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Letters in Mathematical Physics

, Volume 109, Issue 1, pp 11–31 | Cite as

Differential characters and cohomology of the moduli of flat connections

  • Marco Castrillón LópezEmail author
  • Roberto Ferreiro Pérez
Article

Abstract

Let \(\pi {:}\, P\rightarrow M\) be a principal bundle and p an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern–Simons differential characters is exploited to define a homology map \(\chi ^{k} {:}\, H_{2r-k-1}(M)\times H_{k}({\mathcal {F}}/{\mathcal {G}})\rightarrow {\mathbb {R}}/{\mathbb {Z}}\), for \(k<r-1\), where \({\mathcal {F}} /{\mathcal {G}}\) is the moduli space of flat connections of \(\pi \) under the action of a subgroup \({\mathcal {G}}\) of the gauge group. The differential characters of first order are related to the Dijkgraaf–Witten action for Chern–Simons theory. The second-order characters are interpreted geometrically as the holonomy of a connection in a line bundle over \({\mathcal {F}}/{\mathcal {G}}\). The relationship with other constructions in the literature is also analyzed.

Keywords

Characteristic classes Chern–Simons Differential characters Moduli of flat connections 

Mathematics Subject Classification

53C05 55R40 51H25 

Notes

Acknowledgements

M.C.L. was partially supported by MINECO (Spain) under Grant MTM2015–63612-P. R.F.P. was partially supported by “Proyecto de investigación Santander-UCM PR26/16-20305”.

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Álgebra, Geometría y Topología, Facultad de Matemáticas, ICMAT (CSIC-UAM-UC3M-UCM)Universidad Complutense de MadridMadridSpain
  2. 2.Departamento de Economía Financiera y Contabilidad I, Facultad de Ciencias Económicas y EmpresarialesUniversidad Complutense de MadridMadridSpain

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