Isomonodromic τ-functions and WN conformal blocks

We study the solution of the Schlesinger system for the 4-point slN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak{s}{\mathfrak{l}}_N $$\end{document} isomonodromy problem and conjecture an expression for the isomonodromic τ-function in terms of 2d conformal field theory beyond the known N = 2 Painlevé VI case. We show that this relation can be used as an alternative definition of conformal blocks for the WN algebra and argue that the infinite number of arbitrary constants arising in the algebraic construction of WN conformal block can be expressed in terms of only a finite set of parameters of the monodromy data of rank N Fuchsian system with three regular singular points. We check this definition explicitly for the known conformal blocks of the W3 algebra and demonstrate its consistency with the conjectured form of the structure constants.


Introduction
There are two topics in the mathematical physics that remained independent for a long time: the theory of isomonodromic deformations, initiated by R. Fuchs, P. Painlevé and L. Schlesinger in the beginning of 20th century (see [1] and references therein), and the 2d conformal field theory (CFT) founded by A. Belavin, A. Polyakov and A. Zamolodchikov in 1984 [2]. Both theories have wide range of applications. Conformal field theory describes perturbative string theory and second order phase transitions in the 2d systems. The theory of isomonodromic deformations gives rise to non-linear special functions such as Painlevé transcendents, which appear in different problems of mathematical physics: for example, in the random matrix theory and general relativity.
First relations between the theory of isomonodromic deformations and 2d quantum field theory have been established in 1978-80 by M. Sato, M. Jimbo and T. Miwa [3][4][5][6][7]. More recently, O. Gamayun, N. Iorgov and O. Lisovyy have discovered that the τ -function of the Painlevé VI equation (related to the rank two Fuchsian system with four regular singular points on the Riemann sphere) can be expressed as a sum of c = 1 conformal blocks, multiplied by certain ratios of the Barnes functions -a typical expansion of the correlation function in CFT [8]. Their formula gives the general solution of Painlevé VI equation. This conjecture has already been proved in two ways: one proof is purely representation-theoretic and adapted initially for the 4-point τ -function [9] but can provide us with a collection of JHEP09(2015)167 nontrivial bilinear relations for the n-point conformal blocks, whereas another one is based on the computation of monodromies of conformal blocks with degenerate fields and allows to consider an arbitrary number of regular singular points on the Riemann sphere [10]. The correspondence also extends to the irregular case: for instance, it gives exact solutions of the Painlevé V and III equations [11,12], which are known to describe correlation functions in certain massive field theories.
The present paper is concerned with the extension of the isomonodromy-CFT correspondence to higher rank. Already in [8] there was a suggestion that the monodromy preserving deformations of Fuchsian systems of rank N should be related to 2d CFT with central charge c = N − 1. One obvious and natural candidate for such a theory is the Toda CFT with W N algebra of extended conformal symmetry. We show that indeed the N × N isomonodromic problem corresponds to the W N algebra, whose Virasoro part has central charge c = N − 1. These algebras were first introduced by A. Zamolodchikov in [13], and their study was substantially developed in [14] (for the first nontrivial W 3 -case) and [15] (for generic W N ). Other developments in the theory of W -algebras are discussed in the review [16].
The most condensed form of the commutation relations of W 3 is given by the operator product expansions (OPEs) of the energy-momentum tensor T (z) and the W -current W (z): (1.1) where Λ(z) = (T T )(z) − 3 10 ∂ 2 T (z). The representation theory of this algebra is very similar to that of the Virasoro algebra. In the generic case one has the Verma module with the highest vector |∆, w such that L 0 |∆, w = ∆|∆, w , W 0 |∆, w = w|∆, w . Hence the representation space is spanned by the vectors 2) while the set of the highest weight vectors themselves corresponds to primary fields (vertex operators) of the 2d CFT. As in the Virasoro case, these fields can be determined by their OPEs with higher-spin currents T (z) and W (z):

