Principal Bundles and Associated Vector Bundles

Vector bundles are enough for tangent bundles, cotangent bundles, and Bloch bundles. Gauge theory is cleaner if one first separates the bundle of gauge frames from the vector spaces on which matter fields live. This separation is the point of principal bundles and associated vector bundles.

Principal bundles

Let $G$ be a topological group or Lie group. A principal $G$-bundle is a fiber bundle whose fiber is $G$ itself, but with one crucial extra structure: $G$ acts on the fiber in a way compatible with all local product descriptions.

The last condition is often summarized by saying that the trivializations are $G$-equivariant. Written out explicitly, the local model is

$$((x,h),g)\mapsto (x,hg).$$

The base point $x$ is not moved. Only the group coordinate in the fiber is multiplied on the right.

Local sections and transition functions

A local section

$$s_i:U_i\to P$$

of a principal bundle is a local choice of gauge frame. Given a $G$-equivariant trivialization $\phi_i$, the corresponding section is

$$s_i(x)=\phi_i^{-1}(x,e).$$

Every point $p\in P_x$ over $x\in U_i$ can be uniquely written as

$$p=s_i(x)\cdot h$$

for a unique $h\in G$.

On an overlap $U_i\cap U_j$, two local sections are related by a unique smooth map

$$g_{ij}:U_i\cap U_j\to G$$

such that

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

These $g_{ij}$ are the transition functions of the principal bundle. In terms of local trivializations, if

$$\phi_j(p)=(x,h),$$

then

$$p=s_j(x)\cdot h=s_i(x)\cdot g_{ij}(x)h,$$

so

$$\boxed{ (\phi_i\circ\phi_j^{-1})(x,h)=(x,g_{ij}(x)h). }$$

On triple overlaps, consistency gives the cocycle condition

$$\boxed{g_{ij}(x)g_{jk}(x)=g_{ik}(x).}$$

This is the principal-bundle version of patch-gluing consistency.

Associated vector bundles

Matter fields usually transform in a representation of the gauge group. The principal bundle contains the gauge frames; a representation tells us what vector space a matter field lives in.

Let $P\to M$ be a principal $G$-bundle and let

$$\rho:G\to \mathop{\mathrm{GL}}(V)$$

be a representation of $G$ on a real or complex vector space $V$.

The word quotient means that we start with all pairs $(p,v)$ and then declare pairs related by the above rule to represent the same point of $E$. The inverse in $\rho(g^{-1})$ is not decorative: it ensures that changing the frame and transforming the coordinate vector in the opposite way leaves the actual geometric vector unchanged.

Choose a local section $s_i:U_i\to P$. Then every element of $E_x$ for $x\in U_i$ can be written as

$$[s_i(x),v_i]$$

for a unique $v_i\in V$. If $s_j=s_i\cdot g_{ij}$, then

$$[s_j(x),v_j] = [s_i(x)\cdot g_{ij}(x),v_j] = [s_i(x),\rho(g_{ij}(x))v_j].$$

Thus local component vectors obey

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

This is exactly the vector-bundle transition rule.

Connections: Why Gauge Fields Exist

The central problem is simple but fundamental. A section of a vector bundle assigns a vector to each point, but those vectors live in different vector spaces. If $s\in\Gamma(E)$ and $p,q\in M$, then

$$s(p)\in E_p, \qquad s(q)\in E_q.$$

Unless the bundle has been trivialized, the expression

$$s(q)-s(p)$$

is not defined: it tries to subtract vectors in different vector spaces. A connection is the extra structure that makes differentiation of sections meaningful.

The conceptual move

In ordinary calculus, one defines a derivative by subtracting nearby values and taking a limit. For a vector bundle this is not legal until one has a way to compare nearby fibers. One could try to define parallel transport first and then define differentiation, but parallel transport itself needs a rule for what it means to remain parallel. The clean mathematical solution is to define the differentiating machine directly.

This is not a trick. Once $\nabla$ is specified, it produces covariant derivatives, parallel transport, curvature, and holonomy. In a local frame it becomes the familiar expression

$$\nabla=d+A,$$

where $A$ is the gauge potential or connection one-form.

