Second level semi-degenerate fields in W3 Toda theory: matrix element and differential equation

In a recent study we considered W3 Toda 4-point functions that involve matrix elements of a primary field with the highest-weight in the adjoint representation of sl3. We generalize this result by considering a semi-degenerate primary field, which has one null vector at level two. We obtain a sixth-order Fuchsian differential equation for the conformal blocks. We discuss the presence of multiplicities, the matrix elements and the fusion rules.


Introduction
The W N theories [1,2] are 2D conformal field theories (CFT) with current algebra W N , the representations of which are related to the weights of the Lie algebra sl N (for a review, we refer the reader to [3]). The Virasoro algebra coincides with the W 2 algebra and is a sub-algebra of W N >2 . In the case of Virasoro CFT [4], all correlation functions can be expressed in terms of correlation functions of primary fields. This is not true for the W N >2 theory because the symmetry constraints are no longer sufficient to fix the operator product expansion or, equivalently, the matrix elements involving general states. In order to compute W N correlation functions one needs to find additional conditions besides the ones coming from the current algebra. Typically, these additional conditions can originate from the presence of a W N primary field, the null-state, at a certain level of a given representation module. This is the reason why only W N correlation functions involving a set of particular fields can be computed. For a general value of the central charge a W N representation module can contain up to N − 1 null-states: a field is said to be fully-or semi-degenerate if the corresponding module contains N − 1 or less null states.
So far, the studies about W N theories focused mostly on W N correlation functions that contain level-1 semi-degenerate fields [5,6]. These fields are associated to the (anti-)fundamental representation of sl N and we will refer to them as (anti)-fundamental fields. In [7] it was shown that all matrix elements that contain at least one (anti-)fundamental field can be computed explicitly. The interest about this special sector of W N was also greatly motivated by the fact that, via the AGT correspondence [8][9][10], the corresponding conformal blocks are related to the instanton calculus in Ω-deformed N = 2 SUSY SU(N) quiver gauge theories. On the symmetry level this relation is a consequence of the fact the W N (and many others) chiral symmetry algebras can be obtained from a special toroidal algebras having simple action on the cohomologies of instanton moduli spaces [11]. This connection, in turn, reveals a deep integrable structure of W N theory [12,13] (for explicit example see, e.g. [14]). In particular, the computation of the matrix elements of the semi-degenerate fundamental fields in the integrable basis [12] gives a nice combinatorial representation for the conformal blocks. Hence, an interesting problem is whether or not one can extend this connection to a larger set of W N correlation functions. The first question in this direction is whether there exists any generalization of the semi-degenerate fundamental field, such that the corresponding matrix elements can be also constructed explicitly via W N symmetry constraints.
In a recent study [15] we considered a W 3 Toda 4-point functions that involve a fullydegenerate primary field in the fundamental representation of sl 3 and a fully-degenerate primary field in the adjoint representation of sl 3 . This latter field has two null-states at the second level of the associated representation module. In [15], we showed that the associated conformal block satisfy a fourth-order Fuchsian differential equation and we discussed the role of multiplicities that appear in this theory. In this paper we generalize these result by considering the 4-point conformal block that involve, besides the fullydegenerate primary field in the fundamental representation, a semi-degenerate primary field with one null state at level two. The associated local correlation functions have been considered in [6] where their expressions in term of four-dimensional integrals were given. Here we show that this conformal block obeys a sixth-order Fuchsian differential equation. We compute the matrix elements of the semi-degenerate field, between two arbitrary descendant states and we verify that the series expansion of the conformal block constructed from these matrix elements agrees with the differential equation.
The paper is organized as follows. In section 2, we recall basic facts regarding W 3 conformal field theory and introduce the W 3 conformal block function. In 3 we discuss the null-vector conditions for the degenerate and semi-degenerate primary fields. In 4, we focus on a specific 4-point correlation function with one semi-degenerate level=2 and one fully degenerate field in the fundamental sl 3 representation. Here show that the corresponding conformal blocks obeys a sixth-order Fuchsian differential equation. We proceed by discussing the matrix elements of the semi-degenerate level-2 field between two arbitrary descendant states and compare the results of the differential equation with the explicit construction of the conformal blocks in terms of the matrix elements. In 6, we present our conclusion and discuss some open problems. In appendix A, we give some technical details of the derivation of the differential equation.