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However, the W-descendants such as (W −1 φ) and (W −2 φ) are not defined in general (this is to be contrasted with the Virasoro case where one has e.g. (L −1 φ)(w) = ∂φ(w)), which means that the 3-point functions involving such fields are not really defined. As a consequence, one cannot express the matrix elements It was shown in [17] in an elegant way that all such 3-point functions can be expressed in terms of an infinite number of unknown constants The problem is that having this infinite number of constants (which for the 4-point conformal block actually becomes doubly infinite) one can adjust them as to obtain any function as a result. In this paper we show that the isomonodromic approach can fix this ambiguity in such a way that all these parameters become functions on the moduli space of the flat connections on the sphere with 3 punctures. In the sl 3 case this space is 2-dimensional (we denote the corresponding coordinates by µ and ν), so all C k = C k (µ, ν).
Note that for the W N algebra one would have the set of constants C k 1 ,...,k l with l = 1 2 (N −1)(N −2) non-negative indices (e.g., this easily follows from analysis of [17]), which is half of the dimension of the moduli space of flat sl N connections on the 3-punctured sphere. The paper is organized as follows. In section 2 we briefly discuss the origins of the Schlesinger system and the space of flat connections on the punctured Riemann sphere. Then we introduce a collection of convenient local coordinates on this space, which are related to pants decomposition of the sphere. In section 3 an iterative algorithm of the solution of the Schlesinger system is proposed. We then present a set of non-trivial properties of this solution, discovered experimentally, and put forward a conjecture about isomonodromy-CFT correspondence in higher rank, which relates W N conformal blocks to the isomonodromic tau function. In particular, for a collection of known W 3 conformal blocks we present the 3-point functions that can be used to construct the τ -function in the form of explicit expansion. In section 4 we describe the problems of definition of the general W 3 conformal block and discuss how they can be addressed using the global analytic structure induced by crossing symmetry. We conclude with a brief discussion of open questions.

Isomonodromic deformations and moduli spaces of flat connections
The main object of our study will be the Fuchsian linear system Here A ν are traceless matrices with distinct eigenvalues, Φ(z) is the matrix of N independent solutions of the system normalized as Φ(z 0 ) = 1. It is obvious that upon analytic

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continuation of the solutions along a contour γ ν encircling z ν they transform into some linear combination of themselves: where M ν ∈ GL N (C). The relation γ n . . . γ 1 = 1 from π 1 (CP 1 \{z 1 , . . . , z n }, z 0 ) imposes the condition The well-known Riemann-Hilbert problem is to find the correspondence It is easy to see that the conjugacy classes of M ν are The eigenvalues of A ν determine the asymptotics of the fundamental matrix solution near the singularities, so one can fix even this asymptotics and study the corresponding refined Riemann-Hilbert problem. We will work only with traceless matrices A ν since the scalar part trivially decouples.

Schlesinger system
Since it is difficult to solve the generic Riemann-Hilbert problem exactly, one can first ask a simpler question: how to deform simultaneously the positions of the singularities z ν and matrices A ν but preserve the monodromies M ν . The answer follows from the infinitesimal gauge transformation leading to the Schlesinger system of non-linear equations (2.8)

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Note that one can fix z n = ∞, then the corresponding matrix A ∞ = − n−1 ν=1 A ν will be constant. A non-trivial statement is that the relations are compatible and define the τ -function τ (z 1 , . . . , z n ) of the Schlesinger system. It is easy to see that the 3-point τ -function is given by a simple expression: Let us now attempt to solve the Schlesinger system for the 4-point case and compute the corresponding τ -function in the form of certain expansion.

