Preprint typeset in JHEP style- HYPER VERSION The large central charge limit of conformal blocks

We study conformal blocks of conformal field theories with a W3 symmetry algebra in the limit where the central charge is large. In this limit, we compute the four-point block as a special case of an sl3-invariant function. In the case when two of the four fields are semi-degenerate, we check that our results agree with the block's combinatorial expansion as a sum over Young diagrams. We also show that such a block obeys a sixth-order differential equation, and that it has an unexpected singularity at z=-1, in addition to the expected singularities at z=0,1,infinity.


Introduction
Since the work of Belavin, Polyakov and Zamolodchikov [1], the conformal bootstrap method has been an effective tool for studying two-dimensional conformal field theories. That method relies on a systematic exploitation of the symmetries of the theory. These symmetries determine functions called conformal blocks. The simplest conformal blocks are the characters of the representations of the symmetry algebra, which may be called zero-point blocks on the torus. The correlation functions of the theory are then combinations of the conformal blocks. For example, the partition function on the torus is a combination of characters.
Combinatorial expansions for the conformal blocks of the Virasoro algebra have recently been found [2], inspired by the conjecture of Alday, Gaiotto and Tachikawa on the relation between two-dimensional CFTs and four-dimensional gauge theories [3]. Until then, no explicit formulas for four-point blocks on the sphere were known. It would be very interesting to generalize such combinatorial expansions to conformal blocks of other algebras, in particular the W N algebras [4,5,6] which are natural generalizations of the Virasoro algebra. This is however a challenging problem, in particular because the fusion products of W N ≥3 representations in general exhibit infinite fusion multiplicites. (See for instance Section 2.3 of [7].) This feature is at the origin of difficulties in computing the three-point correlation functions in conformal Toda theories. These CFTs have W N symmetry algebras, and their three-point correlation functions are only known in special cases [8,9]. Here we will show how to take infinite fusion multiplicities into account and how to compute conformal blocks in the limit where the central charge c of the W N algebra is large. (The conformal dimensions of the fields are meanwhile kept fixed; this is sometimes called the light asymptotic limit.) It is this limit which was used by Al. Zamolodchikov as the starting point of the characterization of Virasoro (N = 2) conformal blocks by recurrence [10]. In this limit, sℓ N conformal Toda theory reduces to the quantum mechanics of a point particle on SL N (C), the W N algebra reduces to sℓ N , and the W N conformal blocks reduce to special cases of sℓ N -invariant functions. We will study such functions in detail in the cases N = 2 and N = 3.
This will enable us to test a proposal for the combinatorial expansion of a class of W N conformal blocks [11]. The proposed expansion, which we will summarize, is given for all values of c, assuming that all involved fields except two of them are almost fully degenerate. This assumption eliminates the problem of the infinite fusion multiplicities. We will compare our results for large c conformal blocks of that class, with the large c limit of the combinatorial expansion. The two expansions agree up to the order z 5 , which supports the validity of the proposed combinatorial expansion.

