Atiyah-Drinfeld-Hitchin-Manin construction of noncommutativeU(2)k-instantons

Abstract

The basic objects of the ADHM construction are reformulated in terms of elements of the A θ (R 4 ) algebra of the noncommutative R 4 θ space. This new formulation of the ADHM construction makes possible the explicit calculus of the U (2) instanton number which is shown to be the product of a trace of a finite rank projector of the Fock representation space of the algebra A θ (R 4 ) times a noncommutative version of the winding number.

Cohomological gauge theory, quiver matrix models and Donaldson–Thomas theory

“…An explicit construction of the data (4.13) for U (3) gauge group may be done by an elementary extension of the construction of [33] for the four-dimensional noncommutative U (2) ADHM data. The U (3) noncommutative instanton solutions of [29] can be written in the form A = Ψ † dΨ, where Ψ ∈ Hom A (W 0 ⊗ A, E k,3 ).…”