Equivariant differential characters and Chern–Simons bundles

Ferreiro Pérez, Roberto

doi:10.2140/agt.2021.21.1911

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Equivariant differential characters and Chern–Simons bundles

Roberto Ferreiro Pérez

Algebraic & Geometric Topology 21 (2021) 1911–1940

DOI: 10.2140/agt.2021.21.1911

Abstract

We construct Chern–Simons bundles as ${Aut}^{+} P$ –equivariant $U (1)$ –bundles with connection over the space of connections $𝒜_{P}$ on a principal $G$ –bundle $P \to M$ . We show that the Chern–Simons bundles are determined up to isomorphisms by their equivariant holonomy. The space of equivariant holonomies is shown to coincide with the space of equivariant differential characters of order $2$ . Furthermore, we prove that the Chern–Simons theory provides, in a natural way, an equivariant differential character that determines the Chern–Simons bundles. Our construction can be applied in the case in which $M$ is a compact manifold of even dimension and for arbitrary bundle $P$ and group $G$ .

We also generalize the results to the case of the action of diffeomorphisms on the space of Riemannian metrics. In particular, in dimension $2$ we obtain a Chern–Simons bundle over the Teichmüller space.

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Keywords

equivariant differential character, equivariant holonomy, Chern–Simons bundle, space of connections, space of Riemannian metrics

Mathematical Subject Classification 2010

Primary: 55N91, 70S15

Secondary: 53C08, 53C29, 58J28

References

Publication

Received: 8 January 2020

Revised: 23 July 2020

Accepted: 16 August 2020

Published: 18 August 2021

Authors

Roberto Ferreiro Pérez
Departamento de Economía Financiera y Actuarial y Estadística Facultad de Ciencias Económicas y Empresariales, UCM Madrid Spain

Volume 21, issue 4 (2021)