Cyclic source pairings for Penrose--Sparling non-Hausdorff twistor spaces

Abstract

We introduce noncommutative geometry techniques in order to reinterpret the Penrose--Sparling non-Hausdorff twistor space of the anti-self-dual Coulomb field by means of an explicit etale gluing groupoid and its convolution algebra. This algebraic model keeps track of both the identified open part and the two non-separated copies of the source quadric. We compute two kinds of Chern--Connes pairings. The strict tangent-module analogue, obtained from \([T_{\mathbb R}\CP^3\otimes\C]\), vanishes because \[ \operatorname{ch}_3(T^{1,0}\CP^3)+ \operatorname{ch}_3(T^{0,1}\CP^3)=0. \] By contrast, the Penrose--Sparling Coulomb line bundle \(\calC_n\) defines a \(K_0(A_Q)\)-class, and the relative cyclic cocycle supported on the two non-separated copies of a ruling line \(L\subset Q\) gives \[ \mathcal Q(\calC_n)= \frac12\left\langle φ_L^+-φ_L^-,[\calC_n]\right\rangle=n. \] Thus the source-adapted cyclic pairing recovers the Coulomb charge. We also formulate the non-abelian version in principal-bundle language. For a connected complex reductive group \(G\), a maximal torus \(T\subset G\), and a cocharacter \(λ:\C^*\to T\), the source is a principal \(G\)-bundle modification of type \(λ\) along \(Q\).