Cutoff Dependence and Complexity of the CFT${}_2$ Ground State

Abstract

We present the vacuum of a two-dimensional conformal field theory (CFT${}_2$) as a network of Wilson lines in $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ Chern-Simons theory, which is conventionally used to study gravity in three-dimensional anti-de Sitter space (AdS${}_3$). The position and shape of the network encode the cutoff scale at which the ground state density operator is defined. A general argument suggests identifying the `density of complexity' of this network with the extrinsic curvature of the cutoff surface in AdS${}_3$, which by the Gauss-Bonnet theorem agrees with the holographic Complexity = Volume proposal.

4 pages, 1 figure