# Plane partition realization of (web of) W $\mathcal{W}$ -algebra minimal models

Harada, Koichi (0000 0001 2151 536X, Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan) ; Matsuo, Yutaka (0000 0001 2151 536X, Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan)

10 February 2019

Abstract: Recently, Gaiotto and Rapčák (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. Procházka and Rapčák, 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 $\mathcal{N}$ = 2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known character of the minimal models.

Published in: JHEP 1902 (2019) 050