W 3 chiral algebra, representation modules and conformal blocks
We briefly introduce the W 3 chiral symmetry algebra and its representation theory. We use the same notations and normalization conventions as in [15]. W 3 algebra. The W 3 is an associative algebra generated by the modes L n and W n of the spin-2 energy-momentum tensor T (z) and of the spin-3 holomorphic field W(z). The full W 3 algebra is given by the following commutation relations where Λ m are the modes of the quasi-primary field Λ =: T 2 : − 3 10 ∂ 2 T and the colons : : stand for normal-ordering. Explicitly, In (2.2) we assume the following normalisation for the current W(z) 2-point function: The parametrisation of the W 3 central charge c, commonly used in the Toda field theory literature, is W 3 primary fields. A W 3 primary field Φ α (z) is completely characterized by the pair of quantum numbers (h, q), respectively its conformal dimension (h) and W 0 eigenvalue (q). It is labelled by a vector α in the space spanned by the fundamental sl 3 weights, where the standard sl 3 conventions are used: (2.7) In terms of the parameters The fields Φ α are normalized in such a way that the 2-point correlation function In the following we will also use the notation α * def = 2Q ρ − α.
W 3 representation module. The representation module V α associated to Φ α is spanned by the basis states, where the sets of positive integers I = {i m , · · · , i 1 ; j n , · · · , j 1 } with i m · · · i 1 1, j n · · · j 1 1, (2.14) are normal-ordered, The symbol ∅ will be used when no modes L m or W n are present. For instance, set I = {∅; 3, 1}, then Φ α have conformal dimension h + |I|, where |I| = i 1 + i 2 + · · · + j 1 + j 2 + · · · is called the level. Any W 3 highest-weight representation is spanned by the states (2.13) [3]. We also refer to the Appendix of [15] where these properties are reviewed in the same notations and conventions adopted here.
The Shapovalov matrix of inner products. The Shapovalov matrix H, whose ijelement H ij is the scalar product of the states |L I Φ α and where L, M, R is a short notation for α L , α M , α R and Φ Degenerate representations and fusion rules. A W 3 fully-degenerate representation V r 1 r 2 s 1 s 2 is associated with the primary field Φ αr 1 r 2 s 1 s 2 (z) with where r 1 , r 2 , s 1 , s 2 are positive integers. In the following we use the notation: The representation V r 1 r 2 s 1 s 2 exhibits two independent null-states at levels r 1 s 1 and r 2 s 2 and the fusion products of V r 1 r 2 s 1 s 2 with a general W 3 irreducible module Φ α takes the form where h r and h s are the weights of the sl 3 representation with highest-weight (r 1 − 1) ω 1 + (r 2 − 2) ω 2 and (s 1 − 1) ω 1 + (s 2 − 2) ω 2 respectively.
with internal fusion channel Φ M can be represented as the following comb diagram: Invariance under global conformal transformations implies: where the matrix H (i) and the matrix elements Γ I L ,J M ,K R and Γ ′ I L ,J M ,K R were defined in (2.15) and (2.16).
where the notation W (X) −i , i = 1, 2 means that the mode W −i is applied to the field X(= L, 2, 1, R) in the conformal block.
3 Null-states in semi-and fully-degenerate representations 3

.1 Null-state equations for the fully-degenerate fundamental field
We list here the null-state equations for the fully-degenerate fundamental field Φ 2111 (z) that we will need later. The field Φ 2111 (z) = Φ −b ω 1 (z) has quantum numbers This field obeys level-1, level-2 and level-3 null-state conditions: and

Semi-degenerate representation at level-two
In this paper we are interested in the case of general central charge, in particular when b 2 / ∈ Q. Consider a field Φ r 1 r 2 s 1 s 2 with r 1 and s 1 positive integers and r 2 / ∈ N + . In the representation module there is one null-vector at level r 1 s 1 with quantum numbers (h, q) coinciding with the ones of the primary operator Φ r ′ 1 r ′ 2 s 1 s 2 with r ′ 1 = −r 1 and r ′ 2 = r 2 + r 1 , see [16] and references therein. Here we want to focus on the semi-degenerate field at level two Φ −b ω 1 +s ω 2 , which corresponds to Φ r 1 r 2 s 1 s 2 with r 1 = 2 and r 2 = 1 − s. In [17], the explicit expression of the null-vector at level r associated to the field Φ (1−r)b ω 1 +ss ω 2 has been given. For sake of clarity, we derive below, by using our conventions, the same result for the level two null-vector associated to the field Φ −bω 1 +sω 2 .
Let consider a general module V α with quantum number (h, q). At the second level one has five fields Φ (I) with |I| = 2: From (2.13), a general field Ψ at level two can be written as a linear combination of the above five fields: where the global normalisation has been fixed by setting the coefficient of W 2 −1 Φ α to one. We assume Ψ to be a W 3 primary field, i.e.
We will see below that the above relations, in particular the fact that W 0 is diagonalizable, give the correct result. A discussion about the fact that W 0 need not be diagonalizable can be found for instance in [18]. The annihilation by W 1 and W 2 and higher generators follows from the W 3 commutation relations.
The requirement (3.7) is satisfied if The requirement (3.8) is equivalent to the condition We consider now the field Φ −b ω 1 +s ω 2 that has quantum numbers (3.13) We have found that the vector satisfies the conditions (3.9) and (3.11) with the following eigenvalue One can directly verify that q ′ sing is indeed the W 0 eigenvalue of the field Φ 3bω 1 +(−2+s)bω 2 , see the discussion at the beginning of the section. We have therefore shown that the field Φ −b ω 1 +s ω 2 has a singular-state at level two. Decoupling of the singular vector Ψ −b ω 1 +s ω 2 = 0 yields the following null-state condition In a similar way, a similar null-vector condition for the other semi-degenerate field at level two, Φ s ω 1 −b ω 2 can be obtained. The remaining two other cases are obtained by replacing b → 1/b.

