Abstract
Recently an orthogonal basis of $\mathcal{W}_N$-algebra (AFLT basis) labeled by $N$-tuple Young diagrams was found in the context of 4D/2D duality. Recursion relations among the basis are summarized in the form of an algebra SH$^c$ which is universal for any $N$. We show that it has an $\mathfrak{S}_3$ automorphism which is referred to as triality. We study the level-rank duality between minimal models, which is a special example of the automorphism. It is shown that the nonvanishing states in both systems are described by $N$ or $M$ Young diagrams with the rows of boxes appropriately shuffled. The reshuffling of rows implies there exists partial ordering of the set which labels them. For the simplest example, one can compute the partition functions for the partially ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan identities. We also study the description of minimal models by SH$^c$. Simple analysis reproduces some known properties of minimal models, the structure of singular vectors and the $N$-Burge condition in the Hilbert space.
Highlights
Many years ago [1], W-algebra was formulated as a nonlinear generalization of the two dimensional conformal field theory and has been playing significant roles in many branches of physics, such as string theory, quantum gravity, the statistical mechanics, and the exactly solvable systems
We studied some properties of the algebra SHc
We show that the singular vectors of a WN module can be understood from the SHc action on the AFLT basis
Summary
Many years ago [1], W-algebra was formulated as a nonlinear generalization of the two dimensional conformal field theory and has been playing significant roles in many branches of physics, such as string theory, quantum gravity, the statistical mechanics, and the exactly solvable systems. The rank N representation of SHc is spanned by N -tuple Young diagrams and can be realized in terms of N free bosons. It is a duality between minimal models in WN - and WM -algebras with N = M. This is somehow puzzling since we need to find a direct correspondence between two Hilbert spaces spanned by different number of Young diagrams. Another motivation to study the duality is that it would be related to the so-called triality symmetry of W∞[μ] [12, 13]. We study the action of generators of SHc and derive the N -Burge condition
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