Exact Virasoro blocks from Wilson lines and background-independent operators

Aspects of black hole thermodynamics and information loss can be derived as a consequence of Virasoro symmetry. To bolster the connection between Virasoro conformal blocks and AdS3 quantum gravity, we study sl(2) Chern-Simons Wilson line networks and revisit the idea that they compute a variety of CFT2 observables, including Virasoro OPE blocks, exactly. We verify this in the semiclassical large central charge limit and to low orders in a perturbative 1/c expansion. Wilson lines connecting the boundary to points in the bulk play a natural role in bulk reconstruction. Because quantum gravity in AdS3 is rigidly fixed by Virasoro symmetry, we argue that sl(2) Wilson lines provide building blocks for background independent bulk reconstruction. In particular, we show explicitly that they automatically 'know' about the uniformizing coordinates appropriate to any background state.