Grassmannian Sigma-Models

Abstract

We study solutions of Grassmannian sigma-model both in finite-dimensional and infinite-dimensional settings. Mathematically, such solutions are described by harmonic maps from the Riemann sphere $\mathbb C\mathbb P^1$ or, more generally, compact Riemann surfaces to Grassmannians. We describe first how to construct harmonic maps from compact Riemann surfaces to the Grassmann manifold\linebreak $\operatorname{G}_r(\mathbb C^d)$, using the twistor approach. Then we switch to the infinite-dimensional setting and consider harmonic maps from compact Riemann surfaces to the Hilbert-Schmidt Grassmannian $\operatorname{Gr}_{\text{HS}}(H)$ of a complex Hilbert space $H$. Solutions of this infinite-dimensional sigma-model are, conjecturally, related to Yang-Mills fields on $\mathbb R^4$.