Plan of the article.
In Section 2, after a reminder on the large c limit of Virasoro conformal blocks, we study sℓ 3 -invariant functions and their relations with the large c limit of W 3 conformal blocks. Section 3 is devoted to the study of detailed properties of certain conformal blocks: the differential equation they obey, and their critical exponents. Then, in the concluding Section 4, we comment on some aspects of the results, and compare them with the combinatorial expansion. Appendix A is devoted to the study of a quantum particle on SL N (C) (with N = 2, 3), which is at the basis of the computation of the large c limit of correlation functions of sℓ N conformal Toda theory, and provides some justification for a number of the equations of Section 2. Appendix B is devoted to deriving the series expansion (2.66) of certain conformal blocks. be found in [12]. A four-point s-channel Virasoro conformal block on the sphere G ∆s (c|∆ i |z i ) is a function of the positions (z 1 , z 2 , z 3 , z 4 ) ∈ C 4 of four primary fields, which are characterized by their conformal dimensions (∆ 1 , ∆ 2 , ∆ 3 , ∆ 4 ). Such a conformal block also depends on an schannel conformal dimension ∆ s , and on the central charge c of the Virasoro algebra, which is defined by generators L n∈Z and relations [L n , L m ] = (n − m)L m+n + c 12 n(n − 1)(n + 1)δ m+n,0 . (2.1) The conformal block is defined as a sum over the states of a highest-weight representation of the Virasoro algebra. The relevant representation is built from a highest-weight state |∆ s by applying the creation modes L n<0 . If we assume L † n = L −n and ∆ s |∆ s = 1, then we can compute the square norm ||L −n |∆ s || 2 = 2n∆ s + c 12 n(n−1)(n+1). If |n| ≥ 2, this goes to infinity as c → ∞, and L −n descendents do not contribute to the "large c block" Therefore, only the generators (L −1 , L 0 , L 1 ) of the sℓ 2 subalgebra of global conformal transformations survive in the large c limit. Nonetheless, some properties of the blocks are not affected by taking this limit: First, the existence of an analytic expansion in the neighbourhood of z 1 = z 2 , where we use the notation z 12 = z 1 − z 2 . Second, the behaviour under global conformal transformations, which we now review. The sℓ 2 subalgebra of global conformal transformations has the generators (L −1 , L 0 , L 1 ) and commutation relations A representation of the Virasoro algebra with conformal dimension ∆ corresponds to an sℓ 2 representation of spin −∆. Notice that two Virasoro representations whose dimensions are related by the reflection correspond to two isomorphic sℓ 2 representations. A primary field with position z and conformal dimension ∆ behaves as a vector in an sℓ 2 representation of spin −∆ and isospin variable z. The action of the sℓ 2 subalgebra on the primary field is given by the differential operators which are such that D (∆,z) preserves the commutation relations (2.4). The blocks G ∆s (c|∆ i |z i ), and therefore their large c limits F ∆s (∆ i |z i ), are sℓ 2 -invariant four-points functions. What we call an sℓ 2 -invariant n-points function is a function E(∆ i |z i ) of (∆ 1 , ∆ 2 , · · · ∆ n ) and (z 1 , z 2 , · · · z n ) such that For brevity we will sometimes use the notation E(z) = E(∆ i |0, z, 1, ∞). After these reminders on the global conformal symmetry, we are ready to write an explicit integral formula for the large c four-point conformal block, where the normalization factor N (a function of ∆ i , ∆ s ) and integration contour C are determined by the condition (2.3). This expression is justified in Appendix A.1. With a general integration contour, the integral in eq. (2.12) would yield a linear combination of the two "reflected" blocks F ∆s and F ∆ * s . Explicitly, we find where F is the hypergeometric function, and we use the notations and The simplification of the conformal blocks in the large c limit can be interpreted as coming from the elimination of local conformal symmetry, and the survival of only the global symmetry with its finite-dimensional sℓ 2 algebra. This explains why the large c blocks (2.14) coincide with the conformal partial waves which were computed by Ferrara, Gatto and Grillo [13]. Such conformal partial waves are associated to the global conformal symmetry, and can therefore be generalised to higher dimensions [14] 1 . We will be interested in another type of generalisation: staying in two dimensions, we will consider larger symmetry algebras.

sℓ 3 -invariant functions
In preparation for writing the large c conformal blocks of the W 3 algebra, we need to study sℓ 3invariant functions. This is because the W 3 algebra reduces to sℓ 3 in the large c limit [15] 2 , in the same way as the Virasoro algebra reduces to sℓ 2 . (The eight generators L 0 , L ±1 , W 0 , W ±1 , W ±2 of the W 3 algebra which survive in the large c limit can be identified with linear combinations of the generators h i , e i , f i of sℓ 3 which we are about to introduce.) The algebra sℓ 3 is generated by two Cartan elements (h 1 , h 2 ), three generators (f 1 , f 2 , f 3 ) which are eigenvectors of the adjoint actions of h 1 and h 2 for the respective eigenvalues (−2, 1, −1) and (1, −2, −1), and three generators (e 1 , e 2 , e 3 ) which are also eigenvectors but with opposite eigenvalues. The remaining nonzero commutators are In order to parametrize the representations of sℓ 3 , let us introduce its simple roots (e 1 , e 2 ) and the weights of the fundamental representation (h 1 , h 2 , h 3 ) (not to be confused with the sℓ 3 generators e i , h j ). The roots are supposed to be two independent vectors, with a scalar product given by the Cartan matrix, (e i , e j ) = K ij with K = 2 −1 −1 2 . The weights of the fundamental representation are (2.20) A representation is parametrized by a spin vector j in root space, whose coordinates we denote as r = −(e 1 , j) , s = −(e 2 , j) . among which we single out the maximal Weyl reflection j → j * where There is another useful reflection of the root space called the Dynkin diagram automorphism j → j ω where j = (r, s) ⇒ j ω = (s, r) .

