Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities

A connection supplies the comparison between neighboring fibers that a bundle does not possess by itself. In a local frame it is represented by a one-form $A$, but $A$ is not a tensor: its transformation law contains a derivative of the frame change. The curvature

$$F=\mathrm dA-i g_{\mathrm{YM}}A\wedge A$$

does transform homogeneously, measures the failure of covariant derivatives to commute, and obeys

$$D_{\mathrm{ad}}F=0.$$

The last equation is the Bianchi identity. It follows from the definition of curvature and $\mathrm d^2=0$; it is a geometric compatibility identity, not a field equation. None of these constructions requires a metric or a Hodge star.

This page gives the complementary vector- and principal-bundle descriptions, derives the patching and gauge-transformation laws in the site’s conventions, and checks them on a two-patch $U(1)$ field and a non-Abelian potential. Finite transport and holonomy, gauge dynamics, and characteristic-number classification are left to their dedicated treatments.

Required background. Vector, Principal, and Associated Bundles supplies principal right actions, local sections, and associated fields; Differential Forms, Integration, Orientation, and Stokes supplies wedge products, pullbacks, the graded Leibniz rule, and $\mathrm d^2=0$; and Lie Groups, Lie Algebras, and Exponential and Adjoint Maps supplies fundamental vector fields, representations, and the adjoint action.

Bundle setting and gauge-convention translation

Let $M$ be a smooth manifold, let $\pi:P\to M$ be a principal right $G$-bundle, and let

$$E_R=P\times_{\rho_R}V_R$$

be a vector bundle associated through a finite-dimensional representation $\rho_R$. The intrinsic definitions below allow a general finite-dimensional Lie group. Matrix notation will be used locally, but the invariant statements use the adjoint bundle and do not depend on a faithful matrix representation.

For the QFT crosswalk, specialize to a unitary representation of a compact internal gauge group. Its derived Lie-algebra representation is anti-Hermitian, while the site convention extracts Hermitian generators,

$$[T^a,T^b]=if^{abc}T^c, \qquad \operatorname{tr}_R(T^aT^b)=T(R)\delta^{ab},$$

with $T(F)=1/2$ for the fundamental representation of $SU(N)$. The coupling is written $g_{\mathrm{YM}}$ on this page so that it cannot be confused with a transition function $g_{ij}$.

In this unitary realization, mathematical treatments of principal connections usually use an anti-Hermitian connection form. The complete translation to the site’s Hermitian convention is

$$\begin{aligned} \mathcal A&=-i g_{\mathrm{YM}}A, & D_R&=\mathrm d+\rho_{R*}(\mathcal A) =\mathrm d-i g_{\mathrm{YM}}A^aT_R^a, \\ \mathcal F&=\mathrm d\mathcal A+\mathcal A\wedge\mathcal A, & \mathcal F&=-i g_{\mathrm{YM}}F. \end{aligned}$$

Here and below, products of matrix-valued forms mean wedge product of the form parts and matrix multiplication of the coefficients. That convention is why $A\wedge A$ need not vanish.

A connection is a rule for differentiating sections

For a smooth real or complex vector bundle $E\to M$, a connection is an $\mathbb R$- or $\mathbb C$-linear map, respectively,

$$\nabla:\Gamma(E)\longrightarrow\Omega^1(M,E)$$

that satisfies the Leibniz rule

$$\nabla(fs)=\mathrm df\otimes s+f\nabla s, \qquad f\in C^\infty(M),\quad s\in\Gamma(E).$$

Contracting the one-form slot with a vector field $X$ gives the directional covariant derivative $\nabla_Xs$. The derivative of an ordinary function is fixed, but the derivative of a section requires the extra choice $\nabla$ because $s(x)$ and $s(y)$ lie in different vector spaces. Frankel 2012, § 16.3, pp. 428–431 gives this Leibniz definition and its local-frame form; Nakahara 2003, §§ 10.1.1–10.4.2, pp. 375–394 develops the corresponding principal- and associated-bundle construction.

