Local approximation of the Virasoro group and Diff^+(S^1) by finite groups

Determine whether the Virasoro group (the universal central extension of the orientation-preserving diffeomorphism group of the circle Diff^+(S^1)) or the group Diff^+(S^1) itself is locally approximable by finite groups; that is, ascertain whether there exists a family of finite groups and an ultrafilter such that the local ultraproduct (the substructure of the first-order ultraproduct consisting of elements of bounded emerging metric, modulo the infinitesimal pseudo-metric) admits a surjective homomorphism onto the Virasoro group or onto Diff^+(S^1).

Background

The paper develops a rigorous notion of local approximation via the local ultraproduct of emerging-metric structures and demonstrates its applicability to several mathematical physics settings. In particular, it proves that every compact simple Lie group is locally approximated by finite groups, even though global approximation by finite groups is impossible for these groups.

Motivated by these results and by questions from physics about approximating infinite/continuous objects by finite/discrete ones, the authors pose whether similar local approximations exist for certain infinite-dimensional Lie groups. The Virasoro group and the group of orientation-preserving diffeomorphisms of the circle Diff^+(S^1) are natural and important candidates in mathematical physics, prompting the explicit open problem below.