The Wess-Zumino term for a harmonic map (Q2769416)

Let \(G\) be a compact, simple, simply-connected Lie group. If \(B\) denotes the Killing form of \(G\) and \(\omega\) the Maurer-Cartan form of \(G\), then \(\Omega = c B (\omega,[ \omega, \omega ] )\) is a bi-invariant closed 3-form on \(G\), whose de Rham class generates \(H^3 (G, 2 \pi \mathbb Z)\) for some choice of \(c\).NEWLINENEWLINELet \(g : \Sigma \rightarrow G\) be a smooth map of a closed oriented surface \(\Sigma\) to \(G\) and \(M\) some oriented 3-manifold with boundary \(\Sigma\). Then g can be extended to \(\tilde{g} : M \rightarrow G\) and the Wess-Zumino term of \(g\) is defined by \( \Gamma (g) = \int_M \tilde{g}^* \Omega\).NEWLINENEWLINEThis number is independent of the choice of \(M\) modulo \(2 \pi\). Thus \(\Gamma (g) \in \mathbb{T} = \mathbb{R} /{ 2 \pi \mathbb{Z}}\) is uniquely determined.NEWLINENEWLINEThe first part of the paper provides formulas for \(\Gamma (g)\), if \(g\) takes values in some totally geodesic submanifold of \(G\). Starting in section 3 only harmonic maps are considered. In section 3 an explicit formula is given if \(\Sigma = S^2\). For higher genus surfaces \(\Sigma\), a simple formula is proved, which relates the Wess-Zumino term to the holonomy of the natural connection on the Chern-Simons line bundle over \(\mathcal{M}\), where \(\mathcal{M}\) denotes the moduli space of all flat \(G\)-connections on the trivial bundle \(\Sigma x G\). This result is made explicit for harmonic maps from the 2-torus to \(G\) and is illustrated by examples.