A gauge theory is a field theory whose Lagrangian is invariant under a local (spacetime-dependent) group of transformations $G$. For a matter field $\psi(x)$ transforming as $\psi(x) \mapsto U(x)\psi(x)$ with $U(x) \in G$, the ordinary derivative $\partial_\mu\psi$ does not transform covariantly. The gauge covariant derivative $D_\mu = \partial_\mu - igA_\mu$ (where $A_\mu = A_\mu^a T_a$ is the gauge field taking values in the Lie algebra $\mathfrak{g}$, $g$ is the coupling constant, and $T_a$ are generators) transforms as $D_\mu\psi \mapsto U(x)D_\mu\psi$, provided $A_\mu \mapsto UA_\mu U^{-1} + \frac{i}{g}U\partial_\mu U^{-1}$.

A principal $G$-bundle $P \xrightarrow{\pi} M$ over spacetime $M$ with structure group $G$ is a manifold $P$ with a free right $G$-action such that $M = P/G$. A connection on $P$ is a $\mathfrak{g}$-valued 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ satisfying: (1) $\omega(X_\xi) = \xi$ for fundamental vector fields ($\xi \in \mathfrak{g}$), and (2) $R_g^*\omega = \mathrm{Ad}_{g^{-1}}\omega$ (equivariance). In a local trivialization, $\omega$ reduces to the gauge potential $A = A_\mu dx^\mu$. The curvature is $F = d\omega + \omega\wedge\omega$, locally $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - ig[A_\mu, A_\nu]$ (the field strength tensor).

Gauge field configurations are classified by the topology of the bundle. The Chern number (or instanton number) $k = \frac{1}{8\pi^2}\int\mathrm{tr}(F\wedge F) \in \mathbb{Z}$ is a topological invariant measuring the winding of the gauge field. For $G = \mathrm{SU}(2)$ on $S^4$: $\pi_3(\mathrm{SU}(2)) = \mathbb{Z}$ classifies instantons. The BPST instanton is a self-dual ($F = *F$) solution with $k = 1$: $A_\mu = \frac{(x-x_0)_\nu\bar{\sigma}_{\mu\nu}}{|x-x_0|^2 + \rho^2}$ where $\rho$ is the instanton size and $x_0$ its center.