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

Abstract

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.