Geometry and kinematics of two Skyrmions. (English) Zbl 0778.53051
The two-Skyrmion dynamics is modelled by motion on a 12-dimensional space of Skyrme fields. This space can be generated by the gradient flow of the potential energy function or by restricting the gradient flow to the Skyrme fields derived from \(SU(2)\) Yang-Mills instantons of charge two. On both of these spaces, after quotienting by the free action of the group \(R^ 2\times SO(3)_{\text{isospin}}\) there remains a 6- dimensional space. It is shown that the global structure of the quotient space is that of complex projective 3-space, with complex conjugate points on one projective plane identified and the real points in this plane removed.
Reviewer: G.Zet (Iaşi)
MSC:
| 53Z05 | Applications of differential geometry to physics |
| 37C10 | Dynamics induced by flows and semiflows |
| 81V05 | Strong interaction, including quantum chromodynamics |
Keywords:
gradient flow; potential energy function; Yang-Mills instantons; complex projective 3-spaceReferences:
| [1] | Bogomol’nyi, E.B.: The stability of classical solutions. Sov. J. Nucl. Phys.24, 449 (1976) |
| [2] | Weinberg, E.: Parameter counting for multimonopole solutions. Phys. Rev.D20, 936 (1979) |
| [3] | Taubes, C.H.: ArbitraryN-vortex solutions of the first order Ginzburg-Landau equations. Commun. Math. Phys.72, 277 (1980) · Zbl 0451.35101 · doi:10.1007/BF01197552 |
| [4] | Manton, N.S.: A remark on the scattering of BPS monopoles. Phys. Lett.110B, 54 (1982) · Zbl 1190.81087 |
| [5] | Atiyah, M.F., Hitchin, N.J.: The geometry and dynamics of magnetic monopoles, Princeton, NJ: Princeton University Press 1988 · Zbl 0671.53001 |
| [6] | Bates, L., Montgometry, R.: Closed geodesics on the space of stable two-monopoles. Commun. Math. Phys.118, 635 (1988) · Zbl 0820.53058 · doi:10.1007/BF01221112 |
| [7] | Temple-Raston, M.: Closed 2-dyon orbits. Nucl. Phys.B313, 447 (1989) · doi:10.1016/0550-3213(89)90328-3 |
| [8] | Leese, R.: Low energy scattering of solitons in the \(\mathbb{C}\)P 1 model. Nucl. Phys.B344, 33 (1990) · doi:10.1016/0550-3213(90)90684-6 |
| [9] | Samols, T.M.: Vortex scattering. Commun. Math. Phys.145, 149 (1992) · Zbl 0768.53041 · doi:10.1007/BF02099284 |
| [10] | Gibbons, G.W., Manton, N.S.: Classical and quantum dyanmics of BPS monopoles. Nucl. Phys.B274, 183 (1986) · doi:10.1016/0550-3213(86)90624-3 |
| [11] | Schroers, B.J.: Quantum scattering of BPS monopoles at low energy. Nucl. Phys.B367, 177 (1991) · doi:10.1016/0550-3213(91)90047-2 |
| [12] | Manton, N.S.: Unstable manifolds and soltion dynamics. Phys. Rev. Lett.60, 1916 (1988) · doi:10.1103/PhysRevLett.60.1916 |
| [13] | Skyrme, T.R.H.: A unified field theory of mesons and baryons. Nucl. Phys.31, 556 (1962) · Zbl 0106.20105 · doi:10.1016/0029-5582(62)90775-7 |
| [14] | Nyman, E.M., Riska, D.O.: Low energy properties of baryons in the Skyrme model. Reps. Prog. Phys.53, 1137 (1990) · doi:10.1088/0034-4885/53/9/001 |
| [15] | Adkins, G.S., Nappi, C.R., Witten, E.: Static properties of nucleons in the Skyrme model. Nucl. Phys.B228, 552 (1983) · doi:10.1016/0550-3213(83)90559-X |
| [16] | Kopeliovich, V.B., Stern, B.E.: Exotic Skyrmions. JETP Lett.45, 203 (1987) |
| [17] | Verbaarschot, J.J.M.: Axial symmetry of bound baryon number-two solution of the Skyme model. Phys. Lett.195B, 235 (1987) |
| [18] | Schramm, A.J., Dothan, Y., Biedenharn, L.C.: A calculation of the deuteron as a biskyrmion. Phys. Lett.205B, 151 (1988) |
| [19] | Jackson, A.D., Rho, M.: Baryons as chrial solitons. Phys. Rev. Lett.51, 751 (1983) · doi:10.1103/PhysRevLett.51.751 |
| [20] | Wirzba, A., Bang, H.: The mode spectrum and the stability analysis of Skyrmions on a 3-sphere. Nucl. Phys.A515, 571 (1990) · doi:10.1016/0375-9474(90)90273-O |
