Higher rank isomonodromic deformations and W-algebras
aa r X i v : . [ m a t h - ph ] J a n Higher rank isomonodromic deformations and W -algebras P. Gavrylenko a,e,f , N. Iorgov a,c,d , O. Lisovyy b a Bogolyubov Institute for Theoretical Physics, 03143 Kyiv, Ukraine b Institut Denis-Poisson, Universit´e de Tours, Universit´e d’Orl´eans, CNRS,Parc de Grandmont, 37200 Tours, France c Kyiv Academic University, 36 Vernadsky blvd., 03142 Kyiv, Ukraine d Max-Planck-Institut f¨ur Mathematik, 53111 Bonn, Germany e Center for Advanced Studies, Skolkovo Institute of Science and Technology,143026 Moscow, Russia f National Research University Higher School of Economics, Department of Mathematics andInternational Laboratory of Representation Theory and Mathematical Physics,119048 Moscow, Russia
Abstract
We construct the general solution of a class of Fuchsian systems of rank N as well as the asso-ciated isomonodromic tau functions in terms of semi-degenerate conformal blocks of W N -algebrawith central charge c = N −
1. The simplest example is given by the tau function of the Fuji-Suzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respectto intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the min-imal set of matrix elements of vertex operators of the W N -algebra for generic central charge andprove several properties of semi-degenerate vertex operators and conformal blocks for c = N − The theory of monodromy preserving deformations has recently gained new insights from its connec-tions to the two-dimensional conformal field theory. The most relevant for the present work is thesolution of the inverse monodromy problem for rank 2 Fuchsian systems and associated isomonodromictau function with Fourier transforms of c = 1 Virasoro conformal blocks [34]. The simplest instanceof this correspondence expresses the tau function of the Painlev´e VI equation in terms of 4-pointconformal blocks [28]; see also [9] for a different proof of this statement.The last relation has been extended to a number of confluent limits including Painlev´e V, IV andIII [29, 47], as well as to the q -difference [10, 39] and non-commutative [8] setting. On the other hand,the case of Fuchsian systems of higher rank N > W N -algebras[30] with integer central charge c = N − W N = W ( sl N ) algebras appeared in [23, 24, 45, 57] as extensions of the Virasoro algebraincluding chiral currents with higher spins. One of the problems appearing in their investigation isthat, for general N , they do not have known explicit and convenient definition in terms of generatorsand relations. An additional complication is that W N -algebras are not Lie algebras for N >
2. Eventhough in the W case it is possible to proceed in the study of the algebra and its representations [email protected] [email protected] [email protected] W N -algebras with N > W N -algebras for N > W -algebras associating them with pairs formed by a simple Lie algebra and a nilpotent elementtherein [41]. The recent progress in the representation theory of W -algebras [2, 20] was also madein this framework. The present paper uses a bosonic realization of W N -algebras coming from thequantum Miura transformation, but we also need some results from [2].A more significant problem is that, in contrast to the Virasoro ( N = 2) case, the local W -invariancedoes not fix vertex operators uniquely: their descendant matrix elements cannot be reduced to theprimary one. This produces an infinite number of free parameters (even in the 3-point conformalblocks!) which have to be determined or constrained by other means such as crossing symmetry. For N = 3, a characterization of the minimal set of independent matrix elements was given in [15]. Onemay also adopt the point of view that such matrix elements can be fixed arbitrarily. However, theresulting vertex operators may be plagued by divergencies in the multi-point conformal blocks. Eventheir compositions with the fundamental degenerate vertex operator may give rise to a divergentseries. A satisfactory definition of the vertex operator should produce a convergent expansion forthe appropriate 4-point conformal block with consistent global analytic (monodromy) properties withrespect to the position of the degenerate field.In the case c = N −
1, where the CFT/isomonodromy correspondence is expected to hold true, adefinition of the general vertex operator for the W N -algebra was proposed in [30] by employing theisomonodromic tau functions and the corresponding 3-point Fuchsian systems. The elements of thebasis of vertex operators are labeled in this approach by a finite number of moduli parameterizing themonodromy data [32, 54, 31, 18]. For generic central charge, analogous definition is not available sofar, but it is expected to be consistent with an action of the algebra of Verlinde loop operators on thespace of 3-point conformal blocks, see recent work [19].There is a special class of vertex operators of W N -algebras relevant to the construction in thispaper. It consists of the so-called semi-degenerate vertex operators corresponding to 3-point conformalblocks with one field having highest weight labeled by a h , where a ∈ C and h is the highest weightof the first fundamental representation of sl N . It is expected that arbitrary 3-point conformal blocksinvolving one semi-degenerate field (primary or its descendant) are uniquely determined by the primaryones as in the Virasoro case (for N = 3, this statement [42] can be deduced from the results of [15]quoted above). The multi-point conformal blocks of W N -algebras which appear in the AGT-typerelation [1, 56, 22, 46] to Nekrasov instanton partition functions [49] are precisely those obtained bycompositions of semi-degenerate vertex operators.On the differential equations side, the tau function of a Fuchsian system with n regular singularpoints can be written [31, 17] as a Fredholm determinant whose integral kernel is expressed in terms offundamental solutions of 3-point Fuchsian systems which arise upon decomposition of the n -puncturedRiemann sphere into pairs of pants. While for N = 2 these auxiliary systems can be explicitly solvedin terms of Gauss hypergeometric functions, the construction of 3-point solutions (series or integralrepresentations, connection formulas, etc) for N ≥ N − , N F N − . For this reason, the constructions of [31, 17] can be madecompletely explicit for Fuchsian systems with 2 generic singular points and the remaining n − semi-degenerate ; they are Vertex operator corresponding to the highest weight vector of an irreducible representation of W N -algebras associatedto one of the N -dimensional fundamental representations of sl N . N = 2 ones. The specialization of the Schlesinger isomonodromicdeformation equations to semi-degenerate 4-point Fuchsian system (i.e. higher rank semi-degenerateanalog of Painlev´e VI) is known as the Fuji-Suzuki-Tsuda system. It was discovered in [27, 51] as asimilarity reduction of Drinfeld-Sokolov hierarchies, and related to deformations of Fuchsian equationsin [55], where semi-degenerate Fuchsian systems with arbitrary n ≥ W N -algebras with c = N −
1. In particular, we claim that, under suitablenormalization of vertex operators, τ ( z ) = X w ,..., w n − ∈ R e ( β , w )+ ... +( β n − , w n − ) . . . ∞ − θ n − θ σ n − + w n − . . . σ + w z n − z n − z z a n − a n − a a (1.1)where τ ( z ) is the isomonodromic tau function depending on the positions z = { z , . . . , z n − } of thespecial punctures, the generic punctures are located at z = 0 and z n − = ∞ , and the trivalentgraph on the right denotes appropriate n -point conformal block. The parameters θ , − θ n − ∈ C N , a , . . . , a n − ∈ C assigned to external edges describe local monodromy exponents of the Fuchsiansystem. The labels σ , . . . , σ n − ∈ C N of the internal edges as well as Fourier momenta β , . . . , β n − ∈ C N are explicitly related to the remaining moduli of semi-degenerate monodromy; the components ofall vectors in C N sum up to zero. The summation in (1.1) is carried over the root lattice R of sl N .Analogous statement for the fundamental solution involves extra degenerate insertions.In the way of establishing the correspondence, we proved several statements about conformal blocksof the W N -algebras which, to the best of our knowledge, remained so far at the level of folklore for N >
3. For arbitrary central charge, we developed a reduction procedure of matrix elements of vertexoperators to a minimal set, thereby generalizing the results of [15]. For c = N −
1, we proved thatthe descendant 3-point functions involving semi-degenerate field are uniquely expressed in terms ofthe 3-point function of primaries. For the fundamental degenerate field, we found restrictions (fusionrules) to be satisfied to allow for non-vanishing 3-point functions. They are then used to prove thewell-known hypergeometric formulas [21] for the 4-point conformal blocks with one semi-degenerateand one degenerate field using the rigidity property of the associated Fuchsian system.We end this introduction by mentioning a few more relevant papers. A survey of recent results inthe representation theory of W -algebras may be found in [3]. The properties of semi-degenerate W N -conformal blocks describing correlation functions of the Toda CFT and their gauge theory counterpartshave been studied, for instance, in [33, 16]. An interesting direction which is in a sense close to oursis the construction of integral representation of the 4-point conformal blocks of W -algebra involvingone semi-degenerate field of higher level (whose matrix elements cannot be reduced to primary onesonly) and one fundamental degenerate field [4]. This construction is based on the middle convolutionfrom the Katz theory of rigid systems. Connections between Fuchsian systems and W -algebras werealso studied in [5], with further links to topological recursion suggested in [6].The paper is organized as follows. In Section 2, we introduce semi-degenerate Fuchsian systems andprovide an explicit parameterization of their monodromy by suitable coordinates (Proposition 2.4).Section 3 is devoted to W N -algebras and their representations. The minimal set of matrix elementsis described by Theorem 3.1, after which we proceed to the proof of uniqueness of semi-degeneratevertex operator (Proposition 3.2) and fusion rules for completely degenerate fields. Section 4 computesthe operator-valued monodromy of semi-degenerate conformal blocks with respect to positions ofadditional degenerate fields. Diagonalizing this monodromy by Fourier transform, in Section 5 weobtain the fundamental solution (Theorems 5.1 and 5.2) and the tau function (Proposition 5.3) of the3emi-degenerate Fuchsian systems in terms of W N -algebra conformal blocks. Some technical resultsare relegated to appendices. Acknowledgements . We would like to thank M. Bershtein, A. Marshakov, R. Santachiara and G. Wattsfor useful discussions and comments. The present work was supported by the CNRS/PICS project “Isomon-odromic deformations and conformal field theory”. The work of P.G. was partially supported the RussianAcademic Excellence Project ‘5-100’ and by the RSF grant No. 16-11-10160. In particular, odd-numbered for-mulas of Section 3 have been obtained using support of Russian Science Foundation. P.G. is a Young RussianMathematics award winner and would like to thank its sponsors and jury. N.I. thanks Max Planck Institute forMathematics (Bonn), where a part of this research was done, for hospitality and excellent working conditions.
