q -Virasoro/W Algebra at Root of Unity and Parafermions
aa r X i v : . [ h e p - t h ] A ug OCU-PHYS 405August, 2014 q -Virasoro/W Algebra at Root of Unity and Parafermions H. Itoyama a,b ∗ , T. Oota b † and R. Yoshioka b ‡ a Department of Mathematics and Physics, Graduate School of ScienceOsaka City University b Osaka City University Advanced Mathematical Institute (OCAMI)3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
AbstractWe demonstrate that the parafermions appear in the r -th root of unity limit of q -Virasoro/ W n algebra. The proper value of the central charge of the coset model b sl ( n ) r ⊕ b sl ( n ) m − n b sl ( n ) m − n + r is given from theparafermion construction of the block in the limit. ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] Introduction
Ever since the AGT relation [1, 2, 3] (the correspondence between the correlators of 2d QFT andthe 4d instanton sum) was introduced, the both sides of the correspondence have been intensivelystudied by a number of people. For example, in the 2d side, the β -deformed matrix model is usedin order to control the integral representation of the conformal block [4, 5, 6, 7, 8, 9, 10]. Thereare also some proposals for proving the 2d-4d connection [11, 12, 13, 14, 15]. Moreover similarcorrespondence has been found and examined [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Amongthese, we pay our attention, in this paper, to the correspondence between the coset model, b sl ( n ) r ⊕ b sl ( n ) p b sl ( n ) r + p , (1.1)and the N = 2 SU ( n ) gauge theory on R / Z r [20, 23]. Here b sl ( n ) k stands for the affine Liealgebra in the representation of level k and r and p will be specified in this paper.On the 2d CFT side, a quantum deformation ( q -deformation) of the Virasoro algebra [27] andthe W n algebra [28, 29] is known, while the 4d gauge theories can be lifted to five-dimensionaltheories with the fifth direction compactified on a circle. There exists a natural generalizationto the connection between the 2d theory based on the q -deformed Virasoro/W algebra and thefive-dimensional N = 2 gauge theory [30]. For recent developments, see, for example, [31, 32, 33,34, 35, 36, 37]. In the previous paper [32], we proposed a limiting procedure to get the Virasoro/Wblock in the 2d side from that in the q -deformed version. On the other hand, we saw that theinstanton partition function on R / Z r are generated from that on R at the same limit. Thisresult means if we assume the 2d-5d connection, it is automatically assured that the Virasoro/Wblocks generated by using the limiting procedure agree with the instanton partition function on R / Z r . Our limiting procedure corresponds to a root of unity limit in q . A root of unity limit ofthe q -Virasoro algebra was also considered in [38]. Our limit is slightly different from this and issimilar to the one used in order to construct the eigenfunctions of the spin Calogero-Sutherlandmodel from Macdonald polynomials in [39, 40].In the present paper we will elaborate our limiting procedure and show that the Z r -parafermionicCFT which has the symmetry described by (1.1) appears in the 2d side. We clarify also the relationbetween the free parameter p and the omega background parameters in the 4d side.The paper is organized