Sixth-order differential equation
We consider here the conformal block defined in (2.21) with the following identifications We use a shorter notation for the conformal block, In the following we present in full detail the computation of this conformal block. The case in which Φ 2 = Φ s ω 1 −b ω 2 will be briefly discussed in section 5.2.3.
Let us sketch the procedure to obtain the differential equation satisfied by the conformal block defined above. At the beginning we have 9 unknown functions : B M (z) and W (X) We recall that the function W where on the LHS the dots stand for a certain linear combination of differential operators, for instance z −1 ∂ 2 z , (z − 1) −2 ∂ 2 z , (z − 1) 2 ∂ z , (z − 1) −2 etc., acting on B M (z). On the RHS the dots also represent some known linear combination of the factors z −3 , (z − 1) −3 , z −2 (z − 1) −1 and z −1 (z − 1) −2 multiplying W (2) −1 B M (z). We notice that if we had identified Φ 2 = Φ s ω 1 , which is semi-degenerate at level 1, we could have used the relation W −1 Φ s ω 1 (z) ∝ ∂ z Φ s ω 1 (z) in the eq.(4.3) and obtained the third-order generalized hypergeometric differential equation of [5]. For general W N conformal blocks, containing one fully-degenerate and one semi-degenerate fundamental field, one obtains a N-order generalized hypergeometric equation [5]. A detailed study of the W 4 theory can be found in [19]. In the case under consideration here, eq.(4.2), the null-vector condition (3.16) relates states at level 2. We repeat the procedure leading to (4.3), this time for the fields: and the dimension h 2 is given in (3.13). This time we obtain the relation of the type −1 B M (z)+· · · (4.11) Using the system of equations (4.3), (4.10) and (4.11), we finally obtained a sixthorder differential equation for the function B M (z). More details of the calculations are collected in the Appendix A. However, the final and explicit expression of the sixth-order differential is too long to be reported here. We discuss instead the results that follow from it.

Local exponents and matrix elements
Defining the conformal block B M (z) is a function of five parameters, a R 1 , a R 2 , a L 1 , a L 2 and b.

Local exponents from the differential equation
The Fuschsian differential equation of order six has 2 + 1 singularities at 0, 1 and ∞. We refer the reader to [20] for an exhaustive overview of Fuchsian systems. In Riemannsymbol notation the local exponents ρ 0 i , ρ 1 i and ρ ∞ , i = 1, · · · , 6, associated to the 2 + 1 singular points 0, 1 and ∞ can be represented as It is easily checked that the above local exponents satisfy the Fuchs identity: specified in our case where Fuchsian equation has order n = 6 and number of singularity 2(= k)+1. It is important to observe that in our Fuchsian equation the fact that there are local exponent different by integers do not imply logarithmic solutions. Indeed, we argue below that the structure of local exponents does not origin from logarithmic features of the theory but from the undetermined matrix elements in the W 3 algebra. We can verify that the values of the local exponents can be found by using the fusion products (2.20). To this end we note that the solutions of the differential equation with diagonal monodromies around 0, 1, ∞ define correspondingly s, t, u conformal blocks associated with the following diagrams: and that we denote by B Note, that in order to reconstruct s, t, u fusion rules from the differential equation, one has to take into account factors coming form the fields transformation (non trivial only in z → 1/z transformation) . Substituting one of these there functions in the differential equation and keeping leading terms in the expansions around 0, 1, ∞ we find that the exponents α i , β i and γ i , associated respectively to the s−, t−, and u− channels, correspond to the following fusion channels: • s−channel: Channel 5,6 (α 3 ) : α (s) • t−channel: • u−channel: Channel 5,6 (γ 3 ) : α (u) We note that as usual (in W 3 case) in the u-channel we use the conjugated value α * L which is defined below (2.12).