We are interested in the analysis of Fuchsian systems of rank N having n regular singular points z := { z , . . . , z n − , z n − ≡ ∞} on the Riemann sphere CP : d Φ ( y ) dy = Φ ( y ) A ( y ) , A ( y ) = n − X k =0 A k y − z k . (2.1)Here the residues A , . . . , A n − ∈ Mat N × N ( C ) and Φ ( y ) is the fundamental N × N matrix solutionwhich may be normalized as Φ ( y ) = I , with y ∈ CP \ z . It will be assumed that the matrices A , . . . , A n − and A n − := − P n − k =0 A k are diagonalizable and non-resonant, i.e. the pairwise differ-ences of the eigenvalues of each A k are not non-zero integers.The solution Φ ( y ) is a multivalued function on CP \ z . Its monodromy realizes a representation ofthe fundamental group π (cid:0) CP \ z , y (cid:1) in GL ( N, C ). This group is generated by the paths ξ , . . . , ξ n − around the points z , . . . , z n − on CP indicated in Fig. 1, which satisfy one relation ξ · · · ξ n − = 1.In what follows, their orientations will be referred to as positive. Denoting by M k the monodromy ofΦ ( y ) along the loop ξ k , we similarly have M · · · M n − = I . z z n − . . . y ξ n − ξ ξ n − Figure 1: Basis of loops ξ , . . . , ξ n − in π (cid:0) CP \ z , y (cid:1) .The fundamental matrix Φ ( y ) is uniquely fixed by the following properties:(a) Φ ( y ) is holomorphic and invertible on the universal cover of CP \ z and has constant monodromyunder analytic continuation.(b) Φ ( y ) satisfies the normalization condition Φ ( y ) = I .4c) In sufficiently small neighborhoods of z k , k = 0 , . . . , n −
1, the behavior of Φ ( y ) isΦ ( y → z k ) = C k ( z k − y ) Θ k G k ( y ) , (2.2)where G k ( y ) is holomorphic and invertible in the vicinity z k ; C k is a non-degenerate constantmatrix; Θ k is a diagonal matrix conjugate to A k . (The asymptotics at z n − = ∞ should berewritten in terms of a suitable local parameter). Note that M k = C k e πi Θ k C − k . Definition 2.1.
The Riemann-Hilbert problem associated with the Fuchsian system (2.1) is the prob-lem of reconstruction of
Φ ( y ) satisfying the conditions (a)–(c) for a given monodromy data: M k ∈ GL ( N, C ) , k = 0 , . . . , n − , subject to the relation M · · · M n − = I . Instead of normalizing the fundamental solution Φ( y ) by the condition (b), we could also useanother normalization which combines (b) and (c): namely, one may fix the connection matrix C l = I at one of the singular points.Different choices of the normalization point y lead to an overall conjugation of all monodromies.We identify the corresponding monodromy data and consider the space M = Hom (cid:0) π (cid:0) CP \ z , y (cid:1) , GL ( N, C ) (cid:1) /GL ( N, C ) . (2.3)It is often convenient to work with the slice M Θ ⊂ M corresponding to fixed local monodromyexponents Θ = { Θ , . . . , Θ n − } .Besides Φ ( y ), we will be also interested in the isomonodromic tau-function τ ( z ) of Jimbo-Miwa-Ueno [38]. It is defined by the following 1-form d z log τ ( z ) := 12 n − X k =0 res y = z k Tr A ( y ) dz k , (2.4)which is closed provided the monodromy of (2.1) is kept constant. The tau function τ ( z ) ≡ τ ( z | Θ , m )with m ∈ M Θ is a generating function of the Hamiltonians which govern the isomonodromic evolutionof A , . . . , A n − with respect to times z . Let Y be the set of partitions λ = [ λ , . . . λ ℓ ], λ ≥ . . . ≥ λ ℓ >
0, and Y k be the set of all partitionsof k ∈ Z ≥ . One can decompose the space of Fuchsian systems (2.1) according to their spectral type s = ( s (0) , . . . , s ( n − ) ∈ Y nN , where the partition s ( i ) ⊢ N encodes the multiplicities of the eigenvaluesof Θ i or A i . Thus, for example, ℓ ( s ( i ) ) is the number of distinct eigenvalues of Θ i and s ( i )1 is themultiplicity of its most degenerate eigenvalue.The dimension of the space M Θ of monodromy data for irreducible systems of spectral type s coincides with the number of accessory parameters, and is known to be given bydim M Θ = ( n − N + 2 − n − X i =0 ℓ i X j =1 (cid:16) s ( i ) j (cid:17) . (2.5)Generic Fuchsian systems have spectral type s gen = (cid:0)(cid:0) N (cid:1) , . . . , (cid:0) N (cid:1)(cid:1) . It then follows from the lastformula that dim M Θ , gen = 2 ( n −
3) ( N −
1) + ( n −
2) ( N −
1) ( N − . (2.6)This expression has a geometric interpretation. The n -punctured Riemann sphere can be decomposedinto n − n − N −
1) monodromy parameters which play the role of Fenchel-Nielsen-type coordinates(lengths and twists) and give the 1st term in (2.6). The 2nd term comes from ( N −
1) ( N − N ≥ (cid:0) N (cid:1) -punctures at z and z n − , and n − N − ,
1) at z , . . . , z n − . The systems of this type will be called semi-degenerate . The dimension of the relevant space of monodromy data is readily computed to bedim M Θ , s − d = 2 ( n −
3) ( N − . (2.7)For n = 3, this dimension vanishes, meaning that the Fuchsian system with 2 generic puncturesand one puncture of type ( N − ,
1) is rigid. The conjugacy class of monodromy is then completelydetermined by the local exponents Θ, i.e. the pants carry no internal moduli. For n ≥
4, thereexist decompositions of the n -punctured sphere into such semi-degenerate pants, which explains thedifference between (2.6) and (2.7).Our next task is to provide an explicit parameterization of semi-degenerate monodromy. The con-struction of solution of the corresponding Riemann-Hilbert problem and the associated isomonodromictau function constitutes the main goal of the present work. Assumption 2.2.
The monodromy matrices M k ∈ SL ( N, C ) , k = 0 , . . . , n − satisfying the cycliccondition M · · · M n − = I are assumed to be diagonalizable, i.e. M k ∼ exp(2 πi Θ k ) , where Θ k =diag θ k with θ k = ( θ (1) k , . . . , θ ( N ) k ) ∈ C N are traceless diagonal matrices. For k = 1 , . . . , n − , thesematrices are fixed to be θ k = a k (cid:0) N − N , − N , . . . , − N (cid:1) , a k ∈ C . (2.8) It is further assumed that the products M [ k ] := M · · · M k with k = 0 , . . . , n − are also diagonalizableand their eigenvalues Spec M [ k ] are pairwise distinct: M [ k ] ∼ exp (2 πi S k ) , S k = diag σ k , σ k = ( σ (1) k , . . . , σ ( N ) k ) , (2.9) where Tr S k = 0 . Note that M [0] = M , M [ n − = M − n − , so that we can identify S = Θ , S n − = − Θ n − , σ = θ , σ n − = − θ n − . For n = 3, the semi-degenerate monodromy is described by the following result, see e.g. [11]. Lemma 2.3 (Rigidity Lemma) . If M A , M B ∈ GL ( N, C ) are diagonalizable with non-intersecting setsof eigenvalues Spec M A = { α , . . . , α N } , Spec M B = { β , . . . , β N } and M − B M A is a reflection (a rank1 perturbation of the identity matrix) then there exists a unique (up to overall rescaling) basis in which M A = . . . ( − N +1 e N ( A )1 0 0 . . . ( − N e N − ( A )0 1 0 . . . ( − N − e N − ( A ) · · · . . . · . . . e ( A ) , M B = . . . ( − N +1 e N ( B )1 0 0 . . . ( − N e N − ( B )0 1 0 . . . ( − N − e N − ( B ) · · · . . . · . . . e ( B ) , where e k ( A ) and e k ( B ) denote the k -th elementary symmetric polynomials in the eigenvalues of M A and M B , respectively. In this case, Spec M − B M A = (cid:8)Q Nk =1 α k β − k , , . . . , (cid:9) . The matrices W A and W B defined by ( W A ) kl = α l − k and ( W B ) kl = β l − k diagonalize, respectively, M A and M B : W A M A W − A = D A , W B M B W − B = D B , D A = diag ( α , . . . , α N ) and D B = diag ( β , . . . , β N ). The matrix W B W − A relates the eigenvec-tors of M A and M B . Its matrix elements are( W B W − A ) kl = Y s ( = l ) β k − α s α l − α s . (2.10)Note that the general form of a matrix which relates a basis where M A is diagonal to another basiswhere M B is diagonal is given by R B W B W − A R − A , where R A and R B are non-degenerate diagonalmatrices.Now one may use Lemma 2.3 to parameterize recursively the monodromy matrices M k of semi-degenerate Fuchsian systems. To this end observe that it suffices to parameterize instead a related setof matrices M [ k ] = M · · · M k with k = 0 , . . . , n −
2. Indeed, we have M = M [0] , M n − = M − n − and M k = M − k − M [ k ] for k = 1 , . . . , n − Proposition 2.4.
Let M k ∈ SL ( N, C ) , k = 0 , . . . , n − be the monodromy matrices of a semi-degenerate Fuchsian system satisfying genericity conditions of Assumption 2.2. They can be parame-terized as follows: M [ k ] = W − k ] exp (2 πi S k ) W [ k ] , (2.11) W [ k ] = R k W k +1 R k +1 · · · R n − W n − R n − , (2.12) where R k = diag ( r k ) are diagonal matrices from SL ( N, C ) and ( W m ) kl = Y s ( = l ) e πi ( σ ( k ) m − − a m /N ) − e πiσ ( s ) m e πiσ ( l ) m − e πiσ ( s ) m . (2.13) Proof.
