Kac-moody Groups and Integrability of Soliton Equations Boris Feigin and Edward Frenkel
نویسنده
چکیده
Soliton equations describe infinite-dimensional hamiltonian systems. They are closely related to infinite-dimensional Lie groups and algebraic curves. These relations account for complete integrability of soliton equations, i.e. the existence of infinitely many integrals of motion in involution. Recently a new insight has been brought into the theory by the observation that these integrals of motion can be viewed as classical limits of quantum integrals of motion of certain deformations of conformal field theories [Z, EY, HM]. In our previous works [FF1, FF2], using the technique of free field realization of conformal field theories (cf. [F2] for a review), we gave a homological construction of quantum integrals of motion, corresponding to particular deformations. This enabled us to prove the existence of quantum integrals of motion for those deformations. Tracing our construction back to the classical limit, we realized that it provides a new approach to integrability of classical soliton equations, namely, affine Toda equations. Here we will present this approach. We hope that it can be applied to other soliton equations as well.
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