Query complexity and cutoff dependence of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>CFT</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math> ground state

Abstract

We present the vacuum of a two-dimensional conformal field theory as a network of Wilson lines in $SL(2,\mathbb{R})\ifmmode\times\else\texttimes\fi{}SL(2,\mathbb{R})$ Chern-Simons theory, which is conventionally used to study gravity in three-dimensional anti--de Sitter space. The position and shape of the network encode the cutoff scale at which the ground state density operator is defined. In the framework of query complexity, a general argument relates the ``density of complexity'' of the network to the extrinsic curvature of the cutoff surface in ${\mathrm{AdS}}_{3}$, which accords with the holographic $\mathrm{complexity}=\text{volume proposal}$.

Topics & Concepts