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

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title> <jats:p>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 <jats:italic>Y</jats:italic> algebra. Procházka and Rapčák, then proposed to interpret <jats:italic>Y</jats:italic> 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 <jats:italic>W</jats:italic>-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, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \mathcal{N} $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> = 2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known character of the minimal models.</jats:p>