The idea is to use Lemma 2.3 successively for the pairs of matrices M A = e πia k /N M [ k ] , M B = M [ k − , k = n − , . . . , , where the factor e πia k /N ensures that the eigenvalue of M − B M A = e πia k /N M k with multiplicity N − M A = e πia n − /N M [ n − , M B = M [ n − , assuming that M A is diagonal: M A = e πia n − /N exp (2 πi S n − ), where S n − = − Θ n − . Then M B = R − n − W − n − R − n − exp (2 πi S n − ) R n − W n − R n − , where, as follows from (2.10), W n − is given by (2.13) and R k = diag ( r k ) are arbitrary diagonalmatrices from SL ( N, C ). Continuing the recursive procedure, we get the parameterization (2.11) forall M [ k ] . The matrix W [ k ] defined by (2.12) relates the bases which diagonalize M [ n − = M − n − and M [ k ] .Observe that the diagonal matrix R k cancels out in (2.11); however, we keep it for later use.Since R n − only produces an overall conjugation of all monodromies, it does not enter into the pa-rameterization of M Θ , s − d . Thus the semi-degenerate monodromy is parameterized by σ k , r k with k = 1 , . . . , n −
3, which of course agrees with the dimension (2.7).7 .3 Three-point case
This subsection gives an explicit solution of the semi-degenerate Fuchsian system (2.1) with 3 singularpoints z = 0, z = 1 z = ∞ , and the connection A ( y ) having traceless residues A , A , A ∞ = − A − A at these poles. We suppose that A , A , A ∞ are diagonalizable to Θ , Θ , Θ ∞ . Moreover,it is convenient to choose the gauge so that A ∞ is diagonal. Thus A = G − Θ G , A = G − Θ G , A ∞ = Θ ∞ , (2.14)where G , G ∈ GL ( N, C ) andΘ = diag (cid:16) θ (1)0 , . . . , θ ( N )0 (cid:17) , Θ ∞ = diag (cid:16) θ (1) ∞ , . . . , θ ( N ) ∞ (cid:17) , Tr Θ = Tr Θ ∞ = 0 , Θ = a · diag (cid:0) N − N , − N , . . . , − N (cid:1) . (2.15)Recall that while Θ has an eigenvalue of multiplicity N −
1, all eigenvalues of Θ and Θ ∞ are distinct.Such data correspond to a rigid local system and the matrix elements of A can be derived from anadditive variant of Lemma 2.3:( A ) jm = − r j r m · Q k ( θ ( j ) ∞ − a/N + θ ( k )0 ) Q k ( = m ) ( θ ( m ) ∞ − θ ( k ) ∞ ) − δ jm aN , (2.16)where r , . . . , r N are arbitrary non-zero parameters. They appear due to the possibility of overallconjugation of A , A , A ∞ by the diagonal matrix R = diag ( r , . . . , r N ), preserving the diagonal formof A ∞ . Theorem 2.5.
The solution of the Fuchsian system d Φ ( y ) dy = Φ ( y ) A ( y ) , A ( y ) = A y + A y − , (2.17) with A fixed by (2.16) and A = − A − Θ ∞ , which has the asymptotics Φ ( y ) = y − Θ ∞ (cid:0) I + O (cid:0) y − (cid:1)(cid:1) as y → ∞ is Φ jm ( y ) = N jm y − θ ( j ) ∞ − δ jm (cid:16) − y (cid:17) − a/N ×× N F N − (cid:8) − δ jm − a/N + θ ( k )0 + θ ( j ) ∞ (cid:9) k =1 ,N (cid:8) θ ( j ) ∞ − θ ( k ) ∞ + δ mk − δ jm (cid:9) k =1 ,N ; k = j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y , j, m = 1 , . . . , N, (2.18) where N jm = ( A ) jm θ ( m ) ∞ − θ ( j ) ∞ − , j = m, , j = m, and N F N − ( . . . | x ) denotes the generalized hypergeometric function. Let us comment on the computation of the coefficients N jm . They can be derived from theexpansion of Φ ( y ) near infinity. Indeed, as y → ∞ , one has A ( y ) = − A ∞ y + A y + O (cid:0) y − (cid:1) . The solution Φ( y ) can be found iteratively using the expansionΦ ( y ) = y − A ∞ (cid:0) I + H y − + H y − + O (cid:0) y − (cid:1)(cid:1) . (2.19)8ubstituting this expression into the Fuchsian system yields H = [ H , A ∞ ] − A . Under the non-resonance condition for A ∞ , it follows that( H ) jm = ( A ) jm θ ( m ) ∞ − θ ( j ) ∞ − . Comparing (2.18) and (2.19) as y → ∞ , one finds that the coefficients of the off-diagonal leadingterms and diagonal next-to-leading terms are given by the matrix elements of H . In particular, N jm = ( H ) jm for j = m . W N -algebras and their representations It is important for us that W N -algebras with central charge c = N − b sl N at level 1. Since this value of c is exactly what we needfor applications to isomonodromy, the bosonic realization of W N -algebras will be used as their basicdefinition. Thanks to the boson-fermion correspondence, in this case there also exists a fermionicrealization of W N -algebras. W N -algebras with c = N − We are going to define the W N -algebras, W N = W ( sl N ), with c = N −
1, as abstract operator algebrasstarting from their realizations in terms of N free bosonic fields φ k ( z ), k = 1 , . . . , N , subject to onerelation P Nk =1 φ k ( z ) = 0. The currents J k ( z ) = i∂φ k ( z ) have the operator product expansions (OPE)of the form J k ( z ) J l ( z ′ ) = δ kl − N ( z − z ′ ) + regular . (3.1)The modes a ( k ) p of the currents J k ( z ) are defined by J k ( z ) = X p ∈ Z a ( k ) p z p +1 . (3.2)These currents define W N -algebra currents W (2) ( z ) , . . . , W ( N ) ( z ) as sums of normal-ordered mono-mials: W ( j ) ( z ) = X ≤ i < ···
1. The modes W ( j ) p of the currents W ( j ) ( z )are defined by W ( j ) ( z ) = X p ∈ Z W ( j ) p z p + j .
9e will not need the explicit form of all OPEs of the currents W ( j ) ( z ) and the explicit formulasfor the commutation relations of their modes W ( j ) p . However, it will be important for us that thecommutation relations respect two structures on the W N -algebra: Z -gradation with respect to theadjoint action of L = − W (2)0 deg L W ( j ) p = − p, (3.4)and quasi-commutativity with respect to the W -filtration [2] defined bydeg W W ( j ) p = j − . (3.5)Namely, we will need the relationsdeg L [ W ( j ) p , W ( j ) p ] = − ( p + p ) , (3.6)deg W [ W ( j ) p , W ( j ) p ] < ( j −
1) + ( j − . (3.7)The latter inequality means that the modes of W N -algebra currents commute up to elements of smallerdegree with respect to deg W (quasi-commutativity which is commutativity of the corresponding gradedalgebra). Given θ = ( θ , . . . , θ N ) ∈ C N with P Nk =1 θ k = 0, one may introduce the exponential vertex operator V θ ( z ) = : e i ( θ ,φ ( z )) :It has the following OPEs with the currents J k ( z ): J k ( z ) V θ ( z ′ ) = θ k V θ ( z ′ ) z − z ′ + regular , which in turn imply that W ( j ) ( z ) V θ ( z ′ ) = ∞ X k =0 (cid:0) W ( j ) − k V θ (cid:1) ( z ′ )( z − z ′ ) j − k , (3.8)where (cid:0) W ( j )0 V θ (cid:1) ( z ) = e j ( θ ) V θ ( z ) , (3.9)and e j ( θ ) is the j -th elementary symmetric polynomial in the variables θ . The relations (3.8), (3.9)define the primary field V θ ( z ) of the W N -algebra. Due to the state-field correspondence, to every suchprimary field we can associate the highest weight vector | θ i of a Verma module M θ of the W N -algebra.The Verma module M θ is induced from the one-dimensional module with the basis element | θ i ofthe subalgebra of W N -algebra generated by { W ( j ) k ≥ } : W ( j )0 | θ i = e j ( θ ) | θ i , W ( j ) k> | θ i = 0 . (3.10)The W N -algebra admits a Poincar´e-Birkhoff-Witt basis [2]. This means that there is a linear basisin the Verma module M θ consisting of the elements W λ | θ i , where λ = (cid:0) λ (2) , . . . , λ ( N ) (cid:1) ∈ Y N − is an( N − W λ | θ i = W ( N ) − λ ( N ) · · · W (2) − λ (2) | θ i , W ( s ) − λ ( s ) := W ( s ) − λ ( s )1 · · · W ( s ) − λ ( s ) ℓs . (3.11)10e are interested in the matrix elements of descendants (3-point functions) of the vertex operator V θ ( z ): (cid:10) θ ′′ (cid:12)(cid:12) W † λ ′′ ( W λ V θ ) ( z ) W λ ′ (cid:12)(cid:12) θ ′ (cid:11) , λ , λ ′ , λ ′′ ∈ Y N − , (3.12)where † is an anti-linear involutive anti-automorphism of the W N -algebra uniquely defined by (cid:0) W ( j ) k (cid:1) † = W ( j ) − k , and h θ | satisfies the conditions analogous to (3.10): h θ | W ( j )0 = h θ | e j ( θ ) , h θ | W ( j ) k< = 0 . (3.13)It is well-known that in the case of the Virasoro algebra (i.e. for N = 2) thanks to the Wardidentities all matrix elements (3.12) can be expressed in terms of the matrix element of V θ ( z ) betweenthe highest weight vectors, (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) (cid:12)(cid:12) θ ′ (cid:11) (3-point function of primaries). In order to simplify thesematrix elements as much as possible in the N ≥ W ( j ) p , V ( z )] = ∞ X k =1 − j z p − k (cid:18) p + j − k + j − (cid:19) (cid:16) W ( j ) k V (cid:17) ( z ) , (3.14) (cid:16) W ( j ) p V (cid:17) ( z ) = ∞ X k =0 ( − z ) k (cid:18) j + p − k (cid:19) W ( j ) p − k V ( z ) − ∞ X k =0 ( − z ) p + j − k − (cid:18) j + p − k (cid:19) V ( z ) W ( j )1 − j + k , (3.15)valid for any descendant V ( z ) of the primary vertex operator V θ ( z ) and any p ∈ Z (and, in fact, forany central charge c ).The following theorem naturally generalizes the corresponding result for the W -algebra [15, 42]and is valid for any c . Theorem 3.1.
