SH$^c$ Realization of Minimal Model CFT: Triality, Poset and Burge Condition

TL;DR

This paper explores the realization of minimal model conformal field theories through the SH^c algebra, revealing triality symmetry, connections to Young diagrams, poset structures, and reproducing key identities like Rogers-Ramanujan.

Contribution

It demonstrates the SH^c algebra's triality automorphism and its role in describing minimal models, including the structure of singular vectors and partition functions.

Findings

Identification of an  automorphism (triality) in SH^c

Explicit computation of partition functions matching Rogers-Ramanujan identities

Reproduction of minimal model properties and singular vector structures

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.