9.4 Principal bundles

9.4.1 Definitions

A principal bundle has a fibre $F$ which is identical to the structure group $G$ . A principal bundle $P\stackrel{\pi }{\to }M$ is also denoted by $P(M,G)$ and is often called a $G$ bundle over $M$ .

The transition function acts on the fibre on the left as before. In addition, we may also define the action of $G$ on $F$ on the right. Let ${\varphi }_i:U_i\times G\to {\pi }^{-1}(U_i)$ be the local trivialization given by ${\varphi }_i^{-1}(u)=(p,g_i)$ , where $u\in {\pi }^{-1}(U_i)$ and $p=\pi (u)$ . The right action of $G$ on ${\pi }^{-1}(U_i)$ is defined by ${\varphi }_i^{-1}(ua)=(p,g_{i}a)$ , that is (figure 9.8),

Figure 9.8. The right action of $G$ on $P$ .

$$ua={\varphi }_i(p,g_{i}a)\text{\hspace{1em}(9.41)}$$

for any $a\in G$ and $u\in {\pi }^{-1}(p)$ . Since the right action commutes with the left action, this definition is independent of the local trivializations. In fact, if $p\in U_i\cap U_j$ ,

$$ua={\varphi }_j(p,g_{j}a)={\varphi }_j(p,t_{ji}(p)g_{i}a)={\varphi }_i(p,g_{i}a).$$

Thus, the right multiplication is defined without reference to the local trivializations. This is denoted by $P\times G\to P$ or $(u,a)\mapsto ua$ . Note that $\pi (ua)=\pi (u)$ . The right action of $G$ on ${\pi }^{-1}(p)$ is transitive since $G$ acts on $G$ transitively on the right and $F_p={\pi }^{-1}(p)$ is diffeomorphic to $G$ . Thus, for any

$u_1,u_2\in {\pi }^{-1}(p)$ there exists an element $a$ of $G$ such that $u_1=u_{2}a$ . Then, if $\pi (u)=p$ , we can construct the whole fibre as ${\pi }^{-1}(p)=\{ua|a\in G\}$ . The action is also free; if $ua=u$ for some $u\in P$ , $a$ must be the unit element $e$ of $G$ . In fact, if $u={\varphi }_i(p,g_i)$ , we have ${\varphi }_i(p,g_{i}a)={\varphi }_i(p,g_i)a=ua=u={\varphi }_i(p,g_i)$ . Since ${\varphi }_i$ is bijective, we must have $g_{i}a=g_i$ , that is, $a=e$ .

Given a section $s_1(p)$ over $U_i$ , we define a preferred local trivialization ${\varphi }_i:U_i\times G\to {\pi }^{-1}(U_i)$ as follows. For $u\in {\pi }^{-1}(p)$ , $p\in U_i$ , there is a unique element $g_u\in G$ such that $u=s_i(p)g_u$ . Then we define ${\varphi }_i$ by ${\varphi }_i^{-1}(u)=(p,g_u)$ . In this local trivialization, the section $s_i(p)$ is expressed as

$$s_i(p)={\varphi }_i(p,e).\text{\hspace{1em}(9.42)}$$

This local trivialization is called the canonical local trivialization. By definition ${\varphi }_i(p,g)={\varphi }_i(p,e)g=s_i(p)g$ . If $p\in U_i\cap U_j$ , two sections $s_i(p)$ and $s_j(p)$ are related by the transition function $t_{ij}(p)$ as follows

$$\begin{array}{rl}{s}_i(p)& ={\varphi }_i(p,e)={\varphi }_j(p,t_{ji}(p)e)={\varphi }_j(p,t_{ji}(p))\\ & ={\varphi }_j(p,e)t_{ji}(p)=s_j(p)t_{ji}(p).\end{array}\text{\hspace{1em}(9.43)}$$

Example 9.7

Let $P$ be a principal bundle with fibre $\mathrm{U}(1)=S^1$ and the base space $S^2$ . This principal bundle represents the topological setting of the magnetic monopole (section 1.9). Let $\{U_{\mathrm{N}},U_{\mathrm{S}}\}$ be an open covering of $S^2$ , $U_{\mathrm{N}}(U_{\mathrm{S}})$ being the northern (southern) hemisphere. If we parametrize $S^2$ by the usual polar angles, we have

$$\begin{array}{rl}& U_{\mathrm{N}}=\left\{\left(\theta ,\varphi \right)\mid 0\le \theta \le \pi /2+\epsilon ,0\le \varphi <2\pi \right\}\\ & U_S=\left\{\left(\theta ,\varphi \right)\mid \pi /2-\epsilon \le \theta \le \pi ,0\le \varphi <2\pi \right\}.\end{array}$$

