### Abstract

We develop a theory of Chern-Simons classes CS2k-1W∈H2k-1(LM2k-1;R) on the loop space LM of a Riemannian manifold M. These classes are associated to a pair of connections on LM whose connection and curvature forms take values in pseudodifferential operators by [19]. We use the Wodzicki residue of these operators to define and compute the Chern-Simons classes. As an application, we prove that |π1(Diff(M))|=∞ for the total space M of circle bundles associated to high multiples of a Kähler class over integral Kähler surfaces.

Original language | English |
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Pages (from-to) | 485-518 |

Number of pages | 34 |

Journal | Advances in Mathematics |

Volume | 287 |

DOIs | |

Publication status | Published - 2016 Jan 10 |

### Keywords

- Characteristic classes
- Diffeomorphism groups
- Loop spaces
- Pseudodifferential operators
- Wodzicki residue

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Maeda, Y., Rosenberg, S., & Torres-Ardila, F. (2016). The geometry of loop spaces II: Characteristic classes.

*Advances in Mathematics*,*287*, 485-518. https://doi.org/10.1016/j.aim.2015.10.001