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Hall effects in chiral magnets are described in terms of momentum-space and real-space Berry curvatures. 

We present exact results that give insight into how interactions lead to transport and superconductivity in a flat band where the electrons have no kinetic energy. We obtain bounds for the optical spectral weight for flat-band superconductors that lead to upper bounds for the superfluid stiffness and the two-dimensional (2D) $T_c$. We focus on on-site attraction $\left|U\right|$on the Lieb lattice with trivial flat bands and on the π-flux model with topological flat bands. For trivial flat bands, the low-energy optical spectral weight ${\tilde{D}}_{\text{low}}\le \tilde{n}\left|U\right|\mathrm{\Omega }/2$with $\tilde{n}=min\left(n,2-n\right)$, where n is the flat-band density and Ω is the Marzari–Vanderbilt spread of the Wannier functions (WFs). We also obtain a lower bound involving the quantum metric. For topological flat bands, with an obstruction to localized WFs respecting all symmetries, we again obtain an upper bound for ${\tilde{D}}_{low}$linear in $\left|U\right|$. We discuss the insights obtained from our bounds by comparing them with mean-field and quantum Monte Carlo results.