Noncommutative Moduli for Multi-Instantons.pdf

a r X i v : h e p - t h / 0 4 0 4 1 2 7 v 2 2 A u g 2 0 0 4 Noncommutative Moduli for Multi-Instantons Tatiana A. Ivanova? , Olaf Lechtenfeld? and Helge Mu?ller-Ebhardt? ?Bogoliubov Laboratory of Theoretical Physics, JINR 141980 Dubna, Moscow Region, Russia Email: ita@thsun1.jinr.ru ?Institut fu?r Theoretische Physik, Universita?t Hannover Appelstra?e 2, 30167 Hannover, Germany Email: lechtenf, ebhardt@itp.uni-hannover.de Abstract There exists a recursive algorithm for constructing BPST-type multi-instantons on commuta- tive R4. When deformed noncommutatively, however, it becomes difficult to write down non- singular instanton configurations with topological charge greater than one in explicit form. We circumvent this difficulty by allowing for the translational instanton moduli to become noncom- mutative as well. Such a scenario is natural in the self-dual Yang-Mills hierarchy of integrable equations where the moduli of solutions are seen as extended space-time coordinates associated with higher flows. By judicious adjustment of the moduli-noncommutativity we achieve the ADHM construction of generalized ’t Hooft multi-instanton solutions with everywhere self-dual field strengths on noncommutative R4. 1 Introduction In recent years, many important nonperturbative field configurations, like solitons, vortices, monopoles and instantons, have been generalized in various dimensions to Moyal-type noncom- mutative spaces (see e.g. [1]–[13] and reviews [14] for further references). Specializing to instan- tons on R4, the self-dual Yang-Mills equations [15] are solved systematically through the ADHM method [16]. Its noncommutative extension to R4θ, as developed in [1]–[3] and [17]–[20], is straight- forward for self-dual θ but needs modification in case of anti-self-dual θ. A singular subclass of solutions are the ’t Hooft multi-instantons, which have also been deformed noncommutatively by way of the splitting and ADHM approaches [19, 20]. With the help of Murray- von-