Vector, Principal, and Associated Bundles

On a smooth base manifold, locally trivial products assemble into one global bundle when their overlap maps preserve the base point, vary smoothly, and satisfy a cocycle law. A global section is then exactly a compatible family of local representatives. With the convention used throughout this chapter,

$$\boxed{ \begin{aligned} g_{ij}g_{jk}&=g_{ik}, \\ s_i&=g_{ij}\mathbin{\cdot}s_j . \end{aligned} }$$

Linear overlap maps produce vector bundles. A free and transitive right action of a Lie group $G$ on every fiber produces a principal $G$-bundle. A representation $\rho$ of $G$ turns that principal bundle into an associated vector bundle in which matter fields obey $v_i=\rho(g_{ij})v_j$. Different local trivializations change $g_{ij}$ and the local representatives, but not the underlying global object.

This page develops that local-to-global construction and one controlled $U(1)$ patching example. A connection is additional bundle data: its local potentials transform inhomogeneously and are not sections of the charged matter bundle. Connections, curvature, transport, characteristic classes, flux sectors, and gauge dynamics are therefore handed to their dedicated pages.

Required background. Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields supplies smooth maps, charts, tangent and cotangent bundles, and the definition of a section. Those ideas are used here without being reconstructed.

Helpful background. Groups, Actions, Quotients, and Covers supplies group actions and quotient constructions used in the principal-bundle discussion.

Local products and transition functions

The group-action page uses active left actions by default. A principal bundle conventionally carries a right action, so the distinction is stated explicitly below.

Let $M$ be a smooth manifold and let $F$ be a smooth manifold called the typical fiber. A smooth fiber bundle consists of a smooth surjection

$$\pi:E\longrightarrow M$$

and an open cover $\{U_i\}$ of $M$ with fiber-preserving diffeomorphisms

$$\Phi_i:E|_{U_i}\longrightarrow U_i\times F, \qquad E|_{U_i}:=\pi^{-1}(U_i),$$

such that

$$\operatorname{pr}_1\circ\Phi_i=\pi .$$

Thus every restriction $E|_{U_i}$ is a product, but no single product description is assumed over all of $M$. Its transition diffeomorphism $t_{ij}(x):F\to F$ is defined by

$$\Phi_i\circ\Phi_j^{-1}(x,f) = \bigl(x,t_{ij}(x)f\bigr).$$

These induced maps obey the fiber-bundle cocycle in $\operatorname{Diff}(F)$:

$$\begin{aligned} t_{ii}(x)&=\operatorname{id}_F, \\ t_{ji}(x)&=t_{ij}(x)^{-1}, \\ t_{ij}(x)\circ t_{jk}(x)&=t_{ik}(x). \end{aligned}$$

For the bundles used in field theory, one usually chooses a structure group: a Lie group $G$ acting smoothly on $F$ from the left. A $G$-valued atlas includes smooth lifts $g_{ij}:U_i\cap U_j\to G$ such that

$$t_{ij}(x)f=g_{ij}(x)\mathbin{\cdot}f$$

and requires the stricter $G$-valued identities

$$\begin{aligned} g_{ii}(x)&=e, \\ g_{ji}(x)&=g_{ij}(x)^{-1}, \\ g_{ij}(x)g_{jk}(x)&=g_{ik}(x). \end{aligned}$$

The index order is important: $g_{ij}$ converts the $j$-description of a fiber point into its $i$-description. If the $G$-action on $F$ is faithful, the lifts are uniquely determined by $t_{ij}$ and their group identities follow from the diffeomorphism cocycle. If the action has a kernel, the induced maps determine the $g_{ij}$ only modulo that kernel; coherent lifts are then extra data in the chosen $G$-valued atlas. This distinction will matter when a nonfaithful representation forgets some principal-bundle data.

The diffeomorphism cocycle is not optional bookkeeping: without it, the local products do not identify points transitively. Nakahara 2003, §§ 9.2–9.4, develops this convention from general fiber bundles through principal and associated bundles; Husemoller 1994, Chapters 1–4, pp. 11–72 provides a structural treatment independent of a chosen physical example.