Choose a local frame $e_i$ and write $s=e_i\psi_i$. There is a matrix-valued one-form $\Gamma_i$ such that

$$\nabla s=e_i\bigl(\mathrm d\psi_i+\Gamma_i\psi_i\bigr).$$

On an overlap, the chapter convention is

$$e_j=e_i g_{ij}, \qquad \psi_i=g_{ij}\psi_j.$$

Demanding that $\nabla s$ describe the same global one-form in either frame forces

$$\Gamma_i =g_{ij}\Gamma_jg_{ij}^{-1} -(\mathrm dg_{ij})g_{ij}^{-1}.$$

The derivative term is indispensable: applying $\mathrm d$ alone to $\psi_i=g_{ij}\psi_j$ produces $(\mathrm dg_{ij})\psi_j$. A collection of ordinary derivatives therefore fails to glue unless the transition functions are locally constant.

Connections form an affine space

Let $\nabla'$ be another connection on the same vector bundle and set $K=\nabla'-\nabla$. The two Leibniz terms cancel:

$$K(fs)=fK(s).$$

Thus $K$ is $C^\infty(M)$-linear and hence is an $\operatorname{End}(E)$-valued one-form,

$$K\in\Omega^1\bigl(M,\operatorname{End}(E)\bigr).$$

This is why one connection is not itself a tensor, while the difference of two connections is. Equivalently, the space of connections is affine: after one connection has been chosen, every other one is obtained by adding an $\operatorname{End}(E)$-valued one-form, but there is no preferred zero connection on a general bundle.

A principal connection is an equivariant horizontal choice

At $p\in P$, the vertical subspace

$$V_pP:=\ker(\pi_*:T_pP\to T_{\pi(p)}M)$$

contains the directions along the principal fiber. For $\xi\in\mathfrak g$, its fundamental vertical vector is

$$\xi_P(p) := \left.\frac{\mathrm d}{\mathrm dt} \bigl(p\mathbin{\cdot}\exp(t\xi)\bigr)\right|_{t=0}.$$

A principal connection chooses a smooth horizontal complement $H_pP$ such that

$$T_pP=H_pP\oplus V_pP, \qquad H_{p\cdot g}P=(R_g)_*H_pP.$$

The second condition makes the choice compatible with the intrinsic right $G$-action. Equivalently, a principal connection is a $\mathfrak g$-valued one-form $\omega\in\Omega^1(P,\mathfrak g)$ satisfying

$$\omega(\xi_P)=\xi, \qquad R_g^*\omega=\operatorname{Ad}_{g^{-1}}\omega, \qquad H_pP=\ker\omega_p.$$

The first condition identifies vertical directions; the kernel then selects horizontal ones. Conversely, the horizontal–vertical splitting defines $\omega$ by projecting onto the vertical part and identifying it with $\mathfrak g$. This equivalence requires the principal equivariance above; an arbitrary complement to $V_pP$ is not a principal connection.

Nakahara 2003, §§ 10.1.1–10.1.2, pp. 375–377 proves this horizontal-splitting/connection-form equivalence for a principal right action.

The connection on $P$, together with $\rho_R$, induces the vector-bundle connection on $E_R$. Different representations see different matrices $T_R^a$, but they inherit the same principal connection. This is the geometric reason one gauge field can differentiate several matter multiplets.

Local potentials patch inhomogeneously

Let $\sigma_i:U_i\to P$ be the local principal sections used to define the transition functions. Pulling back the global connection form gives the local $\mathfrak g$-valued potential, anti-Hermitian in the unitary realization,

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

On $U_i\cap U_j$, where $\sigma_j=\sigma_i\mathbin{\cdot}g_{ij}$, the reproduction and equivariance properties of $\omega$ give

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

Equivalently,

$$\mathcal A_i =g_{ij}\mathcal A_jg_{ij}^{-1} -(\mathrm dg_{ij})g_{ij}^{-1}.$$

This is Nakahara’s compatibility equation (10.9), pp. 379–380, rewritten in the chapter’s fixed transition direction.

