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We introduce a new approach to find the Tomita-Takesaki modular flow formulti-component regions in general chiral conformal field theory. Our method isbased on locality and analyticity of primary fields as well as the so-calledKubo-Martin-Schwinger (KMS) condition. These features can be used to transformthe problem to a Riemann-Hilbert problem on a covering of the complex plane cutalong the regions, which is equivalent to an integral equation for the matrixelements of the modular Hamiltonian. Examples are considered.

On the modular operator of mutli-component regions in chiral CFT