Reconstructing the global bundle

Conversely, suppose $M$, $\{U_i\}$, $F$, the $G$-action, and smooth maps $g_{ij}$ satisfying the three displayed laws are given. Start from the disjoint union

$$X=\bigsqcup_i(U_i\times F)$$

and impose

$$(x,f)_j \sim \bigl(x,g_{ij}(x)\mathbin{\cdot}f\bigr)_i .$$

Identity makes this relation reflexive, the inverse law makes it symmetric, and the triple-overlap law makes it transitive. The quotient

$$E=X/{\sim}, \qquad \pi\bigl([(x,f)_i]\bigr)=x,$$

inherits the required local trivializations. Hence compatible $G$-valued transition data determine a bundle. Every locally trivial bundle produces the diffeomorphism-valued maps $t_{ij}$; a bundle equipped with a chosen $G$-valued atlas also produces the coherent lifts $g_{ij}$.

This statement is deliberately not a complete classification theorem. Changing the cover, refining it, or changing local trivializations changes the displayed cocycle. Isomorphic bundles can therefore have different transition functions. In particular, merely seeing a nonidentity $g_{ij}$ on one cover does not prove that a bundle is nontrivial.

A useful near miss is a collection of three local products for which, at some $x$ and $f$,

$$\bigl(g_{ij}(x)g_{jk}(x)\bigr)\mathbin{\cdot}f \ne g_{ik}(x)\mathbin{\cdot}f .$$

Equivalently, $t_{ij}(x)t_{jk}(x)\ne t_{ik}(x)$ as diffeomorphisms of $F$. Passing from the $k$-coordinates to the $i$-coordinates directly then gives a different fiber point from passing through the $j$-coordinates, so no underlying fiber bundle results. If the two group products differ only by an element in the action kernel, the induced diffeomorphisms still agree and the underlying $F$-bundle can glue, but those particular lifts do not form the declared coherent $G$-valued atlas.

Likewise, a smooth surjection whose fiber dimension or diffeomorphism type changes from point to point is generally not a fiber bundle: local product structure with one fixed typical fiber is missing.

The product bundle $M\times F$ is the basic valid example. A bundle is trivial when it is isomorphic over $M$ to this product, equivalently when some choice of trivializations makes every transition function the identity. Local triviality is part of every bundle definition; global triviality is a stronger property.

Sections are compatible local representatives

A smooth section is a map

$$s:M\longrightarrow E, \qquad \pi\circ s=\operatorname{id}_M.$$

Its representative in the $i$th trivialization is the smooth map $s_i:U_i\to F$ defined by

$$\Phi_i(s(x))=(x,s_i(x)).$$

On an overlap, the same point $s(x)\in E_x$ must have compatible coordinates:

$$s_i(x)=g_{ij}(x)\mathbin{\cdot}s_j(x).$$

Conversely, any family $\{s_i\}$ obeying this law glues to one global section. A local representative is therefore not an independently defined field, and a global field need not be one $F$-valued function on all of $M$.

Every trivializing patch has local sections. A global section may fail to exist, and when it exists its implications depend on the type of bundle: every vector bundle has a zero section, whereas one global section of a principal bundle trivializes it.

What changes when the presentation changes

Let $h_i:U_i\to G$ be smooth. Define a new local trivialization by declaring that if

$$\Phi_i(e)=(x,f_i),$$

then

$$\Phi_i'(e) = \bigl(x,h_i(x)^{-1}\mathbin{\cdot}f_i\bigr).$$

The same bundle and the same section now have representatives

$$\begin{aligned} g_{ij}'&=h_i^{-1}g_{ij}h_j, \\ s_i'&=h_i^{-1}\mathbin{\cdot}s_i . \end{aligned}$$

A round trip verifies compatibility:

$$\begin{aligned} g_{ij}'\mathbin{\cdot}s_j' &= \bigl(h_i^{-1}g_{ij}h_j\bigr) \mathbin{\cdot} \bigl(h_j^{-1}\mathbin{\cdot}s_j\bigr) \\ &= h_i^{-1}\mathbin{\cdot} \bigl(g_{ij}\mathbin{\cdot}s_j\bigr) =s_i'. \end{aligned}$$

Thus transition functions and local components are presentation-dependent, whereas the bundle and section are invariant. On a fixed common cover, cocycles related by this formula describe isomorphic bundles. A full classification also has to account for cover refinement and the topology of $M$ and $G$.

This fiber-coordinate change is distinct from a change of coordinates on the base. A base chart transition changes the numbers used to describe $x\in M$; a fiber transition changes the numbers used to describe an element of $E_x$. For the tangent bundle the latter is derived from the Jacobian of the former, but for a general internal bundle its transition data are additional geometric input.

A bundle map makes this base–fiber relation explicit. A smooth map $F_E:E\to E'$ covers $f:M\to M'$ when

$$\pi'\circ F_E=f\circ\pi.$$

For vector or principal bundles it must also respect the relevant linear or group action. An arbitrary smooth map between total spaces need not be a bundle map.

