Some characterizations of Dirac type singularity of monopoles. (English) Zbl 1378.70028
The authors study singular monopoles on open subsets of \(3\)-dimensional Euclidean space and give two characterisations of Dirac type singularities. The first characterisation is given in terms of the growth order of the norms of sections invariant under the scattering map (Theorem 1/Theorem 4). The second characterisation (Theorem 2/Theorem 3) is given in terms of the growth order of the norms of the Higgs field. Theorem 2 is established using a deep result on the Dirichlet problem for instantons (see [S. K. Donaldson, J. Geom. Phys. 8, No. 1–4, 89–122 (1992; Zbl 0747.53022)]) and as the authors note, the first characterisation is a direct consequence of the second characterisation.
Reviewer: Alexander Schmeding (Berlin)
MSC:
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 78A99 | General topics in optics and electromagnetic theory |
| 58D27 | Moduli problems for differential geometric structures |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
Citations:
Zbl 0747.53022References:
| [1] | Charbonneau, B.: Analytic aspects of periodic instantons. Thesis (Ph.D.), Massachusetts Institute of Technology (2004) · Zbl 1134.53013 |
| [2] | Charbonneau, B., Hurtubise, J.: Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface. Int. Math. Res. Not. 2011(1), 175-216 (2011) · Zbl 1252.53036 |
| [3] | Cherkis S.A., Kapustin A.: Nahm transform for periodic monopoles and \[{\mathcal{N} = 2}N=2\] super Yang-Mills theory. Commun. Math. Phys. 218, 333-371 (2001) · Zbl 1003.53025 · doi:10.1007/PL00005558 |
| [4] | Cherkis S.A., Kapustin A.: Periodic monopoles with singularities and \[{\mathcal{N} = 2}N=2\] super-QCD. Commun. Math. Phys. 234, 1-35 (2003) · Zbl 1040.53032 · doi:10.1007/s00220-002-0786-0 |
| [5] | Donaldson S.K., Kronheimer P.B.: The Geometry of Four-Manifolds. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1990) · Zbl 0820.57002 |
| [6] | Donaldson S.K.: Boundary value problems for Yang-Mills fields. J. Geom. Phys. 8, 89-122 (1992) · Zbl 0747.53022 · doi:10.1016/0393-0440(92)90044-2 |
| [7] | Hitchin N.: Monopoles and geodesics. Commun. Math. Phys. 83, 579-602 (1982) · Zbl 0502.58017 · doi:10.1007/BF01208717 |
| [8] | Hitchin N.: Construction of monopoles. Commun. Math. Phys. 89, 145-190 (1983) · Zbl 0517.58014 · doi:10.1007/BF01211826 |
| [9] | Kronheimer, P.B.: Monopoles and Taub-NUT metrics. M.Sc. Dissertation, Oxford (1985) · Zbl 1210.53033 |
| [10] | Mochizuki, T.: Notes on periodic monopoles (preprint) · Zbl 1441.53016 |
| [11] | Nash O.: Singular hyperbolic monopoles. Commun. Math. Phys. 277, 161-187 (2008) · Zbl 1134.53013 · doi:10.1007/s00220-007-0368-2 |
| [12] | Norbury P.: Magnetic monopoles on manifolds with boundary. Trans. Am. Math. Soc. 363, 1287-1309 (2011) · Zbl 1210.53033 · doi:10.1090/S0002-9947-2010-04934-7 |
| [13] | Pauly, M.: Monopole moduli spaces for compact 3-manifolds. Math. Ann. 311, 125-46 (1998) · Zbl 0920.58011 |
| [14] | Simpson C.T.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867-918 (1988) · Zbl 0669.58008 · doi:10.1090/S0894-0347-1988-0944577-9 |
| [15] | Yoshino, M.: Master thesis (in Japanese) · Zbl 0502.58017 |
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.
