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We develop the relation between gravitational Wilson line networks, definedas a particular product of Wilson line operators averaged over the cap states,and conformal correlators in the context of the AdS$_2$/CFT$_1$ correspondence.The $n$-point $sl(2, \mathbb{R})$ comb channel global conformal block inCFT$_1$ is explicitly calculated by means of the extrapolate dictionaryrelation from the gravitational Wilson line network with $n$ boundary endpointsstretched in AdS$_2$. Remarkably, the Wilson line calculation directly yieldsthe conformal block in a particularly simple form which up to the leg factor isgiven by the comb function of cross-ratios. It is also found that the combchannel structure constants are expressed in terms of factorials and trianglefunctions of conformal weights whose form determines fusion rules for a given3-valent vertex. We obtain analytic expressions for the Wilson line matrixelements in AdS$_2$ which are building blocks of the Wilson line networks. Weanalyze general cap states and specify those which lead to asymptotic values ofthe Wilson line networks interpreted as boundary correlators of CFT$_1$ primaryoperators. The cases of (in)finite-dimensional $sl(2, \mathbb{R})$ modulescarried by Wilson lines are treated on equal footing that boils down toconsideration of singular submodules and their contributions to the Wilson linematrix elements.

Wilson networks in AdS and global conformal blocks