Open Access
2007 Grassmannian Sigma-Models
Armen Sergeev
J. Geom. Symmetry Phys. 9: 45-65 (2007). DOI: 10.7546/jgsp-9-2007-45-65


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 $\mathbb R^4$.


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Armen Sergeev. "Grassmannian Sigma-Models." J. Geom. Symmetry Phys. 9 45 - 65, 2007.


Published: 2007
First available in Project Euclid: 20 May 2017

zbMATH: 1143.81020
MathSciNet: MR2269884
Digital Object Identifier: 10.7546/jgsp-9-2007-45-65

Rights: Copyright © 2007 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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