The noncommutative instantons were discovered by Nikita Nekrasov and Albert Schwarz in 1998.
Setting B2 and J to zero, one obtains the classical moduli space of nonabelian vortices in a supersymmetric gauge theory with an equal number of colors and flavors, as was demonstrated in Vortices, instantons and branes.
The generalization to greater numbers of flavors appeared in Solitons in the Higgs phase: The Moduli matrix approach.
In both cases the Fayet–Iliopoulos term, which determines a squark condensate, plays the role of the noncommutativity parameter in the real moment map.
Let x be the 4-dimensional Euclidean spacetime coordinates written in quaternionic notation
are equivalent to the factorization condition Then a hermitian projection operator P can be constructed as The nullspace of Δ(x) is of dimension N for generic x.
The basis vectors for this null-space can be assembled into an (N + 2k) × N matrix U(x) with orthonormalization condition U†U = 1.
A regularity condition on the rank of Δ guarantees the completeness condition The anti-selfdual connection is then constructed from U by the formula This quantum mechanics-related article is a stub.