The Existence and Stability of Noncommutative Scalar Solitons

Abstract

We establish existence and stabilty results for solitons in noncommutative scalar field theories in even space dimension $2d$. In particular, for any finite rank spectral projection $P$ of the number operator $\mathcal{N}$ of the $d$-dimensional harmonic oscillator and sufficiently large noncommutativity parameter $\theta$ we prove the existence of a rotationally invariant soliton which depends smoothly on $\theta$ and converges to a multiple of $P$ as $\theta \to \infty$. In the two-dimensional case we prove that these solitons are stable at large $\theta$, if $P=P_N$, where $P_N$ projects onto the space spanned by the $N+1$ lowest eigenstates of $\mathcal{N}$, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to $P=P_0$ for all $\theta$ in its domain of existence. Finally, for arbitrary $d$ and small values of $\theta$, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on $\theta$.

36 pages, 1 figure

References in corpus (5)

• Noncommutative Field Theory
• Quantum Field Theory on Noncommutative Spaces
• Introduction to M(atrix) theory and noncommutative geometry, Part II
• Komaba Lectures on Noncommutative Solitons and D-Branes
• Noncommutative solitons on Kahler manifolds

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• Quantization of Noncommutative Scalar Solitons at finite $\theta$
• On the Moduli Space of Noncommutative Multi-solitons at Finite Theta