A note on noncommutative scalar multisolitons

Abstract

We prove that there do not exist multisoliton solutions of noncommutative scalar field theory in the Moyal plane which interpolate smoothly between $n$ overlapping solitons and $n$ solitons with an infinite separation.Comment: 8 page

On the moduli space of non-commutative multi-solitons at finite θ

“…These mean the following: if we take into account the O(θ −2 )-correction, the multi-soliton configuration cannot be a solution of the equation of motion for generic potential with 1/V (2) (λ) + 1/V (2) (0) = 0, but the single level k soliton configuration can. This seems to be consistent with the argument based on the evaluation of the energy functional in [3,5,7].…”

“…In this case, we need to consider the attractive or repulsive force between solitons. Recently, Durhuus and Jonsson pointed out that there are no multi-soliton solutions which interpolate smoothly between n overlapping solitons and n solitons with an infinite separation at the lowest order perturbation in θ −1 [7]. Therefore multi-solitons at finite θ are in general unstable and decay into infinitely separate or overlapping solitons.…”

Scalar Solitons on the Fuzzy Sphere

“…Assuming that every solution in the Moyal plane descends from solutions on the fuzzy sphere in the limit, one would expect a solution with separated lumps on the Moyal plane to descend from an infinitesimal deformation that breaks the rotation symmetry of a solution on the fuzzy sphere. The fact that no such deformations exist strongly suggests that there are no stable multi-lump solutions at finite theta on the Moyal plane, in agreement with the results of [18] and expectations from perturbative calculations in the limit of infinite non-commutativity [16,17].…”

“…First of all, separated lumps attract each other according to perturbative calculations at large noncommutativity parameter [16,17]. Furthermore, it has recently been proven that there cannot exist a family of static solutions on the Moyal plane at finite noncommutativity that interpolates smoothly between the solution describing two overlapping solitons and a solution with two infinitely separated solitons [18]. Finally, the stability results for scalar solitons on the fuzzy sphere obtained below, combined with a scaling limit that yields the Moyal plane from the fuzzy sphere (see e.g.…”

Noncommutative waves have infinite propagation speed

“…Explicit solitonic solutions have been found in various gauge theories, see, e.g., [1,2,3,4,5] as well as in scalar field theories [6,7,8] at infinite noncommutativity where the existence theory for finite noncommutativity is now rather complete [9,10,11,12] especially for the rotationally invariant case. For general background and reviews of noncommutative field theory we refer to [13,14,15].…”