as follows: In the next section, we review the limiting procedure for q -Virasoro algebra [32]. In section 3, we consider the q -deformed screening current and charge andshow that the Z r -parafermion currents are derived in a natural way. In section 4, we consider thegeneralization to q - W n algebra. q -Virasoro Algebra In this section, we review the root of unity limit [32] of the q -deformed Virasoro algebra [27] whichhas two parameters q and t = q β . The defining relation is f ( z ′ /z ) T ( z ) T ( z ′ ) − f ( z/z ′ ) T ( z ′ ) T ( z ) = (1 − q )(1 − t − )(1 − p ) h δ ( pz/z ′ ) − δ ( p − z/z ′ ) i , (2.1)1here p = q/t and f ( z ) = exp ∞ X n =1 n (1 − q n )(1 − t − n )(1 + p n ) z n ! . (2.2)The multiplicative delta function is defined by δ ( z ) = X n ∈ Z z n . (2.3)Using the q -deformed Heisenberg algebra H q,t :[ α n , α m ] = − n (1 − q n )(1 − t − n )(1 + p n ) δ n + m, , ( n = 0) , [ α n , Q ] = δ n, , (2.4)the q -Virasoro operator T ( z ) can be realized as T ( z ) =: exp X n =0 α n z − n ! : p / q √ βα + : exp − X n =0 α n ( pz ) − n ! : p − / q −√ βα , (2.5)The q -deformed chiral bosons are defined in terms of the q -deformed Heisenberg algebra as e ϕ ( ± ) ( z ) = e ϕ ( ± )0 ( z ) + e ϕ ( ± ) R ( z ) , (2.6)where e ϕ ( ± )0 ( z ) = β ± / Q + 2 r β ± / α log z r + X n =0 (1 + p − nr )(1 − ξ nr ± ) α nr z − nr , e ϕ ( ± ) R ( z ) = r − X ℓ =1 X n ∈ Z (1 + p − nr − ℓ )1 − ξ nr + ℓ ± α nr + ℓ z − nr − ℓ . (2.7)Here ξ + = q , ξ − = t .Let us consider the simultaneous r -th root of unity limit in q and t which is given by q = ω e − √ β h , t = ω e −√ βh , p = e Q E h , h → , (2.8)where ω = e π i r and Q E = √ β − √ β . Since t = q β , this limit is possible if the parameter β takesthe rational number such as β = rm − + 1 rm + + 1 , (2.9)where m ± are non-negative integers. In the limit, we have two types of bosons φ ( w ) and ϕ ( w )[32] respectively given by lim h → e ϕ ( ± )0 ( z ) = r r β ± / φ ( w ) , lim h → e ϕ ( ± ) R ( z ) = r r ϕ ( w ) , (2.10)2here w = z r and φ ( w ) = Q + a log w − X n =0 a n n w − n , (2.11) ϕ ( w ) = r − X ℓ =1 ϕ ( ℓ ) ( w ) , ϕ ( ℓ ) ( w ) = X n ∈ Z ˜ a n + ℓ/r n + ℓ/r w − n − ℓ/r . (2.12)The commutation relations are[ a m , a n ] = mδ m + n, , [ a n , Q ] = δ n, , [ e a n + ℓ/r , e a − m − ℓ ′ /r ] = ( n + ℓ/r ) δ m,m ′ δ ℓ,ℓ ′ . (2.13)The boson φ ( w ) and the twisted boson ϕ ( w ) play an important role for the appearance of the Z r -parafermions. Z r -parafermionic CFT The q -deformed screening current and the charge are defined respectively by S ( ± ) ( z ) =: e ˜ ϕ ( ± ) ( z ) : , Q ( ± )[ a,b ] = Z ba d ξ ± zS ( ± ) ( z ) , (3.1)where the Jackson integral is defined by Z a d ξ ± zf ( z ) = a (1 − q ) ∞ X k =0 f ( aq k ) q k . (3.2)Multiplying the regularization factor, we obtain the screening charge in the root of unity limit,up to normalization, Q ( ± )[ a r ,b r ] ≡ lim h → (1 − q r )(1 − q ) Q ( ± )[ a,b ] = Z b r a r d wψ ( w ) : e √ βφ ( w ) , (3.3)where we have defined [41] ψ ( w ) = A r w ( r − /r r − X k =0 ω k : exp (r r φ ( k ) ( w ) ) : . (3.4)Here A r is the normalization factor and we have introduced φ ( k ) ( w ) ≡ ϕ ( e π i k w ) . (3.5)The correlation function is given by h φ ( k ) ( w ) φ ( k ′ ) ( w ′ ) i = log (1 − ω k ′ − k ( w ′ /w ) /r ) r − w ′ /w = log (1 − w ′ /w ) r − Q r − j =1 (1 − ω k ′ − k + j ( w ′ /w ) /r ) r . (3.6)3ote that φ ( k +1) ( w ) = φ ( k ) ( e π i w ) , φ ( r + k ) ( w ) = φ ( k ) ( w ) , r − X k =0 φ ( k ) ( w ) = 0 . (3.7)For example, we consider the r = 2 case. In the limit, we obtainlim q →− S ( z ) =: e √ βφ ( w ) e ϕ ( w ) : , (3.8)and after the appropriate normalization, we obtain the following screening charge for the super-conformal block [42, 43]: Q [ a ,b ] = Z b a d w ψ ( w ) : e √ βφ ( w ) : , (3.9)where ψ ( w ) ≡ i2 √ w (cid:16) : e ϕ ( w ) : − : e − ϕ ( w ) : (cid:17) , h ψ ( w ) ψ ( w ) i = 1 w − w , (3.10)is the NS fermion.From now on we will show that the Z r -parafermions appear in the general r -th root of unitylimit. In particular, ψ ( w ) will be shown to work as the first parafermion current.The Z r -parafermion algebra consists of ( r −
1) currents ψ ℓ ( w ) ( ℓ = 1 , · · · , r −
1) satisfying thefollowing defining relations [44]: ψ ℓ ( w ) ψ ℓ ′ ( w ′ ) = c ℓ,ℓ ′ ( w − w ′ ) ℓℓ ′ /r { ψ ℓ + ℓ ′ ( w ′ ) + O ( w − w ′ ) } , ℓ + ℓ ′ < r, (3.11) ψ ℓ ( w ) ψ † ℓ ′ ( w ′ ) = c ℓ,r − ℓ ′ ( w − w ′ ) − ℓ ( r − ℓ ′ ) /r { ψ ℓ − ℓ ′ ( w ′ ) + O ( w − w ′ ) } , ℓ ′ < ℓ (3.12) ψ ℓ ( w ) ψ † ℓ ( w ′ ) = ( w − w ′ ) − ℓ (cid:26) ℓ c p ( w − w ′ ) T PF ( w ) + O (( w − w ′ ) ) (cid:27) , (3.13)where ψ † ℓ ( w ) = ψ r − ℓ ( w ) and ∆ ℓ = ℓ ( r − ℓ ) r , c p = 2( r − r + 2 , (3.14)are the conformal dimension of ψ ℓ ( w ) and the central charge of the parafermionic stress tensor T PF . The explicit form of T PF ( w ) is given in [45]. The coefficients c ℓℓ ′ are given by c ℓℓ ′ = s ( ℓ + ℓ ′ )!( r − ℓ )!( r − ℓ ′ )! ℓ ! ℓ ′ !( r − ℓ − ℓ ′ )! r ! . (3.15)The OPE of (3.4) is ψ ( w ) ψ ( w ′ ) ≡ c , ( w − w ′ ) /r { ψ ( w ) + O ( w − w ′ ) } . (3.16)Here we have defined the second parafermion, ψ ( w ) = A r c , w r − /r r − X k,k ′ =0 ω k + k ′ (1 − ω k ′ − k ) : e √ r (cid:16) φ ( k ) ( w )+ φ ( k ′ ) ( w ) (cid:17) : (3.17)4imilarly, the ( ℓ + 1)-th parafermion is obtained from ℓ -th parafermion by ψ ℓ +1 ( w ) ≡ lim w ′ → w ( w − w ′ ) ℓ/r c ,ℓ ψ ( w ′ ) ψ ℓ ( w ) . (3.18)In particular, ψ † ( w ) ≡ ψ r − ( w ) = B r w ( r − /r r − X ℓ =0 ω ℓ exp ( − r r φ ( ℓ ) ( w ) ) , (3.19)where B r is a constant which can be determined by the relation h ψ ( w ) ψ † ( w ′ ) i = 1( w − w ′ ) r − /r . (3.20)After all, we have the chiral boson φ ( w ) coupled to Q E and the Z r -parafermion ψ ℓ ( w ). Therefore,the stress tensor of the whole system is T ( w ) = T B ( w ) + T PF ( w ) , (3.21)where T B ( w ) stands for the usual stress tensor for the chiral boson field. The central charge is c ( r ) = 1 − Q E r + 2( r − r + 2 = 3 rr + 2 − Q E r . (3.22)Because β is restricted to the rational number (2.9), (3.22) is written as c ( r,m,s ) = 3 rr + 2 − rs m ( m + rs ) . (3.23)where we have set m = rm + + 1 and s = m − − m + . Especially, when s = 1, c ( r,m, = 3 rr + 2 − rm ( m + r ) , (3.24)is the central charge of the unitary series of the Z r -parafermionic CFT [46].The form of the screening charge in the case of general r is the same as that of eq. (3.9). q - W n Algebra
