Global conformal blocks via Shadow formalism

We study $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ global conformal blocks on a sphere and a torus, using the shadow formalism. These blocks arise in the context of Virasoro and $\mathcal{W}_3$ conformal field theories in the large central charge limit. In the $\mathfrak{sl}_2$ case, we demonstrate that the shadow formalism yields the known expressions in terms of conformal partial waves. Then, we extend this approach to the $\mathfrak{sl}_3$ case and show that it allows to build simple integral representations for $\mathfrak{sl}_3$ global blocks. We demonstrate this construction on two examples: the four-point block on the sphere and the one-point torus block.


Introduction
An important class of conformal field theories is the class CFTs possessing W N (N ≥ 2) symmetry, the generators of which are N − 1 holomorphic fields of spin 2, ..., N .The simplest case, N = 2, corresponds to the conformal field theory with Virasoro symmetry, where the generator is the spin-2 energy-momentum tensor.Examples of conformal field theories with W N symmetry include the sl N quantum Toda field theory, which is a generalization of Liouville field theory, corresponding to N = 2 and arising in the description of twodimensional quantum gravity.Thus, the sl N quantum Toda field theory is relevant for higherspin generalizations of two-dimensional quantum gravity.
In this work, we describe the CBs of the correlation functions of primary fields in conformal field theories with W 2 (Virasoro) and W 3 symmetries in the large central charge limit, using the so-called shadow formalism [38][39][40][41].This method allows us to express the CBs in integral representations involving the so-called conformal partial waves.In the case of Virasoro CFT, this method has been well-studied in both spherical and torus topologies.Our goal is to generalize this method to the W 3 CFT, particularly on the torus, where no exact expressions for the blocks are known.
The main object of our study is the one-point CB on the torus in the large central charge limit.The one-point correlation function of a primary field ϕ ∆ 1 , ∆1 with holomorphic and antiholomorphic conformal dimensions ∆ 1 , ∆1 on the torus is defined as where Tr ∆ denotes the trace taken over a module of the symmetry algebra A associated with the primary field ϕ ∆, ∆ in the intermediate OPE channel, D is the domain of primary fields of the corresponding conformal field theory, q is the elliptic parameter of the torus q = e 2πiτ , and L 0 is the generator of the algebra satisfying L 0 | ∆⟩ = ∆| ∆⟩.Here, F(∆ 1 , ∆, q) A is the onepoint holomorphic torus CB (for more details, see, e.g., [42][43][44]).In the case of the Virasoro algebra and generic central charge c, an exact expression for F(∆ 1 , ∆, q) A is unknown.When one restricts the analysis to the large central charge limit, exact expressions are known for the sl 2 global one-point 2 torus CBs.Global conformal blocks are defined as the contribution of CBs coming from the sl N subalgebra of the W N algebra.For other W N (N ≥ 3) conformal field theories, no exact expressions are currently known.In [48], a perturbative expression was presented for W 3 CFT in the large central charge limit.In this work, we derive the exact expression for the sl 3 global one-point CBs, which is the main result of this paper.
We consider the light operators relevant for the global CBs, whose conformal dimensions scale as ∆ ∼ o(1) as c → ∞.This behavior allows to restrict the set of generators of W 2 and W 3 to those of sl 2 and sl 3 algebras, respectively, and leads to a significant simplification of the CBs.
W 3 primary fields ϕ j (z, z) are labeled by a vector j belonging to the root space of sl 3 , where w 1 , w 2 are sl 3 fundamental weights.Unlike W 2 , in W 3 CFT, the CBs are not fully determined by the symmetry algebra due to the presence of multiplicities in the OPE of primary fields.In the language of sl 3 representation, this translates into multiplicities in the tensor product of sl 3 representations.Therefore, below, we will restrict our discussion to a class of CBs for which the problem of multiplicities is absent [49], with a number of external fields fulfilling the following condition Here ∆ j , q j are the conformal dimension and W 3 charge of ϕ j , respectively.The paper is organized as follows.In section 2, we briefly review some basic concepts of Virasoro CFT.In section 2.1, we explain the fundamental elements of the shadow formalism in the sl 2 case.We introduce a shadow operator that allows us to express CBs in terms of conformal partial waves.We discuss this construction for the sphere topology in section 2.2 and for the torus topology in section 2.3.In section 3, we recall the basic facts about W 3 CFT, define the sl 3 global one-point torus block and provide its perturbative expression.Section 4 is devoted to generalizing the shadow formalism to the sl 3 case.In section 4.1, we introduce preliminary concepts related to the sl 3 invariant functions theory.In section 4.2, we generalize the shadow formalism to the sl 3 case.In section 4.3, we apply the constructed formalism to the computation of the sl 3 global four-point sphere CB, and in section 4.4, we apply the formalism to compute the sl 3 global one-point torus CB.In section 5, we present our conclusions.Appendices A and B are included to explain some technical details of certain integral computations.In appendix C, we present a short review of sl 3 fields, which are introduced in section 4.2.