The intersection $U_{\mathrm{N}}\cap U_{\mathrm{S}}$ is a strip which is essentially the equator. Let ${\varphi }_{\mathrm{N}}$ and ${\varphi }_{\mathrm{S}}$ be the local trivializations such that

$${\varphi }_{\mathrm{N}}^{-1}(u)=\left(p,{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{N}}}\right){\varphi }_{\mathrm{S}}^{-1}(u)=\left(p,{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{S}}}\right)\text{\hspace{1em}(9.44)}$$

where $p=\pi (u)$ . Take a transition function $t_{\mathrm{NS}}(p)$ of the form ${\mathrm{e}}^{\mathrm{i}n\varphi }$ where $n$ must be an integer so that $t_{\mathrm{NS}}(p)$ may be uniquely defined on the equator. Since $t_{\mathrm{NS}}$ maps the equator $S^1$ to $\mathrm{U}(1)$ , this integer characterizes the homotopy group ${\pi }_1(\mathrm{U}(1))=\mathbb{Z}$ . The fibre coordinates ${\alpha }_{\mathrm{N}}$ and ${\alpha }_{\mathrm{S}}$ are related on the equator as

$${\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{N}}}={\mathrm{e}}^{\mathrm{i}n\varphi }{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{S}}}.\text{\hspace{1em}(9.45)}$$

If $n=0$ , the transition function is the unit element of $\mathrm{U}(1)$ and we have a trivial bundle $P_0=S^2\times S^1$ . If $n\ne 0$ , the $\mathrm{U}(1)$ -bundle $P_n$ is twisted. It is remarkable that the topological structure of a fibre bundle is characterized by an integer. The integer characterizes how two local sections are pasted together at the equator. Accordingly, the integer corresponds to the element of the homotopy group ${\pi }_1(\mathrm{U}(1))=\mathbb{Z}$ .

Since $\mathrm{U}(1)$ is Abelian, the right action and the left action are equivalent. Under the right action $g={\mathrm{e}}^{\mathrm{i}\Lambda }$ , we have

$${\varphi }_{\mathrm{N}}^{-1}(ug)=\left(p,{\mathrm{e}}^{\mathrm{i}\left({\alpha }_{\mathrm{N}}+\Lambda \right)}\right)\text{\hspace{1em}(9.46a)}$$

$${\varphi }_{\mathrm{S}}^{-1}(ug)=\left(p,{\mathrm{e}}^{\mathrm{i}\left({\alpha }_{\mathrm{S}}+\Lambda \right)}\right).\text{\hspace{1em}(9.46b)}$$

The right action corresponds to the U(1)-gauge transformation.

Example 9.8

If we identify all the infinite points of the Euclidean space ${\mathbb{R}}^m$ , the one-point compactification $S^m={\mathbb{R}}^m\cup \{\infty \}$ is obtained. If a trivial $G$ bundle is defined over ${\mathbb{R}}^m$ we shall have a new $G$ bundle over $S^m$ after compactification, which is not necessarily trivial. Let $P$ be an $\mathrm{SU}(2)$ bundle over $S^4$ obtained from ${\mathbb{R}}^4$ by one-point compactification. This principal bundle represents an $\mathrm{SU}(2)$ instanton (section 1.10). Introduce an open covering $\{U_{\mathrm{N}},U_{\mathrm{S}}\}$ of $S^4$ ,

$$\begin{array}{rl}& U_{\mathrm{N}}=\left\{(x,y,z,t)\mid x^2+y^2+z^2+t^2\le R^2+\epsilon \right\}\\ & U_{\mathrm{S}}=\left\{\left(x,y,z,t\right)\mid R^2-\epsilon \le x^2+y^2+z^2+t^2\right\}\end{array}$$

where $R$ is a positive constant and $\epsilon$ is an infinitesimal positive number. The thin intersection $U_{\mathrm{N}}\cap U_{\mathrm{S}}$ is essentially $S^3$ . Let $t_{\mathrm{NS}}(p)$ be the transition function defined at $p\in U_{\mathrm{N}}\cap U_{\mathrm{S}}$ . Since $t_{\mathrm{NS}}$ maps $S^3$ to $\mathrm{SU}(2)$ , it is classified by ${\pi }_3(\mathrm{SU}(2))=\mathbb{Z}$ . The integer characterizing the bundle is called the instanton number. If $t_{\mathrm{NS}}(p)$ is taken to be the unit element $e\in \mathrm{SU}(2)$ , we have a trivial bundle $P_0=S^3\times \mathrm{SU}(2)$ , which corresponds to the homotopy class 0. Non-trivial bundles are obtained as follows. We first note that $\mathrm{SU}(2)\cong S^3$ (example 4.12). An element $A\in \mathrm{SU}(2)$ is written as

