Holonomy representation

Let $\pi:P\to M$ be a principal $G$-bundle with a flat principal connection . Fix a basepoint $x\in M$ and $p\in P_x$.

For a loop $\gamma$ based at $x$, let $g_\gamma\in G$ be the element determined by horizontal lifting as in the definition of the holonomy group :

Flatness implies $g_\gamma$ depends only on the homotopy class $[\gamma]\in \pi_1(M,x)$, and the assignment

is a group homomorphism. This homomorphism is the holonomy representation (also called the monodromy representation) of the flat connection based at $p$.

If one replaces $p$ by $p\cdot h$ for $h\in G$, then $\rho_{p\cdot h}=h^{-1}\rho_p\,h$; thus the holonomy representation is well-defined up to conjugation in $G$. Conversely, a representation $\rho:\pi_1(M,x)\to G$ determines a flat principal bundle (and flat connection) via the standard $(\widetilde M\times G)/\pi_1$ construction.

Examples

1. Circle with $U(1)$-monodromy. Flat principal $U(1)$-bundles over $S^1$ are classified by a single element $e^{i\theta}\in U(1)$; the representation sends the generator of $\pi_1(S^1)\cong \mathbb{Z}$ to $e^{i\theta}$, and sends $n\mapsto e^{in\theta}$.

2. Trivial flat connection. On $M\times G$ with the product flat connection, horizontal lifts return to the same point in the fiber for every loop, so $\rho_p$ is the trivial homomorphism.

3. Constructing a flat bundle from a representation. Given any $\rho:\pi_1(M,x)\to G$, the quotient $P=(\widetilde M\times G)/\pi_1(M)$ with the $\pi_1$-action twisted by $\rho$ has a canonical flat connection whose holonomy representation is $\rho$ (up to conjugacy).