Cyclic monopoles, affine Toda and spectral curves. (English) Zbl 1270.81228
Summary: We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe’s ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha vector) and base point of the linearising flow in the Jacobian of the spectral curve associated to the Nahm equations arise as pull-backs of Toda data. A theorem of Accola and Fay then means that the theta-functions arising in the solution of the monopole problem reduce to the theta-functions of Toda.
MSC:
| 81V22 | Unified quantum theories |
| 81T10 | Model quantum field theories |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
References:
| [1] | Accola Robert, D.M.: Vanishing Properties of Theta Functions for Abelian Covers of Riemann Surfaces. In: Advances in the Theory of Riemann Surfaces: Proceedings of the 1969 Sony Brook Conference, edited by L.V. Ahlfors, L. Bers, H.M. Farkas, R.C. Gunning, I. Kra, H.E. Rauch, Princeton, NJ: Princeton University Press, 1971, pp. 7–18 · Zbl 0222.30026 |
| [2] | Braden H.W., D’Avanzo A., Enolski V.Z.: On charge-3 cyclic monopoles. Nonlinearity 24, 643–675 (2011) · Zbl 1226.14046 · doi:10.1088/0951-7715/24/3/001 |
| [3] | Braden H.W., Enolski V.Z.: Remarks on the complex geometry of 3-monopole, math-ph/0601040 Part I appears as ”some remarks on the Ercolani-Sinha construction of monopoles”. Theor. Math. Phys. 165, 1567–1597 (2010) · Zbl 1255.81196 · doi:10.1007/s11232-010-0131-2 |
| [4] | Braden, H.W., Enolski, V.Z.: Monopoles, Curves and Ramanujan. Reported at Riemann Surfaces, Analytical and Numerical Methods, Max Planck Instititute (Leipzig), #2007. Matem. Sborniki. 201, 19–74 (2010) |
| [5] | Braden H.W., Enolski V.Z.: On the tetrahedrally symmetric monopole. Commun. Math. Phys. 299, 255–282 (2010) · Zbl 1205.81134 · doi:10.1007/s00220-010-1081-0 |
| [6] | Corrigan E., Goddard P.: An n monopole solution with 4n degrees of freedom. Commun. Math. Phys. 80, 575–587 (1981) · doi:10.1007/BF01941665 |
| [7] | Ercolani N., Sinha A.: Monopoles and Baker Functions. Commun. Math. Phys. 125, 385–416 (1989) · Zbl 0707.35128 · doi:10.1007/BF01218409 |
| [8] | Fay, J.D.: Theta functions on Riemann surfaces. Lectures Notes in Mathematics, Vol. 352, Berlin: Springer, 1973 · Zbl 0281.30013 |
| [9] | Hitchin N.J.: Monopoles and Geodesics. Commun. Math. Phys. 83, 579–602 (1982) · Zbl 0502.58017 · doi:10.1007/BF01208717 |
| [10] | Hitchin N.J.: On the Construction of Monopoles. Commun. Math. Phys. 89, 145–190 (1983) · Zbl 0517.58014 · doi:10.1007/BF01211826 |
| [11] | Hitchin N.J., Manton N.S., Murray M.K.: Symmetric monopoles. Nonlinearity 8, 661–692 (1995) · Zbl 0846.53016 · doi:10.1088/0951-7715/8/5/002 |
| [12] | Houghton C.J., Manton N.S., Romão N.M.: On the constraints defining BPS monopoles. Commun. Math. Phys. 212, 219–243 (2000) · Zbl 0983.53016 · doi:10.1007/s002200000208 |
| [13] | Houghton C.J., Sutcliffe P.M.: Octahedral and dodecahedral monopoles. Nonlinearity 9, 385–401 (1996) · Zbl 0886.58115 · doi:10.1088/0951-7715/9/2/005 |
| [14] | Houghton C.J., Sutcliffe P.M.: Tetrahedral and cubic monopoles. Commun. Math. Phys. 180, 343–361 (1996) · Zbl 0879.58084 · doi:10.1007/BF02099717 |
| [15] | Houghton C.J., Sutcliffe P.M.: su(n) monopoles and Platonic symmetry. J. Math. Phys. 38, 5576–5589 (1997) · Zbl 0888.53055 · doi:10.1063/1.532152 |
| [16] | Kostant B.: The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group. Amer. J. Math. 81, 973–1032 (1959) · Zbl 0099.25603 · doi:10.2307/2372999 |
| [17] | Manton N., Sutcliffe P.: Topological Solitons. Cambridge University Press, Cambridge (2004) · Zbl 1100.37044 |
| [18] | Nahm, W.: The construction of all self-dual multimonopoles by the ADHM method. In: Monopoles in Quantum Field Theory, edited by N.S. Craigie, P. Goddard, W. Nahm, Singapore: World Scientific, 1982 |
| [19] | O’Raifeartaigh L., Rouhani S.: Rings of monopoles with discrete symmetry: explicit solution for n = 3. Phys. Lett. 112, 143 (1982) |
| [20] | Sutcliffe P.M.: Seiberg-Witten theory, monopole spectral curves and affine Toda solitons. Phys. Lett. B 381, 129–136 (1996) · Zbl 0979.37507 · doi:10.1016/0370-2693(96)00675-2 |
| [21] | Vanhaecke P.: Stratifications of hyperelliptic Jacobians and the Sato Grassmannian. Acta. Appl. Math. 40, 143–172 (1995) · Zbl 0827.14015 · doi:10.1007/BF00996932 |
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.
