Second level semi-degenerate fields in W 3 Toda theory: matrix element and differential equation

Transcription

1 Second level semi-degenerate fields in W Toda theory: matrix element and differential equation Vladimir Belavin 1, Xiangyu Cao 2, Benoit Estienne, Raoul Santachiara 2 arxiv: v1 [hep-th] 25 Oct I E Tamm Department of Theoretical Physics, P N Lebedev Physical Institute, Leninsky Avenue 5, oscow, Russia, Department of Quantum Physics, Institute for Information Transmission Problems, Bolshoy Karetny per. 19, oscow, Russia oscow Institute of Physics and Technology, Dolgoprudnyi, oscow region, Russia 2 LPTS, CNRS (UR 8626), Université Paris-Saclay, Orsay, France LPTHE, CNRS and Université Pierre et arie Curie, Sorbonne Universités, Paris Cedex 05, France belavin@lpi.ru, xiangyu.cao08@gmail.com, raoul.santachiara@u-psud.fr Abstract: In a recent study we considered W Toda 4-point functions that involve matrix elements of a primary field with the highest-weight in the adjoint representation of sl. 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. Keywords: W N algebra, 2-dimensional conformal field theory, Fuchsian differential equations.

2 Contents 1 Introduction 1 2 W chiral algebra, representation modules and conformal blocks Null-states in semi- and fully-degenerate representations 7.1 Null-state equations for the fully-degenerate fundamental field 7.2 Semi-degenerate representation at level-two 8 4 Sixth-order differential equation 9 5 Local exponents and matrix elements Local exponents from the differential equation ultiplicities and matrix elements ultiplities in the s and u channel ultiplicities in the t channel 14 6 Summary and discussion 15 A Details of the derivation 16 1 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 []). 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. 1

3 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 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, 1] (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 Toda 4-point functions that involve a fullydegenerate primary field in the fundamental representation of sl and a fully-degenerate primary field in the adjoint representation of sl. 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 fully-degenerate primary field in the fundamental representation, a semi-degenerate primary field with one null state at level two. We show that this conformal block obeys a sixth-order Fuchsian differential equation. We compute the matrix elements of the semidegenerate 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 conformal field theory and introduce the W conformal block function. In 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 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 2

4 technical details of the derivation of the differential equation. 2 W chiral algebra, representation modules and conformal blocks We briefly introduce the W chiral symmetry algebra and its representation theory. We use the same notations and normalization conventions as in [15]. W algebra. The W 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- holomorphic field W(z). The full W algebra is given by the following commutation relations and [L m,l n ] = (m n)l m+n + c 12 (m m)δ m+n,0, [L m,w n ] = (2m n)w m+n, (2.1) [W m,w n ] = 1 (m n)λ m+n c m n (2m 2 mn+2n 2 8)L m+n c c (m 2 4)(m m)δ m+n,0, (2.2) 48 5! where Λ m are the modes of the quasi-primary field Λ =: T 2 : 10 2 T and the colons : : stand for normal-ordering. Explicitly, Λ m = p 2 L p L m p + p 1L m p L p 10 (m+2)(m+)l m. (2.) In (2.2) we assume the following normalisation for the current W(z) 2-point function: W(1)W(0) = 22+5c def = η. (2.4) 48 The parametrisation of the W central charge c, commonly used in the Toda field theory literature, is c = 2+24 Q 2, Q = b+ 1 b. (2.5) W primary fields. A W 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 weights, α = α 1 ω 1 +α 2 ω 2, (2.6)

5 where the standard sl conventions are used: ω 1 = In terms of the parameters 2 2 (1,0); ω 2 = ( 1 2, ) ; (2.7) 4 e 1 = 2 ω 1 ω 2 ; e 2 = ω 1 +2 ω 2 ; ρ = ω 1 + ω 2, (2.8) h1 = ω 1 ; h2 = ω 1 e 1 ; h = ω 1 e 1 e 2. (2.9) one has x i = ( α Q ρ) h i, i = 1,2,, (2.10) h = Q 2 +x 1 x 2 +x 1 x +x 2 x and q = i x 1 x 2 x. (2.11) ThefieldsΦ α arenormalizedinsuchawaythatthe2-pointcorrelationfunction Φ α (z)φ α(0), the Φ α (z)def = Φ 2Q ρ α (z) satisfies: lim z z2h Φ α (z)φ α(0) = 1. (2.12) In the following we will also use the notation α def = 2Q ρ α. W representation module. TherepresentationmoduleV α associatedtoφ α isspanned by the basis states, where the sets of positive integers Φ (I) def def α = L I Φ α = L im L i1 W jn W j1 Φ α, (2.1) 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 = { ;,1}, then Φ (I) α = W W 1 Φ α. The descendant fields Φ (I) α have conformal dimension h+ I, where I = i 1 +i 2 + +j 1 +j 2 + is called the level. We refer to the Appendix of [15] for a proof that any W highest-weight representation is spanned by the states (2.1). 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 L J Φ α H IJ α = LI Φ α L J Φ α, (2.15) 4

