Complexity for Conformal Field Theories in General Dimensions

Abstract

We study circuit complexity for conformal field theory states in arbitrary dimensions. Our circuits start from a primary state and move along a unitary representation of the Lorentzian conformal group. Different choices of distance functions can be understood in terms of the geometry of coadjoint orbits of the conformal group. We explicitly relate our circuits to timelike geodesics in anti-de Sitter space and the complexity metric to distances between these geodesics. We extend our method to circuits in other symmetry groups using a group theoretic generalization of the notion of coherent states.

Figures (2)

• Figure 1: Illustration of two nearby timelike geodesics in $\mathrm{AdS}_3$ (blue, red) corresponding to two boundary circuits and the minimal (green) and maximal (brown) perpendicular distance between them. The infinitesimal variation was exaggerated to improve the visualization.
• Figure 2: Illustration of the process of mapping the Lorentzian generators to the Euclidean generators. We start with quantizing the theory with respect to constant $P^0-K^0$ slices on the Lorentzian plane. Those are mapped to constant time slices on the Lorentzian cylinder. The Wick rotation maps those to constant time slices on the Euclidean cylinder. Finally we map the Euclidean cylinder to the Euclidean plane via a radial map.