slides

Construction of Instantons
Instantons on S4 are constructed from monads on C4 , i.e. homomorphisms of free
right A(C4 )-modules
H ⊗ A(C 4 ) σz
−→ K ⊗ A(C 4 )
for complex vector spaces H, K of dimensions k, 2k + 2, such that:
σz is linear in the generators z1, . . . , z4 of A(C 4 );
σz is injective and σ∗ J(z)
the composition σ∗ J(z) σz = 0.
is surjective;
Given such a monad, set V :=
σz σJ(z) ∈ Mat2k+2,2k(A(C4 )) and then
P := 2k+2 − V (V ∗ V ) −1 V ∗ ∈ Mat2k+2(A(S 4 )).
This P is clearly a projection, P 2 = P ∗ = P .
S. Brain (RU) NCG of Self-Dual Gauge Fields Nijmegen, 12th October 2010 8 / 25