General matrix elements of the vertex operators of the W N -algebra can be reduced tothe following linear combinations: (cid:10) θ ′′ (cid:12)(cid:12) W † λ ′′ ( W λ V θ ) ( z ) W λ ′ (cid:12)(cid:12) θ ′ (cid:11) = X µ A µ ( z ) (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) W ( N ) − µ ( N ) · · · W (3) − µ (3) (cid:12)(cid:12) θ ′ (cid:11) , (3.16) where the coefficients A µ ( z ) are labeled by µ = ( µ (3) , . . . , µ ( N ) ) ∈ Y N − , and the corresponding parti-tions are restricted so that µ ( j )1 ≤ j − for j = 3 , . . . , N .Proof. First let us move all W ( j ) p with p > W † λ ′′ .At the next step, use the identities (3.15) to reduce the matrix elements of descendant operators V ( z ) = (cid:0) W ˜ λ V θ (cid:1) ( z ) to those of primary vertex operator V θ ( z ). Note that for p < (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) W λ (cid:12)(cid:12) θ ′ (cid:11) , (3.17)with so far unrestricted vectors (3.11).Finally, let us change the basis (3.11) in the Verma module M θ . We will use new generators of the W N -algebra (see [15] for the W case): w ( j ) p ( z ) = j X k =0 ( − z ) k (cid:18) jk (cid:19) W ( j ) p − k , ˜ w ( j )0 ( z ) = j X k =1 ( − z ) k (cid:18) j − k − (cid:19) W ( j ) − k . p ∈ Z )[ V θ ( z ) , w ( j ) p ( z )] = 0 , [ V θ ( z ) , ˜ w ( j )0 ( z )] = ( W ( j )0 V θ ) ( z ) , which imply the following action formulas: (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) w ( j ) p ( z ) = 0 , p < , (3.18) (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) w ( j )0 ( z ) = e j ( θ ′′ ) (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) , (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) ˜ w ( j )0 ( z ) = e j ( θ ) (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) . (3.19)The new PBW basis in the Verma module is labeled by ( λ , µ , k , ˜ k ), where λ = (cid:16) λ (2) , . . . , λ ( N ) (cid:17) ∈ Y N − , k = ( k , . . . , k N ) ∈ Z N − ≥ , ˜ k = (cid:16) ˜ k , . . . , ˜ k N (cid:17) ∈ Z N − ≥ , µ = (cid:16) µ (3) , . . . , µ ( N ) (cid:17) ∈ Y N − , with µ ( j )1 ≤ j − j = 3 , . . . , N, and is given by the vectors W ( λ , µ , k , ˜ k ) | θ i = w ( N ) − λ ( N ) · · · w (2) − λ (2) (cid:0) w ( N )0 (cid:1) k N (cid:0) ˜ w ( N )0 (cid:1) ˜ k N · · · (cid:0) w (2)0 (cid:1) k (cid:0) ˜ w (2)0 (cid:1) ˜ k W ( N ) − µ ( N ) · · · W (3) − µ (3) | θ i . Thanks to (3.18), (3.19), the matrix elements (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) W ( λ , µ , k , ˜ k ) (cid:12)(cid:12) θ ′ (cid:11) can be expressed in terms ofthe matrix elements (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) W ( ∅ , µ , , ) (cid:12)(cid:12) θ ′ (cid:11) = (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) W ( N ) − µ ( N ) · · · W (3) − µ (3) (cid:12)(cid:12) θ ′ (cid:11) (3.20)labeled by tuples of partitions µ ∈ Y N − which satisfy the above restrictions µ ( j )1 ≤ j −
2. Note that µ ( j ) may be equivalently represented by j − N = 4, the minimal set of matrix elements can be chosen as (cid:10) θ ′′ (cid:12)(cid:12) V θ (cid:0) z (cid:1)(cid:0) W (4) − (cid:1) l (cid:0) W (4) − (cid:1) l − l (cid:0) W (3) − (cid:1) l (cid:12)(cid:12) θ ′ (cid:11) , l ≥ l ≥ , l ≥ . For generic weights θ , θ ′ , θ ′′ the matrix elements (3.20) can not be related by means of the Wardidentities. However, if one of these weights is of semi-degenerate type (to be discussed in the nextsubsection) then all these matrix elements can be expressed in terms of (cid:10) θ ′′ (cid:12)(cid:12) V θ ( z ) (cid:12)(cid:12) θ ′ (cid:11) , just as in thecase of the Virasoro algebra. We will need special reducible Verma modules with θ = a h , where a is a complex number and h s , s = 1 , . . . , N , are the weights of the first fundamental representation of sl N with the components h ( k ) s = δ sk − /N, k = 1 , . . . , N. (3.21)The irreducible quotient with the highest weight θ = a h is called semi-degenerate representation.We have N − (cid:20) W ( r ) − − (cid:18) N − r − (cid:19) (cid:16) − aN (cid:17) r − W (2) − (cid:21) | a h i = 0 , r = 3 , . . . , N, L -gradation in the Verma module. All therelations needed for derivation on the p -th level are given by " W ( r ) − p + ( − r + p p +1 X s =2 (cid:18) N − sr − s (cid:19)(cid:18) r − s − p − s + 1 (cid:19) (cid:16) aN (cid:17) r − s W ( s ) − p | a h i = 0 , ≤ p + 1 < r ≤ N, (3.22)and correspond to factoring out different proper submodules in the Verma module. The derivation ofthese relations is given in Appendix A.The following proposition shows how the relations (3.22) can be used for further reduction of thematrix elements appearing in (3.16). Proposition 3.2.
Matrix elements of the semi-degenerate vertex operator V a h ( z ) and its descendantscan be expressed in terms of the primary matrix element (cid:10) θ ′ (cid:12)(cid:12) V a h ( z ) | θ i = N (cid:0) θ ′ , a h , θ (cid:1) z ∆ θ ′ − ∆ a h − ∆ θ , (3.23) where ∆ θ = − e ( θ ) = θ / .Proof. Theorem 3.1 allows us to start the reduction procedure from the matrix elements of the form (cid:10) θ ′ (cid:12)(cid:12) V a h ( z ) W ( j ) − p W | θ i , ≤ p ≤ j − , (3.24)where W is a product of the generators of the W N -algebra. We will reduce such matrix elements to (cid:10) θ ′ (cid:12)(cid:12) V a h ( z ) f W | θ i with f W having deg W f W < j − W W , cf (3.5), (3.7). Since deg W W ( j ) − p = j − W W .The identity (3.14) can be rewritten for V ( z ) = V θ ( z ) and any p ∈ Z as h W ( j ) − p , V θ ( z ) i = z − p j − X k =0 z k (cid:18) j − p − j − k − (cid:19) (cid:16) W ( j ) − k V θ (cid:17) ( z ) . (3.25)This commutation relation allows to transform (3.24) into a linear combination of matrix elements (cid:10) θ ′ (cid:12)(cid:12) (cid:0) W ( j ) − p V a h (cid:1) ( z ) W | θ i (3.26)with 1 ≤ p ≤ j −
1. Moreover, we can exclude (cid:10) θ ′ (cid:12)(cid:12) (cid:0) W ( j ) − ( j − V a h (cid:1) ( z ) W | θ i from this set of matrixelements using the relation (3.25) with p = 0, which produces one more linear combination of matrixelements (3.26). It can be found since (cid:10) θ ′ (cid:12)(cid:12) (cid:2) W ( j )0 , V θ ( z ) (cid:3) W | θ i may be computed independently using(3.10), (3.13) and the fact that W ( j )0 W = W W ( j )0 + f W , deg W f W < deg W ( W ( j )0 W ) , by the induction assumption. Thus the problem is reduced to finding matrix elements (3.26) for1 ≤ p ≤ j − (cid:10) θ ′ (cid:12)(cid:12) V a h ( z ) f W | θ i for f W with deg W f W < j − W W . These elements are known by the inductionassumption.The starting matrix element of the induction procedure is (cid:10) θ ′ (cid:12)(cid:12) V a h ( z ) | θ i . It can be calculated,up to a normalization factor, from the relation (cid:10) θ ′ (cid:12)(cid:12) (cid:0) L − V a h (cid:1) ( z ) | θ i = ∂ z (cid:10) θ ′ (cid:12)(cid:12) V a h ( z ) | θ i , with L p = − W (2) p . This yields (3.23). 13 .4 Degenerate representations We will need even more special irreducible representation with a = 1, i.e. θ = h . It correspondsto the first fundamental representation of sl N . Another important representation has θ = − h N andcorresponds to the last fundamental representation of sl N . Irreducible representations with highestweights θ = h and θ = − h N are called (completely) degenerate representations.One may expect that all the normalization coefficients N (cid:0) θ ′ , a h , θ (cid:1) (structure constants) arenon-zero for generic a , θ , θ ′ . However, in the case of degenerate vertex operator with a = 1 thereare additional restrictions on the possible values of θ ′ to have non-vanishing N (cid:0) θ ′ , a h , θ (cid:1) due toadditional singular vectors in the Verma module M h . Proposition 3.3.
