Equivariant differential characters and Chern-Simons bundles

• Main Author: Pérez, Roberto Ferreiro
• Format: Journal Article
• Language: English
• Published: 18-09-2020
• Subjects: Mathematics - Mathematical Physics; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.1907.00292

Algebr. Geom. Topol. 21 (2021) 1911-1940 We construct Chern-Simons bundles as $\mathrm{Aut}^{+}P$-equivariant $U(1)$
-bundles with connection over the space of connections $\mathcal{A}_{P}$ on a
principal $G$-bundle $P\rightarrow M$. We show that the Chern-Simons bundles
are determined up to an isomorphisms by means of its equivariant holonomy. The
space of equivariant holonomies is shown to coincide with the space of
equivariant differential characteres of second order. 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 $G$.
The results are also generalized to the case of the action of diffeomorphisms
on the space of Riemannian metrics. In particular, in dimension $2$ a
Chern-Simons bundle over the Teichm\"{u}ller space is obtained.