Noncommutative Instantons via Dressing and Splitting Approaches.pdf

a r X i v : h e p - t h / 0 2 1 1 0 4 1 v 4 2 7 J u n 2 0 0 3 hep-th/0211041 ITP–UH–24/02 Noncommutative Instantons via Dressing and Splitting Approaches Zala?n Horva?th1, Olaf Lechtenfeld2 and Martin Wolf 2,3 1Department of Theoretical Physics Eo?tvo?s University 1117, Budapest, Hungary 2Institut fu?r Theoretische Physik Universita?t Hannover 30167, Hannover, Germany 3Institut fu?r Theoretische Physik Technische Universita?t Dresden 01062, Dresden, Germany Email: zalanh@ludens.elte.hu, lechtenf, wolf@itp.uni-hannover.de Abstract Almost all known instanton solutions in noncommutative Yang-Mills theory have been obtained in the modified ADHM scheme. In this paper we employ two alternative methods for the con- struction of the self-dual U(2) BPST instanton on a noncommutative Euclidean four-dimensional space with self-dual noncommutativity tensor. Firstly, we use the method of dressing transfor- mations, an iterative procedure for generating solutions from a given seed solution, and thereby generalize Belavin’s and Zakharov’s work to the noncommutative setup. Secondly, we relate the dressing approach with Ward’s splitting method based on the twistor construction and red- erive the solution in this context. It seems feasible to produce nonsingular noncommutative multi-instantons with these techniques. 1 Introduction The idea of a noncommutative space-time is more than fifty years old [1]. It offers a mild way to introduce nonlocality into field theories without loosing too much control. Motivated by string theory [2] the investigation of non-Abelian gauge theories defined on noncommutative space-times took center stage during the last couple of years. It is well known that the dynamics of non- Abelian gauge fields involves nonperturbative field configurations, like instantons and monopoles, in an essential way. Hence, before attempting to quantize a gauge theory it is mandatory to study its classical solutions and to characterize their moduli spaces. Nekrasov and