The Wess-Zumino term for a harmonic map

Abstract

We calculate the Wess-Zumino term $\Gamma(g)$ for a harmonic map $g$ of a closed surface to a compact, simply connected, simple Lie group $G$ in terms of the energy and the holonomy of the Chern-Simons line bundle on the moduli space of flat $G$-connections. In the case of the 2-sphere we deduce that $\Gamma(g)$ is 0 or $\pi$ and for the 2-torus and $G=SU(2)$ we give a formula involving hyperelliptic integrals.