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Unitons and their Moduli
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## Unitons and their moduli

### Christopher Kumar Anand

**Abstract.**
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.

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