These are the principal-bundle version of the local-frame law for $\Gamma_i$. In the Hermitian convention, and suppressing $\rho_R$ in matrix notation, the overlap laws become

$$\boxed{ \begin{aligned} \psi_i&=g_{ij}\psi_j, \\ A_i&=g_{ij}A_jg_{ij}^{-1} -\frac{i}{g_{\mathrm{YM}}} (\mathrm dg_{ij})g_{ij}^{-1}, \\ D_{R,i}\psi_i&=g_{ij}D_{R,j}\psi_j. \end{aligned} }$$

For a general associated representation, the first and third lines contain $\rho_R(g_{ij})$, while $A=A^aT^a$ acts through $T_R^a$ in $D_R$. Direct substitution verifies the last line: the derivative of $g_{ij}$ produced by $\mathrm d\psi_i$ cancels the inhomogeneous term in $A_i$.

Passive changes and active gauge transformations

If the same principal bundle is redescribed using

$$\sigma_i'=\sigma_i\mathbin{\cdot}h_i,$$

then

$$\mathcal A_i' =h_i^{-1}\mathcal A_i h_i+h_i^{-1}\mathrm dh_i, \qquad \psi_i'=\rho_R(h_i^{-1})\psi_i.$$

The corresponding curvature representative changes homogeneously, $\mathcal F_i'=h_i^{-1}\mathcal F_i h_i$.

This is a passive change of local section: the global connection and matter section have not changed. By contrast, an active gauge transformation is a principal-bundle automorphism over $\operatorname{id}_M$. Its local functions must satisfy the overlap compatibility required to define one global automorphism. In matrix notation on a fixed cocycle, $U_i=g_{ij}U_jg_{ij}^{-1}$; arbitrary unrelated functions on the patches do not suffice.

Using the site’s active convention

$$\psi^U=U\psi,$$

the requirement $D_R^U\psi^U=U D_R\psi$ determines

$$\boxed{ A^U =UAU^{-1} -\frac{i}{g_{\mathrm{YM}}}(\mathrm dU)U^{-1}. }$$

The passive formula has the same algebraic shape after setting $U=h_i^{-1}$, but the interpretations remain different: one changes a local presentation, the other acts on the fields. Schwartz 2014, § 25.2.2, pp. 491–493 derives this law directly from covariance of the QFT derivative in the same Hermitian-generator convention.

Curvature is the tensorial obstruction

The curvature of the principal connection is the $\mathfrak g$-valued two-form on $P$

$$\Omega =\mathrm d\omega+\frac12[\omega,\omega] =\mathrm d\omega+\omega\wedge\omega.$$

It is horizontal and equivariant, so it descends to an $\operatorname{Ad}P$-valued two-form on $M$. Pulling it back with a local section gives

$$\mathcal F_i :=\sigma_i^*\Omega =\mathrm d\mathcal A_i+\mathcal A_i\wedge\mathcal A_i.$$

Unlike the connection potential, curvature patches without a derivative term:

$$\mathcal F_i =g_{ij}\mathcal F_jg_{ij}^{-1}.$$

Thus the individual matrix-valued forms $\mathcal F_i$ are local representatives of one global $\operatorname{Ad}P$-valued two-form. They are not, in general, one globally defined $\mathfrak g$-valued two-form. For an Abelian group the adjoint action is trivial, and the local curvature forms do agree as ordinary two-forms.

In the Hermitian convention,

$$\boxed{ F=\mathrm dA-i g_{\mathrm{YM}}A\wedge A, \qquad F_i=g_{ij}F_jg_{ij}^{-1}, \qquad F^U=UFU^{-1}. }$$

Nakahara, § 10.3.2, pp. 386–387, gives Cartan’s structure equation, and § 10.3.4, pp. 389–390, gives the local curvature and homogeneous patch law. Schwartz, § 25.2.2, pp. 492–493, equations (25.69)–(25.71), independently checks the Hermitian-generator QFT component formulas.

