Interacting noncommutative solitons (vacua)

Recent development of string theory has shown that noncommutative (NC) models can arise in certain limits of String Theory [1]. Moreover the String Field Theory [2], which is the theory of second quantised string, is based on noncommutative geometry too. Due to their proximity to String Theory noncommutative theories have many features in common with it. It is worthwhile to mention the IR/UV mixing [3], which says that in NC theories there is a correspondence analogous to string theory between contributions at high energies with those at low ones. Another common feature is a number of equivalences/dualities which relate different noncommutative models [4, 5]. This list is continued by the so called noncommutative soliton which was found in [6], in the limit of large noncommutativity θ → ∞ and further generalised to finite θ but nontrivial gauge field background [7]. These solitons are believed to correspond in the string theory language to branes. Noncommutative theories can alternatively be treated as theories of linear operators defined over a (infinite-dimensional separable) Hilbert space. In the operator language, noncommutative solitons look like projectors to finite-dimensional subspaces of the Hilbert space, while the gauge fields split into parts corresponding respectively to the finite dimensional subspace and its orthogonal completion [7]. In this note we are going to consider gauge field “solitons”, i.e. solutions for gauge fields which are projectors to finite-dimensional subspaces of the Hilbert space rather than ones for the scalar field. For more details the reader is referred