$E$-valued one-forms

Before defining a connection, decode the target space of $\nabla$.

Let $E\to M$ be a vector bundle. The tensor product bundle

$$T^*M\otimes E\to M$$

has fiber

$$(T^*M\otimes E)_p=T_p^*M\otimes E_p.$$

A section

$$\eta\in\Gamma(T^*M\otimes E)$$

is called an $E$-valued one-form. At each $p$, it is equivalently a linear map

$$\eta_p:T_pM\to E_p.$$

If locally

$$\eta=\alpha\otimes s,$$

where $\alpha$ is an ordinary one-form and $s$ is a section of $E$, then

$$\eta(X)=\alpha(X)s.$$

For a sum of such terms, extend linearly.

Thus if $\nabla s\in\Gamma(T^*M\otimes E)$, then feeding in a vector field $X$ gives a section of $E$:

$$(\nabla s)(X)\in\Gamma(E).$$

We write

$$\nabla_Xs:=(\nabla s)(X).$$

This is also denoted $\iota_X(\nabla s)$, meaning insertion of $X$ into the one-form slot.

Connection on a vector bundle: formal definition

Equivalently, a connection gives a bilinear operation

$$(X,s)\mapsto \nabla_Xs$$

satisfying

$$\nabla_{fX+gY}s=f\nabla_Xs+g\nabla_Ys,$$ $$\nabla_X(fs)=X[f]s+f\nabla_Xs.$$

The first identity says that $\nabla_Xs$ is tensorial in the direction $X$: if you multiply the direction by a function, the result is multiplied by that function. The second identity says that $\nabla$ differentiates the section slot.

Local frames and the symbolic formula $\nabla=d+A$

Let $e_1,\ldots,e_r$ be a local frame for $E$ over $U\subset M$. Every section over $U$ can be written uniquely as

$$s=\psi^a e_a,$$

where $\psi^a\in C^\infty(U)$ and repeated fiber indices are summed.

A connection must say how to differentiate the local frame itself. Since $\nabla e_b$ is an $E$-valued one-form, it can be expanded as

$$\nabla e_b=A^a{}_b\otimes e_a,$$

where each $A^a{}_b\in\Omega^1(U)$ is an ordinary one-form. The collection

$$A=(A^a{}_b)$$

is a matrix-valued one-form.

Now apply the Leibniz rule:

$$\begin{align*} \nabla s &=\nabla(\psi^b e_b)\\ &=d\psi^b\otimes e_b+\psi^b\nabla e_b\\ &=d\psi^a\otimes e_a+\psi^b A^a{}_b\otimes e_a\\ &=\left(d\psi^a+A^a{}_b\psi^b\right)\otimes e_a. \end{align*}$$

This is the precise $T^*M\otimes E$-valued formula. Some authors swap the tensor factors and write the same object as

$$e_a\otimes \left(d\psi^a+A^a{}_b\psi^b\right).$$

The meaning is the same after the canonical swap $T^*M\otimes E\cong E\otimes T^*M$.

If we evaluate on a vector field $X$, we get

$$\boxed{ \nabla_Xs= \left(X[\psi^a]+A^a{}_b(X)\psi^b\right)e_a. }$$

In matrix notation, if $e$ is the row vector of basis sections and $\psi$ is the column vector of components,

$$s=e\psi, \qquad \nabla_Xs=e\left(X[\psi]+A(X)\psi\right).$$

The shorthand

$$\boxed{\nabla=d+A}$$

means precisely this local formula after choosing a local frame.

Why connection one-forms are Lie-algebra-valued, and how they act on matter

Suppose the vector bundle $E$ is associated to a principal $G$-bundle:

$$E=P\times_G V,$$

where the action of $G$ on the model fiber $V$ is given by a representation

$$\rho:G\to \mathop{\mathrm{GL}}(V).$$

The principal connection itself is locally represented by a $\mathfrak g$-valued one-form

$$\omega_i\in \Omega^1(U_i;\mathfrak g),$$

where

$$\mathfrak g=T_eG.$$

This means that for a tangent vector $Y\in T_xM$,

$$\omega_i(Y)\in\mathfrak g.$$

At first sight this may look puzzling: an element of $\mathfrak g$ is an infinitesimal group element, while a matter field value lies in $V$. To let $\omega_i(Y)$ act on a vector in $V$, one must use the derivative of the representation.

