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

## TL;DR

This paper extends the construction of invariants for families of flat connections using fiber integration of differential characters, covering new parameter ranges and comparing with existing theories.

## Contribution

It introduces a new method for constructing invariants for flat connection families in cases not previously addressed, using fiber integration techniques.

## Key findings

- Invariants coincide with Jaya Iyer's for p > r+1.
- Invariants are trivial for p < r.
- Comparison with existing literature confirms consistency.

## Abstract

Let $E\to B$ 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 (\textit{Letters in Mathematical Physics}, 2016, 106 (1) pp. 131-146) constructed maps $H_r(\mathcal{D}(E)) \to H^{2p-r-1}(B,\mathbb{R}/\mathbb{Z})$ for $p>r+1$ where $\mathcal{D}(E)$ is the simplicial set of relatively flat connections, thereby associating invariants to families of 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.