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 RyuTakayanagi (RT) surface, in terms of an integral over the curvature. Using the GaussBonnet 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.
 Publication:

Fortschritte der Physik
 Pub Date:
 June 2018
 DOI:
 10.1002/prop.201800034
 arXiv:
 arXiv:1710.01327
 Bibcode:
 2018ForPh..6600034A
 Keywords:

 High Energy Physics  Theory;
 Condensed Matter  Statistical Mechanics
 EPrint:
 12 pages, 14 figures, comments welcome, v2: references added, slight clarifications, matches version accepted by journal