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

Generalizing self-duality on R^2 x S^2 to higher dimensions, we consider the Donaldson-Uhlenbeck-Yau equations on R^{2n} x S^2 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 R^{2n}_\theta. For a special S^2-radius R depending on the noncommutativity \theta we find explicit solutions in terms of shift operators. These vortex-like configurations on R^{2n}_\theta determine SO(3)-invariant multi-instantons on R^{2n}_\theta x S^2_R for R=R(\theta). The latter may be interpreted as sub-branes of codimension 2n inside a coincident pair of noncommutative Dp-branes with an S^2 factor of suitable size.


Introduction
Noncommutative deformation is a well established framework for stretching the limits of conventional (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 instantons [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 BPS-type equations in more than four dimensions. In fact, noncommutative instantons in higher dimensions and their brane interpretations have recently been considered in [10,11,12]. Yet already 20 years ago, generalized self-duality equations for YM fields in more than four dimensions were proposed [13,14] and their solutions investigated e.g. in [14,15]. For U(k) gauge theory on a Kähler manifold these equations specialize to the Donaldson-Uhlenbeck-Yau (DUY) equations [16,17]. They are the natural analogues of the 4d self-duality equations.
In this letter we generalize the DUY equations to the noncommutative spaces R 2n θ ×S 2 and construct explicit U(2) multi-instanton solutions even though these equations are not integrable.
Classical field theory on the noncommutative deformation R 2n θ of R 2n may be realized in a star-product formulation or in an operator formalism. While the first approach alters the product of functions on R 2n the second one turns these functions f into linear operatorsf acting on the n-harmonic-oscillator Fock space H. The noncommutative space R 2n θ may then be defined by declaring its coordinate functionsx 1 , . . . ,x 2n to obey the Heisenberg algebra relations with a constant antisymmetric tensor θ µν . The coordinates can be chosen in such a way that the matrix (θ µν ) will be block-diagonal with non-vanishing components We assume that all θ a ≥ 0; the general case does not hide additional complications. For the noncommutative version of the complex coordinates (2.3) we have [ẑ a ,ẑb] = −2δ ab θ a =: θ ab = −θb a ≤ 0 , and all other commutators vanish . (2.10) The Fock space H is spanned by the basis states for k a = 0, 1, 2, . . . , (2.11) which are connected by the action of creation and annihilation operators subject to We recall that, in the operator realization f →f , derivatives of f get mapped according to where θ ab is defined via θ bc θc a = δ a b so that θ ab = −θb a = δ ab 2θ a . Finally, we have to replace 2πθ a Tr Hf . (2.14) Tensoring R 2n θ with a commutative S 2 means extending the noncommutativity matrix θ by vanishing entries in the two new directions. A more detailed description of noncommutative field theories can be found in the review papers [2].
Donaldson-Uhlenbeck-Yau equations. Let M 2q be a complex q=n+1 dimensional Kähler manifold with some local real coordinates x = (x i ) and a tangent space basis ∂ i := ∂/∂x i for i, j = 1, . . . , 2q, so that a metric and the Kähler two-form read ds 2 = g ij dx i dx j and Ω = Ω ij dx i ∧ dx j , respectively. Consider a rank k complex vector bundle over M 2q with a gauge potential A = A i dx i and the curvature two-form Both A i and F ij take values in the Lie algebra u(k). The Donaldson-Uhlenbeck-Yau (DUY) equations [16,17] where Ω is the Kähler two-form, F 0,2 is the (0, 2) part of F, and * is the Hodge operator. In our local coordinates ( Specializing now M 2q to be R 2n ×S 2 , the DUY equations (2.15) in the local complex coordinates (z a , y) take the form where a, b = 1, . . . , n. Using formulae (2.4), we obtain and finally write the Donaldson-Uhlenbeck-Yau equations on R 2n ×S 2 in the alternative form The transition to the noncommutative DUY equations is trivially achieved by going over to operator-valued objects everywhere. In particular, the field strength components in (2.20) then readF ij = ∂xiÂ j − ∂xjÂ i + [Â i ,Â j ], where e.g.Â i are simultaneously u(k) and operator valued. To avoid a cluttered notation, we drop the hats from now on.

Generalized vortex equations on R 2n
θ Noncommutative generalization of Taubes' ansatz. Considering the particular case (2.16) of the SU(2) DUY equations on R 2 ×S 2 , Taubes introduced an SO(3)-invariant ansatz 2 for the gauge potential A which reduces the ASDYM equations (2.16) to the vortex equations on R 2 [5] (see also [21]). Here we extend Taubes' ansatz to the higher-dimensional manifold R 2n ×S 2 and reduce the noncommutative 3 U(2) Donaldson-Uhlenbeck-Yau equations (2.20) to generalized vortex equations on R 2n θ , including their commutative (θ=0) limit. In section 4, we will write down explicit solutions of the generalized noncommutative vortex equations on R 2n which determine multi-instanton solutions of the noncommutative YM equations on R 2n ×S 2 .
For the noncommutative case θ µν = 0 we choose γ = −1. Comparing (3.10) and (3.11), we obtain a constraint equation on the field φ, 17) and the following noncommutative generalization of the vortex equations in 2n dimensions:

18)
Fzāzb These equations and their antecedent DUY equations on R 2n θ ×S 2 are not integrable even for n=1. Therefore, neither dressing nor splitting approaches, developed in [22] for integrable equations on noncommutative spaces, can be applied. The modified ADHM construction [6] also does not work in this case. However, some special solutions can be obtained by choosing a proper ansatz as we shall see next.

Multi-instanton solutions on R 2n
θ ×S 2 Solutions of the constrained vortex-type equations. We are going to present explicit solutions to the noncommutative generalized vortex equations (3.18) - (3.20) subject to the constraint (3.17). The latter can be solved by putting where S N is an order-N shift operator acting on the Fock space H, i.e. 2) with P N being a hermitean rank-N projector: P 2 N = P N = P † N . It is convenient to introduce the operators in terms of which We now employ the shift-operator ansatz (see e.g. [7,23]) for which F z azb = θ ab P N = δ ab P N 2θ a and Fzāzb = 0 (4.6) since θ ab = δ ab 2θ a . After substituting (4.1) and (4.6) into the first vortex equation (3.18), we obtain the condition δ ab θ ab P N = 1 The remaining vortex equations (3.19) and (3.20) are identically satisfied by (4.1) and (4.6).
Hence, for γ = −1 we have established on R 2n a whole class of noncommutative constrained vortex-type configurations we have With this, the topological charge indeed becomes

Concluding remarks
By solving the noncommutative Donaldson-Uhlenbeck-Yau equations we have presented explicit U(2) multi-instantons on R 2n θ ×S 2 which are uniquely determined by abelian vortex-type configurations on R 2n θ . The existence of these solutions required the condition (4.7) relating the S 2 -radius R to θ via R = (2 n a=1 1 θ a ) −1/2 . We see that any commutative limit (θ a →0) forces R → 0 as well, and the configuration becomes localized in R 2n (for n=1) or disappears (for n>1). The moduli space of our N -instanton solutions is that of rank-N projectors in the n-oscillator Fock space.
Since standard instantons localize all compact coordinates in the ambient space they have been interpreted as sub-branes inside Dp-branes [1,2,9,10,11,12]. The presence of an NS background B-field deforms such configurations noncommutatively. In the same vein, the solutions presented in this letter may be viewed as a collection of N sub-branes of codimension 2n, i.e. as D(p−2n)-branes located inside two coincident Dp-branes, with all branes sharing a common two-sphere S 2 R .