Topological Complexity in AdS3/CFT2

TL;DR

The paper develops a curvature-based definition of subregion complexity in AdS3/CFT2 and evaluates it across three complementary perspectives: gravity via Gauss-Bonnet, tensor networks via an Ising-model mapping, and a CFT framework using kinematic space. It reveals that subregion complexity is topological in nature, with discrete, temperature-independent jumps determined by RT-surface topology, and provides explicit formulas for AdS, BTZ, and conical defects. A CFT-based volume expression in terms of entanglement entropies is derived, reproducing the zero-temperature gravity results for the full boundary and suggesting a deep link between complexity and entanglement. The work opens avenues for generalizations to higher dimensions, refined tensor-network mappings, and broader field-theoretic formulations of complexity.

Abstract

We consider subregion complexity within the AdS3/CFT2 correspondence. We rewrite the volume proposal, according to which the complexity of a reduced density matrix is given by the spacetime volume contained inside the associated Ryu-Takayanagi (RT) surface, in terms of an integral over the curvature. Using the Gauss-Bonnet theorem we evaluate this quantity for general entangling regions and temperature. In particular, we find that the discontinuity that occurs under a change in the RT surface is given by a fixed topological contribution, independent of the temperature or details of the entangling region. We offer a definition and interpretation of subregion complexity in the context of tensor networks, and show numerically that it reproduces the qualitative features of the holographic computation in the case of a random tensor network using its relation to the Ising model. Finally, we give a prescription for computing subregion complexity directly in CFT using the kinematic space formalism, and use it to reproduce some of our explicit gravity results obtained at zero temperature. We thus obtain a concrete matching of results for subregion complexity between the gravity and tensor network approaches, as well as a CFT prescription.

Figures (14)

• Figure 1: The subregion complexity is computed from the regularized volume contained in the region $\Sigma$, enclosed by $\gamma_{RT}$ and the segment $\gamma_\epsilon$ of the cutoff surface.
• Figure 2: The two phases of a system with two subregions. For phase I, $\Sigma$ is the union of the colored regions.
• Figure 3: Example of a configuration of RT surfaces for a several entangling intervals ($q=7$) in the vacuum.
• Figure 4: The two phases $a$ and $b$ of $\gamma_{RT}$ for a single interval (red line) in the presence of a black hole.
• Figure 5: Subregion complexity as function of the black hole mass, for a fixed entangling region.