Mathematics – Quantum Algebra
Scientific paper
2006-03-17
Contemp. Math. 434 (2007) 39-72
Mathematics
Quantum Algebra
34 pages; AMS-Latex. v2: Several minor changes. Based on lectures delivered at the 2005 Summer school on ``Geometric and Topol
Scientific paper
10.1143/PTPS.171.211
These notes aim at a pedagogical introduction to recent work on deformation of spaces and deformation of vector bundles over them, which are relevant both in mathematics and in physics, notably monopole and instanton bundles. We first decribe toric noncommutative manifolds (also known as isospectral deformations) and give a detailed introduction to gauge theories on a toric four-sphere. This includes a Yang-Mills action functional with associated equations of motion and self-duality equations. We construct a particular class of instanton solutions on a SU(2) bundle with a suitable use of twisted conformal symmetries. In the second part, we describe a different deformation of an instanton bundle over the classical four-sphere by constructing a quantum group SU_q(2) bundle on a sphere which is different from the toric one.
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