3D Boson representation of affine Yangian of ${\mathfrak{gl}}(1)$ and 3D cut-and-join operators

TL;DR

This paper introduces 3D Bosons and their Fock space, representing affine Yangian generators and defining 3D cut-and-join operators, with applications to matrix model $W$-operators.

Contribution

It constructs a 3D Boson framework and demonstrates its use in representing affine Yangian of ${ m gl}(1)$ and related operators, extending previous 2D theories.

Findings

3D Bosonic Fock space is isomorphic to 3D Young diagrams

Representation of affine Yangian generators using 3D Bosons

Definition of 3D cut-and-join operators and their relation to matrix models

Abstract

We have constructed 3D Bosons. In this paper, we show the 3D Bosonic Fock space, which is isomorphic to the vector space of 3D Young diagrams as graded vector spaces. We use 3D Bosons to represent the generators of the affine Yangian of ${\mathfrak{gl}}(1)$ and define the 3D cut-and-join operators. Then we discuss the 3D Boson representation of $W$-operators in matrix models.