Integer-valued characteristics of solutions of the noncommutative sigma model

Abstract

Any finite-energy solution of a noncommutative sigma model has three nonnegative integer-valued characteristics: the normalized energy e(Φ), canonical rank r(Φ), and minimum uniton number u(Φ). We prove that r(Φ) ≥ u(Φ) and e(Φ) ≥ u(Φ)(u(Φ) + 1)/2. Given any numbers e, r, u ∈ ℕ that satisfy the slightly stronger inequalities r ≥ u and e ≥ r +u(u − 1)/2, we construct a finite-energy solution Φ with e(Φ) = e, r(Φ) = r, and u(Φ) = u.