Moduli spaces with external fields. (English) Zbl 1122.53040
Summary: We consider the geometric structures on the moduli space of static finite energy solutions to the 2+1-dimensional unitary chiral model with the Wess-Zumino-Witten (WZW) term. It is shown that the magnetic field induced by the WZW term vanishes when restricted to the moduli spaces constructed from the Grassmannian embeddings, so that the slowly moving solitons can in some cases be approximated by a geodesic motion on a space of rational maps from \(\mathbb {CP}^1\) to the Grassmannian.
MSC:
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
| 53C80 | Applications of global differential geometry to the sciences |
| 81T99 | Quantum field theory; related classical field theories |
References:
| [1] | Bredon, G. E., Topology and Geometry (1993), Springer-Verlag · Zbl 0159.24602 |
| [2] | Braaten, E.; Curtright, T. L.; Zachos, C. K., Torsion and geometrostasis in nonlinear sigma models, Nuclear Phys. B, 260, 630-688 (1985) |
| [3] | Dai, B.; Terng, C.-L., Bäcklund transformations, (Ward Solitons, and Unitons (2004)) |
| [4] | Dunajski, M.; Manton, N. S., Reduced dynamics of ward solitons, Nonlinearity, 18, 1677-1689 (2005) · Zbl 1083.37055 |
| [5] | Dunajski, M.; Plansangkate, P., Topology and energy of time dependent unitons, Proc. Royal Soc. A, 463, 945-959 (2007) · Zbl 1133.37338 |
| [6] | Fisher, A. E.; Marsden, J. E., The Einstein equations of evolution — A geometric approach, J. Math. Phys., 13, 546 (1971) · Zbl 0233.58009 |
| [7] | Ioannidou, T.; Zakrzewski, W., Lagrangian formulation of the general modified chiral model, Phys. Lett. A, 249, 303-306 (1998) · Zbl 0941.81032 |
| [8] | Klawunn, M.; Lechtenfeld, O.; Petersen, S., Moduli-Space Dynamics of Noncommutative Abelian Sigma-Model Solitons (2006) |
| [9] | Manton, N. S., A remark on the scattering of BPS monopoles, Phys. Lett., 110B, 54-56 (1982) · Zbl 1190.81087 |
| [10] | Manton, N. S.; Sutcliffe, P., Topological Solitons, CUP (2004) |
| [11] | Piette, B.; Stokoe, I.; Zakrzewski, W. I., On stability of solutions of the U(N) chiral model in two dimensions, Z. Phys. C, 37, 449-455 (1988) |
| [12] | Rodnianski, I.; Sterbenz, J., On the Formation of Singularities in the Critical O(3) Sigma-Model (2006) |
| [13] | Ruback, P. J., Sigma model solitons and their moduli space metrics, Comm. Math. Phys., 116, 645-658 (1988) · Zbl 0655.58035 |
| [14] | Stokoe, I.; Zakrzewski, W., Dynamics of solutions of the CP(N−1) models in (2+1) dimensions, Z. Phys. C, 34, 491-496 (1987) |
| [15] | Stuart, D., The geodesic approximation for the Yang-Mills-Higgs equations, Comm. Math. Phys., 126, 106-120 (1994) · Zbl 0814.53052 |
| [16] | Uhlenbeck, K., Harmonic maps into Lie groups, J. Differential Geom., 30, 1-50 (1989) · Zbl 0677.58020 |
| [17] | Ward, R. S., Slowly-moving lumps in the \(C P^1\) model in \((2 + 1)\) dimensions, Phys. Lett., 158B, 424-428 (1985) |
| [18] | Ward, R. S., Soliton solutions in an integrable chiral model in \(2 + 1\) dimensions, J. Math. Phys., 29, 386-389 (1988) · Zbl 0644.58038 |
| [19] | Ward, R. S., Integrability of the chiral equations with torsion term, Nonlinearity, 1, 671-679 (1988) · Zbl 0681.35080 |
| [20] | Ward, R. S., Classical solutions of the chiral model, unitons, and holomorphic vector bundles, Comm. Math. Phys., 128, 319-332 (1990) · Zbl 0707.32007 |
| [21] | Ward, R. S., Nontrivial scattering of localized solitons in a \((2 + 1)\)-dimensional integrable system, Phys. Lett. A, 208, 203-208 (1995) · Zbl 1020.37537 |
| [22] | Witten, E., Nonabelian bosonization in two dimensions, Comm. Math. Phys., 92, 455-472 (1984) · Zbl 0536.58012 |
| [23] | Wu, Y. D.; Zee, A., Abelian gauge structure and anomalous spin and statistics of solitons in nonlinear sigma models, Nuclear Phys. B, 272, 322-328 (1986) |
| [24] | Zakrzewski, W. J., Low-Dimensional Sigma Models (1989), Adam Hilger · Zbl 0787.53072 |
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