Conformal field theory complexity from Euler-Arnold equations

TL;DR

This work develops a unified, geometry-based treatment of state and operator complexity in 1+1D conformal field theories by modeling circuits as Virasoro group elements acting on energy eigenstates. Using a Fubini-Study metric, it derives a regulated, integro-differential variational problem whose solutions describe optimal conformal transformations; it analyzes the resulting geometry via geodesic equations, infinitesimal triangles, and an equivalent invertible metric to extract curvature properties. The study connects these FS-based complexities to Euler-Arnold equations such as KdV, Camassa–Holm, and Hunter–Saxton, and clarifies the role of left- vs right-invariance, differentiating state-based complexity from operator-based variants. These results provide a rigorous QFT framework to compare with holographic complexity proposals and point to fertile directions for extensions to broader symmetry algebras and holographic setups.

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.

Figures (2)

• Figure 1: The space of allowed maps can be described via circular fibres over a base-space. This base space can be described as the space of allowed maps for which $f(0)=0$, while the points along the fibre have $f(\alpha)=0$. The circuit $f(\tau,\sigma)=f(\tau+\sigma)$ is trivially a geodesic for any $\Pi$ and winds around such a fibre as an uncontractible cycle, see section \ref{['sec::fibres']}. Assume for a given $\Pi$, a geodesic circuit $f(\tau,\sigma)$ (solid red curve) connects two points $f_0(\sigma),f_1(\sigma)$ in the base space (shaded area). If the cycles are null, then this circuit can be wiggled along the circle by an arbitrary function $\alpha(\tau)$ (solid, dashed and dotted red lines above), and $f(\tau,\sigma+\alpha(\tau))$ will still be a geodesic of identical length -- the geodesic problem on the degenerate metric has a $U(1)$-gauge redundancy.
• Figure 2: Geodesic triangle in the space of maps. The dashed lines signify the non-geodesic paths $f_1(1,\sigma)=\sigma+\varepsilon\sin(\sigma)$ and $f_2(1,\sigma)=\sigma+\frac{1}{2}\varepsilon\sin(2\sigma)$ parametrised by $\varepsilon$. For small $\varepsilon$, we are able to approximately calculate the geodesics between these endpoints and the identity map $f(\sigma)=\sigma$ via the iterative procedure of section \ref{['sec::approx']}. These geodesics are parametrized by $\tau$, and we can also calculate the approximate geodesic and distance between the points at $0\leq \tau\leq1$ on these geodesics. All geodesics are shown as straight lines. As a function of $\varepsilon$ and $\tau$, we can now calculate the lengths of the sides of this geodesic triangle as well as the angles in its edges. This allows to probe the sectional curvatures of the space in the respective tangent plane.