(2.24)
In the previous Subsection, we represented sℓ 2 transformations in terms of differential operators (2.6), whose isospin variable z could be interpreted as the position of a CFT field on the complex plane. In order to faithfully represent sℓ 3 transformations, we need a triple of variables Z = (w, x, y). (The number of needed variables is the number of creation operators e i ; in the case of sℓ N this would be N (N −1)

2
.) The sℓ 3 generators (h i , e i , f i ) are represented as [12](Section 15.7.4) An sℓ 3 -invariant n-point function associated to n spins j 1 , j 2 , · · · j n is a function Such an invariant will obey additional equations if some representations are degenerate. We will call the representation of spin j 1 semi-degenerate of the first (k = 1) or second (k = 2) kind if (e k , j 1 ) = 0 and d (k) where the differential operators d These formulas for d (1) Z and d (2) Z will be justified in Appendix A.2. Let us write the solutions of the sℓ 3 invariance equation (2.33) in the cases of two-and threepoint invariants. We will write the solutions of these equations in terms of convenient combinations of isospin variables Z i = (w i , x i , y i ), We also introduce the three-point invariant We then find that a nonzero two-point invariant can exist only provided j 1 = j ω 2 (up to Weyl reflections), and the invariant is then Consider now three-point invariants. The function E(j 1 , j 2 , j 3 |Z 1 , Z 2 , Z 3 ) depends on nine variables which are the components of Z 1 , Z 2 , Z 3 , and is subject to the eight equations (2.33). Therefore, there exists an infinite-dimensional space of solutions. This corresponds to the existence of a nontrivial invariant θ 123 (2.39). In the special case when one of the three representations is semi-degenerate, we have an extra equation of the type (2.34), and the space of solutions is onedimensional. Let us start with the case when the first representation is semi-degenerate of the first kind. The three-point invariant should be built from combinations of isospin variables which are killed by the differential operator d We then find the three-point invariant where we introduced the combination of spins Similarly, if the first representation is semi-degenerate of the second kind, we can use combinations which are killed by d and we find the three-point invariant In general, when no representation is semi-degenerate, the most general three-point invariant is This depends on an arbitrary "multiplicity function" g 1 (θ), or on the equivalent function g 2 (θ) = θ −J−r 1 −r 2 +s 3 (θ + 1) −J g 1 (θ). This function encodes the infinite multiplicity of say the third representation in the tensor product of the first two representations. The same feature manifests itself in the fusion products of W 3 representations, we have called this the problem of the infinite fusion multiplicities in the Introduction. Bases of n-point invariants can be built from three-point invariants. For instance, s-channel four-point invariants can be built as where j s is the s-channel spin, g, g ′ are two multiplicity functions, N is a normalization factor which may depend on j s , j i , g, g ′ , and C is an integration domain for Z s ∈ C 3 . The integration measure is the sℓ 3 -invariant measure dZ = dwdxdy.

W 3 conformal blocks in the large c limit
A four-point s-channel W 3 conformal block on the sphere G g,g ′ |αs (c|α i |z i ) is a function of the positions (z 1 , z 2 , z 3 , z 4 ) of four primary fields characterized by their momenta (α 1 , α 2 , α 3 , α 4 ), and of the central charge c of the W 3 algebra. The block also depends on an s-channel momentum α s , and on two multiplicity functions g, g ′ . The presence of such multiplicity functions is in general necessary due to the presence of infinite fusion multiplicities, and we have given a precise definition of such multiplicity functions in the case of sℓ 3 -invariant functions in the previous Subsection. We will however not try to define such functions in the case of W 3 conformal blocks, except in the large c limit. Let us introduce standard notations on W 3 representations. Let b and q = b + b −1 be such that the central charge is c = 2 + 24q 2 . A highest-weight representation of the W 3 algebra is parametrized by its momentum α, a two-dimensional vector which belongs to the root space of sℓ 3 . Such a representation can alternatively be parametrized by its conformal dimension ∆ α and a charge q (3) α which is (up to a normalization factor) the eigenvalue of the spin 3 current, such that where we defined Q = qρ, which involves the Weyl vector ρ = e 1 + e 2 .
The large c limit is defined as This is sometimes called the light asymptotic limit. In the large c limit, the W 3 algebra reduces to sℓ 3 , and the vector j is the spin of an sℓ 3 representation. This spin is related to the limits where r and s are the components of the spin j, see eq. (2.21). Let us define the large c four-point conformal blocks, In analogy with the case of Virasoro conformal blocks, the large c conformal blocks of the W 3 algebra can be computed as special cases of sℓ 3 -invariant functions. We claim that an isospin variable Z = (w, x, y) of an sℓ 3 -invariant function must then be of the type Z = z where we define This relation between the isospin Z and the worldsheet position z comes from the following identities, which hold for any function E(Z): where the sℓ 2 differential operators D (∆,z) (t a ) were defined in eq. (2.6), and the sℓ 3 operators D (j,Z) (t a ) in eq. (2.25)-(2.32). These identities show that the principally embedded sℓ 2 subalgebra of sℓ 3 with generators (h 1 + h 2 , e 1 + e 2 , f 1 + f 2 ) can be identified with the sℓ 2 algebra of global conformal transformations. (See also [15].) Therefore, a large c four-point block is a special case of a four-point sℓ 3 -invariant function (2.50), where the maximal Weyl reflection j → j * and the Dynkin diagram automorphism j → j ω were defined in eqs. (2.23) and (2.24), and the three-point invariant E g was given in eq. (2.48). The normalization factor N and the integration domain C for Z s ∈ C 3 are determined by the condition (2.3). Other choices of integration domains in eq. (2.59) would lead to linear combinations of six conformal blocks whose spins j s are related by Weyl transformations (2.22). Notice that the condition (2.3) of analyticity and normalization of conformal blocks also constrains the multiplicity functions g, g ′ . Applying that condition to a three-point invariant function E g (j 1 , j 2 , j s | z 1 , z 2 , Z s ) leads to the conditions We will now focus on a large c four-point block F js (j i |z i ) such that the fields with numbers 2, 3 are semi-degenerate of the first kind, so that the multiplicity functions g, g ′ disappear and the components r 2 , r 3 of the spins j 2 , j 3 vanish. The assumptions Z 1 = z 1 and Z 2 = z 2 lead to simplifications in the combinations ρ 12 (2.36) and χ 123 (2.38), where the conformal dimensions ∆ i are associated to the spins j i as in eq. (2.53). Then the formula (2.59) implies that z ∆ 1 +∆ 2 F js (j i |0, z, 1, ∞) depends on only four combinations of the six nonvanishing components of the spins j 1 , j 2 , j 3 , j 4 , namely It is actually convenient to use the following four combinations, where we call (r, s) the components of j s , The integral in eq. (2.59) can be expanded near z 1 = z 2 , see Appendix B. This leads to the expansion of the large c four-point block F js (z) = F js (j i |0, z, 1, ∞) near z = 0, where the notation (t) n was defined in eq. (2.15). The second form of this expression is obtained by performing the sums over i and j, and can be helpful in numerical computations. A similar expression can be obtained if the fields with numbers 2, 3 are semi-degenerate of the second kind (instead of the first kind), by exchanging the two components r and s of each spin.
There are five special cases where the block F js (z) (2.66) reduces to a hypergeometric function: This shows that blocks can have a singularity at z = −1, in addition to the physical singularities at z = 0, 1, ∞ which appear when two of the fields at z 1 , z 2 , z 3 , z 4 come together.