The homogeneous transformation follows either by expanding the transformed $A$ or, more economically, from covariant derivatives. To square $D_R$ on a matter zero-form, extend the connection to $E_R$-valued forms by

$$D_R(\alpha\wedge\eta) =\mathrm d\alpha\wedge\eta +(-1)^k\alpha\wedge D_R\eta, \qquad \alpha\in\Omega^k(M).$$

With that extension,

$$D_R^2\psi =-i g_{\mathrm{YM}}F^aT_R^a\psi,$$

and in components,

$$\begin{aligned} [D_\mu,D_\nu] &=-i g_{\mathrm{YM}}F_{\mu\nu}, \\ F_{\mu\nu} &=\partial_\mu A_\nu-\partial_\nu A_\mu -i g_{\mathrm{YM}}[A_\mu,A_\nu]. \end{aligned}$$

Because $D_R^U=U D_RU^{-1}$, its square satisfies $(D_R^U)^2=U D_R^2U^{-1}$, which forces $F^U=UFU^{-1}$. Curvature is therefore the tensorial obstruction to making the local covariant derivatives commute.

Writing $A=A^aT^a$ gives the familiar component formula

$$F^a =\mathrm dA^a +\frac{g_{\mathrm{YM}}}{2} f^{a}{}_{bc}A^b\wedge A^c,$$

or

$$F_{\mu\nu}^a =\partial_\mu A_\nu^a-\partial_\nu A_\mu^a +g_{\mathrm{YM}}f^a{}_{bc}A_\mu^bA_\nu^c.$$

Two near misses clarify the statement. First, $A\ne0$ does not imply $F\ne0$: applying a nonconstant gauge transformation to $A=0$ produces a generally nonzero pure-gauge potential with zero curvature. Second, $F=0$ allows one to remove $A$ on a sufficiently small contractible patch, but it does not by itself settle the global transport data. That global question is developed in Parallel Transport and Holonomy.

Why the Bianchi identity holds

For an adjoint-valued $k$-form $X$, define the exterior covariant derivative in the anti-Hermitian convention by

$$D_{\mathrm{ad}}X =\mathrm dX +\mathcal A\wedge X -(-1)^kX\wedge\mathcal A.$$

This operator acts on the adjoint bundle. It is not the same operator as $D_R$, which acts through the matter representation $R$. For the Hermitian curvature two-form, the corresponding formula is

$$D_{\mathrm{ad}}F =\mathrm dF -i g_{\mathrm{YM}} \bigl(A\wedge F-F\wedge A\bigr).$$

Now start from $\mathcal F=\mathrm d\mathcal A+\mathcal A\wedge\mathcal A$. The graded Leibniz rule gives

$$\mathrm d\mathcal F =\mathrm d\mathcal A\wedge\mathcal A -\mathcal A\wedge\mathrm d\mathcal A.$$

The commutator part is

$$\begin{aligned} \mathcal A\wedge\mathcal F -\mathcal F\wedge\mathcal A &= \mathcal A\wedge\mathrm d\mathcal A -\mathrm d\mathcal A\wedge\mathcal A \\ &\quad +\mathcal A\wedge\mathcal A\wedge\mathcal A -\mathcal A\wedge\mathcal A\wedge\mathcal A. \end{aligned}$$

Every term cancels, proving the Bianchi identity

$$\boxed{ D_{\mathrm{ad}}\mathcal F=0 \qquad\Longleftrightarrow\qquad D_{\mathrm{ad}}F=0. }$$

Nakahara, § 10.3.5, pp. 390–391, and Frankel 2012, § 18.3, pp. 488–489 independently give this graded Bianchi calculation.

In local coordinates this is

$$D_{[\lambda}F_{\mu\nu]}=0.$$

The proof used only associativity, the graded Leibniz rule, and $\mathrm d^2=0$. It did not use an action, a metric, a state, or an approximation. For $U(1)$ all matrix commutators vanish, so

$$F=\mathrm dA, \qquad D_{\mathrm{ad}}F=\mathrm dF=\mathrm d^2A=0$$

on each patch. The potential $A$ may still fail to exist globally even though the Abelian curvature $F$ is a global closed two-form.

The Bianchi identity must not be confused with a dynamical gauge-field equation. An equation of motion involves additional physical input and, typically, a Hodge star and sources. The identity above holds before any such input is chosen.