sl global conformal blocks via shadow formalism
The symmetry of the Virasoro CFT is generated by the energy-momentum tensor T(z)3 (a spin-2 current) whose Laurent series expansion reads (2.1) The modes L n satisfy the Virasoro algebra We denote by φ ∆, ∆(z, z) the Virasoro primary fields with conformal dimensions ∆, ∆.For simplicity, in what follows, we assume that holomorphic and antiholomorphic conformal dimensions are equal ∆ = ∆, (2.3) and denote the primary fields One can show that in the limit c → ∞, in order to have finite inner product of states ⟨∆, n| n, ∆⟩, where |n, ∆⟩ = L −n φ ∆ (0, 0) |0⟩, one needs to restrict the generators L n to the set L 0 , L 1 , L −1 , which form the sl 2 subalgebra The generators L n satisfy the following commutation relations with the primary fields The differential operators L 0 , L 1 , L −1 are the generators of sl 2 transformations on primary fields To compute CBs on sphere and torus, one uses the OPE decomposition of the product of primary fields and the commutation relations (2.6) to compute matrix elements of the type which arise in the decompositions of the CBs.Here |N, ∆ i ⟩ stands for the descendent states of |∆ i ⟩.On the sphere at large c limit, the CBs get contributions only from sl 2 generators; thus, the CBs reduce to the global CBs.While on the torus, besides the global blocks, one also has the so-called light CBs, which contain contributions from the full Virasoro generators.In this work, we will concentrate only on the global blocks.To compute them, we use the shadow formalism.

Basic concepts
Let us introduce the shadow operator of the primary field φ ∆ (z, z).Taking into account (2.3), the shadow operator is defined as where d 2 w = dwd w represents integration over the complex plane, and ∆ * is the holomorphic and antiholomorphic conformal dimension of the shadow operator (which is also a primary field) and N ∆ is a normalization coefficient Below, we will show that the shadow operator has the property where the two-dimensional delta function δ 2 (z − w) is defined according to R 2 f (z, z)δ 2 (z − w)d 2 z = f (w, w).We define the "projector" operator Because of (2.12), we have (2.18) We notice that since φ∆ * (z, z) is a primary field, one could include an extra contribution to the two-point correlation function (2.20)However, this modification does not substantially change (2.19).Indeed, assuming (2.20), we have where in the third line we used that φh can be expressed in terms of its shadow field φ h * .The result (2.21) is essentially the same as (2.19).

Delta function
The relation (2.12) can be established as follows.By inserting the shadow operator (2.9) into the lhs of (2.12), and writing it for z 1 and z 2 coordinates, we have (2.23) By using the parametrization (2.23) can be written more generally as where N ∆ is expressed in terms of h 1 , h 2 as (2.26) By using the relation Γ(∆) we rewrite (2.26) as follows (2.28) After some transformations and integration over x, one can convert this integral to (2.29) For the exponent, we use the Fourier transform By multiplying the rhs of (2.26) by e ik•(x 1 −x 2 ) and performing the two-dimensional integral over the variable (x 1 − x 2 ), we obtain 1.Therefore, by multiplying both sides of (2.29) and integrating over (x 1 −x 2 ), we must obtain the same result.Applying this reasoning, we obtain from (2.29) (2.31) where we used that h 1 + h 2 = 2. Hence in accordance with (2.11) and (2.25).

