:175)(cid:175) c ; P (cid:63) , P ∗ , P σ (cid:162) = t ∆ σ + P ∗ P (cid:63) ∆ σ t + P (cid:63) P ∗ (cid:179) c ∆ σ + (cid:180) − (cid:161) P (cid:63) + P ∗ (cid:162) + (1 + ∆ σ )16 ∆ σ − ∆ σ + c (1 + ∆ σ ) t + O (cid:161) t (cid:162) , (2.17a) F N f = (cid:161) t (cid:175)(cid:175) c ; P ∗ , P σ (cid:162) = t ∆ σ + P ∗ ∆ σ t + P ∗ (cid:179) c ∆ σ + (cid:180) − ∆ σ − ∆ σ + c (1 + ∆ σ ) t + O (cid:161) t (cid:162) , (2.17b) F N f = (cid:161) t (cid:175)(cid:175) c ; P σ (cid:162) = t ∆ σ (cid:34) + t ∆ σ + c ∆ σ + ∆ σ − ∆ σ + c (1 + ∆ σ ) t + O (cid:161) t (cid:162)(cid:35) . (2.17c)The same expansions are also generated by a chain of termwise confluent limits: F N f = (cid:161) t (cid:175)(cid:175) c ; P (cid:63) , P ∗ , P σ (cid:162) = lim Λ →∞ Λ ∆ σ − ∆ − ∆ t t ∆ + ∆ t F N f = (cid:179) t Λ (cid:175)(cid:175) c ; P = Λ − P (cid:63) , P t = Λ + P (cid:63) , P ∗ , P σ (cid:180) , (2.18a) F N f = (cid:161) t (cid:175)(cid:175) c ; P ∗ , P σ (cid:162) = lim P (cid:63) →∞ P ∆ σ (cid:63) F N f = (cid:179) tP (cid:63) (cid:175)(cid:175) c ; P (cid:63) , P ∗ , P σ (cid:180) , (2.18b) F N f = (cid:161) t (cid:175)(cid:175) c ; P σ (cid:162) = lim P ∗ →∞ P ∆ σ ∗ F N f = (cid:179) tP ∗ (cid:175)(cid:175) c ; P ∗ , P σ (cid:180) . (2.18c)The exponentiation of the confluent CBs (2.16) is described by conjectural formulas analogous to (2.13): F N f = (cid:179) − tb (cid:175)(cid:175) c ; i θ (cid:63) b , i θ ∗ b , i σ b (cid:180) b → ∼ const · exp (cid:110) b − W N f = (cid:161) t (cid:175)(cid:175) θ (cid:63) , θ ∗ , σ (cid:162)(cid:111) , (2.19a) F N f = (cid:179) − itb (cid:175)(cid:175) c ; i θ ∗ b , i σ b (cid:180) b → ∼ const · exp (cid:110) b − W N f = (cid:161) t (cid:175)(cid:175) θ ∗ , σ (cid:162)(cid:111) , (2.19b) F N f = (cid:179) tb (cid:175)(cid:175) c ; i σ b (cid:180) b → ∼ const · exp (cid:110) b − W N f = (cid:161) t (cid:175)(cid:175) σ (cid:162)(cid:111) (2.19c)These confluent variants of Conjecture A are of course straightforward to check at low orders in t . We recordbelow only a few terms in the expansions of W N f = ( t ) which easily follow from (2.17), W N f = (cid:161) t (cid:175)(cid:175) θ (cid:63) , θ ∗ , σ (cid:162) = δ σ ln t + θ (cid:63) θ ∗ δ σ t + (cid:195) (cid:161) θ ∗ + δ σ (cid:162)(cid:161) θ (cid:63) + δ σ (cid:162) δ σ (3 + δ σ ) − θ ∗ θ (cid:63) δ σ (cid:33) t + O (cid:161) t (cid:162) , (2.20a) W N f = (cid:161) t (cid:175)(cid:175) θ ∗ , σ (cid:162) = δ σ ln t + θ ∗ δ σ t + (5 δ σ − θ ∗ + δ σ δ σ (3 + δ σ ) t + O (cid:161) t (cid:162) , (2.20b) W N f = (cid:161) t (cid:175)(cid:175) σ (cid:162) = δ σ ln t + t δ σ + δ σ − δ σ (3 + δ σ ) t + O (cid:161) t (cid:162) . (2.20c)Note that the expansions of W N f < ( t ) can also be obtained directly from the regular quasiclassical CB W (cid:161) t (cid:175)(cid:175) { δ k } (cid:162) by the same sequence of termwise limits as in (1.11); in other words, for CBs of the 1st kind the quasiclassicallimit commutes with confluence. 8 .3 Confluent conformal blocks of the 2nd kind Let us recall the construction of a (dual) irregular vertex operator V ∆(cid:126) λ , (cid:126) λ (cid:48) ( t ) : V W , ∗ (cid:126) λ → V W , ∗ (cid:126) λ (cid:48) from [Na1]. It isuniquely determined by the following properties:• Whittaker modules V W , ∗ (cid:126) λ , V W , ∗ (cid:126) λ (cid:48) have the same rank r and satisfy a genericity condition λ r , λ (cid:48) r (cid:54)= V ∆(cid:126) λ , (cid:126) λ (cid:48) ( t ) has the same commutation relations (2.3) with the Virasoro generators as the usualvertex operator: (cid:104) L n , V ∆(cid:126) λ , (cid:126) λ (cid:48) ( t ) (cid:105) = t n (cid:179) t ∂∂ t + ( n + ∆ (cid:180) V ∆(cid:126) λ , (cid:126) λ (cid:48) ( t ). (2.21)This is essentially an expression of the local conformal invariance.