| [21] | Zenkin, S.V., Kopeliovich, V.B., Stern, B.E.: The soliton interaction in the Skyrme model. Sov. J. Nucl. Phys.45, 106 (1987) |
| [22] | Braaten, E., Carson, L.: Deuteron as a toroidal Skyrmion. Phys. Rev.D38, 3525 (1988) · doi:10.1103/PhysRevB.38.3525 |
| [23] | Atiyah, M.F., Manton, N.S.: Skyrmions from instantons. Phys. Lett.222B, 438 (1989) |
| [24] | Manton, N.S.: Skyrme fields and instantons. In: Geometry of low dimensional manifolds: 1 (LMS Lecture Notes 150), Donaldson, S.K., and Thomas, C.B., eds. Cambridge: Cambridge University Press 1990 |
| [25] | Jackson, A., Jackson, A.D., Pasquier, V.: The Skyrmion-Skyrmion interaction. Nucl. Phys.A 432, 567 (1985) · doi:10.1016/0375-9474(85)90002-8 |
| [26] | Vinh Mau, R., Lacombe, M., Loiseau, B., Cottingham, W.N., Lisboa, P.: The static baryon-baryon potential in the Skyrme model. Phys. Lett.150B, 259 (1985) |
| [27] | Braaten, E., Townsend, S., Carson, L.: Novel structure of static multisoliton solutions in the Skyrme model. Phys. Lett.235B, 147 (1990) |
| [28] | Kugler, M., Shtrikman, S.: A new Skyrmion crystal. Phys. Lett.208B, 491 (1988) |
| [29] | Castillejo, L., Jones, P.S.J., Jackson, A.D., Verbaarschot, J.J.M., Jackson, A.: Dense Skyrmion systems. Nucl. Phys.A501, 801 (1989) · doi:10.1016/0375-9474(89)90161-9 |
| [30] | Atiyah, M.F.: Geometry of Yang-Mills fields. Lezioni Fermiane. Pisa: Scuola Normale Superiore, 1979 · Zbl 0435.58001 |
| [31] | Schwartz, A.: On regular solutions of euclidean Yang-Mills equations. Phys. Lett.67B, 172 (1977) |
| [32] | Jackiw, R., Rebbi, C.: Degrees of freedom in pseudoparticle systems. Phys. Lett.67B, 189 (1977) |
| [33] | Brown, L., Carlitz, R., Lee, C.: Massles excitations in pseudoparticle fields. Phys. Rev.D 16, 417 (1977) |
| [34] | Atiyah, M.F., Hitchin, N., Singer, I.: Deformations of instantons. Proc. Natl. Acad. Sci. USA74, 2662 (1977) · Zbl 0356.58011 · doi:10.1073/pnas.74.7.2662 |
| [35] | ’t Hooft, G.: Unpublished |
| [36] | Corrigan, E., Fairlie, D.B.: Scalar field theory and exact solutions to a classicalSU(2) gauge theory. Phys. Lett.67B, 69 (1977) |
| [37] | Jackiw, R., Nohl, C., Rebbi, C.: Conformal properties of pseudoparticle configurations. Phys. Rev.D15, 1642 (1977) |
| [38] | Hartshorne, R.: Stable vector bundles and instantons. Commun. Math. Phys.59, 1 (1978) · Zbl 0383.14006 · doi:10.1007/BF01614151 |
| [39] | Berger, M.: Geometry 2. Berlin, Heidelberg, New York: Springer 1987 |
| [40] | Verbaarschot, J.J.M., Walhout, T.S., Wambach, J., Wyld, H.W.: Symmetry and quantization of the two-Skyrmion system: the case of the deuteron. Nucl. Phys.A468, 520 (1987) · doi:10.1016/0375-9474(87)90181-3 |
| [41] | Walhout, T.S.: Multiskyrmions as nuclei. Nucl. Phys.A531, 596 (1991) · doi:10.1016/0375-9474(91)90742-O |
| [42] | Hosaka, A., Griffies, S.M., Oka, M., Amado, R.D.: Two Skyrmion interaction for the Atiyah-Manton ansatz. Phys. Lett.251B, 1 (1990) |
| [43] | Hosaka, A., Oka, M., Amado, R.D.: Skyrmions and their interactions using the Atiyah-Manton construction. Nucl. Phys.A530, 507 (1991) · doi:10.1016/0375-9474(91)90766-Y |
| [44] | Crutchfield, W.Y., Snyderman, N., Brown, V.R.: The deuteron in the Skyrme model. Phys. Rev. Lett.68, 1660 (1992) · doi:10.1103/PhysRevLett.68.1660 |
| [45] | Massey, W.S.: The quotient space of the complex projective plane under conjugation is a 4-sphere. Geom. Dedicata2, 371 (1973) · Zbl 0273.57019 · doi:10.1007/BF00181480 |
| [46] | Kuiper, N.H.: The quotient space of \(\mathbb{C}\)P 1 by complex conjugation is the 4-sphere. Math. Ann.208, 175 (1974) · doi:10.1007/BF01432386 |
| [47] | Sommerville, D.M.Y.: Non-Euclidean geometry (Chap. 9). London: Bell 1914 · JFM 45.0727.01 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
