Induced differential characters on nonlinear Gra{\ss}mannians

TL;DR

This paper develops a new framework for differential characters on nonlinear Grassmannians, leading to prequantum bundles and group extensions relevant for geometric quantization of submanifold spaces.

Contribution

It introduces a transgression map for differential characters on nonlinear Grassmannians and constructs associated prequantum bundles and group extensions.

Findings

Constructed a transgression map for differential characters.

Established prequantum circle bundles with symplectic structure.

Derived central Lie group extensions of volume-preserving diffeomorphisms.

Abstract

Using a nonlinear version of the tautological bundle over Gra{\ss}mannians, we construct a transgression map for differential characters from $M$ to the nonlinear Gra{\ss}mannians $\mathrm{Gr}^S(M)$ of submanifolds of $M$ of a fixed type $S$. In particular, we obtain prequantum circle bundles of the nonlinear Gra\ss{}mannian 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.