The $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=$ K, Rb, Cs) family of kagome metals hosts unconventional charge density wave (CDW) order whose nature is still an open puzzle. Accumulated evidences point to a time-reversal symmetry breaking (TRSB) orbital flux phase that carries loop currents. Such an order may support anomalous Hall effect. However, the polar Kerr effect measurements that probe the ac anomalous Hall conductivity seem to have yielded contradictory results. We first argue on symmetry grounds that some previously proposed orbital flux order, most notably the one with Star-of-David distortion, shall not give rise to anomalous Hall or polar Kerr effects. We further take the trihexagonal orbital flux phase as an exemplary kagome flux order that does exhibit anomalous Hall response and show that the Kerr rotation angle at two relevant experimental optical frequencies generally reaches microradians to sub-milliradians levels. A particularly sharp resonance enhancement is observed at around $\hslash \omega =1$ eV, suggesting exceedingly large Kerr rotation at the corresponding probing frequencies not yet accessed by previous experiments. Our study provides important guidance to the interpretation of Kerr measurements on the CDW phase of $A{\mathrm{V}}_3{\mathrm{Sb}}_5$. In particular, we highlight the important fact that absence of Kerr signal cannot be equated with the absence of TRSB CDW order in $A{\mathrm{V}}_3{\mathrm{Sb}}_5$.

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• Charge density waves
• Magneto-optical Kerr effect
• Optical conductivity
• Kagome lattice
• Discrete symmetries in condensed matter
• Exact diagonalization
• Linear response theory