Geometry of Complexity in Conformal Field Theory

TL;DR

This work develops a geometric, quantitative framework for complexity in 1+1D CFTs by focusing on unitary circuits generated by the stress-energy tensor (Virasoro group) and comparing two cost-function paradigms: Fubini-Study state distance and direct T-insertion counting. It derives a second-order integro-differential equation for optimal circuits, provides perturbative solutions for small Fourier-mode targets, and analyzes the geometry via sectional curvatures, finding predominantly negative curvature in realistic settings. The study links these geometric insights to Euler-Arnold-type PDEs (KdV, Camassa-Holm, Hunter-Saxton) and discusses implications for holographic complexity while outlining avenues for extending to curved backgrounds and richer operator content.

Abstract

We initiate quantitative studies of complexity in (1+1)-dimensional conformal field theories with a view that they provide the simplest setting to find a gravity dual to complexity. Our work pursues a geometric understanding of complexity of conformal transformations and embeds Fubini-Study state complexity and direct counting of stress tensor insertion in the relevant circuits in a unified mathematical language. In the former case, we iteratively solve the emerging integro-differential equation for sample optimal circuits and discuss the sectional curvature of the underlying geometry. In the latter case, we recognize that optimal circuits are governed by Euler-Arnold type equations and discuss relevant results for three well-known equations of this type in the context of complexity.

Figures (1)

• Figure 1: A graphical representation of the circuit \ref{['sinsolution']}, where for simplicity we have set $c=0, h=1$ and $\epsilon=1$ in order to generate a result that is visible to the eye. For each fixed value of $\tau$, $f(\tau,\sigma)$ is a map of the circle to itself, depicted by the colored points unevenly spread out along the circles according to the function $f(\tau,\sigma)$ with evenly spaced input values for $\sigma$. The radial ordering (also emphasized by colour) corresponds to the progression of the parameter $\tau$ from $-1$ to $1$ in steps of $\Delta\tau=1/4$. The solid circles mark $\tau=-1,0,1$, respectively. At $\tau=0$, $f(\tau,\sigma)$ is the identity map by construction. "Wave breaking" happens when infinitesimally close points are mapped to the same location because $f'(\tau,\sigma)=0$.