$$A=\left(\begin{array}{cc}u& v\\ -\bar{v}& \bar{u}\end{array}\right)$$

where $|u{|}^2+|v{|}^2=1$ . Separating $u$ and $v$ as $u=t+\mathrm{i}z$ and $v=y+\mathrm{i}x$ , we find $t^2+x^2+y^2+z^2=1$ . Thus SU(2) is regarded as the unit sphere $S^3$ and ${\pi }_3(\mathrm{SU}(2))\cong {\pi }_3(S^3)\cong \mathbb{Z}$ classifies maps from $S^3$ to SU(2) $\cong S^3$ . The identity map $f:S^3\to S^3\cong \mathrm{SU}(2)$ is

$$\begin{array}{rl}& f(x,y,z,t)\mapsto \left(\begin{array}{cc}t+\mathrm{i}z& y+\mathrm{i}x\\ -y+\mathrm{i}x& t-\mathrm{i}z\end{array}\right)\\ & =t{I}_2+\mathrm{i}\left(x{\sigma }_x+y{\sigma }_y+z{\sigma }_z\right)\end{array}\text{\hspace{1em}(9.47)}$$

where $I_2$ is the $2\times 2$ unit matrix and the ${\sigma }_{\mu }$ are the Pauli matrices. Let us take a point $p=(x,y,z,t)\in U_{\mathrm{N}}\cap U_{\mathrm{S}}$ . If $R=(x^2+y^2+z^2+t^2{)}^{1/2}$ denotes the radial distance of $p$ , the vector $(x/R,y/R,z/R,t/R)$ has unit length. We assign an element of $\mathrm{SU}(2)$ to the point $p$ as

$$t_{\mathrm{NS}}(p)=\frac{1}{R}\left(t{I}_2+\mathrm{i}\sum _i{x}^i{\sigma }_i\right).\text{\hspace{1em}(9.48)}$$

Let ${\varphi }_{\mathrm{N}}$ and ${\varphi }_{\mathrm{S}}$ be the local trivializations,

$${\varphi }_{\mathrm{N}}^{-1}(u)=\left(p,g_{\mathrm{N}}\right){\varphi }_{\mathrm{S}}^{-1}(u)=\left(p,g_{\mathrm{S}}\right)\text{\hspace{1em}(9.49)}$$

where $p=\pi (u)$ and $g_{\mathrm{N}},g_{\mathrm{S}}\in \mathrm{SU}(2)$ . On $U_{\mathrm{N}}\cap U_{\mathrm{S}}$ we have

$$g_{\mathrm{N}}=\frac{1}{R}\left(t{I}_2+\mathrm{i}\sum _i{x}^i{\sigma }_i\right)g_{\mathrm{S}}.\text{\hspace{1em}(9.50)}$$

While $(t,x)$ scans $S^3$ once, $t_{\mathrm{NS}}(p)$ sweeps $\mathrm{SU}(2)$ once, hence this bundle corresponds to the homotopy class 1 of ${\pi }_3(\mathrm{SU}(2))$ . It is not difficult to see that the transition function corresponding to the homotopy class $n$ is given by

$$t_{\mathrm{NS}}(p)=\frac{1}{R^n}{\left(t1+\mathrm{i}\sum _i{x}^i{\sigma }_i\right)}^n.\text{\hspace{1em}(9.51)}$$

To continue our study of monopoles and instantons, we have to introduce connections (the gauge potentials) on the fibre bundle. We will come back to these topics in the next chapter.

Example 9.9

Hopf has shown that $S^3$ is a U(1) bundle over $S^2$ . The unit three-sphere embedded in ${\mathbb{R}}^4$ is expressed as

$$(x^1{)}^2+(x^2{)}^2+(x^3{)}^2+(x^4{)}^2=1.$$

If we introduce $z^0=x^1+\mathrm{i}{x}^2$ and $z^1=x^3+\mathrm{i}{x}^4$ , this becomes