6 has a block-diagonal structure, H α = diag H (0) α,h (1) α,h (2) α,, where the elements of the i-th block, H (i) α, are the scalar products of the level-i descendants. These elements can be computed using the commutation relations (2.2),(2.1). By definition, H (0) α = 1. The explicit forms of H (1) α and H (2) α can be found for instance in the Appendix of [15]. The matrix elements. The matrix elements of general descendant fields are defined by: Γ I,J,K L,,R (Φ = L )(I) Φ (J) (1)Φ(K) R (0) (Φ L ) Φ, (1)Φ R (0) (2.16) Γ I,J,K L,,R Φ (I) = L Φ(J) (1)Φ(K) R (0), Φ L Φ (1)Φ R (0) (2.17) where L,,R is a short notation for α L, α, α R and Φ (I) X (2.1). (X = L,,R) is defined in Degenerate representations and fusion rules. A W fully-degenerate representation V r1 r 2 s 1 s 2 is associated with the primary field Φ αr1 r 2 s 1 s 2 (z) with α r1 r 2 s 1 s 2 = b (1 r1 ) ω 1 +(1 r 2 ) ω (1 s1 ) ω 1 +(1 s 2 ) ω 2, (2.18) b where r 1,r 2,s 1,s 2 are positive integers. In the following we use the notation: Φ r1 r 2 s 1 s 2 def = Φ αr1 r 2 s 1 s 2. (2.19) The representation V r1 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 r1 r 2 s 1 s 2 with a general W irreducible module Φ α takes the form V r1 r 2 s 1 s 2 V α = hr, hsv α b hr b 1 h s, (2.20) where h r and h s are the weights of the sl 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. W conformal blocks. The conformal block B L,2,1,R ({zi }) with internal fusion channel Φ can be represented as the following comb diagram: 5

7 B L,2,1,R ({zi }) def = Φ L (z L) Φ 2 (z 2 )Φ 1 (z 1 )Φ R (z R ) Φ 2 (z 2 ) Φ 1 (z 1 ) def = Φ L (z L) Φ R (z R ) (2.21) Invariance under global conformal transformations implies: B L,2,1,R ({zi }) = (z L z 1 ) 2h 1 (z L z R ) h 1 h R +h 2 h L (z L z 2 ) h 1+h R h 2 h L (z 2 z R ) h 1 h R h 2 +h L B L,2,1,R (z), (2.22) where z = (z 1 z R )(z 2 z L ) (z 1 z L )(z 2 z R ). (2.2) The function B L,2,1,R (z) is defined by the following expansion z h L+h R h B L,2,1,R (z) = ] 1 1+ z [H i (i) Γ,,K L,2, Γ K K,K,,,1,R, (2.24) i=1 K,K K = K =i where the matrix H (i) and the matrix elements Γ IL,J,K R and Γ I L,J,K R were defined in (2.15) and (2.16). W Ward identities. While the three Ward identities associated with the conserved current T fix the form (2.22), there are other five Ward identities associated with the conserved current W 6

