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

TL;DR

This paper examines the connection between the group of area-preserving diffeomorphisms on a torus and the large-N limit of SU(N), concluding that the limit is ill-behaved and larger than the diffeomorphism group.

Contribution

It clarifies the limitations of associating SDiff(T^2) with the large-N limit of SU(N), highlighting the ill-behaved nature of this limit.

Findings

The limit of SU(N) as N approaches infinity is larger than SDiff(T^2).

The commonly used basis leads to an ill-behaved limit.

The association between SDiff(T^2) and SU(∞) is problematic.

Abstract

There have been various attempts to identify groups of area-preserving diffeomorphisms of 2-dimensional manifolds with limits of SU(N) as $N\to\infty$. 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 $SDiff(T^2)$.