The fusion rule for V h ( z ) is V h ( z ) | θ i = N X s =1 N ( θ + h s , h , θ ) z ∆ θ + h s − ∆ h − ∆ θ (cid:2) | θ + h s i + O ( z ) (cid:3) . (3.27) Similarly, the fusion rule for V − h N ( z ) is V − h N ( z ) | θ i = N X s =1 N ( θ − h s , h , θ ) z ∆ θ − h s − ∆ h − ∆ θ (cid:2) | θ − h s i + O ( z ) (cid:3) . (3.28) Proof.
The derivation of these fusion rules is given in the Appendix B.In what follows, we will use the projector P θ to the irreducible module with the highest weight θ and a special notation for the degenerate vertex operators restricted to a particular fusion channel: ψ s, θ ( y ) = P θ + h s V h ( y ) P θ , ¯ ψ s, θ ( y ) = P θ − h s V − h N ( y ) P θ . (3.29)Sometimes we will use shorthand notations ψ s ( y ) and ¯ ψ s ( y ) if a particular θ is understood. Explicitly,the fusion rules for them are ψ s ( y ) | θ i = N ( θ + h s , h , θ ) y ∆ θ + h s − ∆ h − ∆ θ (cid:2) | θ + h s i + O ( y ) (cid:3) , (3.30)¯ ψ s ( y ) | θ i = N ( θ − h s , h , θ ) y ∆ θ − h s − ∆ h − ∆ θ (cid:2) | θ − h s i + O ( y ) (cid:3) . (3.31)The singular parts of their OPEs become ψ s ( z ) ¯ ψ s ′ ( w ) ∼ δ s,s ′ ( z − w ) ( N − /N , ψ s ( z ) ψ s ′ ( w ) ∼ , ¯ ψ s ( z ) ¯ ψ s ′ ( w ) ∼ , (3.32)provided we choose normalizations so that N ( θ , h , θ + h s ) = N − ( θ + h s , h , θ ). W -algebras and their properties Below we will need special conformal blocks which can be expressed in terms of hypergeometric func-tions, and their properties. In this subsection, the non-vanishing 3-point functions of semi-degeneratevertex operators are normalized by N (cid:0) θ ′ , a h , θ (cid:1) = 1, but later this normalization will be changedto a more convenient one. In what follows, we will use a shorthand notation V a ( z ) := V a h ( z ) for thesemi-degenerate vertex operators. 14 heorem 4.1. The following conformal blocks have hypergeometric expressions: h θ ∞ | V a ( z ) ψ s ( y ) | θ i = z ∆ − ∆ − ∆ − ∆ ( y/z ) ∆ ,s − ∆ − ∆ (1 − y/z ) − a/N G s ( y/z ) , (4.1) h θ ∞ | ψ s ( y ) V a ( z ) | θ i = y ∆ − ∆ − ∆ − ∆ ( z/y ) ∆ ,s − ∆ − ∆ (1 − z/y ) − a/N G ′ s ( z/y ) , (4.2) where, recalling the notation ∆ θ = θ / for conformal weights, we have ∆ = ∆ θ , ∆ = ∆ h = N − N , ∆ = ∆ a h = a · N − N , ∆ = ∆ θ ∞ , ∆ ,s = ∆ θ + h s , ∆ ,s = ∆ θ ∞ − h s , G s ( x ) = N F N − { ( N − a − /N + θ ( s )0 − θ ( k ) ∞ } k =1 ,N { θ ( s )0 − θ ( k )0 } k =1 ,N ; k = s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x , G ′ s ( x ) = N F N − { ( N − a − /N + θ ( k )0 − θ ( s ) ∞ } k =1 ,N { θ ( k ) ∞ − θ ( s ) ∞ } k =1 ,N ; k = s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x . Proof.
The idea of the proof is to use rigidity of the 3-point Fuchsian system (2.17). We claim thatthe solution of the system can be given in terms of 4-point conformal blocks with degenerate fields ψ j ( y ) and a semidegenerate field V a (1) = V a h (1):˜Φ jm ( y ) = K jm h− θ ∞ + h m | ψ j ( y ) V a (1) | θ i = y →∞ K jm y − θ ( j ) ∞ + δ jm − (cid:2) O (cid:0) y − (cid:1)(cid:3) , where we used fusion rules (3.30) to derive the asymptotics as y → ∞ . The constants K jm will befixed later. The leading asymptotics of the matrix elements of ˜Φ ( y ) as y → ∞ can also be rewrittenas ˜Φ jm ( y ) = y − θ ( j ) ∞ (cid:2) δ jm K jj + O (cid:0) y − (cid:1)(cid:3) . Each column m of the matrix ˜Φ ( y ) constitutes a basis in the N -dimensional space V m of conformalblocks with fixed external weights labeled by θ , θ = a h , h , − θ ∞ + h m . This space is invariantunder analytic continuation in y around z = 0, z = 1 and z = ∞ giving the monodromies M ( m )0 , M ( m )1 and M ( m ) ∞ , respectively. The spectra of these monodromy matrices are independent of m and canbe found from the conformal dimensions of fields using (3.30). Namely, all these bases are associatedto different channels corresponding to the fusion of the degenerate field ψ j ( y ) with a generic primaryat z = ∞ .There are two more bases in each of V m associated to the channels corresponding to the fusion ofthe degenerate field ψ j ( y ) with the fields at z = 0 and z = 1, respectively. The leading terms (up toconstant prefactors) of the basis conformal blocks for each V m at z = 0 are y θ = ( y θ (1)0 , . . . , y θ ( N )0 ).Similarly, the leading terms of the basis conformal blocks for each V m at z = 1 are ( y − θ = (cid:0) ( y − θ (1)1 , . . . , ( y − θ ( N )1 (cid:1) . These two bases are distinguished by the property of having diagonalmonodromies exp 2 πi Θ and exp 2 πi Θ under analytic continuation around z and z , respectively.In the initial basis, which is distinguished by the property of having diagonal monodromy around z = ∞ , we have M ( m )0 = C ( m )0 e πi Θ (cid:0) C ( m )0 (cid:1) − , M ( m ) ∞ = e πi Θ ∞ , M ( m )1 = C ( m )1 e πi Θ (cid:0) C ( m )1 (cid:1) − , Since the spectrum of Θ is degenerate of spectral type ( N − , y − k , k ∈ Z > . , Θ and Θ ∞ are given by (2.15).It follows from Lemma 2.3 that the triples (cid:0) M ( m )0 , M ( m )1 , M ( m ) ∞ (cid:1) of monodromy matrices for differ-ent m are related by an overall similarity transformation. Since M ( m ) ∞ are already diagonal for all m and have simple spectrum, the remaining freedom is given by the conjugation by diagonal matrices.Such type of similarity transformations corresponds to choosing the coefficients K jm , j = 1 , . . . , N ,for each m . Fix these coefficients so that all triples (cid:0) M ( m )0 , M ( m )1 , M ( m ) ∞ (cid:1) coincide with the triple (cid:0) M , M , M ∞ (cid:1) of monodromy matrices of the actual fundamental solution Φ( y ) given by (2.18). Thenthe elements of the matrix Φ ( y ) − ˜Φ ( y ) are given by single-valued meromorphic functions with theonly possible poles at 0, 1 or ∞ . However, Φ ( y ) and ˜Φ ( y ) have the same local monodromy exponents,hence this matrix is in fact constant. From the normalization of Φ ( y ) it follows that the constantmatrix is diagonal and may be chosen as the identity matrix, in which case K jj = 1 and Φ ( y ) = ˜Φ ( y ).The diagonal elements of the latter relation give the hypergeometric representation (4.2) with z = 1; arbitrary z may be obtained by a scale transformation of conformal blocks. The proof of (4.1)is completely analogous.Conformal blocks (4.1) are defined as convergent series for | y | < | z | . Their analytic continuationin y to the region with | y | > | z | can be compared to conformal blocks (4.2), given by convergent seriesin the latter domain. The relation between the two sets of conformal blocks may be expressed withthe help of well-known N F N − connection formulas, see e.g. [50]: h θ ∞ | V a ( z ) ψ l ( y.ξ z ) | θ i = N X j =1 e − iπ (( N − /N + θ ( l )0 − θ ( j ) ∞ ) ˜ F [ ∞ , lj ( θ ∞ , a, θ ) h θ ∞ | ψ j ( y ) V a ( z ) | θ i . (4.3)Here y.ξ z stands for the analytic continuation along a contour ξ z going around z in the positivedirection and ˜ F [ ∞ , is the fusion matrix “0 → ∞ ” with the elements˜ F [ ∞ , lj ( θ ∞ , a, θ ) = Q k ( = l ) Γ(1 + θ ( l )0 − θ ( k )0 ) · Q k ( = j ) Γ( θ ( j ) ∞ − θ ( k ) ∞ ) Q k ( = l ) Γ((1 + a ) /N + θ ( j ) ∞ − θ ( k )0 ) · Q k ( = j ) Γ(( N − − a ) /N + θ ( l )0 − θ ( k ) ∞ ) , where Γ ( x ) is the gamma function.In fact, the fusion matrix in (4.3) will not change if instead of the vectors | θ i and | θ ∞ i we useany of their descendants, obtained by the action of the creation operators W ( j ) k , k <
