THE LANDAU-LIFSHITZ EQUATION, ELLIPTIC-CURVES AND THE WARD TRANSFORM

The Landau-Lifshitz (LL) equation is studied from a point of view that is close to that of Segal and Wilson's. work on KdV. The LL hierarchy is defined and shown to exist using a dressing transformation that in volves parameters lambda1, lambda2, lambda3 that live on an elliptic c urve SIGMA. The crucial role of the group K congruent-to Z2 X Z2 of tr anslations by the half-periods of SIGMA and its non-trivial central ex tension K is brought out and an analogue of Birkhoff factorisation for K-equivariant loops in SIGMA is given. This factorisation theorem is given two treatments, one in terms of the geometry of an infinite-dime nsional Grassmannian, and the other in terms of the algebraic geometry of bundles over SIGMA. Further, a Ward-like transform between a class of holomorphic vector bundles on the total space Z of a line-bundle o ver SIGMA and solutions of LL is constructed. An appendix is devoted t o a careful definition of the Grassmannian of the Frechet space C(infi nity)(S1).