Wilson networks in AdS and global conformal blocks

Abstract: We develop the relation between gravitational Wilson line networks, defined as 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 in CFT$_1$ is explicitly calculated by means of the extrapolate dictionary relation from the gravitational Wilson line network with $n$ boundary endpoints stretched in AdS$_2$. Remarkably, the Wilson line calculation directly yields the conformal block in a particularly simple form which up to the leg factor is given by the comb function of cross-ratios. It is also found that the comb channel structure constants are expressed in terms of factorials and triangle functions of conformal weights whose form determines fusion rules for a given 3-valent vertex. We obtain analytic expressions for the Wilson line matrix elements in AdS$_2$ which are building blocks of the Wilson line networks. We analyze general cap states and specify those which lead to asymptotic values of the Wilson line networks interpreted as boundary correlators of CFT$_1$ primary operators. The cases of (in)finite-dimensional $sl(2, \mathbb{R})$ modules carried by Wilson lines are treated on equal footing that boils down to consideration of singular submodules and their contributions to the Wilson line matrix elements.