W-algebras related to parafermion algebras

Abstract

We study a W -algebra of central charge 2(k − 1)/(k + 2), k = 2, 3, . . . , contained in the commutant of a Heisenberg algebra in a simple affine vertex operator algebra L(k, 0) of type A (1) 1 with level k. We calculate the operator product expansions of the W -algebra. We also calculate some singular vectors in the case k 6 and determine the irreducible modules and Zhu's algebra. Furthermore, the rationality and the C 2 -cofiniteness are verified for such k.The argument heavily depends on singular vectors v 0 , v 1 and v 2 of weight 8, 9 and 10, respectively in W and on singular vectors u r = (W 3 1 ) r u 0 of weight k + 1 + r, r = 0, 1, 2, 3 in W. It seems that we can take W 3 1 v 0 and (W 3 1 ) 2 v 0 in place of v 1 and v 2 , respectively. However, we do not verify it. The importance of u 0 is clear from the degenerate case, namely, the case k = 2, 3, 4, for u 0 is a scalar multiple of W 3 , W 4 or W 5 in such a case. It would be difficult to express u r , r = 0, 1, 2, 3 in terms of ω, W 3 , W 4 and W 5 for an arbitrary k. We should take a different approach for a general case.Our notation is fairly standard [12,20]. Let V be a vertex operator algebra and (M, Y M ) be its module. Then Y M (v, x) = n∈Z v n x −n−1 is the vertex operator associated with v ∈ V . The linear operator v n on M is called a component operator. For a subalgebra U of V and a subset S of M, let U · S = span{u n w | u ∈ U , w ∈ S, n ∈ Z}, which is the U -submodule of M generated by S.Part of the results in this paper was announced in [6]. We remark that N 0 (resp. K 0 ) is denoted by W (resp. W) in [6]. In this paper, we distinguish N 0 and W (resp. K 0 and W) clearly to avoid confusion. W and its singular vectorsLet {h, e, f } be a standard Chevalley basis of sl 2 . Thus [h, e] = 2e, [h, f ] = −2 f , [e, f ] = h for the bracket and h, h = 2, e, f = 1, h, e = h, f = e, e = f , f = 0 for the normalized Killing form.We fix an integer k 2. Let V (k, 0) = V sl 2 (k, 0) be a Weyl module for the affine Lie algebra sl 2 = sl 2 ⊗ C[t, t −1 ] ⊕ CC with level k, that is, a generalized Verma module for sl 2 with level k and highest weight 0. Let 1 be its canonical highest weight vector, which is called the vacuum vector. Then sl 2 ⊗ C[t] acts as 0 and C acts as k on 1. We denote by h(n), e(n) and f (n) the operators on V (k, 0) induced by the action of h ⊗t n , e ⊗t n and f ⊗t n , respectively. Thus h(n)1 = e(n)1 = f (n)1 = 0 for n 0 andfor a, b ∈ {h, e, f }. The elements h(−i 1 ) · · · h(−i p )e(− j 1 ) · · · e(− j q ) f (−m 1 ) · · · f (−m r )1, (2.2) i 1 · · · i p 1, j 1 · · · j q 1, m 1 · · · m r 1 form a basis of V (k, 0).Let a(x) = n∈Z a(n)x −n−1 for a ∈ {h, e, f } and

On C2-cofiniteness of parafermion vertex operator algebras

“…Applying this to the parafermion vertex operator algebras, we show that the C 2 -cofiniteness for general parafermion vertex operator algebra K (g, k) is completely determined by the regularity of K (sl 2 , k). It was shown in [4] that K (sl 2 , k) is rational and C 2 -cofinite for k 6, thus the C 2 -cofiniteness for general parafermion vertex operator algebra K (g, k) of small level k follows.…”

“…Some aspects of both the structure and representation theory of parafermion vertex operator algebras have been studied in [3][4][5]11]. In particular, a set of generators for general parafermion vertex operator algebra K (g, k) is obtained.…”

“…On the surface, K (g, k) is the commutant of the Heisenberg vertex operator algebra which is neither rational nor C 2 -cofinite in regular vertex operator algebra L(k, 0). But the parafermion vertex operator algebra K (g, k) can also be realized as the commutant of lattice vertex operator algebra V √ kQ L in L(k, 0), where Q L is the positive definite even lattice generated by the long roots of g (for example, see [6,4]). There is no doubt that the representation theory of K (g, k)…”

The Structure of Parafermion Vertex Operator Algebras: General Case

“…Although the parafermion field theory has been studied for more than two decades, the mathematical investigation of the parafermion vertex operator algebras have been limited by a lack of understanding of the structural theory of these algebras. The goals of this paper and [4][5][6] are to alleviate this situation.…”

“…More importantly, it is widely believed that K (g, k) should give a new class of rational, C 2 -cofinite vertex operator algebras although this can only been proved in the case g = sl 2 and k ≤ 6 [5]. Most well known vertex operator algebras such as lattice vertex operator algebras [2,3,10], the affine vertex operator algebras [7,11,23] and the Virasoro vertex operator algebras [11,25] can be understood well using the underline lattices or Lie algebras.…”