0. This is dueto the fact that all such conformal blocks can be obtained by an action of differential operators in y and z , which commutes (intertwines) with the crossing symmetry transformation (4.3). Combiningthe vertex operators ψ l ( y ), l = 1 , . . . , n , into a column matrix Ψ ( y ), one may therefore rewrite (4.3)as an operator relation P θ ∞ V a ( z ) Ψ ( y.ξ z ) P θ = B − ( θ ) ˜ F [ ∞ , ( θ ∞ , a, θ ) B ′ ( θ ∞ ) · P θ ∞ Ψ ( y ) V a ( z ) P θ , (4.4)where the braiding matrices B and B ′ are diagonal and their non-zero elements are given by B ll ( θ ) = exp iπθ ( l )0 , B ′ jj ( θ ∞ ) = exp iπ (cid:0) θ ( j ) ∞ − N − N (cid:1) . (4.5)Analytic continuation of ψ l ( y ) | θ i in y around 0 in the positive direction leads to multiplicationby the diagonal braiding matrix B ( θ ). Indeed, B ll ( θ ) = exp (cid:8) πi (cid:0) ∆ θ + h l − ∆ θ − ∆ h l (cid:1)(cid:9) = exp 2 πiθ ( l )0 . Taking into account that the braiding matrix is the same for descendants of | θ i , we write the braidingrelation as Ψ (cid:0) ye πi (cid:1) P θ = B ( θ ) · Ψ ( y ) P θ . (4.6)16imilarly, the analytic continuation of h θ ∞ | ψ j ( y ) in y around ∞ in the negative direction leads tomultiplication by the diagonal braiding matrix B ′ ( θ ∞ ), whose non-vanishing elements are B ′ jj ( θ ∞ ) = exp (cid:8) πi (cid:0) ∆ θ ∞ − ∆ θ ∞ − h j − ∆ h j (cid:1)(cid:9) = exp 2 πi (cid:0) θ ( j ) ∞ − N − N (cid:1) . This leads to P θ ∞ Ψ (cid:0) ye πi (cid:1) = B ′ ( θ ∞ ) · P θ ∞ Ψ ( y ) . (4.7)Finally, let us introduce the following formal transformations which are a consequence of thedefinition (3.29): ψ s ( y ) P σ = P σ + h s ψ s ( y ) = ∇ σ ,s P σ ψ s ( y ) , where ∇ σ ,s is the shift operator defined by ∇ σ ,s F ( σ ) = F ( σ + h s ) for any function F depending on σ . We combine the shifts ∇ σ ,s , s = 1 , . . . , N , into the diagonal matrix ∇ σ = diag ( ∇ σ , , . . . , ∇ σ ,N ),to write compactly Ψ ( y ) P σ = ∇ σ P σ Ψ ( y ) . (4.8)Note that there is a useful relation between the two types of braiding, B ( σ ) ∇ σ = ∇ σ B ′ ( σ ) . (4.9) Let us change the normalization of the vertex operators in (3.23): instead of N ( σ ′ , a h , σ ) = 1, wewill use N (cid:0) σ ′ , a h , σ (cid:1) = Q l,j G (1 − a/N + σ ( l ) − σ ′ ( j ) ) Q k The fusion matrix F ( σ ′ , a, σ ) satisfies the following shift transformations: F (cid:0) σ ′ , a, σ ± h m (cid:1) = D N − m F (cid:0) σ ′ , a ± , σ (cid:1) ,F (cid:0) σ ′ ± h m , a, σ (cid:1) = F (cid:0) σ ′ , a ∓ , σ (cid:1) D N − m , (4.17) where D m is a diagonal matrix with matrix elements ( D m ) kj = ( − δ km δ kj and the weight vectors h m are given by (3.21). Compositions of the transformations (4.17) give the following identities: F (cid:0) σ ′ , a, σ + h m − h s (cid:1) = D N − m D N − s F (cid:0) σ ′ , a, σ (cid:1) ,F (cid:0) σ ′ + h m − h s , a, σ (cid:1) = F (cid:0) σ ′ , a, σ (cid:1) D N − m D N − s . (4.18)The main result of the paper is an explicit solution of the Riemann-Hilbert problem for semi-degenerate Fuchsian systems in terms of W N -conformal blocks. We will find the monodromy matricesof the proposed solution and compare them with the monodromy matrices (2.11) parameterized byProposition 2.4 with the help of Lemma 2.3. For such comparison, the following proposition will beuseful. Proposition 4.3. The matrix F ( σ ′ , a, σ ) given by (4.14) and the matrices W k given by (2.13) arerelated by F ( σ k , a k − , σ k − ) = X k W k Y − k , (4.19) where the diagonal matrices X k and Y k are defined by their diagonal elements x ( s ) k and y ( s ) k , s =1 , . . . , N , respectively: (cid:0) x ( s ) k (cid:1) − = e iπ ( N − σ ( s ) k − N Y m =1 sin π (ˆ σ ( m ) k − σ ( s ) k − ) Y m ( = s ) sin π ( σ ( m ) k − − σ ( s ) k − ) , (4.20) (cid:0) y ( s ) k (cid:1) − = e iπ ( N − σ ( s ) k N Y m =1 sin π (ˆ σ ( s ) k − σ ( m ) k − ) Y m ( = s ) sin π (ˆ σ ( m ) k − ˆ σ ( s ) k ) , (4.21) and we use the notation ˆ σ ( s ) k = σ ( s ) k + a k /N . This subsection is devoted to the computation of monodromy of multipoint conformal blocks usingthe transformation properties (4.16), (4.6)–(4.9).Consider the following column F m ( σ | y ), σ = ( σ , . . . , σ n − ) of conformal blocks: F m ( σ | y ) = C m h− θ n − + h m | Ψ ( y ) V a n − ( z n − ) P σ n − V a n − ( z n − ) P σ n − · · · P σ V a ( z ) | θ i , C m is the diagonal matrix with matrix elements ( C m ) jj = ( − Nδ jm . The advantage of incorpo-rating C m into the definition of F m ( σ | y ) is that its monodromy becomes independent of m . Denoteby F m (cid:0) σ | y.ξ [ k ] (cid:1) the analytic continuation of F m ( σ | y ) in y along the path ξ [ k ] := ξ · · · ξ k encircling z , . . . , z k . We have F m (cid:0) σ | y.ξ [ k ] (cid:1) = ˆ M [ k ] F m ( σ | y ) , where the operator-valued monodromy matrix ˆ M [ k ] isˆ M [ k ] = ˆ V ′ [ k ] B ( σ k )( ˆ V ′ [ k ] ) − , (4.22)ˆ V ′ [ k ] = C m T n − ∇ σ n − T n − ∇ σ n − T n − · · · ∇ σ k +1 T k +1 ,T l = B ′ ( σ l ) F − ( σ l , a l , σ l − ) B − ( σ l − ) , and σ n − := − θ n − + h m . Although it may seem that ˆ V ′ [ k ] depends on m , thanks to the identity C m B ′ ( − θ n − + h m ) F − ( − θ n − + h m , a n − , σ n − ) == − B ( − θ n − ) F − ( − θ n − , a n − − , σ n − ) , (4.23)this dependence actually disappears. One may also use the identity B − ( σ l ) ∇ σ l B ′ ( σ l ) = ∇ σ l (cf(4.9)) to simplify the expression (4.22) for the operator-valued monodromy matrix ˆ M [ k ] :ˆ M [ k ] = ˆ V [ k ] B ( σ k ) ˆ V − k ] , (4.24)ˆ V [ k ] = B ( − θ n − ) F − ( − θ n − , a n − − , σ n − ) ∇ σ n − F − ( σ n − , a n − , σ n − ) ××∇ σ n − F − ( σ n − , a n − , σ n − ) ∇ σ n − · · · ∇ σ k +1 F − ( σ k +1 , a k +1 , σ k ) . Recall that there is a simple formula (4.15) for F − .The operator-valued monodromy matrix ˆ M [ k ] has periodic dependence on σ = ( σ , . . . , σ n − ).Namely, ˆ M [ k ] is invariant under the shifts of any σ l , l = 1 , . . . , n − 3, by the vectors w l of root lattice R of sl N embedded into C N , i.e. w l ∈ Z N , P Ns =1 w ( s ) l = 0. This claim is equivalent to the followingproposition. Proposition 4.4. The operator-valued monodromy matrix ˆ M [ k ] has the following property: ∇ σ l ,m ∇ − σ l ,s ˆ M [ k ] = ˆ M [ k ] ∇ σ l ,m ∇ − σ l ,s . Proof. For l < k , ˆ M [ k ] does not depend on σ l . Therefore it is sufficient to prove the commutativityfor the cases with l ≥ k .For l = k , let us extract explicitly the part of ˆ M [ k ] given by (4.24) depending on σ k :ˆ M [ k ] = · · · F − ( σ k +1 , a k +1 , σ k ) B ( σ k ) F ( σ k +1 , a k +1 , σ k ) · · · . Using the identities (4.18) and B ( σ k ) = exp (2 πi diag σ k ), we then obtain ∇ σ k ,m ∇ − σ k ,s ˆ M [ k ] = · · · F − ( σ k +1 , a k +1 , σ k + h m − h s ) B ( σ k + h m − h s ) ×× F ( σ k +1 , a k +1 , σ k + h m − h s ) · · · ∇ σ k ,m ∇ − σ k ,s = ˆ M [ k ] ∇ σ k ,m ∇ − σ k ,s . For l > k , it suffices to show the commutativity property for ˆ V [ k ] : ∇ σ l ,m ∇ − σ l ,s ˆ V [ k ] = ˆ V [ k ] ∇ σ l ,m ∇ − σ l ,s . (4.25)19ndeed, the part of the operator-valued matrix ˆ V [ k ] which depends on σ l is given byˆ V [ k ] = · · · F − ( σ l +1 , a l +1 , σ l ) ∇ σ l F − ( σ l , a l , σ l − ) · · · . The property (4.25) easily follows from (4.18) and the diagonal form of of ∇ σ l .Although F m ( σ | y ) is not invariant with respect to the shifts of σ l by root