slides

A Noncommutative Family of Instantons
From this family of projections P parameterised by the space Mk,θ we obtain a
noncommutative family of instantons.
Theorem (SB-GL) (Generalises k = 1 case of Landi-Pagani-Reina-van Suijlekom)
The finitely generated projective right A(Mk,θ) ⊗ A(S 4 θ )-module
E := P A(Mk,θ) ⊗ A(S 4 θ ) 2k+2
is a noncommutative family of rank two vector bundles over S 4 θ ,
parameterised by the noncommutative space Mk,θ.
The operator ∇ := P ◦ (id ⊗ d) is a noncommutative family of instantons
with topological charge k, parameterised by the noncommutative space Mk,θ.
The latter statement means that the curvature F = ∇ 2 of the family obeys
(id ⊗ ∗θ)F = F,
where ∗θ : Ω 2 (S 4 θ ) → Ω2 (S 4 θ ) is the Hodge operator on S4 θ .
S. Brain (RU) NCG of Self-Dual Gauge Fields Nijmegen, 12th October 2010 18 / 25