$q$-Virasoro/W Algebra at Root of Unity and Parafermions

We 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 $ \frac{\widehat{\mathfrak{sl}}(n)_r \oplus \widehat{\mathfrak{sl}}(n)_{m-n}}{\widehat{\mathfrak{sl}}(n)_{m-n+r}}$ is given from the parafermion construction of the block in the limit.


Introduction
Ever since the AGT relation [1,2,3] (the correspondence between the correlators of 2d QFT and the 4d instanton sum) was introduced, the both sides of the correspondence have been intensively studied by a number of people. For example, in the 2d side, the β-deformed matrix model is used in order to control the integral representation of the conformal block [4,5,6,7,8,9,10]. There are also some proposals for proving the 2d-4d connection [11,12,13,14,15]. Moreover similar correspondence has been found and examined [16,17,18,19,20,21,22,23,24,25,26]. Among these, we pay our attention, in this paper, to the correspondence between the coset model, sl(n) r ⊕ sl(n) p sl(n) r+p , (1.1) and the N = 2 SU(n) gauge theory on R 4 /Z r [20,23]. Here sl(n) k stands for the affine Lie algebra 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] and the W n algebra [28,29] is known, while the 4d gauge theories can be lifted to five-dimensional theories with the fifth direction compactified on a circle. There exists a natural generalization to the connection between the 2d theory based on the q-deformed Virasoro/W algebra and the five-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/W block in the 2d side from that in the q-deformed version. On the other hand, we saw that the instanton partition function on R 4 /Z r are generated from that on R 5 at the same limit. This result means if we assume the 2d-5d connection, it is automatically assured that the Virasoro/W blocks generated by using the limiting procedure agree with the instanton partition function on R 4 /Z r . Our limiting procedure corresponds to a root of unity limit in q. A root of unity limit of the q-Virasoro algebra was also considered in [38]. Our limit is slightly different from this and is similar to the one used in order to construct the eigenfunctions of the spin Calogero-Sutherland model from Macdonald polynomials in [39,40].
In the present paper we will elaborate our limiting procedure and show that the Z r -parafermionic CFT which has the symmetry described by (1.1) appears in the 2d side. We clarify also the relation between 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 and show that the Z r -parafermion currents are derived in a natural way. In section 4, we consider the generalization to q-W n algebra.

Root of Unity Limit of q-Virasoro Algebra
In this section, we review the root of unity limit [32] of the q-deformed Virasoro algebra [27] which has two parameters q and t = q β . The defining relation is where p = q/t and The multiplicative delta function is defined by Using the q-deformed Heisenberg algebra H q,t : the q-Virasoro operator T (z) can be realized as The q-deformed chiral bosons are defined in terms of the q-deformed Heisenberg algebra as (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 Since t = q β , this limit is possible if the parameter β takes the rational number such as where m ± are non-negative integers. In the limit, we have two types of bosons φ(w) and ϕ(w) [32] respectively given by where w = z r and The commutation relations are [a m , a n ] = mδ m+n,0 , [a n , 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 where the Jackson integral is defined by Multiplying the regularization factor, we obtain the screening charge in the root of unity limit, up to normalization, where we have defined [41] Here A r is the normalization factor and we have introduced The correlation function is given by For example, we consider the r = 2 case. In the limit, we obtain lim q→−1 S(z) =: e √ βφ(w) e ϕ(w) :, (3.8) and after the appropriate normalization, we obtain the following screening charge for the superconformal block [42,43]: is the NS fermion. From now on we will show that the Z r -parafermions appear in the general r-th root of unity limit. In particular, ψ 1 (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 the following defining relations [44]: where ψ † ℓ (w) = ψ r−ℓ (w) and 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 The OPE of (3.4) is (3.16) Here we have defined the second parafermion, Similarly, the (ℓ + 1)-th parafermion is obtained from ℓ-th parafermion by In particular, where B r is a constant which can be determined by the relation 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 where T B (w) stands for the usual stress tensor for the chiral boson field. The central charge is 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).

Root of Unity Limit of q-W n Algebra
In this section, we consider the generalization to the q-W n algebra [29]. We denote by h the Cartan subalgebra of sl(n) Lie algebra. The q-W n algebra is expressed in terms of the following h-valued q-deformed boson, and e a (a = 1, · · · , n − 1) are the simple roots and , : h * ⊗ h → C is the canonical pairing. The commutation relations are given by where C ab is the Cartan matrix of A type and The q-number is defined by Similar to the q-Virasoro case, we consider the limit, where ω = e 2πi r and k is a natural number mutually prime to r. The condition to be able to take this limit is that β is a rational number, where m ± are non-negative integers. Taking this limit, lim h→0 ϕ a 0 (z) = we obtain 1 n a a n w −n , (4.11) Here we have normalized as α a nr = −(−1) nk √ rha a n , (4.14) α a nr+ℓ = e iπk(nr+ℓ)/2 − e −iπk(nr+ℓ)/2 √ r(n + ℓ/r) a a n+ℓ/r . The commutation relations are [a a n , a b m ] = nC ab δ n+m,0 , (4.17) [ a a n+ℓ/r , a b −m−ℓ ′ /r ] = n + ℓ r C ab δ n,m δ ℓ,ℓ ′ . (4.18) The correlation functions are For each e a , we define where A r is a normalization factor and φ (ℓ) a (w) ≡ ϕ a (e 2πiℓ w). Let α = n−1 a=1 n a e a ∈ Q, where n a are non-negative integers and Q denotes the root lattice. We obtain the corresponding parafermion, up to its normalization, The independent parafermion can be given only for the case α ∈ Q/rQ. Not of all ψ α are independent; 1 ∼ ψ ea · · · · · · ψ ea r . (4.25) For example, in the the case of sl(3) algebra and r = 4, the corresponding parafermions are drawn in the Fig. 1. We define the parafermion associated with negative of a simple root by ψ −ea ∼ ψ ea ψ ea · · · ψ ea r−1 . (4.26) The normalization can be determined by the correlation functions [47], where α 2 = (α, α). In particular, In the case of the sl(2) algebra, we obtain the first Z r -parafermion, Similar to the case of n = 2 (3.22), the central charge is given by c (r) n = n(n − 1)(r − 1) r + n + (n − 1) 1 − n(n + 1) In the case of s = 0 corresponding to Q E = 0, we have the central charge of the usual Sugawara stress tensor for sl(n) r , c (r,m,0) n = r(n 2 − 1) r + n = c sl(n)r (4.34) It is well-known that the affine Lie algebra sl(n) r is represented by parafermions and an auxiliary boson [47]. In the case of s = 1, because (4.31) becomes c (r,m,1) n = (n 2 − 1)r(m − n)(m + n + r) (r + n)m(m + r) , (4.35) the model gives us the unitary series of the coset, sl(n) r ⊕ sl(n) m−n sl(n) m−n+r . (4.36) We can see how the level p is related with the omega-background parameters ǫ 1 and ǫ 2 in the 4-d side. Since β = −ǫ 1 /ǫ 2 , (4.8) yields the condition to the ratio of these parameters. Therefore, when we introduce the free parameter ǫ, ǫ 1,2 can be written respectively as ǫ 1 = ǫ(p + n + r), ǫ 2 = −ǫ(p + n). (4.37) This result suggests that the Nekrasov-Shatashvili limit ǫ 1 → 0 (resp. ǫ 2 → 0) of the N = 2 gauge theory on the R 4 /Z r corresponds to the critical level limit p + r → −n (resp. p → −n) of the coset model.