Multiplicities and matrix elements
The multiplicities of the local exponents can be argued from the 4-point conformal block expansion (2.24). As explained in [5,7,15,17,[21][22][23] any matrix element Γ I,J,K of three arbitrary W 3 descendant states, Φ functions of (5.18 We have checked up to the second order in the expansion (2.24), that the direct computation of the matrix element is in agreement with the results obtained by using the sixth-order differential equation. The same arguments seen for explaining the multiplicities in the s−channel holds for the u−channel.

Multiplicities in the t−channel
One can notice that in the t−channel the structure of the multiplicities is different. Again this can understood by considering the matrix elements entering in the t−channel expansions. For the fusion channels 1, 2, 3, we have two undetermined matrix elements at first and second order: At the level of the differential equation, this corresponds to the fact that, in order to select a function in the three-dimensional space of solutions with local exponent β 1 , we have to fix two parameters. Fixing these two parameters is equivalent to fixing the values of the above matrix elements. Once these two parameters have been set, all the other matrix elements of higher-order can be computed in terms of these two parameters, as we have seen before. This is consistent with the fact that the field Φ −2b ω 1 +s ω 2 obeys a null-state condition at order three, see the discussion at the beginning of section 3.2. In this respect, the differential equation is the most direct method to determine these matrix elements. Finally, the fact that the space of solutions with local exponent β 2 is uni-dimensional is due to the fact that all the matrix elements, even those at the first level, are known. Indeed in this case α It is a semi-degenerate anti-fundamental field and the corresponding matrix elements (5.15) involve this field can be evaluated for any p.

The case with
We consider now the conformal block: where we consider the other semi-degenerate field at level two, Φ s ω 1 −b ω 2 . In this respect it is convenient to use the invariance of the W 3 conformal blocks under the exchange ω 1 ↔ ω 2 and to consider the following conformal block: Note that we have kept the fields Φ R,L un-exchanged as they are general fields. The computation of the differential equation satisfied by 5.22 is strictly similar to the one for 4.2, the only difference being the fact that q 1 gets an opposite sign with respect to the previous case 3.1. The resulting sixth order equation has the same pattern of local exponents as in 5.2. Their precise values are given by: and correspond to the following fusion rules: in the s, t, u channels: • s−channel: • t−channel: • u−channel: The above values are consistent with the expected fusion rules and the analysis of the degeneracy pattern is strictly analogous to the one done above for the case 4.2.

Summary and discussion
The main motivation of our study lies in the fact that, in W N 3 theories, the general matrix element of a primary field between two descendant states is not expressed solely in terms of the primary 3-point function but involves also an infinite set of new independent basic matrix elements. This greatly limits the available information on correlation functions. Another manifestation of this is that the AGT correspondence [8,9] for W N theories allows to construct matrix elements only for the fields with highest-weights proportional either to ω 1 or to ω N −1 fundamental weights of sl N . In this case, the correspondence between 2-dimensional conformal field theory and 4-dimensional supersymmetric gauge theories, as proposed in [8], is available, and W N conformal blocks are equal to Nekrasov instanton partition functions [9]. See [19,[24][25][26][27] for recent works towards a more general analysis.
In this paper we focused our attention on the field Φ −b ω 1 +s ω 2 of the W 3 Toda conformal field theory. We showed that this field is a second level semi-degenerate field and we found the corresponding null-vector conditions (3.16). These conditions allow for the computation of all except one matrix elements involving this field: the basis elements (5.15) with p = 2, 3, · · · can indeed be computed as a function of the matrix element (5.15) with p = 1. The (5.19) is an example of such relations. Moreover we derived the differential equation obeyed by the conformal block containing a fully-degenerate fundamental fields Φ −b ω 1 , the semi-degenerate field Φ −b ω 1 +s ω 2 and two general fields Φ R and Φ L . We computed the local exponents of this Fuchsian equation and we related the corresponding multiplicities to the number of undetermined matrix elements (5.15). Interestingly we also argued that the field Φ −2b ω 1 +s ω 1 is a semi-degenerate field at level 3 and the associated matrix elements (5.15) with p = 3, 4, · · · can be computed, via the differential equation, in terms of the ones with p = 1, 2.
Our results demand further investigations. To begin with, it would be interesting to find the monodromy group associated to the sixth-order differential systems. This would allow the definition of local correlation functions, to compare with the ones computed in [6] by completely different methods, and the determination of W 3 structure constants that are, at the present, unknown. Moreover, it would be interesting to understand how to recover our results for the semi-degenerate level-2 fields using the AGT correspondence. In this respect, the case of central charge c = 2 can be tackled with the methods proposed in [28].