Moduli spaces of flat connections
The main object of our interest is the τ -function. It depends on monodromy data which provide the full set of integrals of motion for the Schlesinger system. It will be useful to start by introducing a convenient parametrizaton of this space.
One starts with n matrices M ν ∈ SL N , with fixed nondegenerate eigenvalues, i.e. there are n(N 2 − N ) parameters. These matrices are constrained by one equation (2.3) and are considered up to an overall SL N conjugation, which decreases the number of parameters by 2(N 2 − 1). So the resulting number of parameters is Here θ ν ∈ h (h is the Cartan subalgebra) define the conjugacy classes: M ν ∼ e 2πiθν . It is obvious that θ ν is equivalent to θ ν + h ν , such that for all weights of the first fundamental representation e i one has (e i , h ν ) ∈ Z. It means that h ν ∈ ⊕ r i=1 Zα ∨ i , where α ∨ i ∈ h are simple coroots (for the simply-laced case they coincide with the roots).
For the general Lie algebra this formula can be written as In particular, for n = 3 punctures on the sphere This formula gives the number of non-simple roots of g. In the sl N case it specializes to This expression vanishes for sl 2 , which drastically simplifies the study of the corresponding isomonodromic problem. However already for sl 3 this dimension is equal to 2, i.e. it is nonvanishing. One way to simplify the problem is to set θ 2 = ae 1 : in this case the orbit of the adjoint action e 2πiae 1 → g −1 e 2πiae 1 g JHEP09(2015)167 In this calculation the first two terms correspond to the dimensions of orbits: two generic and one with a large stabilizer. The last term corresponds to one equation and one factorization. Hence This case is the best known on the side of W -algebras [18][19][20]. In the mathematical framework, this situation corresponds to rigid local systems.

Pants decomposition of M g 4
We begin our consideration with an arbitrary Lie group G containing a Cartan torus H ⊂ G. The corresponding Lie algebras are g and h, respectively. At some point we will Let us introduce S = M 1 M 2 and consider two triples Note that the products inside each of these triples are equal to the identity. Let us now choose the submanifold with fixed eigenvalues of M 1 , . . . , M 4 , S. One may also use the freedom of the adjoint action to diagonalize S where σ ∈ h. We thereby obtain a submanifold 18) which looks like a relative twist of one part of the sphere with respect to another (in the sl 2 case it will be exactly the geodesic flow). Therefore one can say that The torus action is free, so locally it looks as a product (actually it is true even globally because the fibration ( we can give an algebraic parametrization for one representative from each conjugacy class). Therefore we have the equality for the open subsets (denoted by ≈): The above considerations suggest the following choice of coordinates on M g 4 : • Gluing parameters σ: rank g items.
. They are invariant with respect to the action of "relative twists": we have 2 dim M g 3 such functions.
• Relative twist parameters, which change under the twist (for example, tr , rank g items. These coordinates will be denoted by β ∈ h.
This procedure is schematically depicted in figure 1 (2.21) Iteratively repeating this procedure, one is led to the following choice of coordinates on M g n : -point moduli of flat connections (here we identify σ 0 = θ 1 and σ n−2 = −θ n ).

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Anticipating the result, let us mention that these coordinates are convenient from the CFT point of view: σ i will parametrize intermediate charges in the conformal block and β i will be the Fourier transformation parameters. This description was shown to be valid in the sl 2 case [8,10] and was recently demonstrated to hold for sl N case with dim M g 3 = 0 [21]. From a more conceptual point of view, this decomposition illustrates that all extra parameters in the τ -function expansion come from the 3-point functions.

