Hidden Symmetries of Integrable Systems in Yang-mills Theory and Kähler Geometry

نویسنده

  • Kanehisa TAKASAKI
چکیده

We now have a huge list of “nonlinear integrable systems.” Most of them are more or less related to so called “soliton” phenomena, thereby referred to as “soliton equations.” In the last decade, various soliton equations came to be reorganized in a unified framework, the Kadomtsev-Petviashvili (KP) hierarchy and its family [Sa-Sa] [Se-Wi]. Beside these soliton equations, there are a few exceptional cases that are considered as “higher (or multi) dimensional” nonlinear integrable systems, whereas soliton equations are to describe nonlinear waves in one dimensional space like a canal. Typical and the most important examples are the self-duality equations in the Yang-Mills theory of gauge fields and in the Einstein theory of gravity (see reviews [Ch] [Bo] and references cited). As first pointed out by Ward [Wa1] for the Yang-Mills theory and by Penrose [Pe] for the Einstein theory, the self-duality equations and their family are in close relationship with “twistor theory” [Tw1-6], and this fact lies in the heart of their “integrability” (see [Be-Za] [Ch-Pr-Si] [Fo-Ho-Pa] [Po] for earlier work in that direction.) “Integrability” is a magic word that summarizes a number of aspects observed in these nonlinear systems; we now focus on their “symmetries” on the space of solutions. The existence of a large set of such symmetries is strong evidence of integrability. Suppose that the set of symmetries is so large that its acts, in some sense, transitively on the space of solutions – then all solutions, in principle, can be obtained from some special solution by the action of symmetries. It would be reasonable to call such a system of equations an integrable system. This is indeed the case, literally or approximately, for most nonlinear integrable systems known so far. Further, those symmetries frequently exhibit, at infinitesimal level, some remarkable Lie algebraic structures. For soliton equations, infinite dimensional Lie algebras called “Kac-Moody” algebras and their representation theory [Ka] [Pr-Se] give such structures [Da-Ji-Ka-Mi](see also review [Ta1]). Similar structures are

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تاریخ انتشار 2008