Area-Preserving Diffeomorphisms Groups, the Majorana Representation of Spins, and Su(n)

It is by now common practice to regard large N limits of classical groups, especially SU(N) for large N, as being connected in some way with groups SDiff(M) of area-preserving diffeomorphisms of 2-dimensional manifolds. The limit-taking is problematic since one can rigorously demonstrate that SDiff(M) never has the same de Rham cohomology nor homotopy as any SU(N) or other large N classical group, or products thereof, even with N = ∞. Nevertheless, a connection can be made between SU(n) for any n and a group action on a finite set of points on a sphere using the Majorana representation of spin-frac {n}{2} quantum states by sets of n points on a sphere. This allows for a realization of SU(n) acting on such states, and thus a natural action on the two-dimensional sphere S². There is no need to work with a special basis of the Lie algebra of SU(n), and there is a clear geometrical interpretation of the connection between SU(n) and SDiff(S²).