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.

Figures (1)

• Figure 1: All pictures are equal time snapshots of $(x^+,x^-,u)$-space. Top left: A network of Wilson lines, which computes CFT 6-point functions. Top right: A network which computes multi-point functions. After the red tentacles are amputated, it becomes the ground state at the scale defined by the black line. The density of inputs on the network is $d\mathcal{C} \propto Kd\lambda$ so the network accepts no inputs on geodesic segments. Bottom: Two amputated networks in global AdS$_3$, which differ only by crossing symmetry, are shown in black. Crossing symmetry is implemented by shifting the feature ${\bf -}0\,0{\bf -}$, which implements the projector $|0\rangle \langle 0|$ in computing correlation functions. The blue features---Wilson lines and projections---convert the black network into one at a coarser, uniform cutoff. Projections are necessary to accord with $d\mathcal{C} \propto Kd\lambda$ because coarse-graining eliminates degrees of freedom. The coarse-graining scheme is highly non-unique; it varies with gauge choice for $(A, \bar{A})$ and with crossing symmetry.