Dirac Monopole in Non-Commutative Space

We consider static U(1) monopole in non-commutative space. Up to the second order in the non-commutativity scale θ, we find no non-trivial corrections to the Dirac solution, the monopole mass remains infinite. We argue the same holds for any arbitrary higher order. Some speculation about the nature of non-commutative spacetime and its relation to the cosmological constant is made. Non-commutative geometry arises naturally in string theory when the Neveu-Schwarz B field is turned on. In a certain limit, the low-energy effective theory of the worldvolume is described by gauge theory in non-commutative space [1]. Motivated by the attempt to investigate localized structures, some researchers have recently studied topological field configuration in non-commutative geometry, namely the non-Abelian monopoles [2, 3, 4]. In this letter, we consider another elementary example, the Dirac monopole [5]. It would be helpful to make some comparisons first. Non-Abelian monopole can be visualized as D-string stretched between branes [6, 7]. In the U(2) case, when a background B field is turned on along the branes, the string is tilted because the two endpoints carry opposite charges [2]. This leads to a dipole structure in the magnetic field of the monopole. Explicit calculation of this effect to the O(θ) order has been carried out in [3, 4]. However, U(1) monopole does not admit such a simple geometric picture. First of all, there is no need to introduce a Higgs field φ. The singularity is put in by hand: one simply adds a source term to the Bianchi identity, then the base manifold becomes R \ {0}, which deformation-retracts to S. The Wu-Yang method [8] is applied to yield the quantization of magnetic monopole charge (π1(U(1)) = Z). By contrast the topological invariant of a non-Abelian monopole is defined by the asymptotic behavior of the Higgs field. Second, the energy of Dirac monopole diverges, while that of ’t Hooft-Polyakov monopole [9, 10] is finite (m ∝ 1/gYM). Although it’s difficult to make topological argument in noncommutative space, it has been shown that the BPS bound [11, 12] still exists [2, 3, 4]. One naturally asks whether non-commutativity will render Dirac monopole a finite mass. This is interesting since all the attempts to find magnetic monopole have failed. We can even pose the question: why should one treat a “particle” with infinite mass as a physical entity? The case should be compared to that of electron. Although the field energy due to a point electric charge diverges in the same way, it’s renormalizable as electron has an experimentally measurable mass about half MeV (one simple way to get rid of the infinity is to replace the point source with a smooth compactly-supported charge, while the same procedure is not applicable to monopole [13]). If Dirac monopole became finitely heavy in non-commutative space, it would be of great interests to theorists and experimentalists alike. However, our calculation gives a negative answer. Up to the O(θ) order, we show explicitly that there is no correction to the U(1) gauge connection A (except for θ-dependent gauge transformation), therefore the mass remains infinite. We argue the same is true for any higher order. In the following, we will present our calculation by adopting a mixed notation of tensor and differential form. We also discuss another formulation of the U(1) monopole in non-commutative