[Show abstract][Hide abstract] ABSTRACT: We construct θ-deformations of the classical groups SL(2, H) and Sp(2). Coacting on a basic instanton on a noncommutative four-sphere S4 θ, we construct a noncommutative family of instantons of charge 1. The family is parametrized by the Self-dual (and anti-self-dual) solutions of Yang–Mills equations have played an important role both in mathematics and physics. Commonly called (anti-)instantons, they are connections with self-dual curvature on smooth G-bundles over a four dimensional compact manifold M. In particular, one considers SU(2) instantons on the sphere S4.
International Mathematics Research Notices 11/2007; DOI:10.1093/imrn/rnn038 · 1.07 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We construct a quantum version of the SU(2) Hopf bundle $S^7 \to S^4$. The quantum sphere $S^7_q$ arises from the symplectic group $Sp_q(2)$ and a quantum 4-sphere $S^4_q$ is obtained via a suitable self-adjoint idempotent $p$ whose entries generate the algebra $A(S^4_q)$ of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere $S^4$. We compute the fundamental $K$-homology class of $S^4_q$ and pair it with the class of $p$ in the $K$-theory getting the value -1 for the topological charge. There is a right coaction of $SU_q(2)$ on $S^7_q$ such that the algebra $A(S^7_q)$ is a non trivial quantum principal bundle over $A(S^4_q)$ with structure quantum group $A(SU_q(2))$. Comment: 27 pages. Latex. v2 several substantial changes and improvements; to appear in CMP