×

zbMATH — the first resource for mathematics

Nahm’s transformation for instantons. (English) Zbl 0672.55009
Summary: We describe in mathematical detail the Nahm transformation which maps anti-self dual connections on the four-torus \((S^ 1)^ 4\) onto anti- self-dual connections on the dual torus. This transformation induces a map between the relevant instanton moduli spaces and we show that this map is a (hyper Kähler) isometry.

MSC:
55R10 Fiber bundles in algebraic topology
81T20 Quantum field theory on curved space or space-time backgrounds
53C05 Connections, general theory
58J99 Partial differential equations on manifolds; differential operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atiyah, M. F.: Geometry of Yang-Mills fields, Fermi lectures. Scuola Normale Superiore, Pisa 1979 · Zbl 0435.58001
[2] Atiyah, M. F.: Classical groups and classical differential operators on manifolds: Differential operators on manifolds, pp. 6-48. CIME, Verenna 1975
[3] Atiyah, M. F., Hitchin, N. J., Singer, I. M.: Selfduality in four-dimensional Riemannian geometry. Proc. Roy. Soc. Lond. Ser. A,362, 425-461 (1978) · Zbl 0389.53011 · doi:10.1098/rspa.1978.0143
[4] Atiyah, M. F., Singer, I. M.: The index of elliptic operators IV. Ann. Math.93, 119-138 (1971) · Zbl 0212.28603 · doi:10.2307/1970756
[5] Atiyah, M. F., Hitchin, N. J.: The geometry and dynamics of magnetic monopoles. Porter Lectures, Princeton 1988 · Zbl 0671.53001
[6] Corrigan, E., Goddard, P.: Construction of instanton and monopole solutions and reciprocity. Ann. Phys. (NY)154, 253-279 (1984) · Zbl 0535.58025 · doi:10.1016/0003-4916(84)90145-3
[7] Bismut, J.-M., Gillet, H., Soul?, C.: Analytic torsion and holomorphic determinant bundles. III Quillen metrics on holomorphic determinants. Commun. Math. Phys.115, 301-351 (1988) · Zbl 0651.32017 · doi:10.1007/BF01466774
[8] Donaldson, S. K.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc.50, 1-26 (1985) · Zbl 0547.53019 · doi:10.1112/plms/s3-50.1.1
[9] Donaldson, S. K.: Nahm’s equations and the classification of monopoles. Commun. Math. Phys.96, 387-407 (1984) · Zbl 0603.58042 · doi:10.1007/BF01214583
[10] Donaldson, S. K.: Kronheimer, P. B.: A book on Yang-Mills theory and 4-manifolds. Oxford, University Press (to appear) · Zbl 0904.57001
[11] Donaldson, S. K.: Connections, cohomology and the intersection forms of 4-manifolds. J. Diff. Geom.24, 275-341 (1986) · Zbl 0635.57007
[12] Duistermaat, J. J., Gr?nbaum, B.: Differential equations in the spectral parameter. Commun. Math. Phys.103, 177 (1986) · Zbl 0625.34007 · doi:10.1007/BF01206937
[13] Hitchin, N. J.: On the construction of monopoles. Commun. Math. Phys.89, 145-190 (1983) · Zbl 0517.58014 · doi:10.1007/BF01211826
[14] ’t Hooft, G.: A property of electric and magnetic flux in non-Abelian gauge theories. Nucl. Phys.B153, 141 (1979) · doi:10.1016/0550-3213(79)90595-9
[15] Hurtubise, J., Murray, M.: The moduli space ofSU(N) monopoles and Nahm’s equations. Princeton preprint 1988
[16] Mukai, S.: Duality betweenD(X) andD \(D(\hat X)\) with its application to Picard sheaves. Nagoya Math. J.81, 153-175 (1981) · Zbl 0417.14036
[17] Mukai, S.: Symplectic structure of the moduli space of sheaves on an Abelian or K3 surface. Invent. Math.77, 101-116 (1984) · Zbl 0565.14002 · doi:10.1007/BF01389137
[18] Mukai, S.: Fourier functor and its application to the moduli of bundles on an Abelian variety. Adv. Studies in Pure Math.10 (1987), Alg. Geometry Sendai, 515-550 (1985)
[19] Nahm, W.: Monopoles in quantum theory. In: Proceedings of the Monopole Meeting, Trieste (eds.) Craigie, e.a. Singapore: World Scientific 1982
[20] Nahm, W.: Self-dual monopoles and calorons. In: Lecture Notes in Physics, vol.201. Denardo, G., et al. (eds.). Berlin, Heidelberg, New York: Springer 1984 · Zbl 0573.32030
[21] Schenk, H.: On a generalized fourier transform of instantons over flat tori. Commun. Math. Phys.116, 177-183 (1988) · Zbl 0651.58042 · doi:10.1007/BF01225253
[22] Segal, G., Wilson, G.: Loop groups and the equations of KdV type. Publ. Math. IHES 61, 1985 · Zbl 0592.35112
[23] Taubes, C. H.: Self-dual connections on 4-manifolds with indefinite intersection matrix. J. Diff. Geom.19, 517-560 (1984) · Zbl 0552.53011
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.