Vector bundles and local frames

Let $\mathbb K$ be $\mathbb R$ or $\mathbb C$. A rank-$r$ vector bundle is a fiber bundle whose fibers are $r$-dimensional $\mathbb K$-vector spaces, whose local trivializations are linear on each fiber, and whose transition functions take values in

$$GL(r,\mathbb K).$$

Write $V=\mathbb K^r$. A trivialization determines a local frame, viewed as a linear isomorphism

$$e_i(x):V\longrightarrow E_x, \qquad e_i(x)v:=\Phi_i^{-1}(x,v).$$

On an overlap,

$$e_j=e_i\circ g_{ij}, \qquad s=e_i s_i=e_j s_j, \qquad s_i=g_{ij}s_j.$$

The frame and the component column transform oppositely. Under the presentation change above,

$$e_i'=e_i\circ h_i, \qquad s_i'=h_i^{-1}s_i,$$

so the vector $s(x)\in E_x$ is unchanged.

Sections of a vector bundle can be added and multiplied by smooth functions pointwise. The section space $\Gamma(E)$ is therefore a $C^\infty(M)$-module, not usually a finite-dimensional vector space. Every vector bundle has the distinguished zero section

$$0_E(x)=0\in E_x.$$

Consequently, the existence of one global section says nothing by itself about vector-bundle triviality. A rank-$r$ vector bundle is trivial exactly when it has a global frame: $r$ global sections that are linearly independent at every point. Lee 2013, Chapter 10, pp. 249–268 gives a systematic graduate-level treatment of vector bundles, local and global sections, and bundle homomorphisms.

Tangent components are the first example

For two coordinate systems $x_i^\mu$ and $x_j^\nu$ on an overlap, a tangent vector satisfies

$$\begin{aligned} V &= V_i^\mu\frac{\partial}{\partial x_i^\mu} = V_j^\nu\frac{\partial}{\partial x_j^\nu}, \\ V_i^\mu &= \frac{\partial x_i^\mu}{\partial x_j^\nu} V_j^\nu . \end{aligned}$$

Thus the tangent-bundle transition matrix is

$$(g_{ij})^\mu{}_\nu = \frac{\partial x_i^\mu}{\partial x_j^\nu}.$$

The base-coordinate transition is the nonlinear map $x_i\circ x_j^{-1}$; the fiber transition is its derivative. Conflating the two obscures why a coordinate component is not itself a geometric vector. Cotangent components use the inverse-transpose representation, and tensor bundles use the corresponding tensor representations.

The Möbius line bundle

Let the circle be $S^1=[0,2\pi]/(0\sim2\pi)$. The quotient

$$L_{\mathrm M} = \bigl([0,2\pi]\times\mathbb R\bigr) \Big/ \bigl((2\pi,v)\sim(0,-v)\bigr)$$

is a real line bundle. Locally it is indistinguishable from $U\times\mathbb R$, but going once around the base reverses the fiber coordinate.

A smooth section is represented by a smooth function $f$ whose endpoint jets obey the quotient compatibility conditions and which, in particular, satisfies

$$f(2\pi)=-f(0).$$

This necessary endpoint condition already gives the obstruction. If $f$ were nowhere zero, continuity would force it to keep one sign on the connected interval, contradicting the displayed equality. Hence every section vanishes somewhere. A nowhere-zero section would be a global frame for a real line bundle, so $L_{\mathrm M}$ is not trivial. Nevertheless its zero section exists. This example cleanly separates

$$\text{“has a section”} \quad\text{from}\quad \text{“has a global frame.”}$$

On a two-arc cover of $S^1$, the same twisting appears as transition $+1$ on one connected component of the overlap and $-1$ on the other. Smoothness is not violated because those components are disjoint.

Principal bundles are bundles of frames without an origin

A principal $G$-bundle is a smooth bundle

$$\pi:P\longrightarrow M$$

equipped with a smooth right action

$$P\times G\longrightarrow P, \qquad (p,g)\longmapsto p\mathbin{\cdot}g,$$

that preserves the base point and is free and transitive on each fiber:

$$\pi(p\mathbin{\cdot}g)=\pi(p),$$

and, for $p,p'\in P_x$, there is a unique $g\in G$ such that $p'=p\mathbin{\cdot}g$. The local trivializations are right-equivariant: if $\Phi_i(p)=(x,a)$, then

$$\Phi_i(p\mathbin{\cdot}g)=(x,ag).$$

Each principal fiber is therefore a $G$-torsor. It looks like $G$ after one point has been chosen as an origin, but it has no preferred identity and no intrinsic multiplication of two fiber points. Calling the fiber “canonically the group $G$” loses precisely the information that local sections choose.

