Holonomy representation

Let $\pi:P\to M$ be a principal $G$ -bundle over a connected smooth manifold, equipped with a flat principal connection. Fix $x\in M$ and $p\in P_x$ .

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

\widetilde\gamma(1)=p\cdot g_\gamma.

Use the loop-product convention $\gamma_2*\gamma_1$ for first traversing $\gamma_1$ , then $\gamma_2$ . Equivariance of transport gives $g_{\gamma_2*\gamma_1}=g_{\gamma_2}g_{\gamma_1}$ . Flatness implies $g_\gamma$ depends only on the homotopy class $[\gamma]\in \pi_1(M,x)$ , and the assignment

\rho_p:\pi_1(M,x)\longrightarrow G,\qquad \rho_p([\gamma])\coloneqq g_\gamma

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

Choice of fiber point and reconstruction

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 homomorphism $\rho:\pi_1(M,x)\to G$ determines a flat principal bundle with connection via the standard $(\widetilde M\times G)/\pi_1(M,x)$ construction.

Examples

Circle with $U(1)$ -monodromy. Holonomy representations $\pi_1(S^1)\to U(1)$ are determined by one element $e^{i\theta}\in U(1)$ ; the integer $n$ maps to $e^{in\theta}$ .

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.

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).