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

Bas Janssens, Zhenghan Wang

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.

Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere 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.
Paper Structure (32 sections, 20 theorems, 114 equations)

This paper contains 32 sections, 20 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Main results and structure of the paper
  3. Towards QFTs in 2+1 dimensions
  4. Loop groups with values in $\mathrm{SDiff}(\mathbb{S}^2,\mu)$
  5. Central extensions and 2-Cocycles
  6. Invariant symmetric bilinear forms on $C^{\infty}_0(\mathbb{S}^2)$
  7. The role of the centre
  8. The Poisson algebra of the 2-sphere.
  9. The Koszul map
  10. 2-cocycles on $C^{\infty}_{c}(M, \mathfrak{k})$ and the proof of Theorem \ref{['ThmCocycleLoop']}
  11. Integral classes
  12. Relation between integrality conditions for $L\mathfrak{su}(2)$ and $L\mathfrak{k}$
  13. Projective unitary representations
  14. Twisted loop algebras
  15. The structure of twisted loop algebras
  16. ...and 17 more sections

Key Result

Theorem 1

$H^2(L\mathfrak{k}, \mathbb{R})$ is 1-dimensional. Every continuous 2-cocycle $\psi$ on $L\mathfrak{k}$ is cohomologous to $c\psi_{\infty}$, where $c$ is a real number and $\psi_{\infty}$ is the cocycle

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2: JV16
  • Proposition 3
  • proof
  • Theorem 4
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['Thm:InvBilS2Alg']}
  • Corollary 6
  • Remark 1
  • ...and 29 more