Sphere: Conformal blocks and partial waves
On the sphere, the four-point correlation function can be decomposed into four-point sphere CBs F s (∆ 4 , ∆, z 4 ) as follows4 where ∆ n = ∆ 1 , ∆ 2 , ..., ∆ n , z n = z 1 , z 2 , ..., z n .In the large c limit F s (∆ 4 , ∆, z 4 ) becomes the sl 2 global four-point sphere CB F s (∆ 4 , ∆, z 4 ) sl 2 given by where x is the cross-ratio x = z 12 z 34 z 13 z 24 and 2 F 1 (a, b, c, x) denotes the hypergeometric function.In the large c limit, one can use the shadow formalism to decompose (2.34) into partial waves.This is done by inserting the resolution of identity operator P in (2.34) as follows The first factor of the integrand is just the three-point function, generally given by where where z ij = z i − z j .In the next section, we will prove that the second factor in the integrand ⟨0| φ ∆ * (z, z)φ ∆ 3 (z 3 , z3 )φ ∆ 4 (z 4 , z4 )⟩ is also proportional to the three-point function Thus, (2.36) can be written as where The object on the rhs of (2.39)  [50]), one finds that the four-point partial wave is given by a linear combination of two terms Similar relationships are observed between partial waves and higher-point CBs in the comb channels.In these cases, one has (n + 2)-point correlation functions which are decomposed in the comb channel into (n + 2)-point sphere CBs.Using a decomposition analogous to the one previously discussed, the correlation functions (2.41) can also be expressed in terms of (n + 2)-point partial waves, defined as This implies a direct relation between the CBs and the partial waves (for a more detailed discussion, see [41]).

One-point torus conformal blocks via shadow formalism
The one-point torus correlation function is given by (1.1), where in the case of the Virasoro CFT, CBs F(∆ 1 , ∆, q) A (where A denotes the Virasoro algebra) receive contributions from the full Virasoro algebra.We are interested in the contribution from the sl 2 subalgebra.This contribution is given by the sl 2 global one-point torus block F(∆ 1 , ∆, q) sl 2 , defined as follows: where and 5 For simplicity of writing, we assume that the normalization factor C ∆∆ 1 ∆ = 1.We aim to show that, similar to the spherical case, the block (2.43) can be expressed in terms of one-point torus partial waves defined in some integral representation.Our discussion here follows [51], though the treatment is slightly different.The basic object used to decompose the rhs of (2.43) into partial waves is the resolution of identity operator P (2.15) constructed in the previous section.Let us insert P into (2.43) In the factor ⟨ ∆, m| φ h (z, z) |0⟩ by utilizing the formulas Taking into account (2.47) and writing φh * according to (2.9), (2.45) becomes we obtain . (2.50) To write explicitly the integrand, we use (2.37) (for the moment, we write only the coordinate dependence).Thus, (2.50) becomes (2.51) Integrating over w, w and using the formula (A.5), we obtain where and ) is according to (A.6).Equation (2.52) is the main formula we wanted to prove in this section.The rhs of (2.52), up to the factor C 2 , is the holomorphic contribution in the variable q to the one-point torus partial wave.We will discuss it in more detail below.In order to verify (2.52), we notice first that the lhs is given by Now, the rhs of (2.52) where in the last line, we did a change of variables z → z 1 z, z → z1 z.Expanding in q the integrand we have (2.56) Applying the formula6 and integrating over z, z in (2.56) we obtain where . (2.59) The dependence on z 1 , z1 disappears when we divide (2.43) by the proper normalization We check finally that C 2 C 3 = 1, which confirms the relation (2.52).
To conclude this section, we highlight two observations that will be useful in the following discussions.

Observation I
Aside from an overall constant factor, the expression in (2.52) could have been derived more straightforwardly by directly substituting the result of the three-point correlation function involving the shadow field φh * (z, z).Specifically, employing (2.49) and in (2.45), one can see that the result (2.52) is obtained.This observation is crucial since it allows, in a simple way, generalizations of (2.52) to higher-point CBs (which will be studied in the next section) and higher-spin algebras, as sl 3 which will be studied in section 3.