Define the canonical local section associated with $\Phi_i$ by

$$\sigma_i(x)=\Phi_i^{-1}(x,e).$$

Every $p\in P_x$ is uniquely $p=\sigma_i(x)\mathbin{\cdot}a_i$, and the transition convention gives

$$\sigma_j=\sigma_i\mathbin{\cdot}g_{ij}, \qquad a_i=g_{ij}a_j.$$

The transition functions multiply the local $G$-coordinate on the left, whereas the intrinsic principal action multiplies it on the right. These operations commute, which is why the right action is independent of the chosen trivialization.

If $\sigma:M\to P$ is a global section, then

$$M\times G\longrightarrow P, \qquad (x,g)\longmapsto\sigma(x)\mathbin{\cdot}g$$

is a global principal-bundle trivialization. Conversely, evaluating a global trivialization at $(x,e)$ gives a global section. Therefore

$$P\ \text{is trivial} \quad\Longleftrightarrow\quad P\ \text{has a global section}.$$

This theorem must not be copied to arbitrary vector-bundle sections, because the zero section would make every vector bundle trivial. Nakahara 2003, § 9.4.3, p. 372, proves the principal-bundle criterion and explains this vector-bundle contrast.

Frame bundles

Every rank-$r$ vector bundle $E\to M$ has a principal frame bundle

$$\operatorname{Fr}(E)_x = \operatorname{Iso}_{\mathbb K}(V,E_x), \qquad V=\mathbb K^r.$$

Its points are ordered frames, represented as linear isomorphisms $u:V\to E_x$. The group $GL(V)$ acts on the right by

$$u\mathbin{\cdot}g=u\circ g.$$

This action is free and transitive on each frame fiber. A local frame of $E$ is exactly a local section of $\operatorname{Fr}(E)$, and a global frame is exactly a global section, reproducing the vector-bundle triviality criterion.

For $E=TM$, this gives the full frame bundle $\operatorname{Fr}(M)$. No metric is required. An orthonormal frame bundle is a later reduction of structure group and does require a metric. Frankel 2012, §§ 17.1 and 18.2, independently checks the distinction between left transition maps, the intrinsic right principal action, and associated bundles.

Associated bundles turn frames into fields

Let $P\to M$ be a principal right $G$-bundle and let $G$ act on a space $F$ from the left. Define a right action on $P\times F$ by

$$(p,f)\mathbin{\cdot}g = \bigl(p\mathbin{\cdot}g,g^{-1}\mathbin{\cdot}f\bigr).$$

The inverse is forced by the action law:

$$\bigl((p,f)\mathbin{\cdot}g_1\bigr)\mathbin{\cdot}g_2 = (p,f)\mathbin{\cdot}(g_1g_2).$$

The quotient

$$P\times_G F=(P\times F)/G$$

is the associated fiber bundle. For a representation

$$\rho:G\longrightarrow GL(V),$$

the associated vector bundle is

$$\boxed{ E_\rho=P\times_\rho V = (P\times V) \Big/ \bigl((p,v)\sim (p\mathbin{\cdot}g,\rho(g^{-1})v)\bigr). }$$

Write an equivalence class as $[p,v]$. An equivalent and often more useful identity is

$$[p\mathbin{\cdot}g,v]=[p,\rho(g)v].$$

Indeed,

$$\bigl(p,\rho(g)v\bigr)\mathbin{\cdot}g = \bigl(p\mathbin{\cdot}g,v\bigr).$$

Now use the local principal sections $\sigma_j=\sigma_i\mathbin{\cdot}g_{ij}$. If a section of $E_\rho$ has local representatives $v_i$ defined by

$$s(x)=[\sigma_i(x),v_i(x)],$$

then on an overlap

$$\begin{aligned} [\sigma_j,v_j] &= [\sigma_i\mathbin{\cdot}g_{ij},v_j] \\ &= [\sigma_i,\rho(g_{ij})v_j]. \end{aligned}$$

Therefore

$$v_i=\rho(g_{ij})v_j,$$

which is the chapter-wide overlap convention announced at the beginning. The principal bundle supplies the transition functions; the representation specifies how a particular kind of matter field responds to them.

