Solutions of super Knizhnik–Zamolodchikov equations

Abstract

We establish an explicit bijection between the sets of singular solutions of the (super) KZ equations associated with the Lie superalgebra, of infinite rank, of type $$\\mathfrak {a}, \\mathfrak {b},\\mathfrak {c},\\mathfrak {d}$$ a , b , c , d and with the corresponding Lie algebra. As a consequence, the singular solutions of the super KZ equations associated with the classical Lie superalgebra, of finite rank, of type $$\\mathfrak {a}, \\mathfrak {b},\\mathfrak {c},\\mathfrak {d}$$ a , b , c , d for the tensor product of certain parabolic Verma modules (resp., irreducible modules) are obtained from the singular solutions of the KZ equations for the tensor product of the corresponding parabolic Verma modules (resp., irreducible modules) over the corresponding Lie algebra of sufficiently large rank, and vice versa. The analogous results for some special kinds of trigonometric (super) KZ equations are obtained.