Iterative solution of the Schlesinger system
In order to study the generic Schlesinger system, let us follow the approach proposed in the original paper of M. Jimbo [22] and in [4].
Let us take the 4-point Schlesinger system and fix the singularities to be 0, t, 1, ∞. The system becomes The isomonodromic τ -function is defined by Let us study the solution of the system (3.2) for the case when A 1 (t) is finite in the limit t → 0: . Under this assumption we have If the last term were absent, then the solution would be whereÃ 0 (t) has a well-defined limit as t → 0. We see that in view of its definition B describes the total monodromy around 0 and t in the limit t → 0. Since the deformation is isomonodromic, this monodromy is constant and is given by M 0 M t = M 0t ∼ e 2πiB . This allows to make the identification B = σ . (3.5)

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Our system then becomes (3.6) Here we see an operator t adσ , which produces some fractional powers of t. It is convenient to impose the condition that (σ, σ) ≪ 1, or at least that for all roots α one has |(σ, α)| < 1 2 . This allows to organize the terms of the expansion in powers of t according to their order of magnitude in the asymptotic behavior. If necessary, one can perform an analytic continuation of the solution from the region with small σ. We know that in the Lie algebra the operator t adσ acts by where α ∈ g * is a root and E α , H α are the elements of the Cartan-Weyl basis. Let us define a grading on the space of monomials where w ∈ Q g is an element of the root lattice Q g = rankg i=1 Zα i of g. It is useful to define a filtration on this root lattice, which is recursively constructed as follows: Q 0 g = {0}, Q 1 g is the set of all roots of g and 0, and Also define the double filtration V n,m on the space of all fractional-power series: Each term of the filtration is generated by these monomials. This definition turns out to be useful because of the properties (3.10) One can also see that the degrees present in V n+1,m+k are larger then in V n,m if σ is sufficiently small (∀α ∈ Q 1 g : |(σ, α)| < 1 k ). We also define a slightly ambiguous notation V n,w by t k+(σ,w) ∈ V n,w ⇔ (k ≥ n) . (3.11)

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Now we have all the ingredients that are necessary for the construction of an iterative solution of the system (3.6). Our initial data will be given by the triple of matrices σ, A 0 (0) and A 1 (0). Symbolically, the system (3.6) can be written as where "affine" bilinear (in the sense f (x, y) = axy + bx + cy + d) functions F 0 , F 1 have the following properties: (3.13) Let us substitute into (3.12) the expressions (3.14) From (3.13) we immediately see that (3.6) takes the form .

(3.15)
Because of the ≤ sign our strategy of solving will be to compute first A k 1 (t), and then subsequently determineÃ k 0 (t). One can also write down explicit formulas for bilinears f k 1 and f k 0 , which are immediate (though cumbersome) consequences of the system (3.6). Now let us determine which powers (k, w) will be actually present in the solution. This will be done again iteratively, using only the properties (3.13): • TakingÃ 0 (0) ∈ V 0,0 and A 1 (0) ∈ V 0,0 , and computing F 1 , we get an element of V 1,1 , therefore A 1 ∈ V 0,0 + V 1,1 + . . .

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Continuing this procedure one finds the following structurẽ

(3.16)
It is easy to check that these spaces are stable under the action of (F 0 , F 1 ) described by the rules (3.13). This is somewhat similar to the statement that the cone is stable under the addition operation. Indeed, let us try to find an element ofÃ 0 (t) lying in V k,2k+1 . For this one would need n 0 + n 1 ≤ k, m 0 + m 1 ≥ 2k, so m 0 + m 1 ≥ 2(n 0 + n 1 ). Since m 1 ≤ 2n 1 − 1 for n 1 = 0 (when F 0 vanishes) and m 0 ≤ 2n 0 , such an element cannot exist. Similarly, for A 1 , let us take an element lying in V k,2k . One then needs to satisfy the constraints n 1 ≤ k − 1, m 1 ≥ 2k (impossible) or n 0 +n 1 +1 ≤ k and m 0 +m 1 +1 ≥ 2k, which implies m 0 +m 1 ≥ 2n 0 +2n 1 +1. But m 1 ≤ 2n 1 and m 0 ≤ 2n 0 , therefore one cannot get such an element neither. Now let us compute the τ -function and try to understand in which elements of the filtration does it lie. Since we have where A ν ∼ θ ν . For convenience, let us introduce the notation The terms present in tr (t −σ A 1 (t)t σÃ 0 (t)) that are closest to the boundary originate from the constant part of A 1 (t). These terms belong to ∞ k=0 V k,2k , therefore Note that these estimates are too rough, since we have not taken into account that a number of the commutators actually vanish. The actual result turns out to be the same for all three functions log τ,Ã 0 , A 1 ∈   and it can be checked numerically. Moreover, it turns out that the expansion of the τfunction itself is even more restricted: which in fact provides an evidence for the 2d CFT description: different fractional powers come from t ∆ for the different ∆'s, but the conformal dimension ∆ = 1 2 (σ + w, σ + w) is a quadratic function of w leading to the structure (3.21).