Changing local principal sections

Replace the local sections by

$$\sigma_i'=\sigma_i\mathbin{\cdot}h_i, \qquad h_i:U_i\to G.$$

Then

$$\begin{aligned} \sigma_j' &= \sigma_j\mathbin{\cdot}h_j \\ &= \sigma_i\mathbin{\cdot}g_{ij}h_j \\ &= \sigma_i'\mathbin{\cdot} \bigl(h_i^{-1}g_{ij}h_j\bigr), \end{aligned}$$

so the transition functions and matter representatives become

$$\begin{aligned} g_{ij}'&=h_i^{-1}g_{ij}h_j, \\ v_i'&=\rho(h_i^{-1})v_i . \end{aligned}$$

The overlap law survives:

$$\begin{aligned} \rho(g_{ij}')v_j' &= \rho(h_i^{-1}g_{ij}h_j) \rho(h_j^{-1})v_j \\ &= \rho(h_i^{-1}) \rho(g_{ij})v_j =v_i'. \end{aligned}$$

This is a passive change of local presentation. It is related to, but is not by itself the same as, an active gauge automorphism of a fixed bundle. An active gauge transformation is a globally defined, right-equivariant automorphism of $P$ over $\operatorname{id}_M$; its local functions must obey their own compatibility law. Arbitrary choices of local sections simply redescribe the same bundle.

Sections as equivariant functions

A section of $E_\rho$ can equivalently be represented by a smooth function

$$\widetilde s:P\longrightarrow V$$

obeying

$$\widetilde s(p\mathbin{\cdot}g) = \rho(g^{-1})\widetilde s(p).$$

To derive the inverse, write $s(x)=[p,\widetilde s(p)]$. Replacing $p$ by $p\mathbin{\cdot}g$ must give the same class:

$$\begin{aligned} [p\mathbin{\cdot}g,\widetilde s(p\mathbin{\cdot}g)] &= [p,\rho(g)\widetilde s(p\mathbin{\cdot}g)] \\ &= [p,\widetilde s(p)]. \end{aligned}$$

This forces the displayed equivariance law. For the trivial representation it reduces to ordinary invariance. For general $\rho$, a particular equivariant function can still be invariant when its image lies in the fixed subspace

$$V^G=\{v\in V:\rho(g)v=v\ \text{for every }g\in G\}.$$

Reconstructing a vector bundle from its frames

The three bundle types now meet in one canonical isomorphism:

$$E \simeq \operatorname{Fr}(E)\times_{GL(V)}V, \qquad [u,v]\longmapsto u(v).$$

It is well defined because

$$[u\mathbin{\cdot}g,v]=[u,gv]$$

and both representatives map to $u(gv)$. In particular,

$$TM \simeq \operatorname{Fr}(M) \times_{GL(n,\mathbb R)} \mathbb R^n.$$

One principal bundle can generate many associated bundles, one for each representation. The representation is genuine extra data. A nonfaithful representation can forget part of the principal-bundle transition information; the trivial representation, for example, produces untwisted local matter even when $P$ itself is nontrivial.

Spacetime frames and internal frames are different

The same abstract language applies to tangent and internal bundles, but their geometric roles are not interchangeable.

Object	Transition data	Local representative
$TM$ or $T^*M$	Jacobians or dual Jacobians induced by base coordinates	Tensor components
$\operatorname{Fr}(M)$	Changes of spacetime frame	A chosen frame, not a matter field
$P\to M$	Independent principal $G$-bundle data	A local principal section
$E_\rho=P\times_\rho V$	$\rho(g_{ij})$ induced from $P$	Matter components $v_i$

A spacetime metric, together with the requirement of vanishing torsion, may later select its Levi–Civita connection on the tangent bundle. It does not select an internal principal bundle or an internal gauge connection.

For example, a charged covector field is a section of

$$T^*M\otimes E_\rho.$$

On an overlap its local components transform as

$$\Psi_{i,\mu} = \frac{\partial x_j^\nu}{\partial x_i^\mu} \rho(g_{ij})\Psi_{j,\nu}.$$

The Jacobian acts on the spacetime covector index, while $\rho(g_{ij})$ acts on the internal index. These factors commute because they act on different tensor factors. The full frame bundle is therefore not an internal gauge bundle merely because both are principal bundles.

A connection is additional data on either branch. The symbol $\nabla$ usually denotes a connection acting on tangent or tensor indices, while $D$ denotes an internal gauge-covariant derivative. A field with both kinds of index may need both.

