$A_1^{(1)}$ Admissible Representations -- Fusion Transformations and Local Correlators

TL;DR

This paper revisits solutions to the KZ equations for $A_1^{(1)}$ admissible representations, simplifies the correlator sums, and explores their duality, fusion, and locality properties, revealing a family of monodromy invariants with non-diagonal terms.

Contribution

It provides simplified expressions for 4-point correlators, establishes their duality transformations, and introduces a family of monodromy invariants in the context of $A_1^{(1)}$ representations.

Findings

Effective summation of infinite series for correlators.

Consistency of duality transformations with fusion rules.

Existence of a 1-parameter family of monodromy invariants.

Abstract

We reconsider the earlier found solutions of the Knizhnik-Zamolodchikov (KZ) equations describing correlators based on the admissible representations of $A_1^{(1)}$. Exploiting a symmetry of the KZ equations we show that the original infinite sums representing the 4-point chiral correlators can be effectively summed up. Using these simplified expressions with proper choices of the contours we determine the duality (braid and fusion) transformations and show that they are consistent with the fusion rules of Awata and Yamada. The requirement of locality leads to a 1-parameter family of monodromy (braid) invariants. These analogs of the ``diagonal'' 2-dimensional local 4-point functions in the minimal Virasoro theory contain in general non-diagonal terms. They correspond to pairs of fields of identical monodromy, having one and the same counterpart in the limit to the Virasoro minimal correlators.