In this section, we consider the generalization to the q - W n algebra [29]. We denote by h the Cartansubalgebra of sl ( n ) Lie algebra. The q - W n algebra is expressed in terms of the following h -valued q -deformed boson, h e a , e ϕ ( ± ) ( z ) i ≡ e ϕ ( ± ) a ( z ) = e ϕ ( ± )0 ,a ( z ) + e ϕ ( ± ) R,a ( z ) , (4.1)where e ϕ ( ± )0 ,a ( z ) = β ± Q a + β ± α ,a log z + X n =0 ξ nr/ ± − ξ − nr/ ± α nr,a z − nr , (4.2) e ϕ ( ± ) R,a ( z ) = r − X ℓ =1 e ϕ ( ± ) ℓ,a ( z ) = r − X ℓ =1 X n ∈ Z ξ ( nr + ℓ ) / ± − ξ − ( nr + ℓ ) / ± α nr + ℓ,a z − ( nr + ℓ ) , (4.3)5nd e a ( a = 1 , · · · , n −
1) are the simple roots and h , i : h ∗ ⊗ h → C is the canonical pairing. Thecommutation relations are given by[ Q a , α ,b ] = C ab , [ α n,a , α m,b ] = 1 n ( q n/ − q − n/ )( t n/ − t − n/ ) C ab ( p ) δ n + m, , [ Q a , Q b ] = 0 , [ α ,a , α ,b ] = 0 , (4.4)where C ab is the Cartan matrix of A type and C ab ( p ) = [2] p δ a,b − p / δ a,b − − p − / δ a − ,b . (4.5)The q -number is defined by [ n ] q = q n/ − q − n/ q / − q − / . (4.6)Similar to the q -Virasoro case, we consider the limit, q = ω k e − h √ β , t = ω k e −√ βh , p = q/t = e Q E h , ω = e π i r , h → +0 , (4.7)where ω = e π i r and k is a natural number mutually prime to r . The condition to be able to takethis limit is that β is a rational number, β = rm − + krm + + k , (4.8)where m ± are non-negative integers. Taking this limit,lim h → e ϕ a ( z ) = 1 √ r β / φ a ( w ) , (4.9)lim h → e ϕ aR ( z ) = 1 √ r ϕ a ( w ) , (4.10)we obtain φ a ( w ) = Q a + a a log w − X n =0 n a an w − n , (4.11) ϕ a ( w ) = r − X ℓ =1 ϕ ℓ ( w ) , ϕ ℓ ( w ) = r − X ℓ =1 X n ∈ Z n + ℓ/r e a an + ℓ/r w − ( n + ℓ/r ) , (4.12)Here we have normalized as Q a = 1 √ r Q a , α a = √ ra a , (4.13) α anr = − ( − nk √ rha an , (4.14) α anr + ℓ = e i πk ( nr + ℓ ) / − e − i πk ( nr + ℓ ) / √ r ( n + ℓ/r ) e a an + ℓ/r . (4.15)6he commutation relations are[ Q a , α b ] = C ab , [ Q a , Q b ] = 0 , [ α a , α b ] = 0 , (4.16)[ a an , a bm ] = nC ab δ n + m, , (4.17)[ e a an + ℓ/r , e a b − m − ℓ ′ /r ] = (cid:18) n + ℓr (cid:19) C ab δ n,m δ ℓ,ℓ ′ . (4.18)The correlation functions are h φ a ( w ) φ b ( w ′ ) i = C ab log( w − w ′ ) , (4.19) h ϕ aℓ ( w ) ϕ bℓ ′ ( w ′ ) i = δ ℓ + ℓ ′ ,r C ab r − X k =0 ω − kℓ log " − ω k (cid:18) w ′ w (cid:19) r , (4.20) h ϕ a ( w ) ϕ b ( w ′ ) i = C ab log (cid:20) (1 − ( w ′ /w ) /r ) r − ( w ′ /w ) (cid:21) . (4.21)For each e a , we define ψ e a ( w ) = A r w ( r − /r r − X ℓ =0 ω ℓ : exp "r r φ ( ℓ ) a ( w ) : , (4.22)where A r is a normalization factor and φ ( ℓ ) a ( w ) ≡ ϕ a ( e π i ℓ w ) . (4.23)Let α = P n − a =1 n a e a ∈ Q , where n a are non-negative integers and Q denotes the root lattice. Weobtain the corresponding parafermion, up to its normalization, ψ α ∼ Y ψ n a e a . (4.24)The independent parafermion can be given only for the case α ∈ Q/rQ . Not of all ψ α areindependent; 1 ∼ ψ e a · · · · · · ψ e a | {z } r . (4.25)For example, in the the case of sl (3) algebra and r = 4, the corresponding parafermions are drawnin the Fig. 1. We define the parafermion associated with negative of a simple root by ψ − e a ∼ ψ e a ψ e a · · · ψ e a | {z } r − . (4.26)The normalization can be determined by the correlation functions [47], h ψ α ( w ) ψ − α ( w ′ ) i = ( w − w ′ ) − α r , (4.27)where α = ( α, α ). In particular, h ψ e a ( w ) ψ − e a ( w ′ ) i = ( w − w ′ ) − r − r . (4.28)7ig. 1: The parafermions in the case of sl (3) and r = 4.In the case of the sl (2) algebra, we obtain