$${\left|z^0\right|}^2+{\left|z^1\right|}^2=1.\text{\hspace{1em}(9.52)}$$

Let us parametrize $S^2$ as

$$({\xi }^1{)}^2+({\xi }^2{)}^2+({\xi }^3{)}^2=1.$$

The Hopf map $\pi :S^3\to S^2$ is defined by

$${\xi }^1=2\left(x^1{x}^3+x^2{x}^4\right)\text{\hspace{1em}(9.53a)}$$

$${\xi }^2=2\left(x^2{x}^3-x^1{x}^4\right)\text{\hspace{1em}(9.53b)}$$

$${\xi }^3=(x^1{)}^2+(x^2{)}^2-(x^3{)}^2-(x^4{)}^2.\text{\hspace{1em}(9.53c)}$$

It is easily verified that $\pi$ maps $S^3$ to $S^2$ since

$$({\xi }^1{)}^2+({\xi }^2{)}^2+({\xi }^3{)}^2=[(x^1{)}^2+(x^2{)}^2+(x^3{)}^2+(x^4{)}^2{]}^2=1.$$

Let $(X,Y)$ be the stereographic projection coordinates of a point in the southern hemisphere $U_{\mathrm{S}}$ of $S^2$ from the North Pole. If we take a complex plane which contains the equator of $S^2$ , $Z=X+\mathrm{i}Y$ is within the circle of unit radius. We found in example 8.1 that (figure 9.9)

Figure 9.9. Stereographic coordinates of the sphere $S^2$ . $(X,Y)$ is defined with respect to the projection from the North Pole while $(U,V)$ with respect to the projection from the South Pole.

$$Z=\frac{{\xi }^1+\mathrm{i}{\xi }^2}{1-{\xi }^3}=\frac{x^1+\mathrm{i}{x}^2}{x^3+\mathrm{i}{x}^4}=\frac{z^0}{z^1}(\xi \in U_{\mathrm{S}}).\text{\hspace{1em}(9.54a)}$$

Observe that $Z$ is invariant under

$$\left(z^0,z^1\right)\mapsto \left(\lambda z^0,\lambda z^1\right)$$

where $\lambda \in \mathrm{U}(1)$ . Since $|\lambda |=1$ , the point $(\lambda z^0,\lambda z^1)$ is also in $S^3$ . The stereographic coordinates $(U,V)$ of the northern hemisphere $U_{\mathrm{N}}$ projected from the South Pole are given by

$$W=U+\mathrm{i}V=\frac{{\xi }^1-\mathrm{i}{\xi }^2}{1+{\xi }^3}=\frac{x^3+\mathrm{i}{x}^4}{x^1+\mathrm{i}{x}^2}=\frac{z^1}{z^0}(\xi \in U_{\mathrm{N}}).\text{\hspace{1em}(9.54b)}$$

Note that $Z=1/W$ on the equator $U_{\mathrm{N}}\cap U_{\mathrm{S}}$ .

The fibre bundle structure is given as follows. We first define the local trivializations, ${\varphi }_{\mathbf{S}}^{-1}:{\pi }^{-1}(U_{\mathbf{S}})\to U_{\mathbf{S}}\times \mathrm{U}(1)$ by

$$\left(z^0,z^1\right)\mapsto \left(z^0/z^1,z^1/\left|z^1\right|\right)\text{\hspace{1em}(9.55a)}$$

and ${\varphi }_{\mathrm{N}}^{-1}:{\pi }^{-1}(U_{\mathrm{N}})\to U_{\mathrm{N}}\times \mathrm{U}(1)$ by

$$\left(z^0,z^1\right)\mapsto \left(z^1/z^0,z^0/\left|z^0\right|\right).\text{\hspace{1em}(9.55b)}$$

Observe that these local trivializations are well defined on each chart. For example, $z^0\ne 0$ on $U_{\mathrm{N}}$ , hence both $z^1/z^0=U+\mathrm{i}V$ and $z^0/|z^0|$ are nonsingular. On the equator, ${\xi }^3=0$ , we have $|z^0|=|z^1|=1/\sqrt{2}$ . Accordingly, the local trivializations on the equator are

$${\varphi }_S^{-1}:\left(z^0,z^1\right)\mapsto \left(z^0/z^1,\sqrt{2}{z}^1\right)\text{\hspace{1em}(9.56a)}$$

and

$${\varphi }_{\mathrm{N}}^{-1}:\left(z^0,z^1\right)\mapsto \left(z^1/z^0,\sqrt{2}{z}^0\right).\text{\hspace{1em}(9.56b)}$$

The transition function on the equator is

$$t_{\mathrm{NS}}(\xi )=\frac{\sqrt{2}{z}^0}{\sqrt{2}{z}^1}={\xi }^1+\mathrm{i}{\xi }^2\in \mathrm{U}(1).\text{\hspace{1em}(9.57)}$$

If we circumnavigate the equator, $t_{\mathrm{NS}}(\xi )$ traverses the unit circle in the complex plane once, hence the U(1) bundle $S^3\stackrel{\pi }{\to }{S}^2$ is characterized by the homotopy class 1 of ${\pi }_1(\mathrm{U}(1))=\mathbb{Z}$ . Trautman (1977), Minami (1979) and Ryder (1980) have pointed out that a magnetic monopole of unit strength is described by the Hopf map $S^3\stackrel{\pi }{\to }{S}^2$ .