Observation II
The one-point CB F(∆ 1 , ∆, q) sl 2 can be directly computed from the rhs of (2.52) by performing a one-dimensional integral over z within a certain integration region.Namely, by integrating This integral is much simpler than the original two-dimensional integral.The justification of (2.61) is that the integral (2.57) factorizes into a product of two one-dimensional integrals where C is the contour shown in Fig. 1.The contribution from the integral over z is the relevant contribution that reproduces the block, while the integration over z gives just an overall factor.Hence, to compute the one-point CB from (2.52), one does not need to take the two-dimensional integral but to find the proper integration contour and integrate over it.These two observations substantially simplify the computation of the CBs from the integral representation; below, we will use them in the computation for the sl 3 one-point torus block.

Higher-point torus conformal blocks via shadow formalism
Let us introduce the object which is the holomorphic contribution in q to the two-point torus correlation function Focusing on the sl 2 subsector and using standard techniques of CFT on the torus, one can decompose (2.63) in the s-channel (also called the necklace channel) into global two-point torus CBs F ∆ 2

∆2
(z 2 , q).This is achieved by inserting the identity operator (2.65) in (2.63) between φ ∆ 1 and φ ∆ 2 after that, one obtains the decomposition (for details and explicit form of F ∆ 2

∆2
(z 2 , q) see, e.g., [46,52]) (2.66) On the other hand, one can decompose (2.63) into two-point torus conformal partial waves.This can be done in a similar way to (2.52) by inserting twice the identity operator P in the summand of (2.63): (2.67) Substituting the explicit form of P in the rhs of (2.67), we have where in the fourth and fifth lines we used (2.47,2.60),and ) is a a normalization constant similar to C 2 of the previous section.Thus, (2.63) can be written as

.69)
The factor which arises in (2.68) is the holomorphic contribution in q of the two-point torus partial wave defined as (2.71) Equations (2.66, 2.69) show the one-to-one correspondence that exists between the two-point CBs in the s-channel and the two-point torus partial waves.Therefore, we see that holomorphic two-point CBs can be expressed in terms of the holomorphic two-point conformal partial waves (see [51]).This statement is the generalization of (2.52).
Further generalization to the relation between higher multi-point CBs in the necklace channel and partial waves is possible, as discussed in [51].One can study n-point torus correlation functions: and define the n-point torus partial waves with the identifications ∆n+1 = ∆1 , w n+1 = qw 1 .By inserting the resolution of identity operator P n-times between the external fields φ ∆ i (z i , zi ) in the expression for (2.72), in a similar way to (2.67), we deduce that (2.72) can be expressed as a sum of n-point partial waves.This leads to the conclusion that these partial waves are related in a nontrivial way to the n-point torus CBs in the necklace channel.

Preliminaries: W 3 Conformal Field Theory
The W3 CFT is an extension of the Virasoro CFT.The symmetry of the W 3 CFT is generated by the energy-momentum tensor T(z) and an additional spin-3 current W(z).The Laurent series expansion of W(z) reads: The modes L n and W m generate the W 3 algebra, which reads where The Virasoro algebra (2.2) is a subalgebra of W 3 algebra.In the limit c → ∞ these commutation relations reduce to the ones of the sl 3 algebra, generated by that satisfy W 3 primary fields.We denote the W 3 primary fields characterized by the vector j as where j belongs to root space of sl 3 , generally written in the form (1.2).Alternatively, ϕ j (z, z), can be characterized by two parameters h j , q j corresponding to the conformal dimension and W 3 charge of ϕ j , given by Here ), e i are the weights of the fundamental representation and α j is also a vector on the root space of sl 3 .In the large central charge limit, α j is given by α j = −bj, (3.9) for which the conformal dimension and the charge assume the values The W 3 primary fields satisfy (similarly to (2.6)) the commutation relations where, in contrast to L n , the commutation relations with W n incorporate two additional operators, Ŵ−1 and Ŵ−2 [53], which cannot be expressed as differential operators in terms of the variable z.
A W 3 highest-weight vector |j⟩ given by satisfies the conditions The W 3 module associated with this highest-weight vector is spanned by a basis of descendant states ) The sum is called level of the state L −I |j⟩.We will focus only on the sl 3 module (which is a subspace of (3.15)) spanned by the basis of states where we define I as The level of the states (3.18) is given by (3.20)

