Four-point conformal blocks with three heavy background operators

We study CFT$_2$ Virasoro conformal blocks of the 4-point correlation function $\langle \mathcal{O}_L \mathcal{O}_H \mathcal{O}_H \mathcal{O}_H \rangle$ with three background operators $\mathcal{O}_H$ and one perturbative operator $\mathcal{O}_L$ of dimensions $\Delta_L/\Delta_H \ll1$. The conformal block function is calculated in the large central charge limit using the monodromy method. From the holographic perspective, the background operators create $AdS_3$ space with three conical singularities parameterized by dimensions $\Delta_H$, while the perturbative operator corresponds to the geodesic line stretched from the boundary to the bulk. The geodesic length calculates the perturbative conformal block. We propose how to address the block/length correspondence problem in the general case of higher-point correlation functions $\langle \mathcal{O}_L \cdots \mathcal{O}_L \mathcal{O}_H \cdots \mathcal{O}_H \rangle$ with arbitrary numbers of background and perturbative operators.