Invariants of families of flat connections using fiber integration of differential characters

  • Ishan MataEmail author


Let \(E\rightarrow B\) be a smooth vector bundle of rank n, and let \(P \in I^p(GL(n,{\mathbb {R}}))\) be a \(GL(n,{\mathbb {R}})\)-invariant polynomial of degree p compatible with a universal integral characteristic class \( u \in H^{2p}(BGL(n,{\mathbb {R}}),{\mathbb {Z}})\). Cheeger–Simons theory associates a rigid invariant in \(H^{2p-1}(B,{\mathbb {R}}/{\mathbb {Z}})\) to any flat connection on this bundle. Generalizing this result, Jaya Iyer (Lett Math Phys 106 (1):131–146, 2016) constructed maps \(H_r({\mathcal {D}}(E)) \rightarrow H^{2p-r-1}(B,{\mathbb {R}}/{\mathbb {Z}})\) for \(p>r+1\). Here, \({\mathcal {D}}(E)\) is the simplicial abelian group whose group of r-simplices is freely generated by \((r+1)\)-tuples of relatively flat connections. In this article, we construct such maps for the cases \(p<r\) and \(p>r+1\) using fiber integration of differential characters. We find that for \(p>r+1\) case, the invariants constructed here coincide with those obtained by Jaya Iyer, and that in the \(p<r\) case the invariants are trivial. We further compare our construction with other results in the literature.


Differential characters Flat connections Fiber integration 

Mathematics Subject Classification

53C07 53C08 



I am thankful to my supervisor Dr. Rishikesh Vaidya for discussions and support. I wish to profusely thank the anonymous reviewer whose feedback significantly helped improve the presentation. I am financially supported by the Council of Scientific & Industrial Research-Human Resource Development Group (CSIR-HRDG) under the CSIR-SRF(NET) scheme. I am grateful to CSIR-HRDG for the same.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of PhysicsBirla Institute of Technology and Science- PilaniPilaniIndia

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