Electron-Quasihole Duality and Second Order Differential Equation for Read-Rezayi and Jacks Wavefunctions

TL;DR

This paper derives a second order differential equation for Read-Rezayi quasihole wavefunctions, revealing a duality with electron wavefunctions and providing analytic solutions, extending to Jack states.

Contribution

It introduces a duality between electron and quasihole differential equations and derives a second order PDE for general quasihole wavefunctions in non-abelian quantum Hall states.

Findings

Derived a second order differential equation for quasihole wavefunctions.

Discovered a duality between electron and quasihole equations.

Provided an analytic expression for wavefunctions with excess flux.

Abstract

We consider the quasihole wavefunctions of the non-abelian Read-Rezayi quantum Hall states which are given by the conformal blocks of the minimal model WA_{k-1}(k+1,k+2) of the WA_{k-1} algebra. By studying the degenerate representations of this conformal field theories, we derive a second order differential equation satisfied by a general many-quasihole wavefunction. We find a duality between the differential equations fixing the electron and quasihole wavefunctions. They both satisfy the Laplace-Beltrami equation. We use this equation to obtain an analytic expression for the generic wavefunction with one excess flux. These results also apply to the more general models WA_{k-1}(k+1,k+r) corresponding to the recently introduced Jack states.