slides

The Noncommutative ADHM Construction
Although the space Mk,θ has fewer classical points than Mk, we can nevertheless
work with the whole noncommutative family of monads at once.
The algebra-valued matrix σz =
j M j ⊗ zj is now thought of as a map
σz : H ⊗ A(C 4 θ) → A(Mk,θ) ⊗ K ⊗ A(C 4 θ).
For each character ɛ : A(Mk,θ) → C (i.e. for each classical point of Mk,θ)
there is a corresponding monad
(ɛ ⊗ id) ◦ σz : H ⊗ A(C 4 θ) → K ⊗ A(C 4 θ).
The ADHM construction goes through just as before. This time take
∈ Mat2k+2,2k(A(Mk,θ) ⊗ A(C 4 θ)).
V := σz σ ∗ z
Then set P := 2k+2 − V (V ∗ V ) −1 V ∗ ∈ Mat2k+2(A(Mk,θ) ⊗ A(S 4 θ )).
S. Brain (RU) NCG of Self-Dual Gauge Fields Nijmegen, 12th October 2010 17 / 25