The representation $\rho:G\to\mathop{\mathrm{GL}}(V)$ is a smooth map between Lie groups. Its differential at the identity is

$$\boxed{ d\rho_e:T_eG\to T_I\mathop{\mathrm{GL}}(V). }$$

By definition,

$$T_eG=\mathfrak g.$$

Also,

$$T_I\mathop{\mathrm{GL}}(V)\cong \mathrm{End}(V).$$

Indeed, $\mathop{\mathrm{GL}}(V)$ is an open subset of the vector space $\mathrm{End}(V)$, because invertibility is the open condition $\det\neq0$ in any basis. Therefore its tangent space at the identity is naturally the full vector space of endomorphisms of $V$. We therefore get the induced Lie-algebra representation

$$\boxed{ d\rho:\mathfrak g\to\mathrm{End}(V). }$$

Concretely, if $X\in\mathfrak g$ is represented by a curve $g(t)$ in $G$ with

$$g(0)=e, \qquad \dot g(0)=X,$$

then

$$\boxed{ d\rho(X)=\left.\frac{d}{dt}\right|_{t=0}\rho(g(t)). }$$

This derivative is an element of $\mathrm{End}(V)$, so it acts on $v\in V$ by ordinary linear algebra:

$$d\rho(X)v\in V.$$

Equivalently,

$$\left.\frac{d}{dt}\right|_{t=0}\rho(g(t))v=d\rho(X)v.$$

Thus the associated vector-bundle connection has the local form

$$\nabla_Ys=e\left(Y[\psi]+d\rho(\omega_i(Y))\psi\right),$$

where $e$ is the local frame induced by a local section of $P$, and $\psi$ is the local component vector of $s$. If one writes

$$A_i:=d\rho(\omega_i)\in\Omega^1(U_i;\mathrm{End}(V)),$$

then this becomes the familiar formula

$$\boxed{ \nabla_Ys=e\left(Y[\psi]+A_i(Y)\psi\right). }$$

In many physics examples the notation suppresses $d\rho$. For instance, in the fundamental representation of $U(N)$ on $\mathbb{C}^N$, an element of

$$\mathfrak u(N)=\{X\in M_N(\mathbb{C}):X^\dagger=-X\}$$

is already an $N\times N$ matrix acting on $\mathbb{C}^N$. In that case $d\rho(X)=X$ in the chosen representation, and one simply writes $A(Y)\psi$. In a different representation, however, the same abstract $\mathfrak g$-valued connection form acts by the corresponding matrix $d\rho(A(Y))$.

Why ordinary derivatives do not transform correctly

Suppose on an overlap $U_i\cap U_j$ we have two frames related by

$$e^{(j)}=e^{(i)}g_{ij},$$

where $g_{ij}:U_i\cap U_j\to \mathop{\mathrm{GL}}(r,\mathbb{R})$ or $\mathop{\mathrm{GL}}(r,\mathbb{C})$ is an ordinary smooth matrix-valued function. It is not a Grassmann variable. In path-integral physics, a fermion field may be Grassmann-valued, but the gauge transformation matrix $g_{ij}(x)$ is still an ordinary even matrix acting on the field components.

A section has components

$$s=e^{(i)}\psi_i=e^{(j)}\psi_j.$$

Therefore

$$\psi_i=g_{ij}\psi_j.$$

Taking an ordinary derivative gives

$$d\psi_i=(dg_{ij})\psi_j+g_{ij}d\psi_j.$$

The extra term $(dg_{ij})\psi_j$ means that ordinary derivatives of local components do not transform like the components themselves. A connection is designed so that

$$D_i\psi_i=g_{ij}D_j\psi_j.$$

Gauge transformation law for a vector-bundle connection

Write the covariant derivative in frame $i$ as

$$D_i=d+A_i$$

and in frame $j$ as

$$D_j=d+A_j.$$

The geometric object $\nabla s$ is globally defined, so its components must transform like the components of $s$:

$$D_i\psi_i=g_{ij}D_j\psi_j.$$

Using $\psi_i=g_{ij}\psi_j$,

$$\begin{align*} D_i\psi_i &=d(g_{ij}\psi_j)+A_i g_{ij}\psi_j\\ &=(dg_{ij})\psi_j+g_{ij}d\psi_j+A_i g_{ij}\psi_j. \end{align*}$$

