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Induced differential characters on nonlinear Graßmannians

Tobias Diez, Bas Janssens, Karl-Hermann Neeb, Cornelia Vizman

Abstract

Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from $M$ to the nonlinear Graßmannians $\mathrm{Gr}^S(M)$ of submanifolds of $M$ of a fixed type $S$. In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden-Weinstein symplectic form. The associated Kostant-Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.

Induced differential characters on nonlinear Graßmannians

Abstract

Using a nonlinear version of the tautological bundle over Graßmannians, we construct a transgression map for differential characters from to the nonlinear Graßmannians of submanifolds of of a fixed type . In particular, we obtain prequantum circle bundles of the nonlinear Graßmannian endowed with the Marsden-Weinstein symplectic form. The associated Kostant-Souriau prequantum extension yields central Lie group extensions of a group of volume-preserving diffeomorphisms integrating Lichnerowicz cocycles.

Paper Structure

This paper contains 14 sections, 11 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Differential characters
  3. Basics on differential characters
  4. Stabilizer groups and Lie algebras.
  5. Flux homomorphism.
  6. Tautological bundle over nonlinear Graßmannians
  7. Transgression of differential forms
  8. Tautological bundle
  9. Transgression of differential characters
  10. Integration of Lichnerowicz cocycles
  11. Construction using the nonlinear Graßmannian GrSN(M)
  12. Construction using the embedding space Emb(S, M)
  13. Comparison
  14. The special case H1(M,Z)

Key Result

Theorem A

Let $M$ be a compact manifold of dimension $n$ endowed with a volume form $\mu$ having integral periods, and let $S$ be a closed, oriented manifold of dimension $n-2$. For every choice of a differential character $h \in \widehat{H}^{n-1}(M,\mathbb T)$ with curvature $\mu$, the transgression $\wideti

Theorems & Definitions (26)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Remark 2.2
  • definition 2.3
  • Example 2.4
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 16 more