$q$-Deformation of Corner Vertex Operator Algebras by Miura Transformation

TL;DR

This paper introduces a $q$-deformation of the Miura transformation for corner vertex operator algebras, providing a free field representation of the $q$-deformed algebra derived from quantum toroidal algebra, revealing a simpler structure.

Contribution

It derives a $q$-deformed Miura transformation for corner vertex operator algebras, connecting it to quantum toroidal algebra and demonstrating structural simplifications.

Findings

$q$-deformed Miura transformation provides free field representation

Direct correspondence between Miura operators and quantum toroidal algebra operators

Screening charges of symmetries are identical in the deformed case

Abstract

Recently, Gaiotto and Rapcak proposed a generalization of $W_N$ algebra by considering the symmetry at the corner of the brane intersection (corner vertex operator algebra). The algebra, denoted as $Y_{L,M,N}$, is characterized by three non-negative integers $L, M, N$. It has a manifest triality automorphism which interchanges $L, M, N$, and can be obtained as a reduction of $W_{1+\infty}$ through a "pit" in the plane partition representation. Later, Prochazka and Rapcak proposed a representation of $Y_{L,M,N}$ in terms of $L+M+N$ free bosons through a generalization of Miura transformation, where they use the fractional power differential operators. In this paper, we derive a $q$-deformation of their Miura transformation. It gives the free field representation for $q$-deformed $Y_{L,M,N}$, which is obtained as a reduction of the quantum toroidal algebra. We find that the $q$-deformed version has a "simpler" structure than the original one because of the Miki duality in the quantum toroidal algebra. For instance, one can find a direct correspondence between the operators obtained by the Miura transformation and those of the quantum toroidal algebra. Furthermore, we can show that the screening charges of both the symmetries are identical.