# Wilson loops and holonomies

## 1. Iterated integrals

Before we get onto holonomies, we first need to do some work on iterated integrals. As usual, on an orientable manifold, we define the integral of an $n$-form as, for a coordinate neighbourhood $U \subset M$,

\begin{eqnarray} \int_U \omega &=& \int_{\phi(U)} \omega(x_1, \ldots, x_n) dx_1 \wedge \ldots \wedge dx^n \\ &=&\int_{\phi(U)} \omega(x_1, \ldots, x_n) dx_1 \ldots dx^n \end{eqnarray}where the last integral is just the Lebesgue integral. We will also need the integral of a form along a curve. Given a curve $\gamma : [a,b] \to M$ and a $k$-form $\omega : TM \times \ldots \times TM \to \mathbb{R}$, the pullback of $\omega$ by $\gamma$ (implicitly by the $1$-form of its tangent) is the $k$-form on $[a, b]$ defined by

\begin{eqnarray} (\gamma^* \omega)_t (X_t, Y_t, \ldots) = \omega_{\gamma(t)} (d \gamma_t[X_t], d \gamma_t[Y_t], \ldots) \end{eqnarray}On a curve, the $1$-form is the $n$-form, and therefore any higher pullback will be $0$, so we will only need the pullback by some $1$-form $\omega$ on $M$ :

\begin{eqnarray} (\gamma^* \omega)_t (X_t) &=& \omega_{\gamma(t)} (d_t\gamma[X_t]) \end{eqnarray} [Transgression of differential forms]By using a local basis, with the dual basis $\theta^\mu$ on $M$ and $e$ we obtain

\begin{eqnarray} (\gamma^* \omega)_t (X_t) &=& \omega_{\mu}(\gamma(t)) \theta^\mu ((d \gamma)^\nu_t(t) \theta^t \otimes e_\nu[X_t(t) e^t])\\ &=& \omega_{\nu}(\gamma(t)) (d \gamma)^\nu_t(t) X_t(t)\\ \end{eqnarray}As there is only one component in the curve, this is equivalent to

\begin{eqnarray} (\gamma^* \omega)_t (X_t) &=& \omega_{\nu}(\gamma(t)) (d \gamma)^\nu (t) X(t)\\ &=& (\omega_{\gamma(t)} [\dot{\gamma}(t)]) X(t) \end{eqnarray}Our integral is therefore

\begin{eqnarray} \int_a^b (\gamma^* \omega)_t &=& \int_a^b \omega_\mu(\gamma(t)) \dot{\gamma}^\mu(t) dt \end{eqnarray}Changing the variable of integration to $\phi \circ \gamma = x$, we get

\begin{eqnarray} \int_a^b (\gamma^* \omega)_t &=& \int_a^b \omega_\mu(x(t)) dx^\mu(t) \end{eqnarray}For simplicity, we will just write the line integral of a $1$-form as either one of these forms :

\begin{eqnarray} \int_\gamma \omega = \int_a^b \omega(t) dt \end{eqnarray}This is simply a map from $\omega$ to $\mathbb{R}$. An iterated integral is defined by

\begin{eqnarray} \int_\gamma \omega \omega &=& \int_a^b(\int_a^t \omega(s) ds) \omega(t) dt \end{eqnarray}and so forth.

## Integrating connections

We will be dealing with the integration of connection $1$-forms in particular. A connection $1$-form is an algebra-valued one-form, a map

\begin{eqnarray} \omega : TM \to \mathfrak{g} \end{eqnarray}where $\mathfrak{g}$ is the Lie algebra of our principal bundle. Given a basis of the algebra $T_i$, we can decompose it as

\begin{eqnarray} \omega = \omega^i T_i \end{eqnarray}using the dual basis $\theta^i$, we define the integral of our connection form as

\begin{eqnarray} \int_\gamma \omega = \left[ \int_\gamma \theta^i(\omega) \right] T_i \end{eqnarray}By the linearity of the integral, we can see that, for an iterated integral,

\begin{eqnarray} \int_\gamma \omega \omega &=& \left[ \int_\gamma \left( \left[ \int_\gamma \theta^i(\omega) \right] T_i \right] \theta^j(\omega) \right) T_j\\ &=& \left[ \int_\gamma \left( \left[ \int_\gamma \theta^i(\omega) \right] T_i \right] \theta^j(\omega) \right) T_j \end{eqnarray}## Holonomy of a connection

Let's consider a fiber bundle $\pi : E \to M$ over our spacetime $M$, with tangent bundle $\mathrm{d}\pi : TE \to TM$. We define an Ehresmann connection on the horizontal bundle of $E$

For a curve $\gamma$, take the parallel-transport map $P_\gamma$. If the curve $\gamma$ has a lift $\tilde{\gamma}$ as a map $\tilde{\gamma} : I \to E$, such that $\pi(\tilde{\gamma}(t)) = \gamma(t)$, then the lift is horizontal if $\dot{\tilde{\gamma}}(t)$ is the pullback section of $H$, on the pullback horizontal bundle $H_\gamma$, our connection. The parallel-transport map is then defined by

\begin{eqnarray} P_{\gamma, t, t'} : \pi^{-1}(\{ \gamma(t) \}) &\to& \pi^{-1}(\{ \gamma(t') \})\\ e &\mapsto& \tilde{\gamma}_e(t') \end{eqnarray}such that ...

If we now consider a loop, such that $\gamma(a) = \gamma(b) = p$, the parallel transport of that loop is then a map from the fiber at $p$ to itself. Given a point $p$, with the loop space $\Omega M(p)$ of all loops starting and ending at $p$, $P_\gamma$ forms a group for $\gamma \in \Omega M(p)$. First, for every

[Proof]

The holonomy group of a connection at a point $p$ is then defined by the set of transformations of the fiber by such parallel transport.

\begin{eqnarray} \mathrm{Hol}_p(\nabla) = \{ P_\gamma | \gamma \in \Omega M(p) \} \end{eqnarray}## Bibliography

Last updated :

*2022-03-28 09:13:17*