 Harmonic-curvature unitary line connection

Fix a complex line bundle with a Hermitian metric over a closed oriented Riemannian manifold $X$. For a unitary connection $A$, the real curvature form $i{F}_A$ represents a fixed de Rham cohomology class. The Hodge theorem gives its unique harmonic representative $\eta$. Write $\eta -i{F}_A=d\beta$ with $\beta$ a real one-form. Then $A-i\beta$ is a unitary connection with real curvature form equal to $\eta$.

Two such unitary connections differ by a closed imaginary-valued one-form. A gauge transformation removes exactly those differences whose real periods lie in $2\pi \mathbb{Z}$. Their classes under gauge equivalence of principal connections carry a free transitive action ofwhere the integral cohomology is understood through its image in real cohomology. If the first Betti number is zero, there is just one such equivalence class. The compactness assumption supplies the Hodge theorem used here.^[ref]

// Destination: geometry-and-topology.bigb