Case of Virasoro conformal blocks
The large c limit F ∆s (∆ i |z i ) of a Virasoro four-point block obeys a second-order hypergeometric differential equation, and we now explain how to deduce this equation from the integral expression (2.12) for F ∆s (∆ i |z i ). The large c block is indeed a function of four variables z i , which obeys the three equations (2.7) of global conformal invariance. In addition, the three-point invariant E(∆ 1 , ∆ 2 , ∆ s |z 1 , z 2 , z s ) which appears in the integral expression (2.12) also obeys these equations, and together with the relation (A.5) for the quadratic Casimir C 2 (∆) this implies where the differential operators D (∆,z) (t a ) are defined in eq. (2.6). Thus F ∆s (∆ i |z i ) obeys four differential equations, and F ∆s (∆ i |0, z, 1, ∞) obeys one differential equation, which turns out to be the hypergeometric equation, whose solution (subject to the condition (2.3)) we wrote in eq. (2.14).
The critical exponents of the hypergeometric equation are known, and we deduce the critical exponents λ i of z ∆ 1 +∆ 2 F ∆s (∆ i |0, z, 1, ∞) at the three singularities z = 0, 1, ∞: The number λ (0) 2 = ∆ * s = 1 − ∆ s is not really a critical exponent of the block itself, rather it corresponds to another block with another s-channel dimension ∆ * s . Notice that the exponents at 1 and ∞ are ∆ s -independent; conformal blocks can be expected to behave so simply at these singularities only in the large c limit. The transformations of the hypergeometric functions can be used to rewrite F ∆s (∆ i |0, z, 1, ∞) as a combination of two functions with simple monodromy at say z = 1, but these two functions are not themselves conformal blocks in another channel.

