Spinon Bases, Yangian Symmetry and Fermionic Representations of Virasoro Characters in Conformal Field Theory

TL;DR

This paper constructs a basis for the Hilbert space of an $SU(2)$, level 1 Wess-Zumino-Witten conformal field theory using Yangian symmetry and derives fermionic representations of Virasoro characters, confirming previous conjectures.

Contribution

It introduces a Yangian-based basis for the theory's Hilbert space and explicitly derives fermionic Virasoro character representations, advancing understanding of algebraic structures in conformal field theory.

Findings

Established generalized canonical commutation relations for affine primary fields.

Constructed a Yangian representation-based basis for the Hilbert space.

Derived fermionic representations of Virasoro characters, confirming conjectures.

Abstract

We study the description of the $SU(2)$, level $k=1$, Wess-Zumino-Witten conformal field theory in terms of the modes of the spin-1/2 affine primary field $\phi^\alpha$. These are shown to satisfy generalized `canonical commutation relations', which we use to construct a basis of Hilbert space in terms of representations of the Yangian $Y(sl_2)$. Using this description, we explicitly derive so-called `fermionic representations' of the Virasoro characters, which were first conjectured by Kedem et al.~\cite{kedem}. We point out that similar results are expected for a wide class of rational conformal field theories.