The Hopf map can be understood from a slightly different point of view. We regard $S^3$ as a complex one-sphere

$$S_{\mathbb{C}}^1=\{(z^0,z^1)\in {\mathbb{C}}^2||z^0{|}^2+|z^1{|}^2=1\}.$$

Define a map $\pi :S_{\mathbb{C}}^1\to {\mathbb{CP}}^1$ by

$$\left(z^0,z^1\right)\mapsto \left[\left(z^0,z^1\right)\right]=\left\{\lambda \left(z^0,z^1\right)\mid \lambda \in \mathbb{C}-\{0\}\right\}.\text{\hspace{1em}(9.58)}$$

Under this map, points of $S^3$ of the form $\lambda (z^0,z^1)$ , $|\lambda |=1$ are mapped to a single point of ${\mathbb{CP}}^1=S^2$ . This is the Hopf map $\pi :S^3\to S^2$ obtained earlier. This is easily generalized to the case of the quaternion $\mathbb{H}$ . The quaternion algebra is defined by the product table,

$$\begin{array}{rl}& i^2=j^2=k^2=-1ij=-ji=k\\ & jk=-kj=iki=-ik=j.\end{array}$$

An arbitrary element of $\mathbb{H}$ is written as

$$q=t+ix+jy+kz.$$

Clearly the unit quaternion $|q|=(t^2+x^2+y^2+z^2{)}^{1/2}=1$ represents $S^3\cong \mathrm{SU}(2)$ . The quaternion one-sphere is given by

$$S_{\mathbb{H}}^1=\left\{\left(q^0,q^1\right)\in {\mathbb{H}}^2\mid {\left|q^0\right|}^2+{\left|q^1\right|}^2=1\right\}\text{\hspace{1em}(9.59)}$$

which represents $S^7$ . The Hopf map, in this case, takes the form

$$\pi :S_{\mathbb{H}}^1\to \mathbb{H}{P}^1\text{\hspace{1em}(9.60)}$$

where $\mathbb{H}{P}^1$ is the quaternion projective space whose element is

$$[(q^0,q^1)]=\{\eta (q^0,q^1)\in {\mathbb{H}}^2|\eta \in \mathbb{H}-\{0\}\}.\text{\hspace{1em}(9.61)}$$

Points of $S^7$ with $|\eta |=1$ are mapped under this map to a single point of $\mathbb{H}{P}^1=S^4$ and we have the Hopf map

$$\pi :S^7\to S^4.\text{\hspace{1em}(9.62)}$$

The fibre is the unit quaternion $S^3=\mathrm{SU}(2)$ . The transition function defined by the Hopf map belongs to the class 1 of ${\pi }_3(\mathrm{SU}(2))\cong \mathbb{Z}$ . An instanton of unit strength is described in terms of this Hopf map.

Octonions define a Hopf map $\pi :S^{15}\to S^8$ . This differs from other Hopf maps in that the fibre $S^7$ is not really a group. So far we have not found an application of this map in physics. ¹

Example 9.10

Let $H$ be a closed Lie subgroup of a Lie group $G$ . We show that $G$ is a principal bundle with fibre $H$ and base space $M=G/H$ . Define the right action of $H$ on $G$ by $g\mapsto ga$ , $g\in G$ , $a\in H$ . The right action is differentiable since $G$ is a Lie group. Define the projection $\pi :G\to M=G/H$ by the map $\pi :g\mapsto [g]=\{gh|h\in H\}$ . Clearly, $g,ga\in G$ are mapped to the same point $[g]$ hence $\pi (g)=\pi (ga)(=[g])$ . To define local trivializations, we need to define a map $f_i:G\to H$ on each chart $U_i$ . Let $s$ be a local section over $U_i$ and $g\in {\pi }^{-1}([g])$ . Define $f_i$ by $f_i(g)=s([g]{)}^{-1}g$ . Since $s([g])$ is a section at $[g]$ , it is expressed as $ga$ for some $a\in H$ and accordingly, $s([g]{)}^{-1}g=a^{-1}{g}^{-1}g=a^{-1}\in H$ . Then we define the local trivialization ${\varphi }_i:U_i\times H\to G$ by

