A Closed Form for Unitons
نویسنده
چکیده
Unitons, i.e. harmonic spheres in a unitary group, correspond to ‘uniton bundles’, i.e. holomorphic bundles over the compactified tangent space to the complex line with certain triviality and other properties. In this paper, we use a monad representation similar to Donaldson’s representation of instanton bundles to obtain a simple formula for the unitons. Using the monads, we show that real triviality for uniton bundles is automatic. We interpret the uniton number as the ‘length’ of a jumping line of the bundle, and identify the uniton bundles which correspond to based maps into Grassmannians. We also show that energy-3 unitons are 1-unitons, and give some examples.
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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 ≥ 4), computations of fundamental groups of energy ≤ 4 ...
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