The Moduli Space of Noncommutative Vortices

The abelian Higgs model on the noncommutative plane admits both BPS vortices and non-BPS fluxons. After reviewing the properties of these solitons, we discuss several new aspects of the former. We solve the Bogomoln’yi equations perturbatively, to all orders in the inverse noncommutivity parameter, and show that the metric on the moduli space of k vortices reduces to the computation of the trace of a k × k-dimensional matrix. In the limit of large noncommutivity, we present an explicit expression for this metric. Invited contribution to special issue of J.Math.Phys. on “Integrability, Topological Solitons and Beyond” Introduction and Results Vortices are enigmatic objects. Despite the apparent simplicity of the first order equations, no analytic expression for the solution has been found. Moreover, the metric on the moduli space, encoding the interactions of two or more vortices, remains unknown. This is in stark contrast to higher co-dimension solitons, such as monopoles and instantons, where seemingly more complicated equations readily yield results. Progress may be made in the limit of far separated vortices. By considering the leading order forces experienced by moving vortices, Manton and Speight determined the asymptotic form of the low-energy dynamics [1]. Their expression contains an unknown coefficient that characterizes the exponential return to vacuum of the Higgs field. Although a direct analytic computation of this coefficient appears difficult, a prediction has been given based on T-duality in string theory [2], and is in agreement with previous numerical results [3]. Another approach to understanding the dynamics is to deform the background space on which the vortices live. A cunning choice of deformation may ensure that the Bogomoln’yi equations become tractable. For example, it was discovered long ago that the tricky vortex equation is replaced by Liouville’s equation when the background is taken to be hyperbolic space [4]. Strachan subsequently showed that this simplification is sufficient to allow an explicit calculation of the moduli space metric [5]. More recently, Baptista and Manton considered the case of k vortices interacting on a sphere of area A ∼ 4πk [6]. An analytic expression for the metric was given in the limit as the area of the sphere shrinks to a critical value, A → 4πk. Curiously, in this limit, the vortex motion exhibits a symmetry enhancement, from the underlying SU(2) symmetry of the sphere to SU(k + 1). The physics behind this enhancement remains somewhat puzzling. Here, we shall again deform the background space so that the dynamics of vortices becomes tractable. This time, we take space to be the flat, noncommutative plane. In two spatial dimensions, noncommutivity is rather natural since it breaks only the discrete parity symmetry, leaving the continuous rotational symmetry intact. Solitons in noncommutative geometry have been extensively studied in recent times (see [7] for reviews). In particular, aspects of vortices in the noncommutative abelian Higgs model have been discussed in [8, 9, 10]. As we shall review, noncommutivity yields a one-parameter family of metrics on the vortex moduli space, depending on γ, a dimensionless combination of the gauge coupling constant e, the Higgs expectation value v and the noncommutivity parameter θ, γ = θev (1) It was shown by Bak, Lee and Park [9] that solutions to the Bogomoln’yi equations exist