• Most importantly, there is a “normalization” condition (cid:173) λ r ,..., λ r (cid:175)(cid:175) V ∆(cid:126) λ , (cid:126) λ (cid:48) ( t ) = t α exp (cid:110)(cid:88) rk = β k t k (cid:111)(cid:183)(cid:173) λ (cid:48) r ,..., λ (cid:48) r (cid:175)(cid:175) + ∞ (cid:88) n = t − n (cid:173) w (cid:48) n (cid:175)(cid:175)(cid:184) , (2.22)where the expression in the square brackets is understood as a formal series in V W , ∗ (cid:126) λ (cid:48) ⊗ (cid:67) [[ t − ]].The parameters (cid:126) λ (cid:48) , α , β ,..., β r − on the right of (2.22) as well as all coefficients (cid:173) w (cid:48) n (cid:175)(cid:175) ∈ V W , ∗ (cid:126) λ (cid:48) of the formal seriesare determined by (cid:126) λ , ∆ and β r . In particular, one has λ (cid:48) k = (cid:40) λ k , k = r + r , λ r − r β r , k = r . (2.23) D The canonical pairing between the Virasoro highest weight modules and Whittaker modules of rank 1, alreadyused in the previous subsection, allows to define [Na1] a class of 3-point confluent CBs of the 2nd kind: D N f = (cid:161) t (cid:175)(cid:175) c ; P , P t , P ∗ , P ν (cid:162) : = (cid:68) P ∗ , (cid:175)(cid:175)(cid:175) V ∆ t (cid:161) P ∗ , (cid:162) , (cid:161) P ν , (cid:162) ( t ) (cid:175)(cid:175)(cid:175) ∆ (cid:69) = ∞ t P ∗ , ) ∆ t ( P ν , ) ∆ (2.24)They will be referred to as confluent CBs of type D . The coefficients of the large t expansion of these CBs, D N f = (cid:161) t (cid:175)(cid:175) c ;{ P k } (cid:162) = t P ν ( P ∗ − P ν ) e ( P ∗ − P ν ) t (cid:183) + ∞ (cid:88) n = D n ( c ;{ P k }) t − n (cid:184) , (2.25)as well as the exponent α = P ν ( P ∗ − P ν ), are thus fixed by Virasoro symmetry. Two lowest order coefficientscan be deduced from the expressions for 〈 w (cid:48) | , 〈 w (cid:48) | in [Na1, Appendix A.2.1]. One has D ( c ;{ P k }) = − P ν + P ν P ∗ + P ν (cid:161) ∆ + ∆ t − P ∗ (cid:162) − ∆ P ∗ , (2.26a) D ( c ;{ P k }) = D ( c ;{ P k }) + (cid:161) ∆ + ( P ∗ − P ν ) P ν (cid:162)(cid:161) ∆ t − (2 P ∗ − P ν )( P ∗ − P ν ) (cid:162) + (2.26b) + (cid:161) P ∗ − P ν ) + c − (cid:162) P ν ( P ∗ − P ν ).The algebraic computation of the coefficients { D k } becomes quite lengthy at higher orders. However, all ofthem remain explicitly computable (at least in principle) since D N f = (cid:161) t (cid:175)(cid:175) c ;{ P k } (cid:162) can be identified [LNR] with aparticular limit of the regular u -channel CB suggested in [GT]. Namely,1 + ∞ (cid:88) n = D n ( c ;{ P k }) t − n = lim Λ →∞ (cid:181) Λ t (cid:182) ∆ t − ( P ∗ − P ν )( Λ − P ν ) (cid:181) − Λ t (cid:182) ( P ∗ − P ν )( Λ + P ν ) + ∆ t F (cid:195) Λ + P ∗ P t Λ + P ∗ − P ν P Λ − P ∗ ; Λ t (cid:33) , (2.27)9here the last two factors under the limit are interpreted as formal series in t (note that only the product ofthese series admits a finite limit as Λ → ∞ while the coefficients of each of them diverge when treated sep-arately). Another difference from the confluent limit leading to CBs of the 1st kind is that the intermediatedimension also goes to ∞ . The relation (2.27) of course reproduces (2.26a), (2.26b) and allows to go to anydesired order. For instance, D ( c ;{ P k }) = D ( c ;{ P k }) D ( c ;{ P k }) − D ( c ;{ P k }) − (cid:179) P ν ( P ∗ − P ν ) − P ∗ + ( ∆ + ∆ t ) (cid:180) D ( c ;{ P k }) + (2.28) + (cid:179) ∆ t − ∆ − P ∗ ( P ∗ − P ν ) (cid:180)(cid:179) ( P ∗ − P ν ) ∆ + P ν ∆ t + P ∗ P ν ( P ∗ − P ν ) (cid:180) + ( c − (cid:179) ( P ∗ − P ν ) ∆ − P ν ∆ t (cid:180) ++ (cid:161) P ∗ + c − (cid:162) ( P ∗ − P ν )( P ∗ − P ν ) P ν .The above expressions for D are organized in a way which makes manifest the exponentiation of CBs oftype D at low orders. We thus arrive at the following confluent variant of Conjecture A, cf (2.13). Conjecture 2.1.
