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Magnetic monopoles and the Yang-Baxter equations. (English) Zbl 0757.53038
The aim of this paper is to highlight a coincidence in the mathematical construction of solutions to two sets of physically important equations - - the Yang-Baxter equations and the Bogomolny equations for magnetic monopoles in a three-dimensional hyperbolic space. A common, recognizable, feature of soluble equations from various sources has been the dependence of solutions on algebraic curves, or theta functions associated with them. The most visible ones are solutions involving elliptic functions and for many years these were the only known ones for the Yang-Baxter equations, but in [R. J. Baxter, J. H. H. Perk and H. Au-Yang, New solutions of the star-triangle relations for the chiral Potts model, Phys. Lett. A 128, 138-142 (1988)] curves of higher genus, though highly constrained, appeared. The point made in the lecture, of which this is the written version, is that those constraints are precisely the ones for spectral curves of hyperbolic \(SU(2)\)- monopoles of charge \(N\), invariant under rotation by \(2\pi/N\), as described in [M. F. Atiyah, Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 1-33 (1987; Zbl 0722.53063)]. The paper goes on to indulge in some interesting speculation about the links between the acutal solutions of the two equations, and whether the analogy with monopoles leads to the Yang- Baxter equations being part of a more general soluble family. It is interesting to note that curves with this type of constraint also occur in the classical problem of Poncelet’s theorem.

53Z05 Applications of differential geometry to physics
81T13 Yang-Mills and other gauge theories in quantum field theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
Zbl 0722.53063
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