Abstract
We introduce a class of Higgs bundles called cyclic which lie in the Hitchin component of representations of a compact Riemann surface into the split real form of a simple Lie group. We then prove that such Higgs bundles correspond to a class of solutions to the affine Toda equations. This relationship is further explained in terms of lifts of harmonic maps.
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Aldrovandi, E., Falqui, G.: Geometry of Higgs and Toda fields on Riemann surfaces. J. Geom. Phys. 17(1), 25–48 (1995)
Bolton, J., Pedit, F., Woodward, L.M.: Minimal surfaces and the affine Toda field model. J. Reine Angew. Math. 459, 119–150 (1995)
Bolton, J., Woodward, L.M.: The affine Toda equations and minimal surfaces. In: Fordy, A.P., Wood, J.C. (eds.) Integrable Systems and Harmonic Maps, Aspects of Mathematics E23, pp. 59–82. Vieweg, Braunschweig (1994)
Carter, R.W.: Lie Algebras of Finite and Affine Type. Cambridge University Press, Cambridge (2005)
Doliwa, A., Sym, A.: Nonlinear \(\sigma \)-models on spheres and Toda systems. Phys. Lett. A 185(5–6), 453–460 (1994)
Doliwa, A.: Holomorphic curves and Toda systems. Lett. Math. Phys. 39(1), 21–32 (1997)
Doliwa, A.: Harmonic maps and periodic Toda systems. J. Math. Phys. 38(3), 1685–1691 (1997)
Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55(1), 59–126 (1987)
Hitchin, N.J.: Lie groups and Teichmüller space. Topology 31(3), 449–473 (1992)
Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Am. J. Math. 81, 973–1032 (1959)
Leznov, A.N., Saveliev, M.V.: Exactly and completely integrable nonlinear dynamical systems. Symmetries of partial differential equations. Part II. Acta Appl. Math. 16(1), 1–74 (1989)
Mansfield, P.: Solution of Toda systems. Nucl. Phys. B 208, 277–300 (1982)
Mikhailov, A.V., Olshanetsky, M.A., Perelomov, A.M.: Two-dimensional generalized Toda lattice. Commun. Math. Phys. 79(4), 473–488 (1981)
Olive, D., Turok, N.: The symmetries of Dynkin diagrams and the reduction of Toda field equations. Nucl. Phys. B 215, 470–494 (1983)
Onishchik, A.L., Vinberg, E.B.: Lie Groups and Algebraic Groups. Springer, Berlin (1990)
Onishchik, A.L.: Lectures on Real Semisimple Lie Algebras and Their Representations. European Mathematical Society, Germany (2004)
Simpson, C.T.: Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Am. Math. Soc. 1(4), 867–918 (1988)
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Baraglia, D. Cyclic Higgs bundles and the affine Toda equations. Geom Dedicata 174, 25–42 (2015). https://doi.org/10.1007/s10711-014-0003-2
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DOI: https://doi.org/10.1007/s10711-014-0003-2