Topological Complexity in AdS3/CFT2

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.

Topological Complexity in AdS$_3$/CFT$_2$

The paper "Topological Complexity in AdS$_3$/CFT$_2$" explores the concept of subregion complexity within the framework of the AdS$_3$/CFT$_2$ correspondence. Here, the authors examine the notion that the complexity of a reduced density matrix is proportional to the spacetime volume contained within the associated Ryu-Takayanagi (RT) surface. By rewriting the volume proposal as an integral over curvature, they leverage the Gauss-Bonnet theorem to evaluate this for various configurations of entangling regions and temperatures.

Key Findings and Methodological Advances

1. Curvature Integral Approach: The authors reformulate the complexity measure in terms of an integral of the scalar curvature over the volume, potentially offering a more natural dimensionless framework, particularly beneficial in AdS$_3$ where the simplicity of the local curvature properties provides clear results across various configurations.
2. Gauss-Bonnet Theorem Application: This classical result is utilized to demonstrate that for intervals within these holographic setups, the complexity faces discrete jumps rather than gradual changes when the topology of the RT surface undergoes transitions. This was found to be independent of specific temperatures or the details of the entangling region, which suggests robust topological behavior inherent to the framework.
3. Tensor Network Interpretation: The paper builds a bridge between the holographic view and tensor networks. Through numerical studies, the paper suggests that subregion complexity correlates with the complexity of maps compressing reduced density matrices within random tensor networks. This perspective aligns the discontinuities found in holographic complexity with those in network-based models.
4. CFT Calculations via Kinematic Space: By invoking the kinematic space formalism, the authors propose a direct method to compute subregion complexity within conformal field theory itself. Their approach derives a consistent match between CFT quantities and gravitational results, specifically demonstrating it for zero-temperature scenarios and single entangling intervals.

Implications and Future Directions

The implications of this study offer intriguing insights into the interplay between quantum information measures such as entropy, complexity, and their corresponding holographic duals. The robust nature of topological features observed suggests potential avenues for exploring complexity in theories beyond standard holographic scenarios, perhaps informing definitions of quantum complexity in both condensed matter systems and quantum gravity arenas.

The exploration of tensor networks elucidates an underlying structure that might be key to understanding the emergence of spacetime geometry itself in these theories. Similarly, the suggestion of a method to calculate complexity via kinematic space formalism paves the way for broader applications and insights, particularly into finite temperature and more complex state configurations.

The paper raises several questions that warrant further research. The discrepancy in computed complexity values between analytical results and tensor network simulations highlights a need for deeper exploration of the roles of lattice structures and discretization. Moreover, extending these discussions to higher-dimensional spaces and other areas, like time-dependent or non-AdS scenarios, could further illuminate the universal aspects of complexity in quantum field theories and their gravitational duals.

Overall, this paper contributes significantly to the understanding of holographic complexity, crafting a journey that traverses gravitational calculations, information-theoretic insights, and theoretical constructs within both tensor networks and quantum field theory.