Universality of the central Fréchet–Lie group extension of exact volume-preserving diffeomorphisms

Prove that the central Fréchet–Lie group extension 1 → H^1(M, U(1)) → Diff_ex^1(M, μ) → Diff_ex(M, μ) → 1, constructed via a bundle gerbe on M whose curvature equals μ, is universal among central extensions of the group Diff_ex(M, μ) of exact volume-preserving diffeomorphisms.

Background

By applying the main construction to a bundle gerbe with curvature proportional to μ, the authors build a central extension of the Fréchet–Lie group Diff_ex(M, μ) integrating the well-studied central Lie algebra extension for exact divergence-free vector fields. They discuss conditions under which Neeb’s Recognition Theorem would imply universality at the group level.

The paper explicitly notes that universality of this group extension is conjectured, aligning it with the conjectured universality of its corresponding Lie algebra extension. Proving this would identify the constructed extension as the universal central extension in the Fréchet–Lie category, with implications for the structure and representation theory of groups of exact volume-preserving diffeomorphisms.