On the limiting procedure by which $SDiff(T^2)$ and $SU(\mathrm{\infty })$ are associated

There have been various attempts to identify groups of area-preserving diffeomorphisms of 2-dimensional manifolds with limits of SU(N) as $N\to \mathrm{\infty }$. We discuss the particularly simple case where the manifold concerned is the two-dimensional torus $T^2$ and argue that the limit, even in the basis commonly used, is ill-behaved and that the large-N limit of SU(N) is much larger than $SDiff(T^2)$.