8 X=L,2,1,R X=L,2,1,R X=L,2,1,R X=L,2,1,R N X=L,2,1,R W (X) 2 B L,2,1,R ({zx }) = 0, (2.25) zx W (X) 2 + W (X) 1 B L,2,1,R ({zx }) = 0, (2.26) z 2 X W (X) 2 +2z X W (X) 1 +q B X L,2,1,R ({zx }) = 0, (2.27) z X W (X) 2 +zx 2 W (X) 1 +z X q B X L,2,1,R ({zx }) = 0, (2.28) z 4 X W (X) 2 +4z X W(X) 1 +6z2 X q X B L,2,1,R ({zx }) = 0, (2.29) 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. Null-states in semi- and fully-degenerate representations.1 Null-state equations for the fully-degenerate fundamental field Welistherethenull-stateequationsforthefully-degeneratefundamentalfieldΦ 2111 (z) that we will need later. The field Φ 2111 (z) = Φ b ω1 (z) has quantum numbers def h = h 1 = 1 4b 2 def, q = q1 = i +4b 2 +5b 2. (.1) 2 27b This field obeys level-1, level-2 and level- null-state conditions: and W 1 Φ b ω1 = q 1 L 1 Φ b ω1. (.2) 2h 1 W 2 Φ b ω1 = 1 12q 1 η h 1 (5h 1 +1) L2 1 6q 1(h 1 +1) h(5h 1 +1) L 2 Φ b ω1, (.) W Φ b ω1 = 1 η 16q 1 h 1 (h 1 +1)(5h 1 +1) L 1 where η is given in (2.4). 12q 1 h 1 (5h 1 +1) L 1L 2 q 1(h 1 ) 2h 1 (5h 1 +1) L 7 Φ b ω1, (.4)

9 .2 Semi-degenerate representation at level-two 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: I = 2 : I = {2; },{ ;2},{1,1; },{1;1},{ ;1,1}. (.5) From (2.1), a general field Ψ at level two can be written as a linear combination of the above five fields: def Ψ α = c{2; } L 2 +c { ;2} W 2 +c {1,1; } L 2 1 +c {1;1}L 1 W 1 +W 1 2 Φ α, (.6) where the global normalisation has been fixed by setting the coefficient of W 2 1 Φ α to one. In order Ψ to be a null-state, it has to be a W primary field, i.e. W 0 Ψ α = q Ψ α, (.7) L 1 Ψ α = L 2 Ψ α = 0. (.8) The annihilation by W 1 and W 2 and higher generators follows from the W commutation relations. The requirement (.7) is satisfied if u = q u, (.9) where c is the vector u = (u {2; },u { ;2},u {1,1; },u {1;1},u { ;1,1} ) and the matrix reads q 4h 0 0 2q 2 c+2h 4 q = 0 2 q 2 c+2h 0. (.10) c+2h q q The requirement (.8) is equivalent to the condition Nu = 0, (.11) where N = 4d+ c 2 6q 6h 9q 5h(2 c+2h) (2h + 1) q 2 c+2h (h + 1) 6q. (.12) 8

10 We consider now the field Φ b ω1 +s ω 2 that has quantum numbers def h = h 2 = (s b)+b(s 2b)2 def, q = q 2 = i (b+s)(4b2 2bs+)(5b 2 bs+). b 27b 2 (.1) We have found that the vector ( u sing = (1+2b2 bs)(2+2b 2 bs), i(1+b2 )(+4b 2 2bs), 6b (+b2 2bs)(+7b 2 2bs) 6b 2, i(+4b2 2bs),1 b satisfies the conditions (.9) and (.11) with the following eigenvalue ) (.14) q sing = i(5b s)(+4b2 2bs)( +b 2 +bs) 27b 2. (.15) We have therefore shown that the field Φ b ω1 +s ω 2 has a singular-state at level two. Decoupling of the singular vector Ψ b ω1 +s ω 2 = 0 yields the following null-state condition u sing I I I =2 4 Sixth-order differential equation Φ (I) b ω 1 +s ω 2 = 0. (.16) We consider here the conformal block defined in (2.21) with the following identifications Φ 2 = Φ b ω1 +s ω 2, Φ 1 = Φ b ω1. (4.1) We use a shorter notation for the conformal block, B L,2,1,R (z) B (z): Φ b ω1 +s ω 2 (1) Φ 2111 (z) B (z) def = Φ L ( ) Φ R (0) (4.2) 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 (z) and W (X) i B (z) i = 1,2,X = L,2,1,R. WerecallthatthefunctionW (X) i B (z)istheconformalblockthatinvolvesthedescendant W i Φ X (for some explicit examples see the Appendix A). Using the 8 equations coming from the 5 Ward identities (2.25)-(2.29) plus the null-state conditions (.2), (.) and 9