sl 2 case
In this case we illustrate all procedures, definitions and statements using the exact solution of [8].
The Lie algebra sl 2 is given by 3 generators E α , E −α , H α , such that The root lattice Q sl 2 is shown in figure 2. It is spanned by one root α. Q 0 sl 2 is the empty square, Q 1 sl 2 is the red rectangle, Q 2 sl 2 is green and Q 3 sl 2 is blue. All monomials have the form t n+(σ,w) = t n+m(σ,α) , and therefore can be depicted by the points of a two-dimensional lattice. Note that in our normalization (α, α) = 2. Several examples of the elements of this filtration are presented in figure 3.
Here the blue region represents V 0,0 , red corresponds to V 1,1 and green is V 3,4 . We can also show the "true" and "naive" lattice supports of the quantitiesÃ 0 (t), A 1 (t), log τ (t) and t −χ τ (t). See figure 4: green region is the "naive" support of A 1 (t), the blue region is the true support ofÃ 0 (t), A 1 (t), log τ (t), which can be derived experimentally.  (3.23) which in turn implies

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It looks like a miracle and means that a huge number of terms cancel out when we exponentiate, but this answer confirms the conjecture (3.21). This phenomenon is illustrated in figure 5 in two ways. Upper bold numbers account for the degree in τ (t) (blue region), lower numbers correspond to the degree in log τ (t) (green region). Horizontal coordinate corresponds to the position in the sl 2 root lattice.
Let us take the main formula from [8]: where B(. . . ; t) is the c = 1 Virasoro conformal block and Here (θ ν , −θ ν ) are the eigenvalues of the matrices A ν in the linear system (2.1), (e 2πiσµν , e −2πiσµν ) are the eigenvalues of M µ M ν , s is the only variable depending on σ 1t (in a complicated way). The main properties of (3.25) and (3.26) can be summarized as follows: 1. The support of τ (t) is as indicated in (3.24).
2. Relative twist parameter enters only via the factor s n in the structure constants.
3. The 3-point coefficients C n factorize with respect to the pants decomposition parametrization.
We are now going to check these important properties in the sl 3 case. Figure 6 illustrates the filtration on the sl 3 root lattice. The red hexagon corresponds to Q 1 sl 3 , Q 2 sl 3 is shown in green and Q 3 sl 3 is blue. It is difficult to visualize V m,n , since one would then need a 3d picture. One can however think of ∞ k=0 V k,k as being a cone with hexagonal section.

sl 3 case
Let us perform the numerical study of the 3 × 3 Schlesinger system. We first determine which degrees (k, w) are present in log τ (t) and in τ (t) (figure 7).  Figure 7. Degrees present in t −χ τ (t) and in log τ (t). Number χ is given by (3.19).

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As above, the upper bold numbers correspond to degrees in t −χ τ (t) and the lower ones to log τ (t). We mark with "?" sign those values which are obtained at the limit of machine precision or which are greater then 7 (so that they are not seen in the solution up to the 7th order). Carefully analyzing this picture, one deduces that (3.27) It means that nonzero monomials of τ (t) fill a paraboloid, and not the naively expected cone. In other words, a lot of nontrivial cancellations take place, which provides further evidence for the conjecture (3.21). We now list other nontrivial properties of τ (t) revealed by our experimental study.