Confluent conformal blocks of type D exponentiate as D N f = (cid:179) itb (cid:175)(cid:175) c ; i θ b , i θ t b , i θ ∗ b , i ν b (cid:180) b → ∼ const · exp (cid:110) b − U N f = (cid:161) t (cid:175)(cid:175) θ , θ t , θ ∗ , ν (cid:162)(cid:111) , (2.29) with U N f = (cid:161) t (cid:175)(cid:175) θ , θ t , θ ∗ , ν (cid:162) = ( ν − θ ∗ ) t + ν ( ν − θ ∗ )ln t + ∞ (cid:88) n = U n ( θ , θ t , θ ∗ , ν ) t − n . (2.30)We have checked this claim up to order O (cid:161) t − (cid:162) . The coefficients of the classical CB of type D are readilyobtained from those of D N f = ( t ). E.g. from (2.26) it follows that U ( θ , θ t , θ ∗ , ν ) = ν − ν θ ∗ + ν (cid:161) δ + δ t + θ ∗ (cid:162) − δ θ ∗ , (2.31a) U ( θ , θ t , θ ∗ , ν ) = − (cid:161) δ − ν ( θ ∗ − ν ) (cid:162)(cid:161) δ t + (2 θ ∗ − ν )( θ ∗ − ν ) (cid:162) − ν ( θ ∗ − ν ) (cid:161) ( θ ∗ − ν ) − (cid:162) . (2.31b)Assuming that the quasiclassical limit commutes with confluence for the CBs of the 2nd kind as well, we canalternatively compute these coefficients from a quasiclassical version of (2.27), U n ( θ , θ t , θ ∗ , ν ) = lim Λ →∞ Λ n W n Λ + θ ∗ θ t Λ + θ ∗ − νθ Λ − θ ∗ − δ t − ( θ ∗ − ν )( Λ + ν ) n , (2.32)where W n ’s denote the expansion coefficients of the regular classical CB. G There is no obvious canonical pairing between Whittaker modules of rank r ≥ (cid:173) (cid:126) λ (cid:175)(cid:175) · L k − (cid:175)(cid:175) ∆ (cid:174) of the Whittaker vector and a descendant state in V ∆ for r > (cid:173) (cid:126) λ (cid:175)(cid:175) ∆ (cid:174) since (cid:173) (cid:126) λ (cid:175)(cid:175) is not an eigenstate of L − . A notable exception concerns the pairingof Whittaker modules of rank 2 with an irreducible highest weight module V (as in this case L − | 〉 = G ( t | c ; P , P • , P ν ) = (cid:173) P • ,0, (cid:175)(cid:175) V ∆ (cid:179) P • ,0, 14 (cid:180) , (cid:179) P • − P ν ,0, 14 (cid:180) ( t ) (cid:175)(cid:175) (cid:174) = ∞ t ( P • ,0, ) ∆ ( P • − P ν ,0, ) (2.33)They will be referred to as CBs of the 2nd kind of type G . Setting the eigenvalues λ , λ to 0 and involves noloss of generality thanks to translation and dilatation symmetry of (2.33).The large t expansion of the conformal block G ( t | c ;{ P k }) has the form G ( t | c ;{ P k }) = t ∆ − P ν + P • P ν e P ν t (cid:183) + ∞ (cid:88) n = G n ( c ;{ P k }) t − n (cid:184) . (2.34)10lgebraic definition (2.33) determines the exponent ˜ α = ∆ − P ν + P • P ν in the above and also allows to com-pute the coefficients G n . One has, in particular, G ( c ;{ P k }) = P ν − P • P ν + (cid:161) P • − ∆ − c − (cid:162) P ν + P • ∆ , (2.35a) G ( c ;{ P k }) = G ( c ;{ P k }) + P ν − P • P ν + (cid:161) P • − ∆ − c − (cid:162) P ν + (2.35b) + (cid:161) − P • + ∆ + c − (cid:162) P • P ν + (cid:161) − P • + ∆ + c − (cid:162) ∆ . Conjecture 2.2.
