Gauge theory and the geometry of physics

Under the $U(1)$ gauge transformation $a \mapsto a + d\lambda$ of the electromagnetic potential, which of these quantities are unchanged? (Choose all that apply.)

A $U(1)$-bundle over $S^2$ is built from trivial bundles over the two hemispheres glued along the equator by the transition function $e^{3i\phi}$, so that the potentials satisfy $a_N - a_S = 3\,d\phi$. What is the first Chern number $\frac{1}{2\pi}\int_{S^2} F$?

Writing the electromagnetic field as a $2$-form $F$ on space-time, which Maxwell equations are equivalent to the Bianchi identity $dF = 0$?

In the proof of the instanton bound $\mathrm{YM}(A) \ge 8\pi^2|k|$, we use $\operatorname{tr}(F \wedge F) = \operatorname{tr}(F^+ \wedge F^+) + \operatorname{tr}(F^- \wedge F^-)$. Which fact makes the cross terms vanish?

An $SU(2)$-bundle over the round $S^4$ has $c_2 = -2$. What is the smallest possible Yang–Mills energy $\mathrm{YM}(A)$ of a connection on it, in units of $8\pi^2$?

Electrons of charge $q$ pass on both sides of a thin shielded solenoid of flux $\Phi$. For which values of $\Phi$ is the interference pattern identical to the pattern with $\Phi = 0$?

In the Lean file, gauge g A changes each link variable by U(x → y) ↦ g x * U * (g y)⁻¹. Which statement is the theorem holonomy_gauge for a path p starting at x (with endpoint x p its end)?

By Donaldson's theorem, which of these unimodular forms cannot be the intersection form of a smooth closed simply connected oriented $4$-manifold? (By Freedman, all of them are intersection forms of topological ones.)

Which of these are established theorems? (Choose all that apply.)

A $U(1)$ lattice gauge field on an $n \times n$ torus is given by link angles ax[x][y] (edge $(x,y) \to (x+1,y)$) and ay[x][y] (edge $(x,y) \to (x,y+1)$), indices mod $n$. Write reduce(t) returning the representative of $t$ modulo $2\pi$ in $(-\pi, \pi]$, plaquette(ax, ay, x, y) returning the reduced angle $a_x(x,y) + a_y(x+1,y) - a_x(x,y+1) - a_y(x,y)$, and charge(ax, ay) returning $\frac{1}{2\pi}\sum$ of all plaquettes. The tests check that the charge is a (near-)integer and gauge invariant.