The first term in the expansion of conformal block has the form
This property is new, as compared to the N = 2 case, and we will see later that it is very important.
All these facts tell us that almost all properties of sl 2 case hold in the sl 3 case. This leads us to Main conjecture: The corresponding dimensions and W -charges are given by (3.29) The main advantage of the above definition of conformal block is that it depends only on 4 extra variables instead of a doubly-infinite set.
It is easy to check this definition for the case when W 3 -block can be defined algebraically. This becomes possible when θ t = a t e 1 and θ 1 = a 1 e 1 , where e 1 is the weight of the first fundamental representation. The best way to present this conformal block is to use Nekrasov formulas [23] which can be applied to conformal field theory in view of the extended AGT [24] correspondence, first established in [25,26]. The most convenient (for c = 2) expression for the conformal block can be found in [20]: It yields exactly the same result as our computations using iterative solution of the Schlesinger system. We have also conjectured in this case and checked experimentally a formula for the structure constants, which is a straightforward generalization of (3.26): Here e i denote the weights of the first fundamental representation and α i are all roots of sl N (in our case N = 3). This formula was recently proved [21] for general N . One can also observe a similarity between this formula and Toda 3-point function computed in [18].

General conformal block
Here we consider for simplicity the c = 2 case, but the generalization to arbitrary c is straightforward. First we explain how the W N conformal block is defined algebraically.
For that let us compute the following expression: where |θ 0 and −θ ∞ | are the highest-weight vectors with the charges given by (3.29), P σ is the projector onto the whole Verma module (1.2) with given highest weight. This conformal block can be computed by inserting the resolution of the identity in the Verma module. One can take, for instance, the naive basis (1.2), or (if we do not necessarily want to preserve the L 0 grading) the basis from [17], or (if we wish to add the Heisenberg algebra) the AGT basis from [20,27]. Let us call the vectors of this basis |σ, Y and suppose that Their scalar products will be K σ ( Y , Y ′ ) = σ, Y |σ, Y ′ . This allows to express conformal block as  (1)|θ 0 and Q( Y ) = θ ∞ |φ θ 1 (1)|σ, Y and χ is given by (3.19). The claim of [17] is that Q( Y ) andQ( Y ) are "triangular" "affine" linear functions of infinitely many arbitrary parameters C k ,C k defined by

Degenerate field
Let us consider the case θ t = e 1 (the weight of the first fundamental representation). The fusion rules for such fields are known to be given by Let us also shift θ 0 → θ 0 − e n , multiply the conformal block by t 2 3 = t (e 1 ,e 1 ) , and define the quantity Φ nk (t) = t (e 1 ,e 1 ) B(θ ∞ , θ 1 , θ 0 + e k − e n , e 1 , θ 0 − e n ; t) whereq( Y ) do not contain any free parameters [17]. Now denote the degenerate field φ e 1 (t) by ψ(t) and consider the correlation function In the region t → 0 (s-channel) it can be expanded in the basis of conformal blocks written above. But if we set t → 1 or t → ∞ (t-and u-channel), then we will have the following OPEs (4.7) These formulas suggest that the space of conformal blocks involving ψ(t) is 3-dimensional and near each point we have a basis with asymptotics prescribed by θ ν . It is clear that upon analytic continuation of Ψ 1k (t) around 0, 1, ∞ one gets some linear combinations of JHEP09(2015)167 the basis elements (4.8) In our case M 0 = diag(e 2πi(θ 0 ,e 1 ) , e 2πi(θ 0 ,e 2 ) , e 2πi(θ 0 ,e 3 ) ). That these formulas must hold can be expected on general grounds (crossing symmetry) and from the fact that the space is 3-dimensional. However, looking at the formula (4.6), the freedom in choice of C k can give f l t l with arbitrary f n 's. It means that W -algebra itself does not account for the global structure of conformal blocks and this information should be introduced as an extra input. Now suppose that we have some globally-defined multivalued functions Φ 1k . Then we have three monodromies M 0 , M 1 , M ∞ and one can solve the refined 3-point Riemann-Hilbert problem. Suppose that its solution is given by the matrix F (t) such that

