= Solution
The geometric object underlying an $SU(2)$ <gauge field> is a <principal connection> on a <principal bundle> $P\to M$ with structure group <SU(2)>. The base $M$ is spacetime, or an oriented <Riemannian manifold> in the Euclidean formulation. A choice of local section identifies each fibre with <SU(2)> and produces a local <gauge potential>. Such a choice is a choice of internal frame; changing it produces a <gauge equivalence of principal connections>. The bundle specifies the global gluing of these frames, while the <principal connection> specifies how frames at neighbouring points are compared.
The <SU(2)> group consists of $2\times2$ <unitary matrices> of <determinant> one. Its <Lie algebra> consists of traceless <Skew-Hermitian> matrices. For definiteness use
$$
t_a=-\frac{i}{2}\sigma_a,\qquad
[t_a,t_b]=\epsilon_{abc}t_c,\qquad
-\operatorname{tr}(t_at_b)=\frac12\delta_{ab},
$$
where the $\sigma_a$ are <Pauli matrices>. The <commutator> fixes all signs below. The positive invariant inner product on this <Lie algebra> is $-\operatorname{tr}(XY)$. Absorb the coupling into the <gauge potential>, so the <gauge covariant derivative> in the defining two-dimensional representation is $D=d+A$ with $A=A^at_a$. A convention using Hermitian <Pauli matrices> instead moves factors of $i$ and the coupling into $D$; the underlying <principal connection> is the same geometric data.
A <principal connection> can be given by a <Lie algebra>-valued <differential one-form> $\mathcal A$ on $P$ satisfying
$$
\mathcal A(\xi_P)=\xi,\qquad
R_h^*\mathcal A=\operatorname{Ad}_{h^{-1}}\mathcal A.
$$
Here $\xi_P$ is the vertical vector generated by the right group action. The first identity recovers vertical motion and the second makes the construction independent of a fibre frame. Its kernel is the <horizontal distribution of a principal connection>, complementary to the vertical tangent spaces. A curve is horizontal when $\mathcal A$ annihilates its velocity, so a <principal connection> defines <parallel transport> along curves in $M$.
For a local section $s$, the <local principal connection form> is $A=s^*\mathcal A$. Change the section to $s'=sh$, with $h:U\to SU(2)$. Differentiating $sh$ has a translated horizontal part and a vertical part from $dh$. The two defining identities for the <principal connection> therefore give
$$
\boxed{A'=h^{-1}Ah+h^{-1}dh}.
$$
On overlapping trivializations this is exactly the compatibility law for the <local principal connection forms>. In particular, the <gauge potentials> need not glue as ordinary globally defined <differential one-forms> on $M$. On a nontrivial <principal bundle> there need not be any global section with respect to which one could write a single matrix-valued $A$.
An active <gauge equivalence of principal connections> is generated by a bundle automorphism covering the identity on $M$ and commuting with the right <SU(2)> action. Its local description has the same transformation formula, but its local group-valued functions must satisfy transition compatibility on overlaps. This distinguishes the structure group <SU(2)> from the full <gauge group> of a fixed bundle. Two descriptions related by a passive change of section encode the same <principal connection>; two connections related by such an automorphism lie in the same physical <gauge orbit>. The space of <principal connections> on a fixed bundle is affine: the difference of two connection forms is a <differential one-form> with values in the <adjoint bundle>, since their inhomogeneous transformation terms cancel.
The <curvature of a principal connection> is
$$
\mathcal F=d\mathcal A+\tfrac12[\mathcal A\wedge\mathcal A].
$$
It is horizontal and equivariant, hence descends to a <differential two-form> on $M$ valued in the <adjoint bundle>. In a local section the <gauge curvature> becomes
$$
F=dA+A\wedge A
=\frac12F_{\mu\nu}\,dx^\mu\wedge dx^\nu,\qquad
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu].
$$
The product $A\wedge A$ combines the <exterior product> with matrix multiplication. The <commutator> term is the specifically non-Abelian interaction. Substitution of the transformation of $A$, using $d(h^{-1})=-h^{-1}(dh)h^{-1}$, cancels all derivatives of $h$ and gives
$$
\boxed{F'=h^{-1}Fh}.
$$
Thus the <gauge curvature> transforms tensorially even though the <gauge potential> does not. Geometrically, the <curvature of a principal connection> measures the failure of the <horizontal distribution of a principal connection> to be integrable; physically it is the non-Abelian <gauge field strength>.
Matter fields are sections of <associated vector bundles>. A field in the defining representation of <SU(2)> is a section of $E=P\times_{SU(2)}\mathbb C^2$. Its two local components satisfy $\psi'=h^{-1}\psi$ under the section convention above. The induced <gauge covariant derivative> obeys
$$
D'\psi'=(d+A')(h^{-1}\psi)=h^{-1}(d+A)\psi.
$$
This covariance is why replacing ordinary derivatives by <gauge covariant derivatives> makes the <kinetic terms> compatible with changes of internal frame. Applying the <gauge covariant derivative> twice yields $D^2\psi=F\psi$, so the <gauge curvature> also measures the noncommutativity of covariant differentiation. Other <group representations> give other <associated vector bundles> and replace $A$ by its image under the differential of the representation.
