Plane Partition Realization of (Web of) W-algebra Minimal Models

TL;DR

This paper shows how plane partitions can realize minimal models of W-algebras, connecting geometric combinatorics with algebraic structures in vertex operator algebras, and extends the web of W-algebra framework.

Contribution

It demonstrates that double truncation of plane partitions yields minimal models of W-algebras and refines the rules connecting plane partitions for negative U(1) charges.

Findings

Double truncation of plane partitions produces W-algebra minimal models.

The refined connection rule accurately reproduces known characters of N=2 superconformal minimal models.

The approach extends the web of W-algebra framework to include minimal models.

Abstract

Recently, Gaiotto and Rapcak (GR) proposed a new family of the vertex operator algebra (VOA) as the symmetry appearing at an intersection of five-branes to which they refer as Y algebra. Prochazka and Rapcak, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). They also developed GR's idea to generate a new VOA by connecting plane partitions through an infinite leg shared by them and referred it as the web of W-algebra (WoW). In this paper, we demonstrate that double truncation of PP gives the minimal models of such VOAs. For a single PP, it generates all the minimal model irreducible representations of W-algebra. We find that the rule connecting two PPs is more involved than those in the literature when the U(1) charge connecting two PPs is negative. For the simplest nontrivial WoW, N=2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known character of the minimal models.