$${\varphi }_i^{-1}(g)=([g],f_i(g)).\text{\hspace{1em}(9.63)}$$

It is easy to see that $f_i(ga)=f_i(g)a$ ( $a\in H$ ) hence ${\varphi }_i^{-1}(ga)=(p,f_i(g)a)$ is satisfied. Useful examples are (see example 5.18)

$$\mathrm{O}(n)/\mathrm{O}(n-1)=\mathrm{SO}(n)/\mathrm{SO}(n-1)=S^{n-1}\text{\hspace{1em}(9.64)}$$

$$\mathrm{U}(n)/\mathrm{U}(n-1)=\mathrm{SU}(n)/\mathrm{SU}(n-1)=S^{2n-1}.\text{\hspace{1em}(9.65)}$$

9.4.2 Associated bundles

Given a principal fibre bundle $P(M,G)$ , we may construct an associated fibre bundle as follows. Let $G$ act on a manifold $F$ on the left. Define an action of $g\in G$ on $P\times F$ by

$$(u,f)\to (ug,g^{-1}f)\text{\hspace{1em}(9.66)}$$

where $u\in P$ and $f\in F$ . Then the associated fibre bundle $(E,\pi ,M,G,F,P)$ is an equivalence class $P\times F/G$ in which two points $(u,f)$ and $(ug,g^{-1}f)$ are identified.

Let us consider the case in which $F$ is a $k$ -dimensional vector space $V$ . Let $\rho$ be the $k$ -dimensional representation of $G$ . The associated vector bundle $P{\times }_{\rho }V$ is defined by identifying the points $(u,v)$ and $(ug,\rho (g{)}^{-1}v)$ of $P\times V$ , where $u\in P,g\in G$ and $v\in V$ . For example, associated with $P(M,\mathrm{GL}(k,\mathbb{R}))$ is a vector bundle over $M$ with fibre ${\mathbb{R}}^k$ . The fibre bundle structure of an associated vector bundle $E=P{\times }_{\rho }V$ is given as follows. The projection ${\pi }_E:E\to M$ is defined by ${\pi }_E(u,v)=\pi (u)$ . This projection is well defined since $\pi (u)=\pi (ug)$ implies ${\pi }_E(ug,\rho (g{)}^{-1}v)=\pi (ug)={\pi }_E(u,v)$ . The local trivialization is given by ${\psi }_i:U_i\times V\to {\pi }_E^{-1}(U_i)$ . The transition function of $E$ is given by $\rho (t_{ij}(p))$ where $t_{ij}(p)$ is that of $P$ .

Conversely a vector bundle naturally induces a principal bundle associated with it. Let $E\stackrel{\pi }{\to }M$ be a vector bundle with $\dim E=k$ (i.e. the fibre is ${\mathbb{R}}^k$ or ${\mathbb{C}}^k$ ). Then $E$ induces a principal bundle $P(E)\equiv P(M,G)$ over $M$ by employing the same transition functions. The structure group $G$ is either $\mathrm{GL}(k,\mathbb{R})$ or $\mathrm{GL}(k,\mathbb{C})$ . Explicit construction of $P(E)$ is carried out following the reconstruction process described in section 9.1.

Example 9.11

Associated with a tangent bundle $TM$ over an $m$ -dimensional manifold $M$ is a principal bundle called the frame bundle $LM\equiv {\bigcup }_{p\in M}{L}_{p}M$ where $L_{p}M$ is the set of frames at $p$ . We introduce coordinates $x^{\mu }$ on a chart $U_i$ . The bundle $T_{p}M$ has a natural basis $\{\partial /\partial x^{\mu }\}$ on $U_i$ . A frame $u=\{X_1,\dots ,X_m\}$ at $p$ is expressed as

$$X_{\alpha }=X_{\alpha }^{\mu }\partial /\partial x^{\mu }{|}_{p}1\le \alpha \le m\text{\hspace{1em}(9.67)}$$

where $(X_{\alpha }^{\mu })$ is an element $\mathrm{GL}(m,\mathbb{R})$ so that $\{X_{\alpha }\}$ are linearly independent. We define the local trivialization ${\varphi }_i:U_i\times \mathrm{GL}(m,\mathbb{R})\to {\pi }^{-1}(U_i)$ by ${\varphi }_i^{-1}(u)=(p,(X_{\alpha }^{\mu }))$ . The bundle structure of $LM$ is defined as follows.