Differential equation for large c W 3 conformal blocks
That our large c four-point conformal block F js (j i |z i ) (2.66) with two semi-degenerate fields obeys a differential equation follows from a simple counting of variables and equations. We consider first the corresponding sℓ 3 four-point invariant function E js (j i |Z i ), which depends on 12 isospin variables, namely the components of Z 1 , Z 2 , Z 3 , Z 4 , and is such that F js (j i |z i ) = E js (j i | z i ) where z is defined in eq. (2.55). There are three types of equations for E js (j i |Z i ): 1. The sℓ 3 symmetry condition (2.33) yields 8 equations.
2. As two fields are semi-degenerate, we have two equations d (1) Z was defined in eq. (2.35).
3. The sℓ 3 symmetry condition for the three-point invariant E g (j 1 , j 2 , j s |Z 1 , Z 2 , Z s ) which appears in the integral formula (2.50) will yield two more equations. The sℓ 3 symmetry condition (2.33) applied to E g (j 1 , j 2 , j s |Z 1 , Z 2 , Z s ) indeed implies using the equations (A.23) and (A.24) which involve the the Casimir numbers C 2 (j s ) and C 3 (j s ).
The function E js (j i |Z i ) of 12 variables therefore obeys 12 partial differential equations. Now the function F js (j i |z i ) can be written in terms of a function of just one variable (the cross-ratio of z 1 , z 2 , z 3 , z 4 ), and this function obeys one differential equation. The order of the differential equation can be guessed to be six, the order of the Weyl group of sℓ 3 . This is because our differential equations (3.3) and (3.4) depend on the spin j s through the Weyl invariants C 2 (j s ) and C 3 (j s ). Given a solution, Weyl reflections of j s therefore provide five other solutions. Let us explain how the differential equation for F js (j i |z i ) can be derived in principle. We will not perform the derivation to the end, as the resulting equation would be too complicated to be useful. We will stop at the partial differential equations for E js (j i |Z i ), which imply the equation for F js (j i |z i ) and are much simpler. It is from these partial differential equations that we will derive interesting information like the critical exponents of F js (j i |z i ). To begin with, the first ten equations allow us to rewrite E js (j i |Z i ) in terms of a function of two variablesÊ js (j i |U, V ), where the cross-ratios U, V are solutions of our ten equations when all spins are taken to zero, and the prefactor Q(j i |Z i ) is the product of two three-point invariants of the type (2.44) with one spin set to zero in each invariant (thereby imitating the prefactor P (∆ i |z i ) (2.11) of sℓ 2 -invariant functions), We have Q(j i | z i ) = P (∆ i |z i ) where P (∆ i |z i ) was defined in eq. (2.11) and ∆ i in eq. (2.53), and together with the expression (2.10) for F js (j i |z i ) in terms of F js (z) = F js (j i |0, z, 1, ∞) this implies z ∆ 1 +∆ 2 F js (z) =Ê js (j i |z, z) . (3.10) The two equations (3.3) and (3.4) amount to two partial differential equations forÊ js (j i |U, V ), which we computed with the help of the free mathematical software Sage. The equations are of the type E 2Êjs (j i |U, V ) = E 3Êjs (j i |U, V ) = 0, where the differential operators E 2 and E 3 are where we defined D U = U ∂ ∂U and D V = V ∂ ∂V . (The combinations j 12 , j 34 of the components r i , s i of the spins j i were defined in eq. (3.9).) The differential operators E 2 and E 3 commute, as guaranteed by their origin in the sℓ 3 -invariant differential operators which appear in eqs. (

3.3) and (3.4).
Let us sketch how a sixth-order differential equation forÊ js (j i |z, z) is obtained from the two partial differential equations E 2 and E 3 forÊ js (j i |U, V ). We cannot directly set U = V in E 2 and E 3 , because these differential operators do not keep the line {U = V } invariant. To cure this problem, we take linear combinations of E 2 and E 3 with differential operators as coefficients, so as to eliminate D U − D V while keeping the derivative D U + D V along {U = V }. This yields a differential operator of the type The resulting differential equation is however too complicated to be useful. We were able to compute it explicitly (with the help of a computer) only in special cases when some parameters j i , j s vanish. Even so, the equation is rather complicated, and we do not display it. We will however discuss its singularities and the corresponding critical exponents. Knowing the critical exponents at a given singularity is equivalent to knowing the leading term of the differential equation E 6 near that singularity. To derive this, the algorithm for obtaining the differential equation E 6 from E 2 and E 3 can be applied to the first few leading terms of E 2 , E 3 near the singularity. (Keeping one term of each equation is in general not enough, except at the singularity z = 0 as we shall see.)

