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Monopoles and Baker functions. (English) Zbl 0707.35128

The construction of explicit solutions of Nahm’s equations which satisfy the regularity and reality conditions required for the physical monopoles is presented. A monopole and a spectral curve (i.e. a Riemann surface) are associated together with the data of a line bundle on this curve. The bundle evolves simply, in the Nahm “time” z, on the Jacobian (a torus) of the curve. A class of elliptic solutions of Nahm’s equations for arbitrary monopole number, which correspond to singular curves with elliptic components are discussed.

This study may be used to extend the results on the dynamics of slowly moving monopoles.


MSC:
35Q40PDEs in connection with quantum mechanics
81T13Yang-Mills and other gauge theories
References:
[1]Bogomol’nyi, E. B.: Yad. Fiz.24, 861-870, (1976) [Sov. J. Nucl. Phys.24, 449 (1976)]; Prasad, M. K., Sommerfeld, C. M.: Phys. Rev. Lett.35, 760 (1975)
[2]Nahm, W.: All self-dual multimonopoles for arbitrary gauge groups. (preprint), TH.3172-CERN (1981)
[3]Atiyah, M., Hitchin, N. J.: Geometry and dynamics of magnetic monopoles. Princeton, NJ: Princeton University Press 1988
[4]Donaldson, S. K.: Commun. Math. Phys.96, 387-407 (1984) · Zbl 0603.58042 · doi:10.1007/BF01214583
[5]See e.g. Monopoles in Quantum Field Theory. Craigie, N. S., Goddard, P., Nahm, W. (eds.). Singapore: World Scientific 1982. For an alternative method of solving the Bogomol’nyi equations based on the Atiyah-Ward (Commun. Math. Phys.55, 117-124 (1977)) approach see Ward, R. S.: Commun. Math. Phys.79, 317-325 (1981); Prasad, M. K.: Commun. Math. Phys.80, 137-149 (1981); Prasad, M. K., Sinha, A., Chau-Wang, L-L.: Phys. Rev.D23, 2321-2334 (1981); Prasad, M. K., Rossi, P. [20]; Ward, R. S.: Phys. Lett.102B, 136-138 (1981); Ward, R. S.: Commun. Math. Phys.80, 563-574 (1981); Corrigan, E., Goddard, P.: Commun. Math. Phys.80, 575-587 (1981). See also Forgás, P., Horváth, Z., Palla, L.: Phys. Lett.102B, 131-135 (1981) and Phys. Lett.109B, 200-204 (1982)
[6]Hitchin, N. J.: Commun. Math. Phys.83, 579-602 (1982) · Zbl 0502.58017 · doi:10.1007/BF01208717
[7]Hitchin, N. J.: Commun. Math. Phys.89, 145-190 (1983) · Zbl 0517.58014 · doi:10.1007/BF01211826
[8]Atiyah, M., Hitchin, N., Drinfel’d, V., Manin, Yu.: Phys. Lett.65A, 185 (1978)
[9]Baker, H. S.: Abel’s Theorem and the Allied Theory. Cambridge: Cambridge University Press 1897
[10]Krichever, I. M.: Funct. Anal. Appl.11, 144-146 (1977) · Zbl 0368.35022 · doi:10.1007/BF01135528
[11]Griffiths, P. A.: Am. J. Math.107, 1445-1484 (1985) · Zbl 0585.58028 · doi:10.2307/2374412
[12]Flaschka, H., Newell, A. C., Ratiu, T.: Physica9D, 300-323 (1983)
[13]Flaschka, H., Forest, M. G., McLaughlin, D. W.: Commun. Pure Appl. Math.33, 739-784 (1980) · Zbl 0454.35080 · doi:10.1002/cpa.3160330605
[14]Lax, P., Levermore, D.: Commun. Pure Appl. Math.33, 739-784 (1980) · Zbl 0454.35080 · doi:10.1002/cpa.3160330605
[15]Ercolani, N., Forest, M. G., Montgomery, R., McLaughlin, D. W.: Duke Math. J.55, 949-983 (1987) · Zbl 0668.35064 · doi:10.1215/S0012-7094-87-05548-7
[16]Date, E.: Osaka J. Math.19, 125-158 (1982)
[17]For background information on Algebraic Geometry and Riemann Surface Theory see e.g. Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978; and Springer, G.: Introduction to Riemann surfaces. Reading, MA: Addison-Wesley 1957
[18]Siegel, C. L.: Topics in complex function theory. Vol. II. New York: Wiley 1969
[19]Hartshorne, R.: Commun. Math. Phys.59, 1-15 (1978) · Zbl 0383.14006 · doi:10.1007/BF01614151
[20]Hurtubise, J.: Commun. Math. Phys.92, 195-202 (1983) · Zbl 0548.58040 · doi:10.1007/BF01210845
[21]Fay, J. D.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics, vol.352. Berlin, Heidelberg, New York: Springer 1973
[22]Erdélyi, A et. al.: Higher transcendental functions, Bateman manuscript project, vol. 2. New York: McGraw-Hill (1953)
[23]Prasad, M. K., Rossi, P.: Phys. Rev. Lett.46, 806 (1981); Phys. Rev.D24, 2182 (1981) · doi:10.1103/PhysRevLett.46.806
[24]Brown, S. A., Panagopoulos, H., Prasad, M. K.: Phys. Rev.D26, 854 (1982)
[25]Bidenharn, L. C., Louck, J. D.: Angular momentum in quantum physics theory and application. Reading, MA: Addison-Wesley 1981