Holographic conformal partial waves as gravitational open Wilson networks

Abstract

We propose a method to holographically compute the conformal partial waves in any decomposition of correlation functions of primary operators in conformal field theories using open Wilson network operators in the holographic gravitational dual. The Wilson operators are the gravitational ones where gravity is written as a gauge theory in the first order Hilbert-Palatini formalism. We apply this method to compute the global conformal blocks and partial waves in 2d CFTs reproducing many of the known results.

Holographic dual of the five-point conformal block

“…where β ∞ (·, ·) is defined in (11). Note that setting k 1 = k 2 = k 3 = 0 in (22) recovers the p-adic form of the geodesic bulk diagram given in figure 4.…”

“…Such a task was recently undertaken for four-point global conformal blocks in any dimension with external scalar operators [1] (with further generalizations in Refs. [2,3,4,5,6,7,8,9,10,11,12,13]) and for Virasoro blocks in AdS 3 /CFT 2 [14,15,16,17,18,19,20,21,22,23,24].…”

Late time Wilson lines

“…In this section we recall the basic construction of the Wilson line, and how it provides a representation of Virasoro conformal blocks that admits a convenient expansion at large central charge. See [8][9][10][11][12][13][14][15] for more background and previous results.…”

“…For a numerical study, see [7]. Our approach is via the Wilson representation of conformal blocks, as developed in [8][9][10][11][12][13][14][15]. Here, the conformal block corresponding to OO → stress tensors → O O is expressed as h, h |P e z 2 z 1 (L 1 + 6 c T (z)L 1 )dz |h, h , where T (z) is the CFT stress tensor.…”

Holographic interpretation of 1-point toroidal block in the semiclassical limit

“…Simons formulation [33,34], for recent discussion see, e.g., [35][36][37][38]. Finally, it is interesting to develop the geodesic Witten diagrams technique by analogy with conformal blocks on the complex plane [10].…”