Noncommutative Multi-Instantons on R^{2n} x S^2.pdf

a r X i v : h e p - t h / 0 3 0 5 1 9 5 v 2 1 6 J u n 2 0 0 3 hep-th/0305195 ITP-UH-02/03 Noncommutative Multi-Instantons on R2n×S2 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 Generalizing self-duality on R2×S2 to higher dimensions, we consider the Donaldson-Uhlenbeck- Yau equations on R2n×S2 and their noncommutative deformation for the gauge group U(2). Imposing SO(3) invariance (up to gauge transformations) reduces these equations to vortex-type equations for an abelian gauge field and a complex scalar on R2n θ . For a special S2-radius R depending on the noncommutativity θ we find explicit solutions in terms of shift operators. These vortex-like configurations on R2n θ determine SO(3)-invariant multi-instantons on R2n θ ×S2 R for R = R(θ). The latter may be interpreted as sub-branes of codimension 2n inside a coincident pair of noncommutative Dp-branes with an S2 factor of suitable size. 1 Introduction Noncommutative deformation is a well established framework for stretching the limits of con- ventional (classical and quantum) field theories [1, 2]. On the nonperturbative side, all celebrated classical field configurations have been generalized to the noncommutative realm. Of particular interest thereof are BPS configurations, which are subject to first-order nonlinear equations. The latter descend from the 4d Yang-Mills (YM) self-duality equations and have given rise to instan- tons [3], monopoles [4] and vortices [5], among others. Their noncommutative counterparts were introduced in [6], [7] and [8], respectively, and have been studied intensely for the past five years (see [9] for a recent review). String/M theory embeds these efforts in a higher-dimensional context, and so it is important to formulate BP