Ising-like and Fibonacci-Anyons from KZ-equations

TL;DR

This paper solves KZ equations for Ising- and Fibonacci-Anyons, providing explicit solutions and monodromy representations that confirm their non-Abelian statistics, while uniquely tracking spin basis states.

Contribution

It introduces a method to explicitly solve KZ equations for these anyons and track conformal blocks with spin basis states, enhancing understanding of their non-Abelian braiding.

Findings

Explicit hypergeometric solutions to KZ equations

Derived monodromy representations of braid group

Confirmed non-Abelian statistics of the solutions

Abstract

In this work we present solutions to Knizhnik-Zamolodchikov (KZ) equations corresponding to conformal block wavefunctions of non-Abelian Ising- and Fibonacci-Anyons. We solve these equations around regular singular points in configuration space in terms of hypergeometric functions and derive explicit monodromy representations of the braid group action. This confirms the correct non-Abelian statistics of the solutions. One novelty of our approach is that we explicitly keep track of spin basis states and identify conformal blocks uniquely with such states at relevant points in moduli space.