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Equivariant differential characters and Chern–Simons bundles

Roberto Ferreiro Pérez

Algebraic & Geometric Topology 21 (2021) 1911–1940
Abstract

We construct Chern–Simons bundles as Aut+P–equivariant U(1)–bundles with connection over the space of connections 𝒜P on a principal G–bundle P 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.

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