A bounded QFT bridge: charged fields on two patches

Cover the sphere by

$$U_N=S^2\setminus\{\text{south pole}\}, \qquad U_S=S^2\setminus\{\text{north pole}\}.$$

Their overlap retracts to the equatorial circle. Let $\varphi$ denote its angular coordinate modulo $2\pi$, and choose the principal $U(1)$ transition function

$$g_{NS}(\varphi)=e^{in\varphi}, \qquad n\in\mathbb Z.$$

The real-valued angle $\varphi$ is not one globally defined function on the cylindrical overlap. Its exponential $e^{in\varphi}$ is well defined exactly when $n$ is an integer, and the one-form $\mathrm d\varphi$ is well defined there.

For $n\ne0$, this transition cannot be removed by changing trivializations that extend smoothly over both hemispherical discs. If $g_{NS}=h_Nh_S^{-1}$ on the equator, the restrictions of both $h_N$ and $h_S$ would extend across discs and hence have zero boundary winding; their ratio would also have zero winding. But $e^{in\varphi}$ winds $n$ times. This proves nontriviality for this two-patch construction without claiming a classification of all $U(1)$ bundles. Nakahara 2003, Example 9.7 in § 9.4.1, pp. 364–365, gives the same transition-winding construction.

Choose the integer-weight representation

$$\rho_q(e^{i\alpha})=e^{iq\alpha}, \qquad q\in\mathbb Z\setminus\{0\}.$$

Any compatible pair of charged-field representatives defines one global section of the associated complex line bundle. Its overlap law is

$$\boxed{ \psi_N = e^{iqn\varphi}\psi_S. }$$

After one turn around the equator, the multiplier changes by $e^{2\pi iqn}=1$. The compatible pair $(\psi_N,\psi_S)$ therefore describes a single global section even though neither member is a global complex-valued function on $S^2$. Different physical charge units can be accommodated by changing the normalization of the generator; here the nonzero integer $q$ is the weight in the chosen global $2\pi$-periodic $U(1)$ normalization. The neutral case $q=0$ is the trivial representation and does not test potential patching through matter covariance.

The gauge potential is a different kind of local object. As a preview of the next page, align notation with the site’s Hermitian convention $D=\mathrm d-i g_{\mathrm{YM}}A^aT^a$. For $U(1)$, let $A_i$ be the real coefficient of its generator, let that generator act with weight $q$, and define the coupling-absorbed real potential

$$a_i:=g_{\mathrm{YM}}A_i.$$

Then the local covariant derivative is

$$D_i=\mathrm d-iq\,a_i.$$

If $\chi_{NS}=n\varphi$ locally, covariance of the charged-field overlap requires

$$a_N=a_S+\mathrm d\chi_{NS} = a_S+n\,\mathrm d\varphi .$$

Indeed,

$$\begin{aligned} D_N\psi_N &= \bigl(\mathrm d-iq(a_S+\mathrm d\chi_{NS})\bigr) \bigl(e^{iq\chi_{NS}}\psi_S\bigr) \\ &= e^{iq\chi_{NS}} \bigl(\mathrm d-iq a_S\bigr)\psi_S \\ &= e^{iq\chi_{NS}}D_S\psi_S. \end{aligned}$$

The inhomogeneous $\mathrm d\chi_{NS}$ term shows why $a_i$ is not a local representative of a section of $E_{\rho_q}$. It is local connection data. Tong 2018, § 1.1.2, pp. 6–8, PDF derives the two-patch monopole potentials and the charged-field phase in physical normalization. Nakahara 2003, § 10.5.2, equations (10.90)–(10.91), p. 400, gives an independent bundle treatment. Tong writes $\psi\mapsto e^{ie\omega/\hbar}\psi$ and $A_N=A_S+\mathrm d\omega$. Choosing a basic charge $e_0$, setting

$$a=\frac{e_0}{\hbar}A, \qquad \chi=\frac{e_0}{\hbar}\omega,$$

and taking the field’s physical charge to be $q e_0$ gives the dimensionless formulas above. Tong’s condition $e_0g_{\mathrm m}=2\pi\hbar n$ for magnetic charge $g_{\mathrm m}$ then gives $\chi_{NS}=n\varphi$. Nakahara writes its anti-Hermitian connection as

$$\mathcal A_{\mathrm{Nak}}=+i A_{\mathrm{Nak}},$$

whereas the site’s Hermitian convention corresponds to

$$\mathcal A_{\mathrm{site}}=-i a.$$

After the normalization above, set $a=A_{\mathrm{Nak}}$ and hence $\mathcal A_{\mathrm{site}}=-\mathcal A_{\mathrm{Nak}}$. Nakahara’s real coefficient law $A_{\mathrm{Nak},N}=A_{\mathrm{Nak},S}+\mathrm d\chi$ then becomes exactly $a_N=a_S+\mathrm d\chi$. This sign map is a translation between source conventions, not an additional gauge transformation.