11 (.4), we arrive to an equation of the form [ z + ]B (z) = W (2) 1 B (z), (4.) where on the LHS the dots stand for a certain linear combination of differential operators, for instance z 1 z, 2 (z 1) 2 z, 2 (z 1) 2 z,(z 1) 2 etc., acting on B (z). On the RHS the dots also represent some known linear combination of the factors z, (z 1), z 2 (z 1) 1 and z 1 (z 1) 2 multiplying W (2) 1 B (z). We notice that if we had identified Φ 2 = Φ s ω1, which is semi-degenerate at level 1, we could have used the relation W 1 Φ s ω1 (z) z Φ s ω1 (z) in the eq.(4.) 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 [16]. In the case under consideration here, eq.(4.2), the null-vector condition (.16) relates states at level 2. We repeat the procedure leading to (4.), this time for the fields: W (2) 1 B (z) plus 8 unknown functions : W (X) i W (2) 1 B (z) i = 1,2,X = L,2,1,R. Notice that in total we have 17 unknown functions. Again we use the 8 equations expressing the three null-state conditions (.2), (.) and (.4) and the five Ward identities applied to the function W (2) 1 B (z). With respect to the eqs. (2.25)-(2.29), these five Ward identities have additional terms that originate from the fact that W 1B (2) (z) involves a descendant state. Restoring the dependence on the coordinates z L,z 2,z 1,z R, the modified Ward identities take the form X=L,2,1,R X=L,2,1,R X=L,2,1,R W (X) 2 W (2) 1B ({z X }) = 0, (4.4) z X W (X) 2 + W(X) 1 W (2) 1 B ({z X }) = 0, (4.5) z 2 X W (X) 2 +2z X W (X) 1 +q X W (2) 1B ({z X })+ +κ z2 B ({z X }) = 0, (4.6) z X W (X) 2 +zx 2 W(X) 1 +z X q X W (2) 1B ({z X })+ X=L,2,1,R + z 2 κ z2 B ({z X })+h 2 κb ({z X }) = 0, (4.7) N z 4 X W (X) 2 +4z X W(X) 1 +6z2 X q (2) X W 1 B ({z X })+ X=L,2,1,R +6 z 2 2 κ z 2 B ({z X })+4 h 2 κ B ({z X }) = 0, (4.8) 10

12 where κ = 1 48 (2 c+h 2), (4.9) and the dimension h 2 is given in (.1). This time we obtain the relation of the type [ z + ]W (2) 1 B (z) = [W 2 1] (2) B (z). (4.10) Note that the above manipulations are valid for a general field Φ 2 (x). We can now use the fact that Φ 2 = Φ b ω1 +s ω 2 and use (.16) in order to express [ W 2 1] (2)B (z) in terms of the functions B (z) and W (2) 1B (z) [ W 2 1 ] (2) B (z) = B (z)+ z 1 B (z)+ + zw (2) 1 B (z)+ z 1 W (2) 1 B (z)+ (4.11) Using the system of equations (4.), (4.10) and (4.11), we finally obtained a sixthorder differential equation for the function B (z). ore 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. 5 Local exponents and matrix elements Defining α R = a R1 ω 1 +a R2 ω 2, α L = a L1 ω 1 +a L2 ω 2, (5.1) the conformal block B (z) is a function of five parameters, a R1, a R2, a L1, a L2 and b. 5.1 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 [17] for an exhaustive overview of Fuchsian systems. In Riemannsymbol notation the local exponents ρ 0 i,ρ1 i and ρ, i = 1,,6, associated to the 2+1 singular points 0,1 and can be represented as with 0 1 α 1 β 1 γ 1 α 1 +1 β 1 +1 γ 1 +1 α 2 β 1 +2 γ 2 α 2 +1 β 2 γ 2 +1 α β γ α +1 β +1 γ (5.2)

