Complexity for Conformal Field Theories in General Dimensions

TL;DR

This paper investigates the complexity of conformal field theory states across various dimensions, linking circuit constructions to geometric structures in conformal and anti-de Sitter spaces.

Contribution

It introduces a group-theoretic framework for analyzing circuit complexity in conformal field theories, connecting it to coadjoint orbits and geodesics in anti-de Sitter space.

Findings

Complexity relates to coadjoint orbit geometry.

Explicit connection between circuits and AdS geodesics.

Generalization to other symmetry groups.

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.