Non-linear \(\sigma\)-models on compact Riemann surfaces (Q1088255)

An O(3) model is a theory of a smooth three component real field \(\phi =(\phi^ a)\), \(a=1,2,3\), defined on \(R^ 2\), subject to the constraint \(\phi^ a\phi^ a=1\), with the action given by integrating 1/2 \(\delta\) \({}^{\mu \nu}\partial_{\mu}\phi^ a\partial_{\nu}\phi^ a\) over \(R^ 2\), where \(\delta^{\mu \nu}\) is the Euclidean metric on \(R^ 2\). In this paper a generalization of the O(3) model, in which the field \(\phi\) is a smooth map from a compact Riemann surface into a compact Kähler manifold, is studied. The various results obtained by the author show that such a framework is very convenient for the theory of fields.