13 α 1 = 1 (2a R 1 b+a R2 b), α 2 = 1 ( ar1 b+a R2 b+b 2), α = 1 ( 6 ar1 b 2a R2 b+6b 2), β 1 = 1 ( 2b 2 +bs ), β 2 = 1 ( +4b 2 +bs ), β = 1 ( 6+7b 2 2bs ), γ 1 = 1 ( al1 b a L2 b 5b 2 ), γ 2 = 1 ( 6+aL1 b+2a L2 b 8b 2), γ = 1 ( 2aL1 b a L2 b 2b 2). (5.) It is easily checked that the above local exponents satisfy the Fuchs identity: n ( ρ 0 i +ρ 1 i +ρ i i=1 ) = (k 1) n(n 1) 2 = 15, (5.4) 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 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: s channel t channel u channel α R b ω 1 b ω 1 b ω 1 +s ω 2 α L b ω 1 α (s) α (t) α (u) α L b ω 1 +s ω 2 α L α R α R b ω 1 +s ω 2 andthat we denote by B (s) (z), B(t) (z) and B(u) (z). The crossing symmetry relationrelates B (s) (z) B(t) (1 z) z 2h 1 B (u) (1/z). (5.5) Note, that inorder to reconstruct s,t,ufusion rules fromthe differential equation, one has to take into account factors coming formthe fields transformation (nontrivial 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: 12

14 Channel 1,2 (α 1 ) : α (s) = α R b ω 1 (5.6) Channel,4 (α 2 ) : α (s) = α R +b ω1 ω 2 (5.7) Channel 5,6 (α ) : α (s) = α R +b ω 2 (5.8) t channel: u channel: Channel 1,2, (β 1 ) : α (t) = b ω1 +s ω 2 b ω1 (5.9) Channel 4 (β 2 ) : α (t) = b ω 1 +s ω 2 +b ω 1 ω 2 (5.10) Channel 5,6 (β ) : α (t) = b ω1 +s ω 2 +b ω2 (5.11) Channel 1,2 (γ 1 ) : α (u) = α L b ω 1 (5.12) Channel,4 (γ 2 ) : α (u) = α L +b ω 1 ω 2 (5.1) Channel 5,6 (γ ) : α (u) = α L +b ω 2 (5.14) We note that as usual (in W case) in the u-channel we use the conjugated value α L which is defined below (2.12). 5.2 ultiplicities and matrix elements The multiplicities of the local exponents can be argued from the 4-point conformal blockexpansion (2.24). As explained in[5, 7, 15,18 21] anymatrix element Γ I,J,K ofthree arbitraryw descendant states, Φ (I) L, Φ(J), andφ(k) R canbe written aslinear combinations of the matrix elements Γ { ; },{ ;1,1,1, },{ ; } α L, α, α R, p = 1,2,, (5.15) }{{} p times where the Γ have been defined in (2.16) ultiplities in the s and u channel Consider for instance the conformal block expansion (2.24) in the s channel. To determine the first order coefficient one needs to evaluate the two matrix elements: Γ { ; },{ ;1},{ ; } α (s), b ω 1, α R, Γ αl { ; },{ ;1},{ ; }, b ω 1 +s ω 2, α (s). (5.16) 1

15 If the Γ matrix element, involving the fully-degenerate field Φ b ω1 can be evaluated using the null-state equation (.2): Γ { ; },{ ;1},{ ; } α (s), b ω 1, α R q ( ) 1 = q (s) 2h q R q 1, (5.17) 1 the matrix element Γ { ; },{ ;1},{ ; } αl, b ω 1 +s ω 2, α (s) is determined only if s = b (and h L h (s) ), i.e. when one of the fields entering the matrix elements is a fullydegenerate field in the adjoint representation. This case was fully considered in [15]. For s b instead, the condition at second level (.16) alone is not sufficient to fix this matrix element. This ambiguity is at the origin of the multiplicity of order two of the local exponents α i. Indeed, let us fix αl, b ω 1 +s ω 2, α (s) = λ, (5.18) Γ { ; },{ ;1},{ ; } where λ is an arbitrary constant. This is equivalent, from the point of view of the differential equation, to choose a particular combination in the two-dimensional space of solutions having the same local exponent α i. On the other hand, the semi-degenerate condition (.16) allows to express all the other matrix elements (5.15) with p = 2,.. as functions of (5.18). For instance, ( ) αl, b ω 1 +s ω 2, α (s) = c sing {2; } h L +2h (s) +h 2 + ( ) ( ) q (s) +q L +q 2 +2λ c sing {1,1; } (h L h (s) h 2 1)(h L h (s) h 2) ( ) (h L h (s) h 2 1)λ. (5.19) Γ { ; },{ ;1,1},{ ; } +c sing { ;2} c sing {1;1} 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 ultiplicities 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,, we have two undetermined matrix elements at first and second order: αl, 2b ω 1 +s ω 2, α R, Γ αl { ; },{ ;1,1},{ ; }, 2b ω 1 +s ω 2, α R. Γ { ; },{ ;1},{ ; } (5.20) 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 14