Singularities and critical exponents of large c W 3 conformal blocks
A four-point correlation function 3 conformal Toda theory (or actually in any conformal field theory) is expected to have singularities at z i = z j , which in terms of the cross-ratio amounts to z = 0, 1, ∞. A conformal block like G g,g ′ |αs (c|α i |z i ), and its large c limit F g,g ′ |js (j i |z i ), is therefore also expected to be singular at these points. However, nothing in principle excludes the existence of extra singularities in conformal blocks, and in the case when two fields are semi-degenerate, we will indeed find that F js (j i |z i ) has an unexpected singularity at z = −1, as we already noticed in a special case (2.68).
While it is not clear to us why this singularity appears, we can at least explain why the point z = −1 is special. The set {0, 1, ∞} of the physical singularities is invariant under a set of six P SL 2 (Z) transformations z → (z, 1 − z, 1 z , 1 1−z , z z−1 , 1 − 1 z ). But in our correlation functions the fields 2 and 3 are semi-degenerate, and the only nontrivial transformation which does not mix them with the other fields is z → 1 z . The point z = −1 is characterized as the nontrivial fixed point of that transformation. Notice that this argument is specific neither to s-channel conformal blocks (as opposed to blocks in other channels or to correlation functions), nor to the large c limit (as opposed to generic values of c).
But let us first comment on the singularity at z = 0. In the limit U, V → 0, the partial differential equations E 2 (3.11) and E 3 (3.12) forÊ js (j i |U, V ) are reduced to their respective first lines. Let look for solutions of the typê (assuming c 0,0 = 1) , (3.13) for some exponents (µ, ν). Using the expressions (A.20) and (A.21) for C 2 (j s ) and C 3 (j s ) respectively, we find six solutions which correspond to the pairs (µ, ν) such that j s = −νe 1 − µe 2 up to Weyl reflections. We adopt the solution (µ, ν) = ( 2s+r 3 , 2r+s 3 ) where (r, s) are the components of j s . The equation E 2Êjs (j i |U, V ) = 0 leads to a recursion relation for the coefficients c m,n , where the combinations α, β, γ, δ of spin components were defined in eq. (2.65). This relation has a unique solution such that c 0,0 = 1 (assuming c m,n = 0 unless m, n ≥ 0), which is So, once the critical exponents (µ, ν) are deduced from E 2 and E 3 , the equation E 2 is enough to determine the solution uniquely. The equation E 3Êjs (j i |U, V ) = 0 leads to another recursion relation, which however has the same solution. Setting U = V = z inÊ js (j i |U, V ) as in eq. (3.10), we recover the expression (2.66) for F js (z). It is less straightforward to compute the critical exponents of z ∆ 1 +∆ 2 F js (z) at z = 1, ∞, −1 than at z = 0, and we present only the results. The function z ∆ 1 +∆ 2 F js (z) obeys a differential equation of order six, and therefore has six exponents at each singularity, which we number arbitrarily: 3 where the nontrivial critical exponent at z = −1 is These critical exponents of z ∆ 1 +∆ 2 F js (z) can be compared with those of conformal blocks of the Virasoro algebra (3.2). Notice that the critical exponents at a regular point are (0, 1, 2, 3, 4, 5), so the point −1 is almost regular in that only one exponent is not an integer. The sum of all exponents (with a minus sign for the exponents at ∞) In the five special cases at the end of Subsection 2.3, the blocks reduce to hypergeometric functions, and at each singularity we recover two exponents out of six. Most of the critical exponents (3.16) predicted by the differential equation can thus be confirmed in these special cases.
Finally, the existence of the surprising singularity at z = −1, and the value of the critical exponent λ, can be confirmed using the integral representation (B.1) of the conformal block. After some manipulations which are rather straightforward, we indeed find that the block has the asymptotic behaviour (if λ < 0) As expected, the coefficient of (z + 1) λ vanishes in the four special cases where the singularity at z = −1 disappears, see eq. (2.69)-(2.72).

Comparison with the combinatorial expansion
A combinatorial expansion is proposed in [11] for four-point conformal blocks of the W 3 algebras, such that two fields are semi-degenerate and therefore no infinite fusion multiplicities are present.
If the fields with numbers 2, 3 have momenta along the weight h 1 , that is where the function F λ, λ ′ (α, α ′ , r) is defined as Besides the "W 3 notations" ∆, α, Q, q, b introduced in Subsection 2.3 and the definition (2.24) of the Dynkin diagram automorphism j → j ω , these formulas use notations for Young diagrams which we now review. (See [2] for more details.) The sum in eq. (4.1) is over triples λ = (λ 1 , λ 2 , λ 3 ) of Young diagrams. Each diagram is a collection of boxes, and each box s has an arm length a λ (s) and leg length l λ (s) relative to a diagram λ, which are positive if s ∈ λ. The triple 0 = (0, 0, 0) is the set of three empty diagrams.
We have checked that the large c limit (2.52) of the block (4.1) agrees with the independently derived prediction eq. (2.66) up to the order z 5 , modulo the exchange of the components r and s of the spins due to the use of different kinds of semi-degenerate fields. The agreement is rather non-trivial, because individual terms of the sum over Young diagrams can have spurious poles (as functions of the components of α s ), and these spurious poles cancel when the sum is performed.

