Harmonic maps and integrable systems (Q2867789)

Harmonic maps from a Riemann surface \(\Sigma\) to a Lie group \(G\) or a symmetric space \(G/K\) can be expressed in terms of a loop of flat connections in a principal \(G\)-bundle over the surface \(\Sigma\). For the 2-torus the description of harmonic maps was achieved through the construction of an algebraic curve \(X\), called the spectral curve, together with a line bundle over it and some other algebraic data. The induced linear flow on a sub-torus in the Jacobian of \(X\) relates harmonic maps with algebraic integrable Hamiltonian systems. The paper focusses on the discussion of various forms of periodicity conditions arising in relation to constructing the harmonic maps. A great deal of the background material is reviewed. A relation to the construction of harmonic maps via polynomial Killing fields and multiplier curve is drawn.NEWLINENEWLINEFor the entire collection see [Zbl 1272.57002].