Abstract
We study CFT2 Virasoro conformal blocks of the 4-point correlation function leftlangle {mathcal{O}}_L{mathcal{O}}_H{mathcal{O}}_H{mathcal{O}}_Hrightrangle with three background operators mathcal{O} H and one perturbative operator mathcal{O} L of dimensions ΔL/ΔH ≪ 1. The conformal block function is calculated in the large central charge limit using the monodromy method. From the holographic perspective, the background operators create AdS3 space with three conical singularities parameterized by dimensions ΔH, 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 leftlangle {mathcal{O}}_Lcdots {mathcal{O}}_L{mathcal{O}}_Hcdots {mathcal{O}}_Hrightrangle with arbitrary numbers of background and perturbative operators.
Highlights
We study CFT2 Virasoro conformal blocks of the 4-point correlation function OLOH OH OH with three background operators OH and one perturbative operator OL of dimensions ∆L/∆H 1
The conformal block function is calculated in the large central charge limit using the monodromy method
The background operators create AdS3 space with three conical singularities parameterized by dimensions ∆H, while the perturbative operator corresponds to the geodesic line stretched from the boundary to the bulk
Summary
The large-c conformal blocks (2.2) can be calculated using the monodromy method.1 To this end, one considers the BPZ equation for the 5-point correlation function. Obtained from the original function (2.1) by inserting the degenerate operator Ψ(y, y) of conformal dimension. The 5-point correlation function (2.3) can be expanded into conformal blocks in the OPE channel when the degenerate operator is inserted between two intermediate channels. The monodromy method considers the Fuchsian equation (2.5) with a priori independent accessory parameter having no link to the 4-point conformal block. Comparing monodromy matrices of the original 5-point block and solutions to the Fuchsian equation yields the algebraic equation on the accessory parameter that expresses c2 as a function of coordinates and conformal dimensions.
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