# Solutions of super Knizhnik-Zamolodchikov equations

## TL;DR

This paper establishes a direct correspondence between solutions of super and classical KZ equations for Lie (super)algebras, enabling transfer of solutions between finite and infinite rank cases, including trigonometric variants.

## Contribution

It provides an explicit bijection linking singular solutions of super KZ equations to those of classical KZ equations across different ranks and types.

## Key findings

- Explicit bijection between super and classical KZ solutions.
- Solutions for finite rank super KZ equations derived from infinite rank cases.
- Results extended to trigonometric super KZ equations.

## Abstract

We establish an explicit bijection between the sets of singular solutions of the (super) KZ equations associated to the Lie superalgebra, of infinite rank, of type $\mf{a, b,c,d}$ and to the corresponding Lie algebra. As a consequence, the singular solutions of the super KZ equations associated to the classical Lie superalgebra, of finite rank, of type $\mf{a, b,c,d}$ 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.