Notes on Noncommutative Instantons.pdf

a r X i v : h e p - t h / 0 1 0 8 0 0 7 v 3 1 O c t 2 0 0 1 Preprint typeset in JHEP style. - PAPER VERSION hep-th/0108007 Notes on Noncommutative Instantons Chong-Sun Chua, Valentin V. Khozeb and Gabriele Travaglinib aCentre for Particle Theory, Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK bCentre for Particle Theory and Institute for Particle Physics Phenomenology, Department of Physics, University of Durham, Durham, DH1 3LE, UK E-mail: chong-sun.chu, valya.khoze, gabriele.travaglini@durham.ac.uk Abstract: We study in detail the ADHM construction of U(N) instantons on noncommu- tative Euclidean space-time R4NC and noncommutative space R 2 NC × R2. We point out that the completeness condition in the ADHM construction could be invalidated in certain circum- stances. When this happens, regular instanton configuration may not exist even if the ADHM constraints are satisfied. Some of the existing solutions in the literature indeed violate the com- pleteness condition and hence are not correct. We present alternative solutions for these cases. In particular, we show for the first time how to construct explicitly regular U(N) instanton solutions on R4NC and on R 2 NC × R2. We also give a simple general argument based on the Corrigan’s identity that the topological charge of noncommutative regular instantons is always an integer. Keywords: Noncommutative Geometry, Solitons Monopoles and Instantons. 1. Introduction There has been a lot of interest in gauge theories on noncommutative spaces. One of the reasons for this interest is the natural appearance of noncommutativity [xμ, xν ] = iθμν in the framework of string theory and D-branes [1–4]. Noncommutative gauge theories are also fascinating on their own right [5], mostly due to a mixing between the infrared (IR) and the ultraviolet (UV) degrees of freedom discovered in [6]. While UV/IR mixing arises at the perturbative level, it has been suggested that a resolution of it may lie at t