For an adjoint-valued <differential form> $\Phi$ of degree $r$, set
$$
D_A\Phi=d\Phi+A\wedge\Phi-(-1)^r\Phi\wedge A.
$$
Expansion of $F=dA+A\wedge A$, using $d^2=0$, proves the <gauge-theory Bianchi identity>
$$
\boxed{D_AF=dF+A\wedge F-F\wedge A=0}.
$$
This is an identity for every <principal connection>, independent of any field equation. It should be distinguished from the dynamical <Yang-Mills equations>.
To obtain dynamics, equip $M$ with a <Riemannian metric> and an orientation, and let $*$ be its <Hodge star operator>. With the <Skew-Hermitian> normalization above, the positive Euclidean <Yang-Mills action> is
$$
S[A]=-\frac1{g_{\mathrm{YM}}^2}\int_M\operatorname{tr}(F\wedge *F).
$$
The <matrix trace> and the <Hodge star operator> make this independent of the choice of local section. The minus sign compensates for the negative trace form on <Skew-Hermitian> matrices. Varying the <principal connection> gives $\delta F=D_A\delta A$, and therefore
$$
\delta S=-\frac{2}{g_{\mathrm{YM}}^2}
\int_M\operatorname{tr}(D_A\delta A\wedge *F).
$$
For compactly supported variations, or on a closed base, integration by parts gives the <Yang-Mills equations>
$$
\boxed{D_A*F=0}.
$$
In components these are $D_\mu F^{\mu\nu}=0$. The <gauge-theory Bianchi identity> provides the complementary geometric identity $D_AF=0$. Coupling matter adds the associated gauge current to the dynamical equation; the transformation law of the <gauge covariant derivative> ensures covariance of that current as well.
The <principal connection> also supplies nonlocal observables. Along a path $\gamma$, <parallel transport> is determined by
$$
\frac{d\psi}{dt}+A(\dot\gamma)\psi=0,\qquad
U_\gamma=\mathcal P\exp\left(-\int_\gamma A\right).
$$
The ordering is necessary because the <Lie algebra> matrices at different points need not commute. The resulting <Wilson line> transforms as $U_\gamma'=h(\gamma(1))^{-1}U_\gamma h(\gamma(0))$. For a closed path, its <matrix trace> is a <Wilson loop>, independent of the frame at the basepoint. These observables record the <holonomy of a connection>. A <flat principal connection> has trivial <holonomy of a connection> around sufficiently small contractible loops but may have nontrivial <holonomy of a connection> around noncontractible loops. Thus vanishing local <gauge curvature> need not eliminate global gauge information.
The global topology becomes particularly visible in four dimensions. For a closed oriented four-dimensional base, <Chern-Weil theory> gives the <Second Chern number> of the defining <associated vector bundle>:
$$
k=c_2(E)[M]=\frac1{8\pi^2}\int_M\operatorname{tr}(F\wedge F)\in\mathbb Z.
$$
The plus sign here uses <Skew-Hermitian> curvature and the mathematical second-Chern convention: expanding $\det(I+iF/(2\pi))$ with $\operatorname{tr}F=0$ gives precisely the displayed <Second Chern form>. A convention defining a physics topological charge with a leading minus sign instead calls that charge $-k$. Neither choice changes the absolute-value action bound.
The <Second Chern number> is independent of the <principal connection> on a fixed bundle. Indeed, the <gauge-theory Bianchi identity> implies
$$
\delta\operatorname{tr}(F\wedge F)
=2d\,\operatorname{tr}(\delta A\wedge F),
$$
whose integral vanishes on a closed base by <Stokes theorem>. A nonzero <Second Chern number> obstructs a global trivialization: for a globally defined $A$, $\operatorname{tr}(F\wedge F)=d\,\operatorname{tr}(A\wedge dA+\tfrac23A\wedge A\wedge A)$, so its integral on a closed base would be zero. On $S^4$, gluing two trivial bundles over four-balls uses a transition map $S^3\to SU(2)$. Since <SU(2) as the three-sphere> identifies the target with $S^3$, such gluing maps have integer <degree of a map between oriented manifolds>, illustrating how distinct topological sectors arise.
In four Euclidean dimensions, the <Hodge star operator> squares to one on <differential two-forms>. Decompose $F=F_++F_-$ with $*F_\pm=\pm F_\pm$. Define the nonnegative squared norms $a_\pm=-\int\operatorname{tr}(F_\pm\wedge *F_\pm)$. Orthogonality gives
$$
S=\frac{a_++a_-}{g_{\mathrm{YM}}^2},\qquad
8\pi^2k=-a_++a_-,
\qquad
\boxed{S\ge\frac{8\pi^2}{g_{\mathrm{YM}}^2}|k|}.
$$
Equality means that one component vanishes, giving the <self-dual Yang-Mills equations> or the <Anti-self-dual Yang-Mills equations>. Such finite-action solutions are <Yang-Mills instantons>. They solve the full <Yang-Mills equations> because $D_A*F=\pm D_AF=0$. With the second-Chern convention just specified, anti-self-dual curvature has $k\ge0$ and self-dual curvature has $k\le0$. On noncompact spacetime one must impose suitable decay or boundary conditions before treating the topological integral as an integer and using the same bound without boundary terms.
\b[Local <gauge potentials>, matter <gauge covariant derivatives>, <gauge field strength>, <parallel transport> and <Second Chern number> are different manifestations of one <principal connection>.] The <principal bundle> preserves the global information that a single local <gauge potential> cannot capture; the <Yang-Mills action> then selects its physical dynamics.
Back to article page