A two-patch U(1) connection on the sphere

Continue the two-patch bundle from the prerequisite page. On

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

let $\theta$ and $\varphi$ be the usual polar angles. For $n\in\mathbb Z$, take the coupling-absorbed real potentials

$$\begin{aligned} a_N&=\frac n2(1-\cos\theta)\,\mathrm d\varphi, \\ a_S&=-\frac n2(1+\cos\theta)\,\mathrm d\varphi. \end{aligned}$$

Each expression is regular on its own patch. On the overlap,

$$a_N-a_S=n\,\mathrm d\varphi.$$

This is exactly the inhomogeneous law associated with $g_{NS}=e^{in\varphi}$. For a field of integer weight $q$,

$$\psi_N=e^{iqn\varphi}\psi_S, \qquad D_i=\mathrm d-iq a_i.$$

Choose the local real lift $\chi_{NS}=n\varphi$. The lift itself is not single-valued on the full cylindrical overlap, while $e^{i\chi_{NS}}$ and $\mathrm d\chi_{NS}$ are. The covariance check is

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

The local curvatures are

$$\begin{aligned} f_N=\mathrm da_N &=\frac n2\sin\theta\, \mathrm d\theta\wedge\mathrm d\varphi, \\ f_S=\mathrm da_S &=\frac n2\sin\theta\, \mathrm d\theta\wedge\mathrm d\varphi. \end{aligned}$$

Thus two distinct local potentials define one global curvature two-form. The curvature is closed, $\mathrm df=0$, while no single regular potential was assumed over all of $S^2$. Tong 2018, § 1.1.2, pp. 6–8, PDF derives these northern and southern potentials in physical monopole normalization. Here $a=g_{\mathrm{YM}}A$ is the dimensionless, coupling-absorbed potential used on the preceding page; this rescaling converts Tong’s charged-field phase to the integer-weight convention above.

This calculation establishes potential patching, matter covariance, curvature gluing, and the Abelian Bianchi identity. It does not derive flux quantization or classify the bundle by a characteristic number. After the required de Rham background, that step belongs to Characteristic Classes and Chern–Weil Theory.

A non-Abelian check that cannot be hidden in dA

The sphere example is Abelian, so it cannot test the commutator term. On a coordinate patch with coordinates $(x,y)$, take constant real numbers $a,b$ and Hermitian generators satisfying

$$[T^1,T^2]=iT^3.$$

Set

$$A=aT^1\,\mathrm dx+bT^2\,\mathrm dy.$$

Although $\mathrm dA=0$,

$$\begin{aligned} A\wedge A &=ab[T^1,T^2]\,\mathrm dx\wedge\mathrm dy \\ &=iabT^3\,\mathrm dx\wedge\mathrm dy, \end{aligned}$$

so

$$F=g_{\mathrm{YM}}abT^3\, \mathrm dx\wedge\mathrm dy.$$

This is the smallest calculation showing why non-Abelian field strength is not $\mathrm dA$. Under a change of gauge it is conjugated, so its individual matrix components depend on the local frame while invariant expressions such as $\operatorname{tr}_R(F\wedge\star F)$, when a metric and an action are later introduced, do not. The developed interpretation of gauge redundancy, observables, and gauge-field dynamics belongs to Gauge Fields, Redundancy, and Observable Content.

Internal and spacetime connections are not the same datum

The abstract connection language applies to both internal bundles and tangent bundles, but their roles differ.

Feature	Internal gauge connection	Spacetime connection
Bundle acted on	A principal $G$-bundle and its associated matter bundles	$TM$, frame bundles, and tensor bundles
Local indices	Internal representation indices	Tangent, cotangent, or frame indices
Source of the data	An independent connection on the internal bundle	A connection on spacetime geometry
Metric dependence	None is required kinematically	The Levi–Civita choice is selected by metric compatibility and zero torsion
Curvature	$F\in\Omega^2(M,\operatorname{Ad}P)$	Riemann curvature acting on tangent data

A metric does not determine an internal gauge connection. Conversely, a general spacetime connection need not be Levi–Civita. The differential Bianchi identity $D_{\mathrm{ad}}F=0$ is also distinct from the algebraic first Bianchi identity for a torsion-free spacetime connection. A field carrying both a spacetime index and an internal charge generally needs both connections, acting on different tensor factors.