(i) If $u=\{X_1,\dots ,X_m\}$ is a frame at $p$ , we define ${\pi }_L:LM\to M$ by ${\pi }_L(u)=p$ .
(ii) The action of $a=(a_j^i)\in \mathrm{GL}(m,\mathbb{R})$ on the frame $u=\{X_1,\dots ,X_m\}$ is given by $(u,a)\mapsto ua$ , where $ua$ is a new frame at $p$ , defined by

$$Y_{\beta }=X_{\alpha }{a}_{\beta }^{\alpha }.\text{\hspace{1em}(9.68)}$$

Conversely, given any frames $\{X_{\alpha }\}$ and $\{Y_{\beta }\}$ there exists an element of $\mathrm{GL}(m,\mathbb{R})$ such that (9.68) is satisfied. Thus, $\mathrm{GL}(m,\mathbb{R})$ acts on $LM$ transitively.

(iii) Let $U_i$ and $U_j$ be overlapping charts with the coordinates $x^{\mu }$ and $y^{\mu }$ , respectively. For $p\in U_i\cap U_j$ , we have

$$X_{\alpha }=X_{\alpha }^{\mu }\partial /\partial x^{\mu }{|}_p={\tilde{X}}_{\alpha }^{\mu }\partial /\partial y^{\mu }{|}_p\text{\hspace{1em}(9.69)}$$

where $(X_{\alpha }^{\mu })$ $({\tilde{X}}_{\alpha }^{\mu })\in \mathrm{GL}(m,\mathbb{R})$ . Since $X_{\alpha }^{\mu }=(\partial x^{\mu }/\partial y^{\nu }{)}_p{\tilde{X}}_{\alpha }^{\mu }$ , we find the transition function $t_{ij}^L(p)$ to be

$$t_{ij}^L(p)=\left({\left(\partial x^{\mu }/\partial y^{\nu }\right)}_p\right)\in \mathrm{GL}(m,\mathbb{R}).\text{\hspace{1em}(9.70)}$$

Accordingly, given $TM$ , we have constructed a frame bundle $LM$ with the same transition functions.

In general relativity, the right action corresponds to the local Lorentz transformation while the left action corresponds to the general coordinate transformation. It turns out that the frame bundle is the most natural framework in which to incorporate these transformations. If $\{X_{\alpha }\}$ is normalized by introducing a metric, the matrix $(X_{\alpha }^{\mu })$ becomes the vierbein and the structure group reduces to $\mathrm{O}(m)$ ; see section 7.8.

Example 9.12

A spinor field on $M$ is a section of a spin bundle which we now define. Since $\mathrm{GL}(k,\mathbb{R})$ has no spinor representation, we need to introduce an orthonormal frame bundle whose structure group is $\mathrm{SO}(k)$ . As we mentioned in example 4.12, $\mathrm{Spin}(k)$ is the universal covering group of $\mathrm{SO}(k)$ . [To define a spin bundle, we have to check whether the $\mathrm{SO}(k)$ bundle lifts to a $\mathrm{SPIN}(k)$ bundle over $M$ . The obstruction to this lifting is discussed in section 11.6.]

To be specific, let us consider a spin bundle associated with the four-dimensional Lorentz frame bundle $LM$ , where $M$ is a four-dimensional Lorentz manifold. We are interested in a frame with a definite spacetime orientation as well as a time orientation. The structure group is then reduced to

$${\mathrm{O}}_{\uparrow }^{+}(3,1)\equiv \left\{\Lambda \in \mathrm{O}(3,1)\mid \det \Lambda =+1,{\Lambda }_0^0>0\right\}.\text{\hspace{1em}(9.71)}$$

The universal covering group of ${\mathrm{O}}_{\uparrow }^{+}(3,1)$ is $\mathrm{SL}(2,\mathbb{C})$ , see example $5.16(c)$ . The homomorphism $\phi :\mathrm{SL}(2,\mathbb{C})\to O_{\uparrow }^{+}(3,1)$ is a $2:1$ map with $\ker \phi =\{I_2,-I_2\}$ The Weyl spinor is a section of the fibre bundle $(W,\pi ,M,{\mathbb{C}}^2,\mathrm{SL}(2,\mathbb{C}))$ . The Dirac spinor is a section of

$$(D,\pi ,M,{\mathbb{C}}^4,\mathrm{SL}(2,\mathbb{C})\oplus \overline{\mathrm{SL}(2,\mathbb{C})}).\text{\hspace{1em}(9.72)}$$

A section of $W$ is a $(1/2,0)$ representation of ${\mathrm{O}}_{\uparrow }^{+}(3,1)$ and a section of $(\bar{W},\pi ,M,{\mathbb{C}}^2,\overline{\mathrm{SL}(2,\mathbb{C})})$ is a $(0,1/2)$ representation, see Ramond (1989) for example. A Dirac spinor belongs to $(1/2,0)\oplus (0,1/2)$ .