Confluent conformal blocks of type G exponentiate as G (cid:181) t (cid:113) ib (cid:175)(cid:175) c ; i θ b , i θ • b , i ν b (cid:182) b → ∼ const · exp (cid:169) b − ˜ U (cid:161) t (cid:175)(cid:175) θ , θ • , ν (cid:162)(cid:170) , (2.36) with ˜ U (cid:161) t (cid:175)(cid:175) θ , θ • , ν (cid:162) = − ν t + (cid:161) δ + ν − θ • ν (cid:162) ln t + ∞ (cid:88) n = ˜ U n ( θ , θ • , ν ) t − n . (2.37)Let record the first coefficients of the classical CB of type G :˜ U ( θ , θ • , ν ) = − (cid:163) ν − θ • ν + (cid:161) θ • + δ + (cid:162) ν − δ θ • (cid:164) , (2.38a)˜ U ( θ , θ • , ν ) = − (cid:163) ν − θ • ν + (cid:161) θ • + δ + (cid:162) ν − (cid:161) θ • + δ + (cid:162) θ • ν + (cid:161) θ • + δ + (cid:162) δ (cid:164) . (2.38b)In the next section, they will be compared with the coefficients of the strong-coupling expansion of accessoryparameter function of the biconfluent Heun equation. Consider the Schroedinger equation (cid:126) ψ (cid:48)(cid:48) ( z ) = U ( z ) ψ ( z ), (3.1)with a rational potential U ( z ) which can also analytically depend on (cid:126) , so that U ( z ) = (cid:80) ∞ n = (cid:126) n U n ( z ) with U ( z ) (cid:54)=
0. It admits a formal WKB solution ψ ( z ) = exp (cid:189) ∞ (cid:88) n =− (cid:126) n (cid:90) z S n ( z ) d z (cid:190) , (3.2)where S − ( z ) ≡ p ( z ) = (cid:112) U ( z ) and all other S k ’s are determined by the recurrence relations S (cid:48) n ( z ) + n + (cid:88) k =− S k ( z ) S n − k ( z ) = n ≥ −
1. (3.3)In particular, one has S = (cid:161) − ln p (cid:162) (cid:48) , S = pp (cid:48)(cid:48) − (cid:161) p (cid:48) (cid:162) p (3.4a) S = (cid:195) (cid:161) p (cid:48) (cid:162) p − p (cid:48)(cid:48) p (cid:33) (cid:48) , S = − (cid:161) p (cid:48) (cid:162) − p (cid:161) p (cid:48) (cid:162) p (cid:48)(cid:48) + p (cid:161) p (cid:48)(cid:48) (cid:162) + p p (cid:48) p (cid:48)(cid:48)(cid:48) − p p (cid:48)(cid:48)(cid:48)(cid:48) p , (3.4b)and so on. All even contributions S n ( z ) are given by total derivatives. In fact, denoting S odd ( z ) : = ∞ (cid:88) n = (cid:126) n − S n − ( z ), S even ( z ) : = ∞ (cid:88) n = (cid:126) n S n ( z ), (3.5)it can be shown that S (cid:48) odd + S even S odd =
0, and therefore ψ ( z ) = (cid:112) S odd ( z ) exp (cid:82) z S odd ( z ) d z .11iven a cycle γ on the Riemann surface p = U ( z ), one may then introduce the Bohr-Sommerfeld (BS)period , describing a formal monodromy of ψ ( z ) along γ : ν : = π i (cid:73) γ S odd ( z ) d z = ∞ (cid:88) n = (cid:126) n − ν n − , (3.6)where we denote ν n − : = π i (cid:73) γ S n − ( z ) d z . Let us now consider the potential of H V , V ( z ) = − δ z − δ t ( z − t ) + + θ ∗ z − E z ( z − t ) = P ( z ) z ( z − t ) (3.7)and explain how a small parameter playing the role of (cid:126) appears in our setup. Suppose that t is large and that E / t = κ + o (1). It is not difficult to understand that in this case two roots z of the 4th degree polynomial P ( z )are of order O (1) and two other roots z are close to t . The former two turning points are asymptotically closeto the roots of − δ z + + θ ∗ + κ z =
0, the latter are asymptotic to t + ξ , where ξ satisfies − δ t ξ + − κξ =
0. To makethe differentials S n ( z ) d z single-valued on (cid:67)(cid:80) , we introduce two branch cuts: one connecting z and z andanother connecting z and z .Choose the cycle γ to be the circle of radius (cid:112) t , so that the branch points z = z (and eventually the pole z =
