Abstract
We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the mathfrak{sl}(2)/mathfrak{u}(1) coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from mathfrak{sl}(2) Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type mathfrak{sl}left(N+1right)/left(mathfrak{sl}(N)times mathfrak{u}(1)right) and investigate the equivalence to a theory with an mathfrak{sl}left(N+left.1right|Nright) structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for mathfrak{sl}(N) and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].
Highlights
The level k is related to the inverse of curvature and the sigma model description is suitable for large k
We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the sl(2)/u(1) coset and sine-Liouville theory
Recall that at the corner of interfaces of four dimensional supersymmetric gauge theories with gauge groups U(N1), U(N2) and U(N3) there is a vertex operator algebras (VOAs), called the YN1,N2,N3-algebra [9]. This vertex algebra is parameterized by a parameter ψ, which is the level of the associated W-algebra shifted by the dual Coxeter number
Summary
As explained in the introduction, a class of VOA denoted by YN1,N2,N3[ψ] was introduced in [9] through brane junctions. The second one is realized by a truncation of a W1+∞[λ]algebra, which will be explained in the subsection. The third one is given by an intersection of kernels of screening charges in a free field theory, which is the definition of WN1,N2,N3 mentioned in the introduction. This screening realization has been proven to coincide with the coset definition in a special series of cases [14]. An ideal forms at λ = L with integer L ≥ 2 and WL-algebra with the truncation of spin as s = 2, 3, . For the symmetry algebra of the coset model (1.3), the non-negative integers Nj should be set as.
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