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

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.