The Holographic Geometry of Conformal Blocks

Conformal blocks are the building blocks of correlation functions in conformal field theory. They figure prominently in the study of quantum gravity in light of the AdS/CFT correspondence which identifies conformal field theory as a holographic representation of quantum gravity in Anti de-Sitter space. Thus while conformal blocks are fundamentally CFT objects, it promises to be both conceptually and computationally useful to obtain for them a description on the AdS side. We lay out that description here, extending the holographic dictionary relating CFT objects to AdS objects by adding to it an entry for the conformal blocks themselves. The holographic dual to a global conformal block is a Feynman diagram-like object we call a geodesic Witten diagram, and a similar picture applies to semiclassical Virasoro blocks as well. We discuss ways in which this new picture for an old tool can help to further our understanding of how spacetime emerges from the quantum information structure of conformal field theory