16 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 suggests that the field Φ 2b ω1 +s ω 2 obeys probably some null-state condition at higher orders. 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 (t) = (s b) ω 2. It is a semidegenerate anti-fundamental field and the corresponding matrix elements (5.15) involve this field can be evaluated for any p. 6 Summary and discussion The main motivation of our study lies in the fact that, in W N theories, the general matrix element of a primary field between two descendant states is not expressed solely in terms of the primary -point function but involves also an infinite set of new independent basic matrix elements. This greatly limits the available information on correlation functions. AnothermanifestationofthisisthattheAGTcorrespondence[8,9]forW 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 [16, 22 25] for recent works towards a more general analysis. InthispaperwefocusedourattentiononthefieldΦ b ω1 +s ω 2 ofthew Todaconformal field theory. We showed that this field is a second level semi-degenerate field and we found the corresponding null-vector conditions (.16). These conditions allow for the computation of all except one matrix elements involving this field: the basis elements (5.15) with p = 2,, 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. oreover we derived the differential equation obeyed by the conformal block containing a fully-degenerate fundamental fields Φ b ω1, the semi-degenerate field Φ b ω1 +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 ω1 +s ω 1 is a semi-degenerate field at level and the associated matrix elements (5.15) with p =,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 and the determination of W structure constants that are, at the present, unknown. oreover, it would be interesting 15

17 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 [26]. A Details of the derivation Besides the conformal block B L,2,1,R ({zx }), we need to consider the two functions: W (2) 1B L,2,1,R ({zx }) = Φ L (z L)[W 1 Φ b ω1 +s ω 2 ](z 2 )Φ b ω1 (z 1 )Φ R (z R ), [ W 2 1 ] (2)B L,2,1,R ({zx }) = Φ L(z L ) [ W 2 1Φ b ω1 +s ω 2 ] (z2 )Φ b ω1 (z 1 )Φ R (z R ). (A.1) Analogously to (2.22), the global conformal invariance fixes the coordonate dependence: W (2) 1 B L,2,1,R ({z X }) = (z L z 1 ) 2h 1 (z L z R ) h 1 h R +h 2 +1 h L (z L z 2 ) h 1+h R h 2 1 h L (z 2 z R ) h 1 h R h 2 1+h L H(z), [ W 2 1 ] (2)B L,2,1,R ({zx }) = (z L z 1 ) 2h 1 (z L z R ) h 1 h R +h 2 +2 h L where H(z) def = W (2) 1B (z), (z L z 2 ) h 1+h R h 2 2 h L (z 2 z R ) h 1 h R h 2 2+h L H 1 (z), H 1 (z) def = [ W 2 1] (2)B (z), (A.2) and z is given in (2.2). We recall that the values of (h 1,q 1 ) and (h 2,q 2 ), characterizing the fields at position z 1 and z 2, are given respectively in (.1) and in (.1). Using the (.16) for the semi-degenerate field at z 2, one has the relation c sing {2; } L(2) 2 B (z)+c sing { ;2} W(2) 2 B [ ] (z)+c sing {1,1; } L 2 (2) 1 B (z)+c sing {1;1} L(2) 1 H(z)+H 1(z) = 0. (A.) where c sing I are the components of the vector c sing given in (.14). The functions L (2) 2B (z) and [ L 1] 2 (2)B (z), related to conformal blocks involving pure Virasoro descendants, can be expressed in terms of differential operators acting on B (z): [ ] L (2) (z 2)z h 1 2B (z) = z 1 B (z)+ h 1 +2h R +h 2 h L + B (z 1) 2 (z), [ ] L 2 (2)B 1 (z) = z 2 B (z)+2z(h 1 +h R +h 2 h L +1)B (z)+ (A.4) (h 1 +h R +h 2 h L )(h 1 +h R +h 2 h L +1)B (z), (A.5) 16

