Tags: ? Refs: curvature
flat bundle
A flat bundle is a vector bundle with constant transition functions. Any flat bundle admits a canonical flat connection, i.e. a connection with vanishing curvature form.
flat bundle
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curvature
The covariant exterior derivative satisifes \(d_\nabla^2 s = F_\nabla \wedge s\) for \(s\in { { {\Omega}^{\scriptscriptstyle \bullet}} }_M(E)\) an \(E{\hbox{-}}\)valued form, and thus \(d^2=0\) when the curvature form vanishes. This yields a flat bundle.
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Talk Surface bundles, mapping class groups, moduli spaces, and cohomology
Such bundles are called flat \(\Sigma_g\) bundles, exactly those which admit a flat connection.
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Mirror Symmetry
For \(X\) a Calabi-Yau - A-side: the Fukaya category of \(X\) corresponds to \(A{\hbox{-}}\)branes on \(X\), so roughly Lagrangian submanifolds equipped with a flat bundle. - B-side: DCoh of \(X\) corresponds to \(B{\hbox{-}}\)branes on \(X\).