The right-hand side is

$$g_{ij}D_j\psi_j=g_{ij}d\psi_j+g_{ij}A_j\psi_j.$$

Canceling $g_{ij}d\psi_j$ and using arbitrary $\psi_j$ gives

$$dg_{ij}+A_i g_{ij}=g_{ij}A_j.$$

Multiplying by $g_{ij}^{-1}$ on the left,

$$\boxed{A_j=g_{ij}^{-1}A_i g_{ij}+g_{ij}^{-1}dg_{ij}.}$$

The term $g_{ij}^{-1}dg_{ij}$ appears because the change of frame depends on position.

For a $U(1)$ line bundle, with $g_{ij}=e^{i\chi_{ij}}$,

$$g_{ij}^{-1}dg_{ij}=i\,d\chi_{ij}.$$

Depending on whether one uses Hermitian or anti-Hermitian gauge potentials, this becomes the familiar physics formula $A\mapsto A+d\chi$ up to sign and factors of $i$.

Local gauge transformations versus transition functions

There are two closely related but distinct uses of $G$-valued functions.

First, transition functions

$$g_{ij}:U_i\cap U_j\to G$$

are part of the gluing data of a bundle. They tell us how the frame on patch $U_j$ is compared with the frame on patch $U_i$.

Second, a local gauge transformation is a change of local frame inside the same bundle. On a patch $U_i$, choose

$$h_i:U_i\to G.$$

If the old frame is $e^{(i)}$, the new frame may be written

$$\widetilde e^{(i)}=e^{(i)}h_i.$$

Then local components and gauge potentials change by

$$\widetilde\psi_i=h_i^{-1}\psi_i, \qquad \widetilde A_i=h_i^{-1}A_i h_i+h_i^{-1}dh_i.$$

On overlaps, the transition functions also change:

$$\widetilde g_{ij}=h_i^{-1}g_{ij}h_j.$$

Thus a gauge transformation is not literally the same thing as a transition function. Rather, both are $G$-valued functions describing changes of local frame. In QFT on flat spacetime $\mathbb{R}^d$, one often assumes the bundle is trivial and uses one global patch; then a gauge transformation is just a smooth map

$$h:M\to G.$$

On a nontrivial bundle there may be no global frame, so the same idea is described by compatible local functions $h_i:U_i\to G$.

Connection on the tangent bundle and Christoffel symbols

A connection on $TM$ differentiates vector fields:

$$\nabla_XY\in\Gamma(TM).$$

Here $X$ is the direction of differentiation and $Y$ is the vector field being differentiated.

Choose coordinates $x^1,\ldots,x^n$ on $U\subset M$. The coordinate vector fields

$$\partial_1,\ldots,\partial_n, \qquad \partial_\mu:=\frac{\partial}{\partial x^\mu},$$

form a local frame for $TM$. Since $\nabla_{\partial_\mu}\partial_\nu$ is again a vector field, it can be expanded in this frame:

$$\boxed{ \nabla_{\partial_\mu}\partial_\nu = \Gamma^\lambda{}_{\mu\nu}\partial_\lambda. }$$

The coefficients $\Gamma^\lambda{}_{\mu\nu}$ are the Christoffel symbols, or connection coefficients, of this connection in this coordinate frame.