the first Z r -parafermion, ψ ( w ) = ψ e ( w ) . (4.29)Similar to the case of n = 2 (3.22), the central charge is given by c ( r ) n = n ( n − r − r + n + ( n − (cid:18) − n ( n + 1) Q E r (cid:19) = r ( n − r + n − n ( n − Q E r . (4.30)When we set m = rm + + k , m − = m + + s in (4.8), this central charge becomes c ( r,m,s ) n = r ( n − r + n − rs n ( n − m ( m + rs )= ( n − r ( ms − n )( ms + n + r )( r + n ) ms ( ms + r ) , (4.31)which is the same as that of the coset model, b sl ( n ) r ⊕ b sl ( n ) ms − n b sl ( n ) ms − n + r . (4.32)Compared with (1.1) we find p = ms − n. (4.33)In the case of s = 0 corresponding to Q E = 0, we have the central charge of the usual Sugawarastress tensor for b sl ( n ) r , c ( r,m, n = r ( n − r + n = c b sl ( n ) r (4.34)8t is well-known that the affine Lie algebra b sl ( n ) r is represented by parafermions and an auxiliaryboson [47]. In the case of s = 1, because (4.31) becomes c ( r,m, n = ( n − r ( m − n )( m + n + r )( r + n ) m ( m + r ) , (4.35)the model gives us the unitary series of the coset, b sl ( n ) r ⊕ b sl ( n ) m − n b sl ( n ) m − n + r . (4.36)We can see how the level p is related with the omega-background parameters ǫ and ǫ in the4-d side. Since β = − ǫ /ǫ , (4.8) yields the condition to the ratio of these parameters. Therefore,when we introduce the free parameter ǫ , ǫ , can be written respectively as ǫ = ǫ ( p + n + r ) , ǫ = − ǫ ( p + n ) . (4.37)This result suggests that the Nekrasov-Shatashvili limit ǫ → ǫ →
0) of the N = 2 gaugetheory on the R / Z r corresponds to the critical level limit p + r → − n (resp. p → − n ) of thecoset model. Acknowledgments
We thank D. Serban for valuable discussions. The authors’ research is supported in part bythe Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture,Japan(23540316).
References [1] L. F. Alday, D. Gaiotto and Y. Tachikawa, “Liouville Correlation Functions from Four-dimensional Gauge Theories,” Lett. Math. Phys. , 167-197 (2010) [arXiv:0906.3219 [hep-th]].[2] N. Wyllard, “ A N − conformal Toda field theory correlation functions from conformal N = 2 SU ( N ) quiver gauge theories,” JHEP , 002 (2009) [arXiv:0907.2189 [hep-th]].[3] A. Mironov and A. Morozov, “On AGT relation in the case of U(3),” Nucl. Phys. B ,1-37 (2010) [arXiv:0908.2569 [hep-th]].[4] R. Dijkgraaf and C. Vafa, “Toda Theories, Matrix Models, Topological Strings, and N = 2Gauge Systems,” [arXiv:0909.2453 [hep-th]].[5] H. Itoyama, K. Maruyoshi and T. Oota, “The Quiver Matrix Model and 2d-4d ConformalConnection,” Prog. Theor. Phys. , 957-987 (2010) [arXiv:0911.4244 [hep-th]].[6] A. Mironov, A. Morozov and Sh. Shakirov, “Matrix Model Conjecture for Exact BS Periodsand Nekrasov Functions,” JHEP , 030 (2010) [arXiv:0911.5721 [hep-th]].97] A. Mironov, A. Morozov and Sh. Shakirov, “Conformal blocks as Dotsenko-Fateev IntegralDiscriminants,” J. Mod. Phys. A , 3173-3207 (2010) [arXiv:1001.0563 [hep-th]].[8] H. Itoyama and T. Oota, “Method of generating q -expansion coefficients for conformal blockand N = 2 Nekrasov function by β -deformed matrix model,” Nucl. Phys. B , 298-330(2010) [arXiv:1003.2929 [hep-th]].[9] A. Mironov, A. Morozov and And. Morozov, “Matrix model version of AGT conjecture andgeneralized Selberg integrals,” Nucl. Phys. B , 534-557 (2011) [arXiv:1003.5752 [hep-th]].