0) are inside γ , while the branch points z = z (and eventually the poles z = t , ∞ ) are outside. Equivalently,we could set from the very beginning z = λ (cid:112) t , so that in the λ -plane the first two branch points would become λ = O (cid:179) (cid:112) t (cid:180) and two others λ = O (cid:161) (cid:112) t (cid:162) . The cycle γ maps to the unit circle | λ | = tV (cid:161) λ (cid:112) t (cid:162) = (cid:126) − U ( λ ), with (cid:126) = (cid:112) t and U ( λ ) = + (cid:126) θ ∗ λ + (cid:126) E λ (1 − (cid:126) λ ) − (cid:126) δ λ − (cid:126) δ t (1 − (cid:126) λ ) . (3.8)Recall that (cid:126) E = κ + o (1), and therefore U ( λ ) = + O ( (cid:126) ). In order to calculate different contributions to the BSperiod ν , it now suffices to expand the integrands in (cid:126) and compute the residue at 0 or ∞ . Substituting E = κ t + ∞ (cid:88) n = E n t − n , (3.9)we find that ν − = π i (cid:73) | λ |= p ( λ ) d λ = ( κ + θ ∗ ) (cid:126) + (cid:161) E − κ − κθ ∗ (cid:162) (cid:126) ++ (cid:104) E − E ( θ ∗ + κ ) + κ (cid:161) κ + θ ∗ + κθ ∗ + δ + δ t (cid:162) + δ t θ ∗ (cid:105) (cid:126) ++ (cid:104) E − E ( θ ∗ + κ ) + E (cid:161) − E + δ + δ t + θ ∗ + κθ ∗ + κ (cid:162) − κ ( θ ∗ + κ ) − θ ∗ + κ )( κδ + κδ t + δ t θ ∗ ) − θ ∗ κ ( θ ∗ + κ ) − δ δ t (cid:105) (cid:126) + O (cid:161) (cid:126) (cid:162) , (3.10a) ν = π i (cid:73) | λ |= pp (cid:48)(cid:48) − (cid:161) p (cid:48) (cid:162) p d λ = − κ ( κ + θ ∗ ) (cid:126) + O (cid:161) (cid:126) (cid:162) , (3.10b) ν = − κ ( κ + θ ∗ ) (cid:126) + O (cid:161) (cid:126) (cid:162) , ... (3.10c)Note in particular that the leading contribution to the period is of 0th order: ν = κ + θ ∗ + O ( (cid:126) ).Inverting the relation ν = ν ( E , t ) to express E = E [BS] ( t | ν ) as a function of ν and t is equivalent to asking allquantum ( (cid:126) -dependent) corrections to the BS period to vanish. It follows that E = κ ( κ + θ ∗ ) = ν − θ ∗ ) ν , E = − U ( θ , θ t , θ ∗ , ν ), E = − U ( θ , θ t , θ ∗ , ν ), ..., (3.11)where U coincide with those given in (2.31). It now becomes straightforward to formulate Conjecture B forconfluent CBs of type D . 12 onjecture 3.1. Let ν be the Bohr-Sommerfeld period along the cycle described above. Then E [BS] ( t | θ , θ t , θ ∗ , ν ) = t ∂∂ t U N f = (cid:161) t (cid:175)(cid:175) θ , θ t , θ ∗ , ν (cid:162) , (3.12) where U N f = (cid:161) t (cid:175)(cid:175) θ , θ t , θ ∗ , ν (cid:162) is the classical confluent conformal block of type D . Let us now follow a similar approach for the biconfluent HE. The potential of H IV is given by V ( z ) = − δ z − E z + θ • + ( z + t ) = Q ( z ) z . (3.13)Assume again that t is large and that E / t = κ + o (1). Two zeros z of Q ( z ) are asymptotic to ξ t , where ξ satisfy the equation − δ ξ − κξ + =
0. Two other zeros z are asymptotic to − t + ξ , where ξ satisfy theequation κ + θ • + ξ =
0. Introducing branch cuts going from z to z and from z to z , we choose γ definingthe BS period to be the unit circle. Also, we rewrite the potential as V ( z ) = (cid:126) − U ( z ), with (cid:126) = t − and U ( z ) = + (cid:126) z − (cid:126) E z + (cid:126) (cid:181) z − δ z + θ • (cid:182) . (3.14)By the above assumption, (cid:126) E = κ + o (1). Making the ansatz (3.9), one obtains ν − = − κ (cid:126) − E (cid:126) + (cid:161) − E + δ + κθ • + κ (cid:162) (cid:126) + ( − E + (2 θ • + κ ) E ) (cid:126) ++ (cid:179) − E + E (2 θ • + κ ) + E − κ (2 θ • + κ ) − δ ( θ • + κ ) − θ • κ (cid:180) (cid:126) + O (cid:161) (cid:126) (cid:162) , (3.15a) ν = − κ (cid:126) − E (cid:126) + (cid:161) − E + δ + θ • κ + κ (cid:162) (cid:126) + O (cid:161) (cid:126) (cid:162) , (3.15b) ν = − κ (cid:126) + O (cid:161) (cid:126) (cid:162) , ... (3.15c)Requiring the quantum corrections to the BS period to vanish, we find that E = E = E = ... = κ = − ν , E = δ + ν − θ • ν , (3.16a) E = ν − θ • ν + (cid:161) θ • + δ + (cid:162) ν − δ θ • , ... (3.16b)Compraison of these expressions with (2.37)–(2.38) finally leads to a variant of Conjecture B for confluent CBsof type G . Conjecture 3.2.