This equation should be read exactly like the vector-bundle formula $\nabla e_b=A^a{}_b\otimes e_a$. For $TM$, the local frame is $e_\nu=\partial_\nu$, and

$$(A_\mu)^\lambda{}_{\nu}=\Gamma^\lambda{}_{\mu\nu}.$$

If

$$Y=Y^\nu\partial_\nu,$$

then

$$\begin{align*} \nabla_{\partial_\mu}Y &=\nabla_{\partial_\mu}(Y^\nu\partial_\nu)\\ &=(\partial_\mu Y^\nu)\partial_\nu+Y^\nu\nabla_{\partial_\mu}\partial_\nu\\ &=(\partial_\mu Y^\lambda+\Gamma^\lambda{}_{\mu\nu}Y^\nu)\partial_\lambda. \end{align*}$$

Therefore

$$\boxed{(\nabla_\mu Y)^\lambda=\partial_\mu Y^\lambda+\Gamma^\lambda{}_{\mu\nu}Y^\nu.}$$

Covariant derivative of covectors and tensors

A one-form

$$\alpha=\alpha_\nu dx^\nu$$

is a section of $T^*M$. A connection on $TM$ induces a connection on $T^*M$ by requiring the natural pairing to obey the product rule:

$$X[\alpha(Y)]=(\nabla_X\alpha)(Y)+\alpha(\nabla_XY).$$

Set $X=\partial_\mu$ and $Y=\partial_\nu$. Since $\alpha(\partial_\nu)=\alpha_\nu$,

$$\begin{align*} \partial_\mu\alpha_\nu &=(\nabla_\mu\alpha)(\partial_\nu)+\alpha(\nabla_{\partial_\mu}\partial_\nu)\\ &=(\nabla_\mu\alpha)_\nu+\Gamma^\lambda{}_{\mu\nu}\alpha_\lambda. \end{align*}$$

Thus

$$\boxed{(\nabla_\mu\alpha)_\nu=\partial_\mu\alpha_\nu-\Gamma^\lambda{}_{\mu\nu}\alpha_\lambda.}$$

The minus sign is forced by compatibility with the pairing between vectors and covectors.

For a tensor, one gets one $+\Gamma$ term for each upper index and one $-\Gamma$ term for each lower index. This is just repeated use of the product rule.

Torsion

The Lie bracket $[X,Y]$ measures the infinitesimal failure of the flows of $X$ and $Y$ to close as a coordinate parallelogram. The expression

$$\nabla_XY-\nabla_YX$$

measures an antisymmetrized change of vector fields using the chosen connection. These are not automatically the same.

Torsion is tensorial in $X$ and $Y$, so it depends only on the values $X_p,Y_p$ at a point, not on how the vector fields are extended nearby.

In a coordinate frame, $[\partial_\mu,\partial_\nu]=0$, so

$$T(\partial_\mu,\partial_\nu) = (\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu})\partial_\lambda.$$

Thus

$$T^\lambda{}_{\mu\nu}=\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu}.$$

A connection is torsion-free if

$$\nabla_XY-\nabla_YX=[X,Y].$$

In coordinates, torsion-free means

$$\Gamma^\lambda{}_{\mu\nu}=\Gamma^\lambda{}_{\nu\mu}$$

for coordinate vector fields.

Levi-Civita connection

A general connection on $TM$ is not determined by the smooth manifold alone. If $M$ has a Riemannian metric $g$, there is a distinguished connection.

In coordinates, its Christoffel symbols are

$$\boxed{ \Gamma^\lambda{}_{\mu\nu} =\frac12 g^{\lambda\rho} \left( \partial_\mu g_{\nu\rho} +\partial_\nu g_{\mu\rho} -\partial_\rho g_{\mu\nu} \right). }$$

This formula is not the definition of every connection. It is the special formula for the unique torsion-free, metric-compatible connection determined by a metric.

Curvature as noncommuting covariant derivatives

The curvature of a vector-bundle connection is the obstruction to commuting covariant derivatives, corrected by the bracket of the directions.

Why is the final term needed? Moving along $X$ then $Y$ and moving along $Y$ then $X$ do not generally end at the same point. The endpoint mismatch is of order area and is governed by $[X,Y]$. To compare the transported vectors fairly, one must correct by the covariant derivative in the gap direction. This is exactly the role of $-\nabla_{[X,Y]}s$.