Concluding remarks
In this article, we have given an integral formula (2.59) for the large c limits of arbitrary W 3 conformal blocks on the sphere. This formula is a special case of an sℓ 3 -invariant function, where the isospin variables take special values determined by the positions of the fields. This result implies that the large c conformal block depends nontrivially on only eight combinations of the ten components of the spins (j s , j 1 , j 2 , j 3 , j 4 ), in the same way as the large c Virasoro conformal block (2.14) depends nontrivially on only three combinations of the five conformal dimensions.
We have thus shown how to take infinite fusion multiplicities into account in this limit. This might be helpful for solving the problem of the infinite fusion multiplicities in general, and allow us to deal with arbitary conformal blocks in conformal field theories with W N symmetries. So far, we know combinatorial expansions only for blocks with no fusion multiplicities. For fusion multiplicities to be absent, it is necessary to restrict the momenta of the fields, such that they are all almost fully degenerate except two of them. (See also [11].) While we can deal with large c blocks in general, imposing such restrictions brings important simplifications. Thus we studied the detailed properties a certain class of W 3 large c four-point conformal blocks with two semidegenerate fields, and in particular we derived their series expansion (2.66). We found that, for generic values of the parameters, such blocks have a singularity at z = −1, in addition to the expected singularities at z = 0, 1, ∞. We believe that the singularity at z = −1 is absent for non-infinite values of the central charge c. It would be interesting to confirm this expectation, and to understand how the singularity disappears for finite values of c.
We expect that our results can be generalized to sℓ N -invariant functions and W N conformal blocks in the large c limit. In particular, we expect the large c limit of a W N four-point conformal block with two almost fully degenerate fields to obey a differential equation of order N !, as we observed in the cases N = 2 and N = 3 in Section 3.

A. Quantum mechanics of a point particle on SL N (C)
For any integer N ≥ 2, the algebra W N is the symmetry algebra of a CFT called sℓ N conformal Toda theory. In the case N = 2 for example, the Virasoro (W 2 ) algebra is the symmetry algebra of Liouville (sℓ 2 conformal Toda) theory. The functional integral representation of the correlation functions of sℓ N conformal Toda theory can be used for studying their large c limits, which turn out to be correlation functions of the quantum mechanics of a point particle on SL N (C). (See for instance [8].) This will provide some justification for our identification of large c conformal blocks with sℓ N -invariant functions. We will start with the case of the point particle on SL 2 (C), before dealing with the technically more complicated case of SL 3 (C).

A.1 Point particle on SL 2 (C)
Let us call V ∆ (z,z) a primary vertex operator of Liouville theory. This depends on the complex coordinates (z,z) of a point on the complex plane, and on the conformal dimension ∆ of the corresponding Virasoro representation. The functional integral representation of a correlation function n i=1 V ∆ i (z i ,z i ) of n such vertex operators leads to the large c limit where Φ ∆ z,z (g) is the classical limit of the field V ∆ (z,z), evaluated on a solution of the Liouville equation labelled by g ∈ SL 2 (C), (Real solutions of the Liouville equation actually correspond to Hermitian matrices g. We will neglect this subtlety.) The functions Φ ∆ z,z (g) and v z have a simple behaviour under sℓ 2 tranformations of the matrix g: where the differential operators D (∆,z) (t a ) were introduced in eq. (2.6), and the generators (t a ) = (L 1 , L 0 , L −1 ) of sℓ 2 are realized as the matrices Now the differential operators D (∆,z) (t a ) obey the relation where we defined Together with eq. (A.3), this implies that Φ ∆ z,z (g) is an eigenvector of the sℓ 2 -invariant Laplacian on SL 2 (C). Actually, the functions Φ ∆ z,z (g) provide a basis of functions of SL 2 (C), whose completeness can be expressed as the decomposition of the Dirac delta function δ(g, g ′ ), where the reflected dimension ∆ * = 1 − ∆ was defined in eq. (2.5). The equivalence between representations with labels ∆ and ∆ * manifests itself in the relation C 2 (∆ * ) = C 2 (∆), and in the reflection relation where R(∆) is a reflection coefficient, and the two-point invariant E(∆, ∆|z, z ′ ) was given in eq. (2.8).
Let us go back to Liouville theory, of which SL 2 (C) quantum mechanics is the large c limit. An important axiom of conformal field theory is the assumption that a four-point function can be decomposed into four-point conformal blocks, where the structure constant C(c|∆ 1 , ∆ 2 , ∆ 3 ) depends on the central charge c and on the conformal dimensions ∆ i , but not on the field positions z i . (We will not study how the c-dependent contour of integration must be manipulated in the large c limit.) We now check that our formula (2.12) for the large c four-point block F ∆s (∆ i |z i ) is compatible with the large c limit of this axiom. To do this, we insert the identity 1 = SL 2 (C) dg ′ δ(g, g ′ ) together with the formula (A.7) for δ(g, g ′ ) in the large c limit (A.1) of the four-point function, The integrals over g and g ′ produce z-dependent factors proportional to the three-point invariant E(∆ 1 , ∆ 2 , ∆ 3 |z 1 , z 2 , z 3 ) (2.9), and z-independent factors which we call B(∆ 1 , Decomposing the integral C d (2) z s into a combination of contour integrals over z s andz s yields a linear combination of the two terms |F ∆s (∆ i |z i )| 2 and F ∆ * s (∆ i |z i ) 2 where F ∆s (∆ i |z i ) is given by eq. (2.12). But these two terms give the same contribution to lim , because the integration contour in 1 2 +iR d∆ s is invariant under the reflection ∆ s → ∆ * s . Absorbing any remaining prefactors into the B-factors, we obtain This formula can be interpreted as the large c limit of the decomposition (A.9) of a four-point function into four-point conformal blocks, provided we have B(∆ 1 , This provide a justification for the formula (2.12) for F ∆s (∆ i |z i ).