Common pitfalls

Using $\mathrm d$ as though local field components were global. If $\psi_i=g_{ij}\psi_j$, then $\mathrm d\psi_i$ contains $(\mathrm dg_{ij})\psi_j$. The inhomogeneous connection term is what makes $D_R\psi$ patch like $\psi$.

Calling the potential a tensor. A local connection potential transforms inhomogeneously. Its curvature and the difference between two connections transform homogeneously.

Dropping the commutator in a non-Abelian theory. The wedge of a matrix-valued one-form with itself need not vanish. The constant-potential example above has $\mathrm dA=0$ but $F\ne0$.

Replacing $D_{\mathrm{ad}}F=0$ by $\mathrm dF=0$. The latter is correct for an Abelian group. In a non-Abelian theory the commutator terms are required for a gauge-covariant statement.

Treating Bianchi as an equation of motion. Bianchi follows identically from the definition of $F$. Dynamics requires an action or other physical law and introduces assumptions absent from this page.

Inferring global triviality from $F=0$. Vanishing curvature is a local integrability statement. Global transport can retain information that local curvature does not; finite transport and holonomy are the next topic.

Identifying an internal connection with Levi–Civita. Levi–Civita acts on spacetime tensor data and is selected by a metric plus zero torsion. An internal connection is independent principal-bundle data.

Exercises

These checks test the defining axioms, the local-to-global law, the affine nature of connections, the non-Abelian term, and the Bianchi identity.

Connection retrieval. State the Leibniz definition of a vector-bundle connection, the two axioms for a principal connection form, and the qualitative difference between the patch laws for a connection and its curvature.

Retrieval answer

A vector-bundle connection is a linear map $\nabla:\Gamma(E)\to\Omega^1(M,E)$ satisfying

$$\nabla(fs)=\mathrm df\otimes s+f\nabla s.$$

A principal connection form obeys

$$\omega(\xi_P)=\xi, \qquad R_g^*\omega=\operatorname{Ad}_{g^{-1}}\omega.$$

Local connection forms patch inhomogeneously, with a term containing $\mathrm dg_{ij}$. Curvature forms patch homogeneously by conjugation and therefore represent one global $\operatorname{Ad}P$-valued two-form.

Overlap round trip. Starting from $\psi_i=g_{ij}\psi_j$ and $D_i=\mathrm d+\mathcal A_i$, derive the required relation between $\mathcal A_i$ and $\mathcal A_j$, then verify that the curvature transforms without a derivative term.

Overlap answer

Covariance requires

$$(\mathrm d+\mathcal A_i)(g_{ij}\psi_j) =g_{ij}(\mathrm d+\mathcal A_j)\psi_j.$$

Canceling the common $g_{ij}\mathrm d\psi_j$ term gives

$$\mathcal A_i =g_{ij}\mathcal A_jg_{ij}^{-1} -(\mathrm dg_{ij})g_{ij}^{-1}.$$

Equivalently, $\mathcal A_j=g_{ij}^{-1}\mathcal A_i g_{ij} +g_{ij}^{-1}\mathrm dg_{ij}$. Substitution into $\mathcal F_j=\mathrm d\mathcal A_j+\mathcal A_j\wedge\mathcal A_j$ makes all terms containing $\mathrm dg_{ij}$ cancel, leaving

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

Tensorial difference. Prove directly from the Leibniz rule that the difference of two connections is an $\operatorname{End}(E)$-valued one-form. Why is the difference of two local potential matrices homogeneous?

