Noncommutative Instantons on CP^n.pdf

a r X i v : h e p - t h / 0 6 0 3 1 2 5 v 2 1 D e c 2 0 0 6 hep-th/0603125 ITP–UH–05/06 Noncommutative Instantons on CP n Tatiana A. Ivanova? and Olaf Lechtenfeld? ?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@itp.uni-hannover.de Abstract We construct explicit solutions of the Hermitian Yang-Mills equations on the noncommutative space Cn θ . In the commutative limit they coincide with the standard instantons on CPn written in local coordinates. 1 Introduction and summary Noncommutative deformations of gauge field theory provide a controlled theoretical framework beyond locality [1]. Of particular importance are noncommutative instantons (see e.g. [2, 3] and references therein), which are BPS configurations in four dimensions solving the Yang-Mills self- duality equations. In the string context, these solutions describe arrangements of noncommutative branes (see e.g. [4] and references therein). Natural BPS-type equations for gauge fields in more than four dimensions [5, 6] appear in superstring compactification as the conditions for the survival of at least one supersymmetry [7]. Various solutions to these first-order equations were found e.g. in [8, 9], and their noncommutative generalizations have been considered e.g. in [10, 11]. For U(n) gauge theory on a Ka?hler manifold these BPS-type equations specialize to the Hermitian Yang-Mills equations [6]. In this Letter we consider the noncommutative space Cnθ and construct an explicit u(n)-valued solution of the Hermitian Yang-Mills equations. In the commutative limit our configuration coin- cides with the instanton solution on CPn given in local coordinates on a patch Cn of CPn. We also describe a noncommutative deformation of a local form of the Abelian configuration on CPn. 2 Noncommutative space R2nθ Classical field theory on the