Description of Solutions with the Uniton Number 3 in the Case of One Eigenvalue: Counterexample to the Dimension Conjecture

Abstract

We explicitly describe solutions of the noncommutative unitary U (1) sigma model that represent finitedimensional perturbations of the identity operator and have only one eigenvalue and the minimum uniton number 3. We also show that the solution set M(e, r, u) of energy e and canonical rank r with the minimum uniton number u = 3 has a complex dimension greater than r for e = 4 n - 1 and r = n+1, where n ≥ 3. This disproves the dimension conjecture that holds in the case u ∈ {1, 2}.