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We sketch the proof that unitons (harmonic spheres in $U(N)$) correspond to holomorphic `uniton bundles', and that these admit monad representations analogous to Donaldson's representation of instanton bundles. We also give a closed-form expression for the unitons involving only matrix operations, a finite-gap result (two-unitons have energy $\ge4$), computations of fundamental groups of energy $\le 4$ components, new methods of proving discreteness of the energy spectrum and of Wood's Rationality Conjecture, a discussion of the maps into complex Grassmannians and some open problems.

Copyright American Mathematical Society 1996

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Article Info

• ERA Amer. Math. Soc. 02 (1996), pp. 7-16
• Publisher Identifier: S 1079-6762(96)00002-5
• 1991 Mathematics Subject Classification. Primary 58E20, 58D27, 58G37
• Key words and phrases. Uniton, harmonic map, chiral field, sigma model
• Received by the editors September 19, 1995
• Communicated by Eugenio Calabi
• Comments

Christopher Kumar Anand

Mathematics Research Centre, University of Warwick, Coventry CV4 7AL, UK

E-mail address: anand@math.warwick.ac.uk

Research supported by NSERC and FCAR scholarships.