Affine and Yangian Symmetries in $SU(2)_1$ Conformal Field Theory

TL;DR

This paper compares affine and Yangian algebraic structures in $SU(2)_1$ conformal field theory, demonstrating how the Yangian approach offers new methods for computing correlation functions and characters, with potential for broader application.

Contribution

It introduces a Yangian-based formulation of $SU(2)_1$ CFT, providing alternative tools for correlation functions and character calculations beyond the affine symmetry approach.

Findings

Yangian formulation reproduces known correlation functions.

New expressions for Virasoro and affine characters derived.

Yangian approach generalizes to other rational CFTs.

Abstract

In these lectures, we study and compare two different formulations of $SU(2)$, level $k=1$, Wess-Zumino-Witten conformal field theory. The first, conventional, formulation employs the affine symmetry of the model; in this approach correlation functions are derived from the so-called Knizhnik-Zamolodchikov equations. The second formulation is based on an entirely different algebraic structure, the so-called Yangian $Y(sl_2)$. In this approach, the Hilbert space of the theory is obtained by repeated application of modes of the so-called spinon field, which has $SU(2)$ spin $j=\thalf$ and obeys fractional (semionic) statistics. We show how this new formulation, which can be generalized to many other rational conformal field theories, can be used to compute correlation functions and to obtain new expressions for the Virasoro and affine characters in the theory. [Lectures given at the 1994 Trieste Summer School on High Energy Physics and Cosmology, Trieste, July 1994.]