On noncommutative multi-solitons

Abstract

We find the moduli space of multi-solitons in noncommutative scalar field theories at large θ, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/θ is a consequence of a Bogomolnyi bound obeyed by the kinetic energy of the θ=∞ solitons. In two spatial dimensions, the parameter space for k solitons is a Kahler de-singularization of the symmetric product (R²)^k/S_k. We exploit the existence of this moduli space to construct solitons on quotient spaces of the plane: R²/Z_k, cylinder, and T². However, we show that tori of area less than or equal to 2πθ do not admit stable solitons. In four dimensions the moduli space provides an explicit Kahler resolution of (R⁴)^k/S_k. In general spatial dimension 2d, we show it is isomorphic to the Hilbert scheme of k points in C^d, which for d>2 (and k > 3) is not smooth and can have multiple branches.