This page stops at the compatibility check. The construction and transformation of connections, covariant derivatives, curvature, and the Bianchi identity belong to Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities. Parallel Transport and Holonomy develops transport along curves and around loops. Local Potentials and Global Gauge Configurations develops the physical treatment of patchwise gauge configurations, bundle sectors, flux quantization, and global gauge observables.

Common pitfalls

Reversing only one overlap convention. If the transition order is changed from $g_{ij}g_{jk}=g_{ik}$ to the opposite convention, the local field law must be inverted at the same time. Keeping one formula and reversing the other makes triple-overlap patching inconsistent.

Calling every locally trivial bundle a product. Local products are part of the definition. A global product requires a compatible global trivialization, which can be obstructed by transition data such as the Möbius sign or the $U(1)$ winding.

Treating nonidentity transitions as proof of twisting. A change of local trivialization can create nonidentity transition functions even for a product bundle. Nontriviality requires showing that no compatible change removes them.

Confusing a global section with a global frame. Every vector bundle has the zero section; a rank-$r$ vector bundle needs $r$ pointwise independent global sections to be trivial. One global section does trivialize a principal bundle because the free transitive right action generates every fiber from it.

Giving a principal fiber a preferred identity. A principal fiber is a $G$-torsor. A local section chooses an origin locally; changing that choice changes the local $G$-coordinate.

Dropping the inverse in the associated quotient. For a principal right action and a left representation, the diagonal right action is $(p,v)\mathbin{\cdot}g=(p\mathbin{\cdot}g,\rho(g^{-1})v)$. Without the inverse, the action law fails for a non-Abelian group.

Calling a change of local section an active gauge transformation. The formula $\sigma_i'=\sigma_i h_i$ changes the local presentation. An active gauge transformation is a globally compatible principal-bundle automorphism; related local formulas do not erase this distinction.

Identifying a gauge potential with a charged field. Matter representatives transform homogeneously by $\rho(g_{ij})$. Local connection forms have an additional derivative term. They belong to different geometric objects.

Identifying internal and spacetime bundles. A Jacobian acts on a tangent or cotangent index. An internal transition function acts through $\rho$ on an independent fiber. Equal ranks do not make the bundles the same.

Exercises

These checks test the cocycle, the two triviality criteria, the associated quotient, and the controlled QFT transfer.

Cocycle retrieval. Assuming the $G$-action on $F$ is faithful, start from $\Phi_i\Phi_j^{-1}(x,f)=(x,g_{ij}\mathbin{\cdot}f)$ and derive the triple-overlap law and the local section law. What changes for a nonfaithful action?

Cocycle answer

On a triple overlap,

$$\Phi_i\Phi_k^{-1} = \Phi_i\Phi_j^{-1}\Phi_j\Phi_k^{-1},$$

so acting on $f$ gives

$$g_{ik}\mathbin{\cdot}f = (g_{ij}g_{jk})\mathbin{\cdot}f.$$

Faithfulness then implies $g_{ij}g_{jk}=g_{ik}$. If $\Phi_j(s(x))=(x,s_j(x))$, applying $\Phi_i\Phi_j^{-1}$ gives

$$\Phi_i(s(x)) = (x,g_{ij}\mathbin{\cdot}s_j(x)),$$

so $s_i=g_{ij}\mathbin{\cdot}s_j$, or $s_i=\rho(g_{ij})s_j$ in an associated vector bundle.

For a nonfaithful action, composition proves equality of the induced maps on $F$, namely $t_{ij}t_{jk}=t_{ik}$, not uniqueness of the elements of $G$. The chosen $G$-valued lifts are therefore included in the $G$-valued atlas and required to satisfy the stricter group-valued cocycle law.

Möbius obstruction. Why does the zero section of $L_{\mathrm M}$ not trivialize it, and why would a nowhere-zero section do so?

Möbius answer

A rank-one vector bundle is trivial exactly when it has one nowhere-zero section, because that section supplies a basis in every fiber. A section of $L_{\mathrm M}$ is represented by $f:[0,2\pi]\to\mathbb R$ with $f(2\pi)=-f(0)$. If $f$ never vanished, continuity on the connected interval would keep its sign fixed, contradicting the endpoint relation. The zero section satisfies the relation but supplies no fiber basis.

