Geometry and Complexity of Path Integrals in Inhomogeneous CFTs

TL;DR

The paper develops a path integral optimization framework for inhomogeneous 2d CFTs defined on curved backgrounds with metric $ds^2 = f(x)^2 d\tau^2 + dx^2$, and shows optimal path-integral geometries are hyperbolic and embed as slices of $AdS_3$ where Einstein's equations coincide with the minimal complexity condition. By solving the curved-background Liouville equation, it derives explicit optimized metrics for vacuum, primary states, and thermofield double states, and computes the on-shell Liouville action to obtain the path integral complexity, finding that leading UV divergences are universal while finite terms depend on the inhomogeneity through $f(x)$ and a position-dependent cutoff. The work also connects entanglement entropy in these inhomogeneous CFTs to Hill's equations and kinematic space, revealing how energy density and local scales are modulated by the background, and analyzes concrete deformations (Möbius/SSD, Rainbow, constant curvature) and boundary effects within BCFTs. Together, these results establish a geometric and holographic interpretation of path integral complexity in inhomogeneous systems and suggest broad applications to quantum circuits, tensor networks, and dynamical holography. The framework provides a tractable route to study time-dependent and driven inhomogeneous CFTs, with implications for understanding holographic duals and the role of geometry in quantum state preparation.

Abstract

In this work we develop the path integral optimization in a class of inhomogeneous 2d CFTs constructed by putting an ordinary CFT on a space with a position dependent metric. After setting up and solving the general optimization problem, we study specific examples, including the Möbius, SSD and Rainbow deformed CFTs, and analyze path integral geometries and complexity for universal classes of states in these models. We find that metrics for optimal path integrals coincide with particular slices of $AdS_3$ geometries, on which Einstein's equations are equivalent to the condition for minimal path integral complexity. We also find that while leading divergences of path integral complexity remain unchanged, constant contributions are modified in a universal, position dependent manner. Moreover, we analyze entanglement entropies in inhomogeneous CFTs and show that they satisfy Hill's equations, which can be used to extract the energy density consistent with the first law of entanglement. Our findings not only support comparisons between slices of bulk spacetimes and circuits of path integrations, but also demonstrate that path integral geometries and complexity serve as a powerful tool for understanding the interesting physics of inhomogeneous systems.

Figures (1)

• Figure 1: The spacetime boundary of the optimized path integral for the SSD deformed background on the interval $x\in [ -L, L ]$. If one treats the boundary conditions carefully, and solves for the boundary cosmological constant $\mu_B$, one finds that the spacelike portions of the boundary $\tau=F_-(x)$ (orange) and $\tau=F_+(x)$ (green) intersect with each other, yielding a finite domain of path integration.