Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces (Q1901337)

A smooth map \(f: M\to \text{SU} (2)\) of a compact Riemann surface of genus \(\geq 2\) pulls back the Maurer-Cartan form to a flat connection \(\omega_f\) on the trivial bundle \(E\to M\). By a result of \textit{N. J. Hitchin} [J. Differ. Geom. 31, No. 3, 627-710 (1990; Zbl 0725.58010)], the loop \[ t\to \sigma_f (t)= {\textstyle {1\over 2}} (\omega_f+ (\cos t) \omega_f+ (\sin t) (*\omega_f)), \qquad 0\leq t\leq 2\pi, \] lies in the space \({\mathcal F}\) of flat connections iff \(f\) is harmonic. On the other hand, any loop \(\sigma: s^1\to {\mathcal F}\) extends to a map \(\widetilde {\sigma}\) of the disk \(D^2\) to the space of all connections \({\mathcal C}\) and thereby defines a connection on the SU(2)-bundle \(E\times D^2\to M\times D^2\). The second Chern polynomial evaluated on the curvature of this connection gives a closed 4-form on \(M\times D^2\). This, integrated along \(D^2\), gives a closed 2-form \(\chi (\sigma)\in H^2 (M, \mathbb{R})\approx \mathbb{R}\). The author shows that \(\chi (\sigma_f )=- (1/4 \pi) E(f)\), where \(E\) is the energy of the harmonic map \(f: M\to \text{SU} (2)\) with respect to the Poincaré metric on \(M\) and the Killing form on SU(2).