The ADHM Construction of Instantons on Noncommutative Spaces

Abstract

We present an account of the ADHM construction of instantons on Euclidean space-time ${\mathbb{R}}^4$ from the point of view of noncommutative geometry. We recall the main ingredients of the classical construction in a coordinate algebra format, which we then deform using a cocycle twisting procedure to obtain a method for constructing families of instantons on noncommutative space-time, parameterised by solutions to an appropriate set of ADHM equations. We illustrate the noncommutative construction in two special cases: the Moyal-Groenewold plane ${\mathbb{R}}_{\hslash }^4$ and the Connes-Landi plane ${\mathbb{R}}_{\theta }^4$.

Latex, 40 pages

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• Warped Products and Yang-Mills equations on non commutative spaces
• Finitely-Generated Projective Modules over the Theta-deformed 4-sphere
• The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples
• The Noncommutative Topology of Anti-Self-Dual Gauge Fields