Associated-bundle derivation. Starting from

$$(p,v)\mathbin{\cdot}g = (p\mathbin{\cdot}g,\rho(g^{-1})v),$$

derive the useful class identity, the equivariance law, and the passive change-of-presentation formulas.

Associated-bundle answer

Acting on $(p,\rho(g)v)$ by $g$ gives $(p\mathbin{\cdot}g,v)$, so

$$[p\mathbin{\cdot}g,v]=[p,\rho(g)v].$$

Requiring $[p,\widetilde s(p)] =[p\mathbin{\cdot}g,\widetilde s(p\mathbin{\cdot}g)]$ then gives

$$\widetilde s(p\mathbin{\cdot}g) = \rho(g^{-1})\widetilde s(p).$$

Finally, if $\sigma_i'=\sigma_i\mathbin{\cdot}h_i$, then

$$g_{ij}'=h_i^{-1}g_{ij}h_j, \qquad v_i'=\rho(h_i^{-1})v_i.$$

Substitution verifies $v_i'=\rho(g_{ij}')v_j'$.

QFT transfer. For $g_{NS}=e^{in\varphi}$ and $\rho_q(e^{i\alpha})=e^{iq\alpha}$ with $q\ne0$, find the charged-field overlap, check single-valuedness, and determine the potential overlap required by $D=\mathrm d-iq\,a$.

QFT-transfer answer

The associated representation gives

$$\psi_N=e^{iqn\varphi}\psi_S.$$

After $\varphi\mapsto\varphi+2\pi$, the multiplier gains $e^{2\pi iqn}=1$ because $q,n\in\mathbb Z$. Writing $\chi_{NS}=n\varphi$, covariance requires

$$a_N=a_S+\mathrm d\chi_{NS}.$$

Then

$$D_N\psi_N = e^{iq\chi_{NS}}D_S\psi_S.$$

The homogeneous phase identifies $\psi$ as an associated-bundle section; the derivative term identifies $a_i$ as local connection data.

Synthesis and next steps

A fiber bundle is a global space reconstructed from local products and transition functions satisfying identity, inverse, and cocycle laws. A global section is the compatible family $s_i=g_{ij}\mathbin{\cdot}s_j$. Vector bundles make the fibers linear and are trivialized by global frames. Principal bundles replace a fiber origin by a free transitive right $G$-action and are trivialized by one global principal section. A representation then forms $P\times_\rho V$, where local matter fields obey $v_i=\rho(g_{ij})v_j$. The frame-bundle reconstruction $E\simeq\operatorname{Fr}(E)\times_{GL(V)}V$ ties the three constructions together.

Continue according to the additional structure required:

• Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities adds connections, covariant derivatives, local potential laws, curvature, and Bianchi identities;
• Parallel Transport and Holonomy develops transport along curves and holonomy around loops;
• Local Potentials and Global Gauge Configurations develops the physical gauge-patching example, bundle sectors, flux quantization, and global observables;
• Spin Structures and Dirac Operators uses lifted principal bundles and associated spinor bundles after its additional prerequisites.

References

• Theodore Frankel, The Geometry of Physics: An Introduction, third edition, Cambridge University Press, 2012, § 16.1, pp. 415–419, § 17.1, pp. 451–455, and § 18.2a, pp. 481–483. These sections treat vector cocycles, principal and frame bundles, and associated bundles.
• Dale Husemoller, Fibre Bundles, third edition, Graduate Texts in Mathematics 20, Springer, 1994, “The General Theory of Fibre Bundles”: “Generalities on Bundles,” pp. 11–23; “Vector Bundles,” pp. 24–39; “General Fibre Bundles,” pp. 40–60; and “Local Coordinate Description of Fibre Bundles,” pp. 61–72. These sections develop the local-to-global construction and its invariant meaning.
• John M. Lee, Introduction to Smooth Manifolds, second edition, Graduate Texts in Mathematics 218, Springer, 2013, Chapter 10, pp. 249–268. This supplies the independent smooth-manifold treatment of vector bundles, local and global sections, and bundle homomorphisms.
• Mikio Nakahara, Geometry, Topology and Physics, second edition, Institute of Physics Publishing, 2003, §§ 9.2.1–9.4.3, pp. 350–372, and § 10.5.2, pp. 400–401. These sections develop bundle reconstruction, vector and principal bundles, associated bundles, and the two-patch monopole application.
• David Tong, Gauge Theory — Open PDF, Cambridge Part III lecture notes, 2018, § 1.1.2, pp. 6–8. This supplies the monopole patches, overlap gauge transformation, charged-field phase, and single-valuedness condition.