9)
A 0 = diag ((θ 0 , e 1 ), (θ 0 , e 2 ), (θ 0 , e 3 )) and F (t) is normalized in such a way that F (t) = This vector has the trivial monodromies around all singular points, it is regular there and R(0) = (1, 0, 0), so that R(t) = (1, 0, 0). It means that Φ 1k (t) = F 1k (t) . (4.10) This formula allows us to fix all constants C k . This is done in the following way: we solve the 3-point Riemann-Hilbert problem, take F 11 (t) and read the coefficients of conformal block from its series. These coefficients are triangular linear combinations (4.3) of C k (i.e., kth term of the conformal block expansion involves only C j≤k ). This construction thus expresses C k via the moduli (µ, ν) of flat connections on the 3-punctured sphere. (4.11) All constants are expressed in terms of only two parameters. If we now recall the 5th experimental property of the τ -function, its origin can be easily understood: the first term of the conformal block (with the structure constants fixed above) depends only on C 1 (µ, ν) andC 1 (µ, ν) and this dependence is at most bilinear. define the action of the Verlinde loop operators on this space in the same way as it was done in [28]. This action is given by some operatorsV (γ) depending on the loop γ.
The question about interpretation in c = N −1 case is still open: the problem is caused by non-commutativity of the algebra ofV (γ). Moreover, even in the minimal model-like cases c = N − 1 − N (N 2 − 1) (k−1) 2 k for integer k, when the algebra is commutative again, the relation to the isomonodromic deformations becomes unclear (see "concluding remarks" in [9] for discussion of the Virasoro case).

Conclusions
We have discovered several important properties of the isomonodromic τ -functions in higher rank, which can be interpreted as signatures of the isomonodromy-CFT correspondence for the W N case. This allows to give a definition of the general W N conformal block, depending only on a finite number of parameters. It is also possible to prove [21] that the algebraic way to define well-known conformal blocks for semi-degenerate fields agrees with the above definition.
We have also considered a particular conformal block with degenerate field and shown that its global structure is not fixed algebraically. The requirement of the correct global behavior of this object yields an expression for the whole infinite series of constants in the W 3 conformal block in terms of the solution of the 3-point Riemann-Hilbert problem.
These expressions can be written in terms of coordinates on the moduli space of flat connections on sphere with 3 punctures. This is expected to be universal and work for any conformal block (not only for those with degenerate fields). We have checked experimentally some properties, which support this conjecture.
Finally, let us list some remaining open problems: • One needs to check that the procedure of fixing C k is self-consistent.
• If the constants C k can be fixed in such a way, we may try to prove that the τ -function can be given as a sum of the general W N conformal blocks.
• A constructive solution of the 3-point Riemann-Hilbert problem is still missing.
• It would be interesting to understand the meaning of Z bif (θ t , σ, θ 0 ; µ, ν| Y , Y ′ ) in the context of isomonodromy-CFT correspondence. It can be done for the case Y ′ = 0 and it is interesting what happens for the arbitrary Young diagrams.

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• There as an approach to the definition of conformal blocks of the light fields in the limit c → ∞ [29]. In that case explicit integral expression for the conformal block was derived. All the information about the 3-point functions enters this definition via several functions of one variable. The open problem is to obtain the monodromy properties of such conformal blocks and to identify the choice of 3-point functions that gives the conformal blocks arising in our approach.
• It is also important to understand the meaning of the results [30] about partition functions of T N theories without lagrangian description (which are believed to be the counterparts of the general W N 3-point functions) from the isomonodromic point of view.