vectors, the aboveproposition suggests to consider Fourier transform of F m ( σ | y ) with respect to such shifts. It willbe shown in the next section that this Fourier transform gives the solution of the Riemann-Hilbertproblem we are interested in. As above, let R denote the root lattice of sl N embedded into C N , i.e. w ∈ R if and only if w ∈⊕ N − k =1 Z ( h k − h k +1 ), where h k are the weights of the first fundamental representation of sl N , cf (3.21).Equivalently, w ∈ R if and only if w ∈ Z N and the sum of its components vanishes. Theorem 5.1. Let Φ ( y ) be the solution of the semi-degenerate Fuchsian system (2.1) having theasymptotics Φ ( y ) = y − Θ n − (cid:2) I + O (cid:0) y − (cid:1)(cid:3) as y → ∞ and monodromies { M k } k =0 ,...,n − , described byProposition 2.4 with parameters θ , θ n − ; { r k , a k } k =1 ,...,n − , { σ l } l =1 ,...,n − . The matrix elements of Φ( y ) can be written in terms of conformal blocks of the W N -algebra: Φ jm ( y ) = ( − N ( δ jm − N m D − θ n − + h m | ψ j ( y ) | Θ D jm E h− θ n − | Θ D i , (5.1) where | Θ jm ( σ , . . . , σ n − ) i :== P − θ n − − h j + h m V a n − ( z n − ) P σ n − V a n − ( z n − ) P σ n − · · · P σ V a ( z ) | θ i , (cid:12)(cid:12) Θ D jm (cid:11) := X w ,..., w n − ∈ R e ( β , w )+ ··· +( β n − , w n − ) | Θ jm ( σ + w , . . . , σ n − + w n − ) i , (5.2) (cid:12)(cid:12) Θ D (cid:11) := | Θ D11 i = . . . = (cid:12)(cid:12) Θ D NN (cid:11) , and R is the root lattice of sl N . The relation between the monodromy parameters r k and conjugateFourier momenta β k is given by the formulas (4.20), (4.21) of Proposition 4.3 and R k = Y − k H k X k +1 , ≤ k ≤ n − , R n − = Y − n − B ( θ n − ) ,R k = diag r k , H k = diag (cid:0) e β (1) k , . . . , e β ( N ) k (cid:1) . (5.3) The normalization coefficients N m follow from the normalization (4.12) of ψ m ( y ) : N m = ˇ N ( − θ n − + h m , h , − θ n − ) . Proof. We have to show that proposed matrix Φ( y ) solves the initial Riemann–Hilbert problem withthe monodromy matrices specified above. The monodromy matrix M n − around z n − = ∞ in thepositive direction on the Riemann sphere is diagonal and has matrix elements( M n − ) jj = (cid:0) B ′− ( − θ n − + h m ) (cid:1) jj = exp n πi (cid:16) θ ( j ) n − − h ( j ) m + N − N (cid:17)o = e πiθ ( j ) n − . M [ n − = B ( − θ n − ) F − ( − θ n − , a n − − , σ n − ) B ( σ n − ) F ( − θ n − , a n − − , σ n − ) B − ( − θ n − ) . The next one, however, has operator-valued entries:ˆ M [ n − = B ( − θ n − ) F − ( − θ n − , a n − − , σ n − ) ∇ σ n − F − ( σ n − , a n − , σ n − ) ×× B ( σ n − ) F ( σ n − , a n − , σ n − ) ∇ − σ n − F ( − θ n − , a n − − , σ n − ) B − ( − θ n − ) . This operator expression is invariant with respect to the shifts of intermediate weights σ n − and σ n − by root vectors h m − h s of sl N (see Proposition 4.4) and therefore it is the same for all conformalblocks appearing in (5.2).Let us move the components of ∇ σ to the right to act on conformal blocks. We need a few identitieswhich follow from (4.17) and (4.18) written in components, ∇ σ ,l F lj (cid:0) − σ ′ , a, − σ (cid:1) = ( − N − F lj (cid:0) − σ ′ , a − , − σ (cid:1) ∇ σ ,l , ∇ σ ,m F jl (cid:0) σ , a, σ ′ (cid:1) = ( − ( N − δ ml F jl (cid:0) σ , a − , σ ′ (cid:1) ∇ σ ,m , ∇ σ ,m ∇ − σ ,l F lp (cid:0) σ ′′ , a, σ (cid:1) = ( − ( N − δ ml − F lp (cid:0) σ ′′ , a, σ (cid:1) ∇ σ ,m ∇ − σ ,l , to obtainˆ M [ n − = B ( − θ n − ) F − ( − θ n − , a n − − , σ n − ) ˜ ∇ σ n − F − ( σ n − , a n − − , σ n − ) ×× B ( σ n − ) F ( σ n − , a n − − , σ n − ) ˜ ∇ − σ n − F (cid:0) − θ n − , a n − − , σ n − ) B − ( − θ n − (cid:1) , where the components of ˜ ∇ σ n − act directly on conformal blocks. Therefore, the action on the Fouriertransformed conformal blocks diagonalizes producing a numerical monodromy matrix, M [ n − = B ( − θ n − ) F − ( − θ n − , a n − − , σ n − ) H − n − F − ( σ n − , a n − − , σ n − ) ×× B ( σ n − ) F ( σ n − , a n − − , σ n − ) H n − F ( − θ n − , a n − − , σ n − ) B − ( − θ n − ) , where H k is the diagonal matrix with the elements ( H k ) mj = e β ( j ) k δ mj .Analogous procedure may be applied to other monodromy matrices. We note that their structureas products of elementary building blocks reproduces the structure of the corresponding products(2.11), (2.12) in Proposition 2.4. The exact relation (5.3) between the parameters is obtained using(4.19).There is a related theorem for an alternative normalization of the fundamental solution. Theorem 5.2. The solution of the semi-degenerate Fuchsian system with the same monodromies,normalized as Φ ( y ) = I , can be written in terms of conformal blocks as Φ jm ( y ) = ( − N ( δ jm − ( y − y ) ( N − /N D − θ n − (cid:12)(cid:12) ¯ ψ m ( y ) ψ j ( y ) (cid:12)(cid:12) Θ D jm E h− θ n − | Θ D i , (5.4) where (cid:12)(cid:12)(cid:12) Θ D jm E and (cid:12)(cid:12) Θ D (cid:11) are given by the same formulas (5.2). The normalizations of ψ j ( y ) and ¯ ψ m ( y ) are fixed by (4.12) and (4.13). The proof of Theorem 5.2 goes along the same lines as the proof of Theorem 5.1. Note that thesolution (5.1) can be obtained from the solution (5.4) by fusion of h− θ n − | ¯ ψ m ( y ) in the limit y → ∞ .By the arguments given in [28], one obtains the isomonodromic tau function (2.4) as the Fouriertransform of the W N -conformal blocks which appears in the denominators of the fundamental solutions(5.1) and (5.4): 21 roposition 5.3. The isomonodromic tau function of Jimbo-Miwa-Ueno for semi-degenerate Fuch-sian systems is given by τ ( z ) = (cid:10) − θ n − | Θ D (cid:11) , (5.5) where | Θ D i is given by (5.2). Let us exemplify the above results with the simplest example of n = 4, i.e. two generic puncturesand two punctures of spectral type ( N − , z = 0 and z = ∞ . Itcan be also assumed that z = 1, so that the only remaining time parameter is z ≡ t . The Fuchsiansystem (2.1) then reduces to d Φ ( y ) dy = Φ ( y ) (cid:18) A y + A t y − t + A y − (cid:19) . (5.6)It should be stressed that the matrices A , A t , A are traceless (this assumption involves no loss ofgenerality but is crucial from the CFT perspective). Their respective diagonalizations arediag θ , diag (cid:0) N − N , − N , . . . , − N (cid:1) a t , diag (cid:0) N − N , − N , . . . , − N (cid:1) a , with a t , a ∈ C . We also have A ∞ = − A − A t − A = diag θ ∞ . The monodromy preservingdeformations for (5.6) are described by the equations dA dt = [ A , A t ] t , dA dt = [ A , A t ] t − , (5.7)equivalent to the polynomial Hamiltonian Fuji-Suzuki-Tsuda system [55]. The corresponding taufunction is defined, up to arbitrary non-zero constant factor, by d ln τ FST ( t ) dt = Tr A A t t + Tr A A t t − . (5.8)Proposition 5.3 states that τ FST ( t ) is nothing but the Fourier transform of the 4-point semi-degenerate conformal block, τ FST ( t ) = const · X w ∈ R e ( β , w ) σ + w − θ ∞ θ a a t t ∞ σ = (cid:0) σ (1) , . . . , σ ( N ) (cid:1) ∈ C N with P Ns =1 σ ( s ) = 0 are related to diagonalized composite monodromy of Φ ( y )around 0 and t : M M t ∼ e πi diag σ .There is a freedom in the choice of σ , which can be shifted by any vector in R . Obviously, sucha shift does not affect the result (5.9). From the point of view of asymptotic analysis of τ FST ( t ) as t → 0, it is convenient to choose σ so that (cid:12)(cid:12) ℜ σ ( i ) − ℜ σ ( j ) (cid:12)(cid:12) ≤ i, j = 1 , . . . , N . Indeed, τ FST ( t ) ≃ t ( σ − θ − a t h ) X w ∈ R C ( σ , w ) e ( β , w ) t w +( σ , w ) h O ( t ) i , (5.10)as t → 0, where C ( σ , w ) = N ( − θ ∞ , a h , σ + w ) N ( σ + w , a t h , θ ) N ( − θ ∞ , a h , σ ) N ( σ , a t h , θ ) . (5.11)22he normalization coefficients N ( σ ′ , a h , σ ) are given by (4.10), which implies that the structureconstants C ( σ , w ) can be expressed in terms of gamma functions. In the generic case of strict inequality (cid:12)(cid:12) ℜ σ ( i ) − ℜ σ ( j ) (cid:12)(cid:12) < 1, the leading asymptotic contribution to (5.10) is determined by w = 0, and thesubleading terms correspond to the roots, w = h i − h j . It follows that τ FST ( t ) ≃ t ( σ − θ − a t h ) N X i = j C ( σ , h i − h j ) e β ( i ) − β ( j ) t σ ( i ) − σ ( j ) + E ( t ) , (5.12) E ( t ) = O ( t ) + N X i,j,k,l =1 O (cid:16) t ℜ ( σ ( i ) − σ ( j ) + σ ( k ) − σ ( l ) ) (cid:17) . (5.13)The asymptotics (5.12) is a specialization to the semi-degenerate case of the Proposition 3.9 of [36],which provides a higher-rank analogue of the Jimbo’s asymptotic formula [37] for Painlev´e VI. We conclude with some open research directions. As already mentioned in the introduction, thefundamental solutions of 3-point Fuchsian systems allow to construct Fredholm determinant represen-tation of the n -point tau function [17, 31]. The principal minor expansion of the determinant givesa series representation for τ ( z ), which in the semi-degenerate case may be expected to coincide withNekrasov formulas [49] for instanton partition functions of linear quiver gauge theories in the self-dualΩ-background. This would produce a new (direct) proof of the AGT-W relation [1, 56, 22, 46] for c = N − W N -algebras.It might be possible to adapt the technique developed in [36] to compute the connection coefficientfor the tau function of the Fuji-Suzuki-Tsuda system (relative normalization of the asymptotics as t → t → ∞ ). The CFT counterpart of this quantity is the fusion kernel relating the s - and u -channel semi-degenerate c = N − N = 2 case): σθ ∞ θ z z a a ∞ Z d σ ′ F h a a θ ∞ θ ; σ ′ σ i σ ′ θ ∞ θ z a a z ∞ F [ . . . ], it would be interesting to clarify the relation of this quantityto symplectic geometry of the moduli space of semi-degenerate monodromy data.Monodromy preserving deformations and Fuchsian systems are also related to quasi-classical ( c →∞ ) limit of CFT [52, 48, 44, 53]. One may wonder whether a direct connection between the quasi-classical and c = N − W N -conformal blocks can be established. Quantitative aspects of this relationfor N = 2 have been the subject of recent study [43]. A Singular vectors of semi-degenerate Verma modules For c = N − 1, we are going to use a free-boson realization of the W N -algebra to find the singularvectors of the semi-degenerate Verma modules. It will be helpful to extend W N = W ( sl N ) to W ( gl N )23y introducing one more free-boson field J ( z ) with the OPE J ( z ) J ( w ) = 1 /N ( z − w ) + regular , and regular OPEs with the currents J k ( z ) entering the definition of W N : J k ( z ) J ( w ) = regular.Introduce the currents ˜ J k ( z ) = J k ( z ) + J ( z ) which have the following OPEs:˜ J k ( z ) ˜ J l ( w ) = δ kl ( z − w ) + regular , and define the generators of the W ( gl N )-algebra: f W ( s ) ( z ) = X ≤ i <... . For the semi-degenerate representation of the W -algebra, we use θ = ( a, , . . . , θ will be used. For the action of modes of f W ( s ) ( z ) defined by f W ( s ) ( z ) = X p ∈ Z f W ( s ) p z p + s , we have f W ( s ) − p | θ i = 0 , ≤ p ≤ s − . (A.3)On the other hand, we have a relation which can be obtained from the expansion of (A.1) with theuse of (3.3): f W ( s ) ( z ) = s X r =0 (cid:18) N − rN − s (cid:19) : J s − r ( z ) : W ( r ) ( z ) . (A.4)In order to find the relations for the elements of W N = W ( sl N ) in the Fock module F , let us usethe relation (A.4) modulo vectors from the submodule F ′ generated by a − p | θ i , p > 0. We have, for p > f W ( s ) − p | θ i = s X r =0 (cid:18) N − rN − s (cid:19) ( a/N ) s − r W ( r ) − p | θ i mod F ′ . (A.5)These relations are linear and can be inverted: W ( r ) − p | θ i = r X s =0 (cid:18) N − sN − r (cid:19) ( − a/N ) r − s f W ( s ) − p | θ i mod F ′ . (A.6)24sing (A.3), rewrite (A.6) as W ( r ) − p | θ i = p +1 X s =2 (cid:18) N − sN − r (cid:19) ( − a/N ) r − s f W ( s ) − p | θ i (cid:0) mod F ′ (cid:1) == p +1 X s =2 (cid:18) N − sN − r (cid:19) ( − a/N ) r − s s X t =2 (cid:18) N − tN − s (cid:19) ( a/N ) s − t W ( t ) − p | θ i . (A.7)After summation over s and changing summation index t to s , we finally get W ( r ) − p + ( − r + p p +1 X s =2 (cid:18) N − sr − s (cid:19)(cid:18) r − s − p − s + 1 (cid:19) (cid:16) aN (cid:17) r − s W ( s ) − p ! | θ i = 0 , ≤ p + 1 < r ≤ N. (A.8) B Null vectors and fusion rules for completelydegenerate fields This appendix uses a free-fermionic realization of the extension of W N = W ( sl N ) to W ( gl N ). Itis convenient since the fermionic fields realize completely degenerate fields for W ( gl N ) and theirproperties can be studied easily.The algebra of N -component free-fermionic fields is generated by ψ + α ( z ), ψ − α ( w ), α = 1 , . . . , N ,with the standard singular part of the OPEs: ψ + α ( z ) ψ − β ( w ) ∼ δ α,β z − w , ψ + α ( z ) ψ + β ( w ) ∼ , ψ − α ( z ) ψ − β ( w ) ∼ . (B.1)The W ( gl N )-algebra is a subalgebra of the free-fermionic algebra generated by the fieldsˆ W ( k ) ( z ) = N X α =1 : ∂ k − ψ + α ( z ) ψ − α ( z ) : , k = 1 , . . . , N. (B.2)It is convenient to extend this definition to all integer k > 0. Using the bosonization formulas ψ ± α ( z ) = : exp( ± iφ α ( z )) :, α = 1 , . . . , N , we get a free-boson realization of ˆ W ( k ) ( z ) as differentialpolynomials in bosonic currents ˜ J α ( z ) = i∂φ α ( z ):ˆ W ( k ) ( z ) = N X α =1 : ∂ k e iφ α ( z ) · e − iφ α ( z ) : . (B.3)This free-boson realization of the currents ˆ W ( k ) ( z ) does not coincide with the currents f W ( k ) ( z ) givenby the formula (A.1) of the Appendix A, but they generate the same algebra (a proof of this fact canbe found in e.g. [7]).The OPE ˆ W ( k ) ( z ) ψ + α ( w ) ∼ ∂ k − ψ + α ( w ) z − w (B.4)means that ˆ W ( k )1 − k (cid:12)(cid:12) ψ + α (cid:11) = L k − − (cid:12)(cid:12) ψ + α (cid:11) , ˆ W ( k ) m | ψ + α i = 0 , m > − k, (B.5)which gives us the list of convenient null-vectors for the degenerate fields ψ + α ( z ) (they can also berewritten in terms of standard generators). Note that the OPEs (B.4) are the same for all α and, infact, give the OPEs of the W N -algebra currents with the completely degenerate fields.25o find the fusion rules for ψ + α ( z ), consider the conformal blockΩ ( k ) α ( z ) = h θ ∞ | ˆ W ( k ) ( z ) ψ + α (1) | θ i . (B.6)Since h θ ∞ | ψ + α ( t ) | θ i = t ∆ ∞ − ∆ − ∆ t = t ( θ ∞ − θ − ) , (B.7)we have h θ ∞ | ∂ k − ψ + α (1) | θ i = (cid:2) θ ∞ − θ − (cid:3) k − =: A k , (B.8)where [ x ] k = x ( x − · · · ( x − k + 1) denotes the falling factorial. This fixes the singular part ofΩ ( k ) α ( z ) near z = 1 because of (B.4):Ω ( k ) α ( z ) = A k z − O (1) as z → . (B.9)In order to compute the asymptotics of Ω ( k ) α ( z ) near z = 0 and z = ∞ , one may use the modeexpansion ˆ W ( k ) ( z ) = X n ∈ Z ˆ W ( k ) n z n + k , considered for | z | < | z | >