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}

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

  1. F. Brown, Iterated integrals in quantum field theory
  2. K. Chen, Iterated Integrals of Differential Forms and Loop Space Homology
  3. R. Giles, Reconstruction of gauge potentials from Wilson loops

Last updated : 2020-09-30 13:21:13
Tags : physics , general-relativity , gauge-theory