sl 3 global one-point torus conformal blocks
The holomorphic sl 3 global one-point CB F(j 1 , j, q) sl 38 is defined as where the states |N, j⟩ belong to the sl 3 module (3.18), and is the inverse of the Shapavalov matrix Unlike the sl 2 one-point torus block, to date, there is no known exact expression for (3.21).
One reason for this is the lack of a general expression for the matrix elements In fact, for a general ϕ j 1 (3.23) is not uniquely defined.In this work, we will concentrate on the case when ϕ j 1 is a degenerate field at the first level, satisfying (1.3).This condition is fulfilled when j 1 consists of a single component.In the parameterization a perturbative expression of (3.21) up to the second level (up to q 2 ) was computed in [48], and looks as follows r(r − 1) + − 3a(3a + 3) 3a(3a + 3)(r − 1) + 36(r + 1) (r + 1)s + + 6a(3a + 3)(3a − 3r)(3a + 3r + 3) (r + 1)r(r + s + 1) q 2 + . . . .In this paper, we will provide an exact expression for (3.21) using the Shadow formalism.

sl 3 conformal blocks via shadow formalism
This section aims to extend the shadow formalism theory from sections (2.1, 2.3) to the sl 3 case, enabling us to compute the sl 3 global one-point torus block (3.21).For sl 3 global four-point sphere CBs, a similar approach to the shadow formalism was presented in [55].We will reformulate this approach using the language of shadow formalism to apply it to torus topology.We begin by reviewing the theory of sl 3 invariant functions.

sl 3 invariant functions
The sl 3 algebra can be represented in the Chevalley basis, which includes two Cartan elements (h 1 , h 2 ), along with creation and annihilation generators (e 1 , e 2 , e 3 ) and (f 1 , f 2 , f 3 ), respectively.These elements satisfy the following relations with all other commutators being zero.The generators in (4.1) can be expressed as linear combinations of the generators in (3.5).The generators (2.7) correspond to the sl 2 transformation for the variable z (representing the physical coordinates of the fields).To represent the generators of sl 3 in terms of differential operators, we introduce three-component isospin variables Z Z = (w, x, y).(4.2) For a given vector j = rw 1 + sw 2 , the generators (4.1) can be constructed as differential operators acting on monomials of the form x a y b w c for a + c ≤ r, b ≤ s, as follows To construct sl 3 invariant functions, we define the following notations: where j * i and j w i correspond to the maximal Weyl transformation and Dynkin automorphism of the spin j i , respectively.An sl 3 invariant n-point function associated with n spins j 1 , j 2 , ..., j n is a function ξ The function ξ will satisfy additional equations if some representations are labeled by only one fundamental weight: if for some k (j, w k ) = 0, then, where d (1) For two spins j 1 and j 2 , the sl 3 invariant two-point function is given by For general representations j 1 , j 2 , j 3 , the sl 3 invariant three-point function is not uniquely determined by (4.5).It can be specified up to a general function g(θ 123 ), where θ 123 = ρ 12 ρ 23 ρ 31 ρ 21 ρ 32 ρ 13 .In the case of interest, where one of the representations has only one component, say j 2 = (0, s 2 ), and the others, j 1 = (r 1 , s 1 ) and j 3 = (r 3 , s 3 ), have two components, the sl 3 invariant three-point function ξ is given by where J = (w 2 − w 1 ) • (j 1 + j 2 + j 3 ), in our case J = 1 3 (s 1 + s 2 + s 3 − r 1 − r 3 ).Two and threepoint sl 3 invariant functions (4.8, 4.9) are the analogs of the two-point function (z 1 − z 2 ) −2∆ and three-point function (2.38).