The general structure of the spin bundle will be worked out in section 11.6.

9.4.3 Triviality of bundles

A fibre bundle is trivial if it is expressed as a direct product of the base space and the fibre. The following theorem gives the condition under which a fibre bundle is trivial.

Theorem 9.2

A principal bundle is trivial if and only if it admits a global section.

Proof

Let $(P,\pi ,M,G)$ be a principal bundle over $M$ and let $s\in \Gamma (M,P)$ be a global section. This section may be used to show that there exists a homeomorphism between $P$ and $M\times G$ . If $a$ is an element of $G$ , the product $s(p)a$ belongs to the fibre at $p$ . Since the right action is transitive and free, any element $u\in P$ is uniquely written as $s(p)a$ for some $p\in M$ and $a\in G$ . Define a map $\Phi :P\to M\times G$ by

$$\Phi :s(p)a\mapsto (p,a).\text{\hspace{1em}(9.73)}$$

It is easily verified that $\Phi$ is indeed a homeomorphism and we have shown that $P$ is a trivial bundle $M\times G$ .

Conversely, suppose $P\cong M\times G$ . Let $\varphi :M\times G\to P$ be a trivialization. Take a fixed element $g\in G$ . Then $s_g:M\to P$ defined by $s_g(p)=\varphi (p,g)$ is a global section.

Is there a corresponding theorem for vector bundles? We know that any vector bundle admits a global null section. Thus, we cannot simply replace $P$ by $E$ in theorem 9.2. Let us consider the associated principal bundle $P(E)$ of $E$ . By definition, $E$ and $P(E)$ share the same set of transition functions. Since the twisting of a bundle is described purely by the transition functions, we obtain the following corollary.

Corollary 9.2

A vector bundle $E$ is trivial if and only if its associated principal bundle $P(E)$ admits a global section.

9.1 Let $L$ be the real line bundle over $S^1$ (i.e. $L$ is either the cylinder $S^1\times \mathbb{R}$ or the Möbius strip). Show that the Whitney sum $L\oplus L$ is a trivial bundle. Sketch $L\oplus L$ to confirm the result.

9.2 Let ${\Omega }_n$ be the volume element of $S^n$ normalized as ${\int }_{S^n}{\Omega }_n=1$ . Let $f:S^{2n-1}\to S^n$ be a smooth map and consider the pullback $f^{\ast }{\Omega }_n$ .

(a) Show that $f^{\ast }{\Omega }_n$ is closed and written as $\mathrm{d}{\omega }_{n-1}$ , where ${\omega }_{n-1}$ is an $(n-1)$ -form on $S^{2n-1}$ .
(b) Show that the Hopf invariant

$$H(f)\equiv {\int }_{S^{2n-1}}{\omega }_{n-1}\wedge \mathrm{d}{\omega }_{n-1}$$

is independent of the choice of ${\omega }_{n-1}$

(c) Show that if $f$ is homotopic to $g$ , then $H(f)=H(g)$ .
(d) Show that $H(f)=0$ if $n$ is odd. [Hint: Use ${\omega }_{n-1}\wedge \mathrm{d}{\omega }_{n-1}=\frac{1}{2}\mathrm{d}({\omega }_{n-1}\wedge {\omega }_{n-1})$ .]
(e) Compute the Hopf invariant of the map $\pi :S^3\to S^2$ defined in example 9.9.

Endnotes

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF

来源版本：second-edition-local

来源 PDF SHA-256：fcd24c3371d092a8c699ccf5cf0765462924b6c48412804231076e8084b1940d

OCR 来源 SHA-256：0b6324f01172a3548cd34382e82fd8bfae574602b6b2cfaa357ba96bc2c35478

OCR 产物 SHA-256：2674add18853aa5dad5747cf618769b3a479c92afaaeb33117fe74676a8206b1

Footnotes

1. Octonions are also known as Cayley numbers. The set of octonions is a vector space over $\mathbb{R}$ but not a field. The product is neither commutative nor associative. See John C Baez, The Octonions math.RA/0105155 for a recent review. ↩