[10] H. Itoyama, T. Oota and N. Yonezawa, “Massive scaling limit of the β -deformed matrixmodel of Selberg type,” Phys. Rev. D , 085031 (2010) [arXiv:1008.1861 [hep-th]].[11] A. Mironov, A. Morozov and Sh. Shakirov, “A direct proof of AGT conjecture at β = 1,”JHEP , 067 (2011) [arXiv:1012.3137 [hep-th]].[12] S. Kanno, Y. Matsuo and H. Zhang, “Extended Conformal Symmetry and Recursion Formulaefor Nekrasov Partition Function,” JHEP , 028 (2013) [arXiv:1306.1523 [hep-th]].[13] A. Morozov and A. Smirnov, “Towards the Proof of AGT Relations with the Help of theGeneralized Jack Polynomials,” Lett. Math. Phys. , 109-113 (2014) [arXiv:1312.5732 [hep-th]].[15] Y. Matsuo, C. Rim and H. Zhang, “Construction of Gaiotto states with fundamental multi-plets through Degenerate DAHA,” arXiv:1405.3141 [hep-th].[16] V. Belavin and B. Feigin, “Super Liouville conformal blocks from N = 2 SU (2) quiver gaugetheories,” JHEP , 079 (2011) [arXiv:1105.5800 [hep-th]].[17] T. Nishioka and Y. Tachikawa, “Central charges of para-Liouville and Toda theories fromM-5-branes,” Phys. Rev. D , 046009 (2011) [arXiv:1106.1172 [hep-th]].[18] A. Belavin, V. Belavin and M. Bershtein, “Instantons and 2d Superconformal field theory,”JHEP , 117 (2011) [arXiv:1106.4001 [hep-th]].[19] G. Bonelli, K. Maruyoshi and A. Tanzini, “Instantons on ALE spaces and super Liouville con-formal field theories,” JHEP , 056 (2011) [arXiv:1106.2505 [hep-th]]; “Gauge Theorieson ALE Space and Super Liouville Correlation Functions,” Lett. Math. Phys. , 103-124(2012) [arXiv:1107.4609 [hep-th]].[20] N. Wyllard, “Coset conformal blocks and N = 2 gauge theories,” arXiv:1109.4264 [hep-th].[21] B. Estienne, V. Pasquier, R. Santachiara and D. Serban, “Conformal blocks in Virasoro andW theories: Duality and the Calogero-Sutherland model,” Nucl. Phys. B , 377-420 (2012)[arXiv:1110.1101 [hep-th]]. 1022] Y. Ito, “Ramond sector of super Liouville theory from instantons on an ALE space,” Nucl.Phys. B , 387-402 (2012) [arXiv:1110.2176 [hep-th]].[23] M. N. Alfimov and G. M. Tarnopolsky, “Parafermionic Liouville field theory and instantonson ALE spaces,” JHEP , 036 (2012) [arXiv:1110.5628 [hep-th]].[24] A. A. Belavin, M. A. Bershtein, B. L. Feigin, A. V. Litvinov and G. M. Tarnopolsky, “Instan-ton moduli spaces and bases in coset conformal field theory,” Commun. Math. Phys. ,269-301 (2013) [arXiv:1111.2803 [hep-th]].[25] A. A. Belavin, M. A. Bershtein and G. M. Tarnopolsky, “Bases in coset conformal field theoryfrom AGT correspondence and Macdonald polynomials at the roots of unity,” arXiv:1211.2788[hep-th].[26] M. N. Alfimov, A. A. Belavin and G. M. Tarnopolsky, “Coset conformal field theory andinstanton counting on C / Z p ,” JHEP , 134 (2013) [arXiv:1306.3938 [hep-th]].[27] J. Shiraishi, H. Kubo, H. Awata and S. Odake, “A quantum deformation of the Virasoro al-gebra and the Macdonald symmetric functions,” Lett. Math. Phys. , 33-51 (1996) [arXiv:q-alg/9507034].[28] B. Feigin and E. Frenkel, “Quantum W -Algebras and Elliptic Algebras,” Commun. Math.Phys. , 653-678 (1996) [arXiv:q-alg/9508009].[29] H. Awata, H. Kubo, S. Odake and J. Shiraishi, “Quantum W N Algebras and MacdonaldPolynomials,” Commun. Math. Phys. , 401-416 (1996) [arXiv:q-alg/9508011].