Let ν be the Bohr-Sommerfeld period along the cycle described above. Then E [BS] ( t | θ , θ • , ν ) = ∂∂ t ˜ U (cid:161) t (cid:175)(cid:175) θ , θ • , ν (cid:162) , (3.17) where ˜ U (cid:161) t (cid:175)(cid:175) θ , θ • , ν (cid:162) denotes the classical confluent conformal block of type G . We conclude by mentioning a few questions that have been left outside the scope of this note.• All-order Bohr-Sommerfeld relation used to define the accessory parameter functions is an analog of theHill determinant evaluation of the Mathieu characteristic exponent. Although E [BS] ( t ) was so far de-fined only as a formal asymptotic series, it can be promoted to an actual function of t by relating theresummed BS period to genuine monodromy/Stokes data with the help of exact WKB techniques, seee.g. [IN, AJJRT]. Classical irregular conformal blocks may then be interpreted as generating functionsof canonical transformations between ( E , t ) and a pair of suitable coordinates on confluent Heun mon-odromy manifolds. 13 It would be interesting to identify conformal blocks whose quasiclassical limit describes accessory pa-rameters of the confluent Heun equations other than H V and H IV . A recent work [Na2] contains a fewpromising candidates for this role for H III and H II (see also [NU]). Even in the case of H V and H IV , thereexist other possibilities to define BS accessory parameter function which lead to ansätze different from(3.9). On the gauge theory side, they correspond to different strongly coupled expansion points. An alge-braic construction of the corresponding CBs of the 2nd kind does not seem to be known.• There is a well-known correspondence between Heun and Painlevé equations under which Heun cou-plings t are mapped to special points of Painlevé functions. It was realized in [NC, CCN, LN] that thisrelation can be used to compute asymptotic expansions of Heun accessory parameters. In the termi-nology of the present paper, [NC, CCN, LN] deal with the weak coupling expansions of Floquet charac-teristics of H VI and H V which can in fact be derived more easily as explained in Appendices A and B.However the method of loc. cit. can be extended to strong coupling using the long-distance Painlevéasymptotic expansions from [BLMST]; this was effectively done for Mathieu equation H III in [GMS]. TheHeun/Painlevé correspondence in the latter case admits an interpretation in terms of the quasiclassicallimit of the Nakajima-Yoshioka blowup equations for pure SU(2) gauge theory [GG], whose generaliza-tions to the regular case were recently studied in [JN, Nek2]. It should be possible to obtain (and perhapseven prove!) similar identities (implicitly present in the approach of [NC, CCN, LN]) relating BS accessoryparameters and Painlevé functions in other cases, including those corresponding to Argyres-Douglas the-ories. We plan to return to these questions in a future work. Acknowledgements . The authors are grateful to Pavlo Gavrylenko and Nikolai Iorgov for illuminating discussions, and toHajime Nagoya for sharing a Mathematica code computing higher order contributions to confluent conformal blocks oftype G . A Heun accessory parameter from continued fractions
It will be convenient for us here to transform the Heun equation (1.1) into its canonical form by the substitution ψ ( z ) = z − θ ( z − t ) − θ t ( z − − θ φ ( z ). (A.1)The resulting equation for φ ( z ) is given by φ (cid:48)(cid:48) ( z ) + (cid:181) γ z + δ z − + (cid:178) z − t (cid:182) φ (cid:48) ( z ) + αβ z − qz ( z − z − t ) φ ( z ) =
