Logarithmic intertwining operators and W(2,2p−1) algebras

Abstract

For every p ജ 2, we obtained an explicit construction of a family of W͑2,2p −1͒ modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual W͑2,2p −1͒ modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic W͑2,2p −1͒ modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been referring to as "logarithmic conformal field theory" of central charge c p,1 =1 − ͓6͑p −1͒ 2 / p͔, p ജ 2. Our explicit construction can be easily applied for computations of correlation functions. Techniques from this paper can be used to study the triplet vertex operator algebra W͑2,͑2p −1͒ 3 ͒ and other logarithmic models.

W-algebras, higher rank false theta functions, and quantum dimensions

“…In many ways this paper is a continuation of [10] (see also [11,14,30]) where the authors thoroughly studied properties of certain false theta functions coming from characters of modules of the singlet vertex operator algebra. The starting point in this line of work was an observation [10,11] that the classical Rogers' false function and other related false theta functions show up as "numerators" of characters of the (1, p)-singlet W -algebra (see [1]). …”

“…where W (2, 2 p − 1) is the (1, p)-singlet vertex algebra [1]. It turns out that the most interesting q-series come from considerations of characters of W 0 ( p) Q -modules.…”

The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector

“…By using the same arguments as in [2] and [5], we can conclude that every irreducible Z ≥0 -gradable σ-twisted SM (1)-module is isomorphic to an irreducible subquotient of M (1, λ) ⊗ M ± . By using the structure of twisted Zhu's algebra A σ (SM (1)) and the methods developed in [3], we can also construct logarithmic σ-twisted SM (1)-modules.…”

An explicit realization of logarithmic modules for the vertex operator algebra $\mathcal {W}_{p,p^{\prime }}$Wp,p′

“…In [AM3], Theorem 9.1, we already constructed examples of logarithmic intertwining operators [AM1], [HLZ] among triples of logarithmic W p,p ′ modules, involving at most linear logarithmic factor log(z). Shortly after, Huang in [H], Theorem 5.8, connected our construction with his notion of generalized twisted modules, and based on [AM1] provided examples coming from automorphisms e 2πiQ and e 2πiQ .…”

“…For p ′ = 2 a strong generating set was found in [AM4]. Several W p,p ′ -modules in this paper will be logarithmic, that is they are not diagonalizable with respect to the L(0) Virasoro generator [AM1], [HLZ]. We also say that a…”