Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere limits

TL;DR

This paper classifies central extensions of the loop group of area-preserving diffeomorphisms on the 2-sphere and demonstrates their relation to fuzzy sphere limits of Kac-Moody cocycles in the large k limit.

Contribution

It provides a classification of central extensions for LSDiff(S^2) and links these to fuzzy sphere limits of Kac-Moody cocycles, revealing new connections between geometric and algebraic structures.

Findings

Classified central extensions for LSDiff(S^2)

Established fuzzy sphere limits of Kac-Moody cocycles

Connected geometric group extensions to algebraic limits

Abstract

We classify central extensions for the loop group LSDiff(S^2) of area-preserving diffeomorphisms of the 2-sphere, and of related twisted loop groups. We then show that the corresponding Lie algebra cocycles are `fuzzy sphere limits' of Kac-Moody cocycles for (twisted) loop algebras Lsu(k+1) for the limit of large k, provided that the cocycles are rescaled by 6/k^3.