= Harmonic-curvature unitary line connection
{title2=$iF_A=\eta$}
The real curvature $iF_A$ of a unitary line connection has a fixed de Rham class. Let $\eta$ be its harmonic representative and write $\eta-iF_A=d\beta$. 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
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