Quadratic Relations of the Deformed $W$-Algebra for the Twisted Affine Lie Algebra of Type $A_{2N}^{(2)}$

TL;DR

This paper develops a free field construction for higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$, establishing quadratic relations and duality.

Contribution

It introduces a new free field construction for higher $W$-currents of the deformed algebra for $A_{2N}^{(2)}$, including quadratic relations and duality.

Findings

Established quadratic relations for the deformed $W$-algebra.

Defined the algebra ${\mathcal W}_{x,r}(A_{2N}^{(2)})$ via generators and relations.

Extended the free field construction to higher $W$-currents.

Abstract

We revisit the free field construction of the deformed $W$-algebra by Frenkel and Reshetikhin [Comm. Math. Phys. 197 (1998), 1-32], where the basic $W$-current has been identified. Herein, we establish a free field construction of higher $W$-currents of the deformed $W$-algebra associated with the twisted affine Lie algebra $A_{2N}^{(2)}$. We obtain a closed set of quadratic relations and duality, which allows us to define deformed $W$-algebra ${\mathcal W}_{x,r}\big(A_{2N}^{(2)}\big)$ using generators and relations.