A.2 Point particle on SL 3 (C)
A basis of functions on SL 3 (C) can be defined as where we recall that Z = (w, x, y) is a three-component isospin vector, and that the components (r, s) of the spin j are defined in eq. (2.21). The vectors u Z and v Z are such that (A.14) where the differential operators D (j,Z) (t a ) are defined in eqs. It follows that the function Φ j Z,Z (g) behaves under sℓ 3 tranformation as As a result, Φ j Z,Z (g) is an eigenvector of the quadratic (Laplacian) and cubic invariant differential operators on SL 3 (C), with the respective eigenvalues These Casimir numbers can be derived by computing sℓ 3 -invariant combinations of the differential operators D (j,Z) (t a ), using the covariant tensors Then we have the identities The Casimir numbers C 2 (j) and C 3 (j) are invariant under the six Weyl reflections (2.22), which is a manifestation of the equivalence of two representations whenever their spins are related by a reflection. At the level of the function Φ j Z,Z (g), this equivalence manifests itself as a relation which we now write in the case of the maximal reflection j → j * , where R(j) is a reflection coefficient, and the two-point invariant E(j, j ω |Z, Z ′ ) was given in eq. (2.42). This generalizes the SL 2 (C) reflection relation eq. (A.8). Now, the function Φ j Z,Z (g) simplifies if either r = 0 or s = 0, and then it obeys the differential equations, where the operators d (1) Z and d (2) Z were defined in eq. (2.35). This is a consequence of the vectors u Z and v Z obeying d (2) Z u Z = d (1) Z v Z = 0, and this justifies our definitions of d (1) Z and d (2) Z . Notice that these operators obey the remarkable property (We recall that e k are the simple roots of sℓ 3 ; for instance the coordinates of the spin j = −e 1 are (r, s) = (2, −1).) Finally, we can write the analog in sℓ 3 conformal Toda theory of the expression (A.1) for large c Liouville theory correlation functions, by using the functions Φ j Z,Z (g) on SL 3 (C). The isospin variable Z must be specialized as Z =z = ( 1 2 z 2 , z, z) (as in eq. (2.55)), and we must remember the relation (2.52) between W 3 momenta α and sℓ 3 spins j. Calling V α (z,z) the vertex operator of sℓ 3 conformal Toda theory with the momentum α, we have [8] (A.28)

B. Derivation of the expansion of a four-point block
Here we derive the expansion eq. (2.66) of the large c four-point block F js (z) = F js (j i |0, z, 1, ∞), starting from the integral formula eq. (2.59). We propose two possible ways to perform the calculation. The first way is more straightforward, but it leads to a formula (B.4) which is less symmetric than eq. (2.66) and has spurious poles. The second way is less straightforward as it starts with a six-dimensional (instead of three-dimensional) integral, but the symmetry (α, δ, s) ↔ (β, γ, r) of eq. (2.66) is manifest throughout the calculation. (We recall that r and s are the components of j s , and that the combinations of spin components α, β, γ, δ are defined in eq. (2.65).)

(B.2)
Let us denote the integral (B.1) as F js (z) = (1 − yz) −α (1 − xz + wz 2 ) −γ , and expand the integrand in powers of z. This reduces the problem to computing expectation values of monomials w n x k y m , and we find w n x k y m = z −∆ s 12 (−1) n (δ) m+n (r − β) n (r + s − 1) m+n m ℓ=0 C ℓ m (β − ℓ) k (s) ℓ (r) k+n−ℓ , (B.3) where the notation (t) n was introduced in eq. (2.15), and we write C ℓ m = m! ℓ!(m−ℓ)! . This leads to This can be seen to agree with eq. (2.66) by an automatic calculation of the first few orders in z. This agreement is non-trivial, as it involves the cancellation of the spurious poles which are present in eq. (B.4).