[30] H. Awata and Y. Yamada, “Five-Dimensional AGT Relation and the Deformed β -Ensemble,”Prog. Theor. Phys. , 227-262 (2010) [arXiv:1004.5122 [hep-th]].[31] F. Nieri, S. Pasquetti and F. Passerini, “3d & 5d gauge theory partition functions as q -deformed CFT correlators,” arXiv:1303.2626 [hep-th].[32] H. Itoyama, T. Oota and R. Yoshioka, “2d-4d Connection between q -Virasoro/W Block atRoot of Unity Limit and Instanton Partition Function on ALE Space,” Nucl. Phys. B ,506-537 (2013) [arXiv:1308.2068 [hep-th]]; “q-Virasoro algebra at root of unity limit and2d-4d connection,” J. Phys. Conf. Ser. , 012022 (2013).[33] M.-C. Tan, “An M-Theoretic Derivation of a 5d and 6d AGT Correspondence, and Relativisticand Elliptized Integrable Systems,” JHEP , 031 (2013) [arXiv:1309.4775 [hep-th]].[34] D. Orlando, “A stringy perspective on the quantum integrable model/gauge correspondence,”arXiv:1310.0031 [hep-th].[35] L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi, “Non-Lagrangian theories from branejunctions,” JHEP , 175 (2014) [arXiv:1310.3841 [hep-th]].[36] F. Nieri, S. Pasquetti, F. Passerini and A. Torrielli, “5D partition functions, q -Virasorosystems and integrable spin-chains,” arXiv:1312.1294 [hep-th].1137] H. Itoyama, A. Mironov and A. Morozov, “Matching branches of non-perturbative conformalblock at its singularity divisor,” arXiv:1406.4750 [hep-th].[38] P. Bouwknegt and K. Pilch, “The Deformed Virasoro Algebra at Roots of Unity,” Commun.Math. Phys. , 249-288 (1998) [arXiv:q-alg/9710026].[39] K. Takemura and D. Uglov, “The orthogonal eigenbasis and norms of eigenvectors in the spinCalogero-Sutherland model,” J. Phys. A , 3685-3718 (1997).[40] D. Uglov, “Yangian Gelfand-Zetlin Bases, gl N -Jack Polynomials and Computation of Dynam-ical Correlation Functions in the Spin Calogero-Sutherland Model,” Commun. Math. Phys. , 663-696 (1998).[41] G. Cristofano, G. Maiella and V. Marotta, “A twisted conformal field theory description ofthe quantum Hall effect,” Mod. Phys. Lett. A , 547-555 (2000) [arXiv:cond-mat/9912287].[42] Y. Kitazawa, N. Ishibashi, A. Kato, K. Kobayashi, Y. Matsuo and S. Odake, “Operatorproduct expansion coefficients in N = 1 superconformal theory and slightly relevant pertur-bation,” Nucl. Phys. B , 425-444 (1988).[43] L. Alvarez-Gaum´e and Ph. Zaugg, “Structure constants in the N = 1 superoperator algebra,”Annals Phys. , 171-230 (1992) [arXiv:hep-th/9109050].[44] A. B. Zamolodchikov and V. A. Fateev, “Nonlocal (parafermion) currents in two-dimensionalconformal quantum field theory and self-dual critical points in Z N -symmetric statistical sys-tems,” Zh. Eksp. Teor. Fiz. , 380-399 (1985) [Sov. Phys. JETP , 215-225 (1985)]; “Repre-sentations of the algebra of “parafermion currents” of spin 4/3 in two-dimensional conformalfield theory. Minimal models and the tricritical potts Z model,” Teor. Mat. Fiz. , 163-178(1987) [Theor. Math. Phys. , 451-462 (1987)].[45] V. Marotta, “Stress-tensor for parafermions from winding subalgebras of affine algebras,”Mod. Phys. Lett. A , 853-860 (1998) [arXiv:hep-th/9712031].[46] A. B. Zamolodchikov, “Exact solutions of conformal field theory in two dimensions and criticalphenomena,” Rev. Math. Phys. , 197-234 (1989).[47] D. Gepner, “New Conformal Field Theories Associated with Lie Algebras and Their PartitionFunction,” Nucl. Phys. B290