On the geometry of noncommutative gauge fields

Our goal is to find a proper formulation of noncommutative gauge field theories; we do this by studying an example of a noncommutative 4-dimensional sphere S4θ; it is the base space of a noncommutative principal bundle: SU(2) → S7θ → S4θ This is a quantization of the Hopf fibration on the commutative 4-sphere, that plays a central role in the construction of (charge 1) instantons on S4. Recall: moduli space M1 of charge 1 instantons on S4 is isomorphic to the homogeneous space: M1 ' SL(2,H) / Sp(2,H) conformal transformations gauge transformations Question: What does the moduli space of (charge 1) instantons on S4θ look like? To answer this question, we first need a notion of gauge fields on S4θ. Toric noncommutative manifolds Let M a compact Riemannian spin manifold (dim M = m) with a smooth isometrical action of the n-torus Tn. action σs of Tn on C∞(M) by automorphisms: σs(f)(x) = f (s −1 · x). decompose f = fr in homogeneous elements of degree r: σs(fr) = e 2piir ·s fr. C∞(M) is represented on Hilbert space H = L2(M,S) of spinors by pointwise multiplication: pi: C∞(M) → B(H). There is a representation U of Tn on H such that U(s)DU(s)−1 = D, U(s)pi(f)U(s)−1 = pi(σs(f)). Given any real n × n anti-symmetric matrix θ = (θµν) a twisted representation Lθ of C ∞(M) is defined by Lθ(f) = r frU(rµθµ1,..., rµθµn). and set C∞(Mθ): = Lθ(C∞(M)); quantization map Lθ: C ∞(M) → C∞(Mθ) satisfying Lθ(f ×θ g) = Lθ(f)Lθ(g) with on homogeneous elements: fr ×θ gr ′ = frσr ·θ(gr ′) = e2piir ·θ·r ′frgr ′. the triple (C∞(Mθ),H,D) satisfies all properties of Connes’ noncommutative spin geometry of dim m (i.e. an m-summable spectral triple) Noncommutative integral given by Dixmier trace:∫ − Lθ(f) = TrωLθ(f)|D|−m. Lemma (GIV05) If f ∈ C∞(M) the