 Harmonic-curvature unitary line connection

The real curvature $i{F}_A$ of a unitary line connection has a fixed de Rham class. Let $\eta$ be its harmonic representative and write $\eta -i{F}_A=d\beta$. Then $A-i\beta$ has curvature $\eta$. Any two such connections differ by a closed imaginary one-form. If $b_1=0$, that form is exact and comes from a circle-valued gauge transformation. In general their gauge classes form a torsor for $H^1(X;\mathbb{R})/(2\pi H^1(X;\mathbb{Z}))$.

// Destination: geometry-and-topology.bigb