18 while the function L (2) 1H(z) is easily expressed as L (2) 1 H(z) = (h 1 +h R +h 2 h L +1)H(z) zh (z). (A.6) LessdirectisthecomputationofW (2) 2 B (z)term. UsingfiveW-WardidentitiesforB (z) together with the three null-vector conditions (.2), (.) and (.4) for the field Φ b ω1 (z 1 ) allows to express W (2) 2 B (z) in terms of B (z) and H(z). The resulting expressions is W (2) 2 B (z) = 2H(z) 12q 1z 2 B (z) h 1 (5h 1 +1) q 1z(h 1 (9z 7)+5z )B (z) h 1 (5h 1 +1)(z 1) [ q1 (5h 1 11h 1 z 6h L z +7z 1) + 6q 1(h 1 h R +h R h 2 z h 1 h 2 z h L z) (5h 1 +1)(z 1) h 1 (5h 1 +1)(z 1) + 2h 1q 1 (11z 2 16z +5) + 6h ] 2(h 1 +1)q 1 (z 2)z (5h 1 +1)(z 1) 2 h 1 (5h 1 +1)(z 1) +q 2 R +q 2 q L B (z). (A.7) Finally we have to express the function H(z) and H 1 (z) in terms of the differential operator acting on B (z). This can be done by using the Ward identities (4.4)-(4.8). We obtain the following two identities: 1 (z 1) 2 z 2H(z)+g 0B (z)+g 1 B (z)+g 2 B (z)+g B (z) = 0, (A.8) and 1 (z 1) 2 z 2H 1(z)+ g 0 B (z)+ g 1 B (z)+l 0H(z)+l 1 H (z)+l 2 H (z)+l H (z) = 0. (A.9) The coefficients in the above equations read respectively: g 0 = (h 1 1)h R q 1 (2 z) 2h 1 (5h 1 +1)(z 1) 2 z (q R(z 1) 2 q L z(z 1) 2 q 2 (2z 1)z) (z 1) z + h 2q 1 (h 1 (9z 7) z ) + h Lq 1 (h 1 (10z 7)+2z ) 2h 1 (5h 1 +1)(z 1) z 2 2h 1 (5h 1 +1)(z 1) 2 z 2 + q 1(h 1 (1 40z) 8z +11) 2(5h 1 +1)(z 1) 2 z 2, (A.10) g 1 = q 1(44z 2 5z +12) 2(5h 1 +1)(z 1) 2 z 12q 1(h R (z 1) h 2 z) 2 h 1 (5h 1 +1)(z 1) 2 z 2 q 1(20z 2 25z +4) 2h 1 (5h 1 +1)(z 1) 2 z h L q 1 h 1 (5h 1 +1)(z 1)z, g 2 = 12q 1(1 z)(5z ) h 1 (5h 1 +1)(z 1) 2 z, g = 17 16q 1 (1 z) h 1 (h 1 +1)(5h 1 +1)(z 1), (A.11) (A.12)

19 and g 0 = 9q 1q 2 (h 1 (9z 8) z 8) 2h 1 (5h 1 +1)(z 1) 4 z h 2κ(z 2 z +1) g 1 = + (h 1 +h R )κ(1 2z) h Lκ(1 2z) (z 1) z 2 (z 1) z 2, (A.1) (z 1) 4 z 2 6q 1 w h 1 (5h 1 +1)(z 1) + κ(1 2z) (z 1) z, (A.14) l 0 = q 1(h 1 z(1 40z)+9h R (2 z)) 9(h 2 +1)q 1 (z +1) 2(5h 1 +1)(z 1) 2 z 2h 1 (5h 1 +1)(z 1) z 2 (q R(z 1) 2 q L z(z 1) 2 +q 2 z(1 2z)) h 2 q 1 (9z 7) + (z 1) z 2(5h 1 +1)(z 1) z 2 l 1 = l 2 = q 1(4z 2 2z +16) (5h 1 +1)(z 1) z 2 + q 1(h R (z 2)+h L (2z )z +h 1 h L (10z 7)z) 2h 1 (5h 1 +1)(z 1) 2 z, 12q 1 (h L z h R ) h 1 (5h 1 +1)(z 1)z 2 q 1(44z 2 5z +12) 2(5h 1 +1)(z 1) 2 z 2 q 1(20z 2 z +4) 2h 1 (5h 1 +1)(z 1) 2 z + 12h 2 w 1 2 h 1 (5h 1 +1)(z 1) 2 z, 12q 1 ( 5z) h 1 (5h 1 +1)(z 1)z, l 16q 1 = h 1 (h 1 +1)(5h 1 +1). (A.15) (A.16) (A.17) Using (A.8) and (A.9), we are able to express H(z) and H 1 (z) in terms of B (z) and its derivatives (up to 6th order). Using these results in (A.), we finally get the 6th order differential equation for B (z). Acknowledgements We thank the Institut Henri Poincare, Paris, where this work was initiated, and the Poncelet Laboratory (oscow) where this work was ended, for excellent hospitality and financial support. The work of V.B. was performed at the Landau Institute for Theoretical Physics, with the financial support from the Russian Science Foundation (Grant No ). We greatly thank O. Foda for contributions to the early stages of this project. We thank P. Gavrylenko, N. Iorgov, Y. Ikhlef, Y. atsuo and S. Ribault for discussions. References [1] A. B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. ath. Phys. 65 (1985) [2] V. A. Fateev and A. B. Zamolodchikov, Conformal quantum field theory models in two dimensions having Z symmetry, Nucl. Phys. B280 (1987)