0, (A.2)with α = − θ − θ − θ t − θ ∞ , β = − θ − θ − θ t + θ ∞ , (A.3a) γ = − θ , δ = − θ , (cid:178) = − θ t , (A.3b) q = (1 − t ) E + γ(cid:178) + αβ t − (cid:161) γ + δ (cid:162) (cid:178) t α + β + = γ + δ + (cid:178) . In spite of loosing a symmetry θ k (cid:55)→ − θ k ( k = t , ∞ ) presentin the normal form (1.1), the canonical form turns out to be better adapted for perturbative calculation below.A similar and very much related phenomenon is known from the AGT correspondence: 4D instanton partitionfunctions have simpler series reperesentations than 2D conformal blocks but loose certain of their manifestsymmetries because of the presence of the so-called U(1) factor.Choose a basis in the space of solutions of (A.2) which diagonalizes the composite monodromy aroundthe pair of singular points 0, t . Assuming that 0 < | t | <
1, its elements can be represented inside the annulus | t | < | z | < φ ( z ) = (cid:88) n ∈ (cid:90) c n z n + ω . (A.4)Substituting this series into (A.2), one obtains a linear 3-term recurrence relation for the coefficients, A n c n − − B n c n + C n tc n + =
0, (A.5)14here A n = ( ω + n − + α ) (cid:161) ω + n − + β (cid:162) , (A.6a) B n = q + ( ω + n ) (cid:161) (cid:178) + δ t + (cid:161) ω + n − + γ (cid:162) (1 + t ) (cid:162) , (A.6b) C n = ( ω + n + (cid:161) ω + n + γ (cid:162) . (A.6c)The recurrence relation above is the main reason we started with (A.2) instead of (1.1). As we will see in amoment, the crucial point for the computation of the small t expansion of the accessory parameter, q ( t ) = ∞ (cid:88) n = q n t n , (A.7)is how the time t appears in (A.5).For n ≥
0, define u n = c n + c n . This change of variables transforms (A.5) into a nonlinear 2-term “Riccati”equation u n − = A n B n − tC n u n . Under assumption that B n = O (1) as t → n ≥
1, one can then express u as aninfinite continued fraction u = A B − tC A B − tC A B − ... . (A.8)For n ≤
0, we first define rescaled coefficients d − n = c n t n and rewrite the 3-term relation (A.5) as t A − n d n + − B − n d n + C − n d n − =
0, (A.9)Introducing v n = d n + d n , we have v n − = C − n B − n − t A − n v n , so that v = C − B − − t A − C − B − − t A − C − B − − ... , (A.10)under assumption that B n = O (1) as t → n ≤ −
1. From A c − − B c + tC c = t ( C u + A v ) = B , which ultimately gives an equation determining the accessory parameter q as a functionof t for given ω : tC A B − tC A B − tC A B − ... + t A C − B − − t A − C − B − − t A − C − B − − ... = B . (A.11)Note that { A n } and { C n } are just some monodromy-dependent constants; thus q enters into (A.11) only through{ B n }. Substituting into (A.6b) the expansion (A.7), the coefficients q k can be recursively determined from (A.11)by truncating the continued fraction ladder at the desired order in t .For example, at order O (1) we have B = O ( t ), and therefore q = − ω (cid:161) ω + γ + (cid:178) − (cid:162) . (A.12a)At order O ( t ), one has B = t (cid:179) C A B + A C − B − (cid:180) + O (cid:161) t (cid:162) , which gives q = − ω (cid:161) ω + γ + (cid:178) − (cid:162) + ( ω + ω + α ) (cid:161) ω + β (cid:162)(cid:161) ω + γ (cid:162) ω + γ + (cid:178) − ω ( ω + α − (cid:161) ω + β − (cid:162)(cid:161) ω + γ − (cid:162) ω + γ + (cid:178) − E = E [F]VI ( t | σ ): E [F]VI ( t | σ ) = ( δ σ − δ − δ t ) + W ({ δ k }) t + W ({ δ k }) t + O (cid:161) t (cid:162) , (A.13)where W ({ δ k }) are given by (2.6a)–(2.6b), { δ k } are defined by (1.10), { θ k } are related to α , β , γ , δ , (cid:178) by (A.3a)–(A.3b), and the Floquet exponent appears only in σ = ω − θ − θ t + . One thus easily recognizes in (A.13) theexpansion of the logarithmic derivative t ∂∂ t W (cid:161) t (cid:175)(cid:175) { δ k } (cid:162) of the classical regular conformal block (1.8).15 Floquet characteristics for confluent Heun equations