sl 3 shadow formalism
To extend the Shadow formalism to the sl 3 algebra, we introduce sl 3 fields Φ j (Z, Z, g).These fields depend on isospin variables Z, rather than on the two-dimensional complex coordinates (z, z), and are labeled by an element g ∈ SL 3 and a representation j.Detailed expressions for these fields are provided in [55] and Appendix C. They behave as primary fields under sl 3 transformations of g, satisfying Φ j (Z, Z, (1 + ϵt a )g) = (1 + ϵD (j,Z) (t a ))Φ j (Z, Z, g). (4.10) where D (j,Z) (t a ) are from (4.4).In the large c limit, the correlation functions of primary fields ϕ j i (z i , zi ) in sl 3 conformal Toda theory become [49] lim where we define and Clearly, the rhs of (4.12) is invariant under (4.10), hence the "correlation function" is an sl 3 invariant n-point function.This implies that (4.8) is interpreted as the two-point correlation function of sl 3 fields Similarly for the sl 3 three-point function (4.9).Since the integration over the group in (4.12) does not interfere with the calculation procedure we will apply, we will omit the label g in Φ j and use the following notation keeping in mind that the correlation functions of Φ j (Z, Z) are given by (4.12) by definition.
The construction of the shadow formalism for sl 3 starts with the construction of the shadow field.We define the shadow field as9 where d 2 Z ′ = dZ ′ d Z′ , and the kernel K j * (Z, Z ′ ) is given by the two-point function Analogously to (2.13, 2.15), we define the operators and Let us verify that the operator P acts as the identity operator, namely This property (4.21) will play an important role, similar to (2.16) in the sl 2 case.Let us verify (4.21) using two different methods.The first check follows directly.Writing explicitly P, we have By using (4.22) becomes where in the last equality we applied the definition (4.17), and the fact that the field Φ j 1 can be written as the shadow of Φ j * 1 .The same procedure can be applied to show ⟨0| Φ j 1 (Z 1 , Z1 )P = ⟨0| Φ j 1 (Z 1 , Z1 ).A more rigorous proof is as follows.If P acts as the identity operator, namely, ⟨0| Φ j 1 (Z 0 , Z0 )P = ⟨0| Φ j 1 (Z 0 , Z0 ), then Let us check (4.25).Writing explicitly P we have By inserting Φ j * from (4.17), and denoting the expression by I j 1 , we have One can show that the last integral is given by the rhs of (4.25).Indeed, by writing this integral in terms of the components of the isospin variables, we have where x ij = x i − x j , w ij = w i − w j .Performing the integration over y 1 , we obtain a delta function where C i (i = 1, 2, 3) are constants present in the delta function formula.Performing the integration over w 1 and y 2 , we obtain  Performing the integration over w 2 , we get and then, after integrating over x 1 and x 2 , we finally obtain This proves (4.25) (up to the normalization constant C).Thus, the above proof shows that P plays the role of the resolution of the identity operator.Since there is such a resolution of identity in sl 3 , one can proceed as in the sl 2 case.

