Conformal field theory complexity from Euler-Arnold equations

TL;DR

This paper explores the complexity of states and operators in 1+1 dimensional conformal field theories using Euler-Arnold equations, providing new mathematical tools and insights into the geometry of quantum complexity.

Contribution

It introduces a comprehensive framework based on Euler-Arnold equations for analyzing complexity in conformal field theories beyond free models.

Findings

Solution of integro-differential equations for Fubini-Study state complexity

Application of differential regularization techniques

Probing the geometric structure underlying complexity

Abstract

Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.