Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}\bigl(A(M,N)\bigr)$

TL;DR

This paper constructs free field representations of the deformed $W$-superalgebra for Lie superalgebras of type $A(M,N)$, establishing quadratic relations among higher $W$-currents that are independent of Dynkin-diagram choices.

Contribution

It provides a novel free field construction and quadratic relations for the deformed $W$-superalgebra ${\

Findings

Established free field realization of basic $W$-current and screening currents.

Derived quadratic relations among higher $W$-currents.

Showed relations are independent of Dynkin-diagram choices.

Abstract

We find the free field construction of the basic $W$-current and screening currents for the deformed $W$-superalgebra ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$ associated with Lie superalgebra of type $A(M,N)$. Using this free field construction, we introduce the higher $W$-currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin-diagrams for the Lie superalgebra $A(M,N)$, though the screening currents are not. This allows us to define ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$ by generators and relations.