sl 3 global four-point sphere conformal blocks via shadow formalism
In W 3 CFT, analogous to (2.34), one can decompose the four-point correlation function involving external fields free of multiplicities into CBs.On the other hand, one can use shadow formalism to obtain an integral representation of the conformal block.For this, one needs to compute the four-point correlation function of sl 3 fields To avoid the problem of multiplicities, one chooses j 1 = (r 1 , s 1 ), j 2 = (0, s 2 ), j 3 = (0, s 3 ), j 4 = (r 4 , s 4 ).(4.34) Then, one inserts the resolution of identity operator P as follows The result of this procedure is that one decomposes (4.33) in terms of the following object (4.36) To simplify the above integral and identify this object with the sl 3 global four-point sphere CB F s (j 4 , j, z 4 ) sl 3 (where j 4 = j 1 , j 2 , j 3 , j 4 ), one needs to find the proper integration contour C s sl 3 over Z (this is given in appendix B.1 of [55]).After this simplification, one obtains the CB from (4.36) as follows where N s is a normalization constant chosen properly to have the correct asymptotic behavior, and the function ξ(j 1 , j 2 , j 3 |Z 1 , Z 2 , Z 3 ) is given by (4.9).The above result (4.37) reproduces the known result of the sl 3 global four-point CB [55], which has also been obtained from the AdS 3 holographic perspective [56].
4.4 sl 3 global one-point torus conformal blocks via shadow formalism In this section, we study the sl 3 global one-point torus block using the shadow formalism developed in section 4.2 and provide an expression for the CB.We start by presenting the result.We have found that the sl 3 global one-point torus conformal block (3.21), for j = (r, s) and j 1 = (3a, 0), is given by the following integral representation and relations where the final expression for F (j 1 , j, q) is given below by (4.51).N (r, s, a) is a normalization factor chosen properly so that the expansion in q starts from 1, as in (3.25).ξ is from (4.9), j * w = (2 − r, 2 − s)10 , dZ = dwdxdy, C sl3 is a proper integration contour defined below in (4.49), and Z • q = (q 2 w, qx, qy).
In the remaining part of this section, we will justify (4.38) along the lines of the sphere case and compute the rhs of the first line of (4.38).To proceed, we redefine (3.21) in terms of sl 3 fields, namely The external and intermediate fields Φ j w 1 (⃗ z 1 , ⃗ z1 ), Φ j (Z, Z) are sl 3 fields that depend on isospin variables.The descendant states are given as follows The operators L 0 , W −2 , L −1 , W −1 satisfy the algebra (3.5) and are given by a linear combination of differential operators (4.4).We use the conjectured holomorphic sl 3 operator product expansion From this expansion, we have We insert P into (4.40),obtaining Using (4.43) in the factor ⟨j, M | Φ j (Z, Z) |0⟩, and q L 0 |N, j⟩ = q ∆(j)+|N | |N, j⟩ and after the cancelation of the matrices B −1 and B, we obtain Since |N | = n 1 + n 2 + 2n 3 , we can write q |N | = q n 1 +n 2 +2n 3 , and applying again (4.43), we obtain For the coordinate dependence of the three-point correlation function of sl 3 fields, according to the above discussion, we use the formula hence F (j 1 , j, q) = q ∆(j) d 2 Zξ(j * w , j w 1 , j|Z, ⃗ z 1 , Z.q)ξ(j * w , j w 1 , j| Z, ⃗ z1 , 0). (4.48) Equation (4.48) is the direct analog to (2.52).As explained in observation II of section 2.3, to obtain the CB from (4.48), one needs to find the proper integration contour in order to simplify the above integral.Furthermore, after replacing z 1 in (4.48) by using (4.9), we notice that it is possible to take out z 1 as an overall factor.Since z 1 does not play an important role in our discussion, we set z 1 = 1.Under this consideration, we found that this contour is Hence, (4.48) can be simply written as the first line of (4.38), where in the normalization coefficient N (r, s, a) of (4.38) we absorbed the factor q ∆(j) .Finally, we compute the first line of (4.38).Our final result precisely reproduces (3.25), namely, we found the relation given in the second line of (4.38).The expression we found for F (j 1 , j, q) from (4.38) is given by the integral In appendix B, we explain how we compute this integral; here we present the final result

Conclusions
In this work, we have studied global conformal blocks using shadow formalism.Our study focused on sl 2 and sl 3 global CBs, which arise in the large central charge limit of Virasoro and W 3 conformal field theories.In sections 2.

(B.3)
To integrate over y, we expand the factor (y − a) k in y.This expansion adds an extra sum from 0 to k present in the final result (4.51).The result of the integration over y is given by a function of w, x which contains a hypergeometric function 2 F 1 with the argument 1 − 1 2x .Then, we perform the integration over w.The integration over w has the form where α is some power.The result of (B.4) is given by a ratio of gammas functions.Finally, to compute the integration over x, we expand the above-mentioned hypergeometric function in 1 − 1 2x (the sum from this expansion was already considered in (4.51) and produces the factor 3 F 2 ).Thus, the integration over x has the form where α, β are again some powers.After integrating over y, w, x, the final result is given in (4.51).

C sl 3 Fields
In this appendix, we briefly describe the sl 3 fields denoted above as Φ j (Z, Z, g).For a more detailed description, see [55].The fields Φ j (Z, Z, g), for g ∈ SL 3 , are basis functions on SL 3 defined as Φ j (Z, Z, g) = u Z P g −1T P u T

(3. 25 )
The computation of higher-level contributions to (3.25) is a very challenging task, specially when one computes them directly from(3.23).The expression (3.25) was verified by two other methods, using the AGT relation and the Wilson lines interpretation of CBs in AdS 3 .