20 [] P. Bouwknegt and K. Schoutens, W symmetry, Adv.Ser.ath.Phys. 22 (1995) [4] A. A. Belavin, A.. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241 (1984) 80. [5] V. A. Fateev and A. V. Litvinov, Correlation functions in conformal Toda field theory I, JHEP 11 (2007) 002, [ ]. [6] V. A. Fateev and A. V. Litvinov, Correlation functions in conformal Toda field theory II, JHEP 01 (2009) 0, [ ]. [7] S. Kanno, Y. atsuo and S. Shiba, Analysis of correlation functions in toda theory and the alday-gaiotto-tachikawa-wyllard relation for s u () quiver, Physical Review D 82 (2010) [8] L. F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. ath. Phys. 91 (2010) , [ ]. [9] N. Wyllard, A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002, [ ]. [10] A. ironov and A. orozov, On agt relation in the case of u (), Nuclear physics B 825 (2010) 1 7. [11] V. Belavin and B. Feigin, Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories, JHEP 07 (2011) 079, [ ]. [12] V. A. Alba, V. A. Fateev, A. V. Litvinov and G.. Tarnopolsky, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. ath. Phys. 98 (2011) 64, [ ]. [1] B. Estienne, V. Pasquier, R. Santachiara and D. Serban, Conformal blocks in Virasoro and W theories: Duality and the Calogero-Sutherland model, Nucl.Phys. B860 (2012) , [ ]. [14] A. Belavin and V. Belavin, AGT conjecture and Integrable structure of Conformal field theory for c=1, Nucl.Phys. B850 (2011) , [ ]. [15] V. Belavin, B. Estienne, O. Foda and R. Santachiara, Correlation functions with fusion-channel multiplicity in {mathcal{w}} Toda field theory, Journal of High Energy Physics 6 (June, 2016) 17, [ ]. [16] P. Furlan and V. Petkova, On some -point functions in the w 4 cft and related braiding matrix, arxiv preprint arxiv: (2015). [17]. Yoshida, Fuchsian differential equations. Springer, [18] P. Bowcock and G.. T. Watts, Null vectors of the W() algebra, Phys. Lett. B297 (1992) , [hep-th/ ]. [19] Z. Bajnok, L. Palla and G. Takacs, A(2) Toda theory in reduced WZNW framework and the representations of the W algebra, Nucl. Phys. B85 (1992) 29 60, [hep-th/ ]. 19

21 [20] P. Bowcock and G.. T. Watts, Null vectors, three point and four point functions in conformal field theory, Theor. ath. Phys. 98 (1994) 50 56, [hep-th/909146]. [21] G.. T. Watts, Fusion in the W() algebra, Commun. ath. Phys. 171 (1995) 87 98, [hep-th/94016]. [22] G. Bonelli, A. Tanzini and J. Zhao, Vertices, vortices & interacting surface operators, Journal of High Energy Physics 2012 (2012) [2] J. Gomis and B. Le Floch, 2-brane surface operators and gauge theory dualities in Toda, [ ]. [24] V. itev and E. Pomoni, Toda -Point Functions From Topological Strings, JHEP 06 (2015) 049, [ ]. [25]. Isachenkov, V. itev and E. Pomoni, Toda -Point Functions From Topological Strings II, [ ]. [26] P. Gavrylenko and A. arshakov, Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations, Journal of High Energy Physics 2 (Feb., 2016) 181, [ ]. 20