Tensorial-difference answer

For $K=\nabla'-\nabla$,

$$\begin{aligned} K(fs) &=\mathrm df\otimes s+f\nabla's -\mathrm df\otimes s-f\nabla s \\ &=fK(s). \end{aligned}$$

Therefore $K$ is $C^\infty(M)$-linear and is a section of $T^*M\otimes\operatorname{End}(E)$. Locally, if the two potentials are $\Gamma_i'$ and $\Gamma_i$, their identical inhomogeneous terms cancel:

$$\Gamma_i'-\Gamma_i =g_{ij}(\Gamma_j'-\Gamma_j)g_{ij}^{-1}.$$

Non-Abelian failure mode. For $A=aT^1\mathrm dx+bT^2\mathrm dy$ with $[T^1,T^2]=iT^3$, compute $F$ and identify the false step in the claim “$\mathrm dA=0$, therefore the connection is flat.”

Non-Abelian answer

Antisymmetry of the form factors combines the two matrix orders into a commutator:

$$A\wedge A =ab[T^1,T^2]\,\mathrm dx\wedge\mathrm dy =iabT^3\,\mathrm dx\wedge\mathrm dy.$$

Hence

$$F=\mathrm dA-i g_{\mathrm{YM}}A\wedge A =g_{\mathrm{YM}}abT^3\, \mathrm dx\wedge\mathrm dy.$$

The false step is replacing the non-Abelian curvature by its Abelian special case $\mathrm dA$.

Bianchi cancellation. Expand $D_{\mathrm{ad}}\mathcal F$ from $\mathcal F=\mathrm d\mathcal A+\mathcal A\wedge\mathcal A$ and show which terms cancel. What remains for $U(1)$?

Bianchi answer

The graded Leibniz rule gives

$$\mathrm d\mathcal F =\mathrm d\mathcal A\wedge\mathcal A -\mathcal A\wedge\mathrm d\mathcal A.$$

Meanwhile,

$$\mathcal A\wedge\mathcal F-\mathcal F\wedge\mathcal A =\mathcal A\wedge\mathrm d\mathcal A -\mathrm d\mathcal A\wedge\mathcal A;$$

the two cubic terms cancel each other. Adding the two lines gives zero. For $U(1)$ the commutator term is absent, so the identity reduces to $\mathrm dF=\mathrm d^2A=0$ on every patch, with the local closed forms gluing to a global two-form.

Synthesis and continuations

A vector-bundle connection is a Leibniz derivative, while a principal connection is an equivariant horizontal distribution or, equivalently, a connection one-form. Pullback to a local section produces a potential that patches inhomogeneously, exactly canceling the derivative of the transition function in $D_R\psi$. Curvature is the square of that derivative, patches by conjugation, and obeys $D_{\mathrm{ad}}F=0$ by a graded-algebra identity. The two-patch $U(1)$ example shows how local potentials can yield one global curvature; the constant non-Abelian example shows why the commutator term is essential.

Continue according to the additional question:

• Parallel Transport and Holonomy integrates the infinitesimal comparison rule along open and closed paths and explains flat but globally nontrivial transport;
• Gauge Fields, Redundancy, and Observable Content develops the physical interpretation of gauge potentials, field strengths, redundancy, observables, and dynamics;
• after de Rham theory, Characteristic Classes and Chern–Weil Theory turns invariant curvature polynomials into topological information.

References

• Theodore Frankel, The Geometry of Physics: An Introduction, third edition, Cambridge University Press, 2012, § 16.3, pp. 428–431, and §§ 18.1–18.3, pp. 475–489. This independently develops connections on vector and principal bundles, associated-bundle differentiation, adjoint-valued curvature, and the Bianchi identity.
• Mikio Nakahara, Geometry, Topology and Physics, second edition, Institute of Physics Publishing, 2003, §§ 10.1.1–10.4.2, pp. 375–394. These sections treat principal connections, local connection forms, curvature, associated-bundle covariant derivatives, and the Bianchi identity.
• Matthew D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, § 25.2.2, pp. 490–493, especially equations (25.60)–(25.71). This supplies the QFT convention check for $D_\mu$, the finite gauge transformation of $A_\mu$, the commutator definition of $F_{\mu\nu}$, and homogeneous curvature transformation.
• David Tong, Gauge Theory — Open PDF, Cambridge Part III lecture notes, 2018, § 1.1.2, pp. 6–8. This supplies the physical two-patch monopole potentials and their overlap law.