On Noncommutative Multi-solitons

TL;DR

This paper characterizes the moduli space of multi-solitons in noncommutative scalar field theories across various dimensions, revealing geometric structures and stability conditions, and constructing solitons on specific quotient spaces.

Contribution

It provides a detailed description of the moduli space of multi-solitons in noncommutative theories, including explicit constructions and stability analysis in different dimensions.

Findings

Moduli space at large theta is a Kahler de-singularization of symmetric products.

Constructs solitons on quotient spaces like R^2/Z_k, cylinder, T^2.

Tori with area ≤ 2πθ do not support stable solitons.

Abstract

We find the moduli space of multi-solitons in noncommutative scalar field theories at large theta, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/theta is a consequence of a Bogomolnyi bound obeyed by the kinetic energy of the theta=infinity solitons. In two spatial dimensions, the parameter space for k solitons is a Kahler de-singularization of the symmetric product (R^2)^k/S_k. We exploit the existence of this moduli space to construct solitons on quotient spaces of the plane: R^2/Z_k, cylinder, and T^2. However, we show that tori of area less than or equal to (2 pi theta) do not admit stable solitons. In four dimensions the moduli space provides an explicit Kahler resolution of (R^4)^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.