We theoretically derive the sum rule for the negative first moment of the absorptive optical conductivity with excitonic effects and establish its connection to the quantum weight $K$ and Chern number $C$ of the ground state. Applying this framework, we investigate the excitonic optical response of the Chern insulator at a hole filling factor $\nu =1$ in twisted bilayer ${\mathrm{MoTe}}_2$. A single chiral exciton state, which selectively absorbs circularly polarized light of a specific handedness, dominates the optical sum rule. The chiral exciton state comprises two types of interlayer electron-hole transitions, which cancel out the total out-of-plane dipole moment. The absorption spectrum shows nearly perfect magnetic circular dichroism, which can be attributed to the nearly saturated bound $K\ge \left|C\right|$ of the Chern insulator under study. Our work illustrates the potential of using excitonic optical responses to probe quantum geometry encoded by $K$ and $C$ of Chern insulators in moiré superlattices.

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• Chern insulators
• Excitons
• Optical conductivity
• Sum rules
• Twisted heterostructures
• Bethe-Salpeter equation
• Linear response theory