Derivation of $F=dA+A\wedge A$

In a local frame, write

$$\nabla=d+A.$$

Let $\psi$ be the column vector of components of a section. Then

$$\nabla\psi=d\psi+A\psi.$$

Apply $\nabla$ again. Here $d\psi$ is a vector-valued one-form, and $A$ is a matrix-valued one-form. Using the graded product rule,

$$d(A\psi)=dA\,\psi-A\wedge d\psi.$$

Therefore

$$\begin{align*} \nabla^2\psi &=(d+A)(d\psi+A\psi)\\ &=d^2\psi+d(A\psi)+A\wedge d\psi+A\wedge A\psi\\ &=0+(dA\,\psi-A\wedge d\psi)+A\wedge d\psi+A\wedge A\psi\\ &=(dA+A\wedge A)\psi. \end{align*}$$

Thus

$$\boxed{F=dA+A\wedge A.}$$

For matrix-valued forms,

$$(A\wedge A)^a{}_b=A^a{}_c\wedge A^c{}_b.$$

Write

$$A=A_\mu dx^\mu.$$

Then

$$dA=(\partial_\mu A_\nu)dx^\mu\wedge dx^\nu =\frac12(\partial_\mu A_\nu-\partial_\nu A_\mu)dx^\mu\wedge dx^\nu.$$

Also

$$A\wedge A =\frac12[A_\mu,A_\nu]dx^\mu\wedge dx^\nu.$$

Therefore

$$\boxed{ F=\frac12F_{\mu\nu}dx^\mu\wedge dx^\nu, \qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu]. }$$

For an Abelian $U(1)$ connection, the commutator vanishes, so $F=dA$.

Riemann curvature tensor

For a connection on $TM$, the curvature acts on vector fields:

$$R(X,Y)Z = \nabla_X\nabla_YZ- \nabla_Y\nabla_XZ- \nabla_{[X,Y]}Z.$$

This is the Riemann curvature tensor of the connection. In coordinates,

$$R(\partial_\mu,\partial_\nu)\partial_\sigma = R^\rho{}_{\sigma\mu\nu}\partial_\rho.$$

Using $A_\mu{}^\rho{}_{\sigma}=\Gamma^\rho{}_{\mu\sigma}$ in the formula for $F_{\mu\nu}$ gives

$$\boxed{ R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}. }$$

For the Levi-Civita connection, this is the usual Riemann curvature tensor of Riemannian geometry.

How curvature transforms under gauge transformations

Under a frame change,

$$A_j=g_{ij}^{-1}A_i g_{ij}+g_{ij}^{-1}dg_{ij}.$$

A direct computation using $d(g^{-1})=-g^{-1}(dg)g^{-1}$ gives

$$\boxed{F_j=g_{ij}^{-1}F_i g_{ij}.}$$

Thus the connection one-form $A$ transforms inhomogeneously, but the curvature transforms covariantly.

For $U(1)$, conjugation is trivial, so

$$F_j=F_i.$$

The local curvatures glue to a globally defined ordinary two-form $F$. The potentials $A_i$ may not glue to one global one-form.

Parallel transport and holonomy

Let

$$\gamma:[0,1]\to M$$

be a path. A section of $E$ along $\gamma$ is a map

$$v(t)\in E_{\gamma(t)}.$$

More formally, it is a section of the pullback bundle $\gamma^*E\to[0,1]$.

In a local frame along the path, write $v(t)$ as a component vector. The covariant derivative along the path is

$$\boxed{ D_t v:=\nabla_{\dot\gamma(t)}v = \frac{dv}{dt}+A_{\gamma(t)}(\dot\gamma(t))v. }$$

Here inserting $\dot\gamma(t)$ into the one-form $A$ means

$$A_{\gamma(t)}(\dot\gamma(t)) = A_\mu(\gamma(t))\dot\gamma^\mu(t).$$

A vector is parallel along $\gamma$ if

$$D_t v=0.$$

Thus parallel transport is the solution of the ordinary differential equation

$$\frac{dv}{dt}=-A_\mu(\gamma(t))\dot\gamma^\mu(t)v.$$

The formal solution is

$$v(1)=\mathcal P\exp\left(-\int_\gamma A\right)v(0).$$

The path-ordering symbol $\mathcal P$ is needed because matrices $A_\mu(\gamma(t_1))$ and $A_\nu(\gamma(t_2))$ may not commute at different times.