Throughout this section we refer to the notations of Table 1. The perturbative computation of accessory pa-rameter functions of Floquet type presented here allows to check confluent Conjecture B (1.12) at any desiredorder in t . Assuming the conjecture is true, this technique provides the most elementary method of computingclassical CBs of the 1st kind. B.1 Equation H V The change of variables ψ ( z ) = z − θ ( z − t ) − θ t e z φ ( z ) transforms the confluent HE into the canonical form, φ (cid:48)(cid:48) ( z ) + (cid:181) β z + γ z − t + (cid:182) φ (cid:48) ( z ) + α z − qz ( z − t ) φ ( z ) =
0, (B.1)with α = − θ − θ t − θ ∗ , β = − θ , γ = − θ t , (B.2a) q = − E + α t − (cid:161) β + t (cid:162) γ . (B.2b)Looking for the solutions of (B.1) in the Floquet form (A.4), we arrive at the same continued fraction equation(A.11) for q ( t ), except that the coefficients { A n }, { B n }, { C n } are now given by A n = ω + n − + α , (B.3a) B n = q − ( ω + n ) (cid:161) ω + n − + β + γ − t (cid:162) , (B.3b) C n = − ( ω + n + (cid:161) ω + n + β (cid:162) . (B.3c)The expansion of q ( t ) can now be computed to any desired order. Its first terms read q ( t ) = ω (cid:161) ω + β + γ − (cid:162) + (cid:183) − ω + ( ω + ω + α ) (cid:161) ω + β (cid:162) ω + β + γ − ω ( ω + α − (cid:161) ω + β − (cid:162) ω + β + γ − (cid:184) t + O (cid:161) t (cid:162) . (B.4)The expansion of the Floquet characteristic E is then obtained from (B.2b) (note that, in contrast with non-confluent HE, the coefficients of expansions of E and − q coincide starting from the quadratic term). If wedenote σ = ω − θ − θ t + as before, then E [F]V ( t | σ ) = ( δ σ − δ − δ t ) − θ ∗ ( δ σ − δ + δ t )2 δ σ t ++ (cid:34) θ ∗ (cid:161) δ σ − ( δ − δ t ) (cid:162) δ σ − (cid:161) θ ∗ + δ σ (cid:162)(cid:161) δ σ + δ σ ( δ + δ t ) − δ − δ t ) (cid:162) δ σ (3 + δ σ ) (cid:35) t + O (cid:161) t (cid:162) . (B.5)It is instructive to check that this indeed agrees with the limit (1.11a) and reproduces the expansion of t ∂∂ t W N f = ( t ), cf (2.14). B.2 Equation H
III After the transformation ψ ( z ) = (cid:112) z e z + t z φ ( z ), the equation H III becomes φ (cid:48)(cid:48) ( z ) + (cid:181) + z − tz (cid:182) φ (cid:48) ( z ) + (cid:195) t (cid:161) − θ (cid:63) (cid:162) z + E − t − z + − θ ∗ z (cid:33) φ ( z ) =
0. (B.6)This is not the canonical form of doubly confluent Heun equation given in http://dlmf.nist.gov/31.12.E2, whichis obtained by a slightly different change of variables; yet it is more convenient for us to continue with (B.6).The Floquet substitution (A.4) with ω = σ yields the equation (A.11) with A n = θ ∗ − σ − n + , B n = E − t − + ( σ + n ) , C n = θ (cid:63) + σ + n + , (B.7)which then allows to compute the small t expansion of E . Its first few terms are given by E [F]III ( t | σ ) = δ σ + θ ∗ θ (cid:63) δ σ t + (cid:195) (cid:161) θ ∗ + δ σ (cid:162)(cid:161) θ (cid:63) + δ σ (cid:162) δ σ (3 + δ σ ) − θ ∗ θ (cid:63) δ σ (cid:33) t + O (cid:161) t (cid:162) . (B.8)16 .3 Equation H III In this case, the relevant change of variables is ψ ( z ) = (cid:112) z e z φ ( z ). It transforms the equation H III into φ (cid:48)(cid:48) ( z ) + (cid:181) + z (cid:182) φ (cid:48) ( z ) + (cid:195) − tz + E − z + − θ ∗ z (cid:33) φ ( z ) =
0. (B.9)The Floquet substitution (A.4) with ω = σ again gives the equation (A.11), whose coefficients are now given by A n = θ ∗ − σ − n + , B n = E − + ( σ + n ) , C n =
1, (B.10)The expansion of accessory parameter function reads E [F]III ( t | σ ) = δ σ + θ ∗ δ σ t + (5 δ σ − θ ∗ + δ σ δ σ (3 + δ σ ) t + θ ∗ (cid:161) (7 δ σ − δ σ + (cid:161) δ σ − δ σ + (cid:162) θ ∗ (cid:162) δ σ (3 + δ σ )(2 + δ σ ) t + O (cid:161) t (cid:162) . (B.11) B.4 Equation H
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