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 Kähler 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 T 2. 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 Kähler resolution of (ℝ4)k/S k . In general spatial dimension 2d, we show it is isomorphic to the Hilbert scheme of k points in ℂd, which for d>2 (and k>3) is not smooth and can have multiple branches.
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Received: 29 May 2001 / Accepted: 16 August 2002 Published online: 7 November 2002
Communicated by R.H. Dijkgraaf
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Gopakumar, R., Headrick, M. & Spradlin, M. On Noncommutative Multi-Solitons. Commun. Math. Phys. 233, 355–381 (2003). https://doi.org/10.1007/s00220-002-0734-z
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DOI: https://doi.org/10.1007/s00220-002-0734-z