A precise definition is

$$\boxed{ \mathcal P\exp\left(-\int_\gamma A\right) := \lim_{N\to\infty} \prod_{n=N}^{1} \left(I-A_{\gamma(t_n)}(\dot\gamma(t_n))\Delta t\right), }$$

where $0=t_0<t_1<\cdots<t_N=1$ and $\Delta t=1/N$. The factor with larger time stands to the left because it acts later on the vector.

If $\gamma$ is a closed loop, the resulting linear map from $E_{\gamma(0)}$ to itself is called holonomy:

$$\mathop{\mathrm{Hol}}_\gamma(A)=\mathcal P\exp\left(-\oint_\gamma A\right).$$

For $U(1)$, path ordering is unnecessary, and holonomy is simply a phase, up to convention:

$$\mathop{\mathrm{Hol}}_\gamma(A)=\exp\left(-\oint_\gamma A\right).$$

This is the bundle language for Berry phase and Aharonov-Bohm phase.

Infinitesimal holonomy and curvature

Let us derive the standard small-loop formula. Work in local coordinates and consider the small rectangle based at $x$ with sides $dx^\mu$ and $dx^\nu$. Parallel transport in the $+\mu$ direction contributes

$$I-A_\mu(x)dx^\mu.$$

Transport in the $+\nu$ direction at the shifted point contributes

$$I-\left(A_\nu(x)+\partial_\mu A_\nu(x)dx^\mu\right)dx^\nu.$$

Transport back in the $-\mu$ direction contributes

$$I+\left(A_\mu(x)+\partial_\nu A_\mu(x)dx^\nu\right)dx^\mu.$$

Transport back in the $-\nu$ direction contributes

$$I+A_\nu(x)dx^\nu.$$

Multiplying these four factors in the order in which they act and keeping only terms of order $dx^\mu dx^\nu$ gives

$$\mathop{\mathrm{Hol}}_{\square} = I- \left( \partial_\mu A_\nu- \partial_\nu A_\mu+[A_\mu,A_\nu] \right)dx^\mu dx^\nu +O(|dx|^3).$$

Therefore

$$\boxed{\mathop{\mathrm{Hol}}_{\square}\approx I-F_{\mu\nu}dx^\mu dx^\nu.}$$

Curvature is the infinitesimal holonomy per unit area.

Principal connections

Let $P\to M$ be a principal $G$-bundle with Lie algebra $\mathfrak g$. At $p\in P$, the vertical subspace is

$$V_pP:=\ker(d\pi_p)\subset T_pP.$$

It consists of tangent directions that move only along the fiber. A principal connection can be described as a smooth choice of horizontal subspaces

$$T_pP=H_pP\oplus V_pP$$

compatible with the right $G$-action.

Equivalently, a principal connection is a $\mathfrak g$-valued one-form

$$\omega\in\Omega^1(P,\mathfrak g)$$

satisfying:

1. on vertical vectors generated by $X\in\mathfrak g$, it returns $X$,

2. under the right action $R_g:P\to P$, it transforms as

$$R_g^*\omega=\mathop{\mathrm{Ad}}_{g^{-1}}\omega.$$

A local gauge field is obtained by choosing a local section $s_i:U_i\to P$ and pulling back:

$$A_i=s_i^*\omega\in\Omega^1(U_i,\mathfrak g).$$

Thus the physics object $A_i=A_{i,\mu}dx^\mu$ is the local expression of the global principal connection $\omega$.

Bianchi identity

The curvature satisfies the Bianchi identity

$$d_AF=0,$$

where

$$d_AF=dF+A\wedge F-F\wedge A.$$

For $U(1)$, this reduces to

$$dF=0.$$

In electromagnetism, $dF=0$ contains the homogeneous Maxwell equations.

Lie derivative versus connection

Now the difference can be stated cleanly.

For functions,

$$\mathcal{L}_Xf=X[f]=\nabla_Xf.$$

For vector fields, if $\nabla$ is torsion-free,

$$\mathcal{L}_XY=[X,Y]=\nabla_XY-\nabla_YX.$$

Thus $\mathcal{L}_XY$ is not simply $\nabla_XY$. For sections of an arbitrary associated vector bundle, there is no canonical Lie derivative unless the base-space flow is lifted to the bundle. A connection, however, always gives $\nabla_Xs$.