10.2 Holonomy

10.2.1 Definitions

Let $P(M,G)$ be a principal bundle and let $\gamma :[0,1]\to M$ be a curve whose horizontal lift through $u_0\in {\pi }^{-1}(\gamma (0))$ is $\tilde{\gamma }$ . In the last section, we defined a map $\Gamma (\tilde{\gamma }):{\pi }^{-1}(\gamma (0))\to {\pi }^{-1}(\gamma (1))$ which maps a point $u_0=\tilde{\gamma }(0)$ to $u_1=\tilde{\gamma }(1)$ . Let us consider two curves $\alpha ,\beta :[0,1]\to M$ with $\alpha (0)=\beta (0)=p_0$ and $\alpha (1)=\beta (1)=p_1$ . Take horizontal lifts $\tilde{\alpha }$ and $\tilde{\beta }$ of $\alpha$ and $\beta$ such that $\tilde{\alpha }(0)=\tilde{\beta }(0)=u_0$ . Then $\tilde{\alpha }(1)$ is not necessarily equal to $\tilde{\beta }(1)$ . This shows that if we consider a loop $\gamma :[0,1]\to M$ at $p=\gamma (0)=\gamma (1)$ , we have $\tilde{\gamma }(0)\ne \tilde{\gamma }(1)$ in general. A loop $\gamma$ defines a transformation ${\tau }_{\gamma }:{\pi }^{-1}(p)\to {\pi }^{-1}(p)$ on the fibre. This transformation is compatible with the right action of the group,

$${\tau }_{\gamma }(ug)={\tau }_{\gamma }(u)g\text{\hspace{1em}(10.21)}$$

which follows immediately from (10.18). We note that ${\tau }_{\gamma }$ depends not only on the loop $\gamma$ but also on the connection.

Example 10.3

Consider an $\mathbb{R}$ -bundle over $M={\mathbb{R}}^2-\{0\}$ . The connection one-form $\omega$ and the loop $\gamma$ in example 10.2 define a map ${\tau }_{\gamma }:{\pi }^{-1}((1,0))\to {\pi }^{-1}((1,0))$ given by $g\mapsto g+2\pi$ , $g\in \mathbb{R}$ .

Take a point $u\in P$ with $\pi (u)=p$ and consider the set of loops $C_p(M)$ at $p$ ; $C_p(M)\equiv \{\gamma :[0,1]\to M|\gamma (0)=\gamma (1)=p\}$ . The set of elements

$${\Phi }_u\equiv \{g\in G|{\tau }_{\gamma }(u)=ug,\gamma \in C_p(M)\}\text{\hspace{1em}(10.22)}$$

is a subgroup of the structure group $G$ and is called the holonomy group at $u$ . The group property of ${\Phi }_u$ is easily derived from exercise 10.3. If $\alpha ,\beta$ and $\gamma =\alpha \ast \beta$ are loops at $p$ , we have ${\tau }_{\gamma }={\tau }_{\beta }\circ {\tau }_{\alpha }$ , hence

$${\tau }_{\gamma }(u)={\tau }_{\beta }\circ {\tau }_{\alpha }(u)={\tau }_{\beta }(u{g}_{\alpha })={\tau }_{\beta }(u)g_{\alpha }=u{g}_{\beta }{g}_{\alpha }$$

where ${\tau }_{\alpha }(u)=u{g}_{\alpha }$ etc. This shows that

$$g_{\gamma }=g_{\beta }{g}_{\alpha }.\text{\hspace{1em}(10.23)}$$

The constant loop $c:[0,1]\mapsto p$ defines the identity transformation ${\tau }_c:u\mapsto u$ . The inverse loop ${\gamma }^{-1}$ of $\gamma$ induces the inverse transformation ${\tau }_{{\gamma }^{-1}}={\tau }_{\gamma }^{-1}$ , hence $g_{{\gamma }^{-1}}=g_{\gamma }^{-1}$ .

Exercise 10.5

(a) Let ${\tau }_{\alpha }(u)=u{g}_{\alpha }$ . Show that

$${\tau }_{\alpha }(ug)=ug\left({\mathrm{ad}}_g{g}_{\alpha }\right)=ug\left(g^{-1}{g}_{\alpha }g\right).\text{\hspace{1em}(10.24)}$$

Verify that

$${\Phi }_{ua}\cong a^{-1}{\Phi }_{u}a.\text{\hspace{1em}(10.25)}$$

(b) Let $u,u^{\prime }\in P$ be points on the same horizontal lift $\tilde{\gamma }$ . Show that ${\Phi }_u\cong {\Phi }_{u^{\prime }}$ .
(c) Suppose that $M$ is connected. Show that all ${\Phi }_u$ are isomorphic to each other.

Exercise 10.6

Let ${\mathcal{A}}_i={\mathcal{A}}_{i\mu }\mathrm{d}{x}^{\mu }$ be a gauge potential over $U_i$ and $\gamma$ a loop in $U_i$ . Let ${\tau }_{\gamma }(u)=u{g}_{\gamma }$ , $u\in P$ , $g_{\gamma }\in G$ . Use (10.14) to show that

$$g_{\gamma }=\mathcal{P}\exp \left(-{\oint }_{\gamma }{\mathcal{A}}_{i\mu }\mathrm{d}{x}^{\mu }\right).\text{\hspace{1em}(10.26)}$$

Let $C_p^0(M)$ denote the set of loops at $p$ , which are homotopic to the constant loop at $p$ . The group

$${\Phi }_u^0\equiv \left\{g\in G\mid {\tau }_{\gamma }(u)=ug,\gamma \in C_p^0(M)\right\}\text{\hspace{1em}(10.27)}$$

is called the restricted holonomy group.

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