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.

Complexity in the presence of a boundary

“…We are going to show that this is indeed the case. This phase transition is very similar to the one found in the case of two or more intervals in [57,58] in absence of boundaries. Similar finite jumps of the complexity have been found in [59][60][61].…”

Aspects of the first law of complexity

“…Hence it is reasonable to interpret the holographic results in this way, i.e., the holographic circuits invoke a new set of gates in preparing coherent states. 30 It is noteworthy that the κ = 1 cost function (4.36) is an exception to the above property. That is, δC κ=1 is first order in the small amplitudes of the coherent state, as shown in eq.…”

On subregion action complexity in AdS3 and in the BTZ black hole

“…Note that in this limit the log ε divergence disappears because it is suppressed by the segment length l. For comparison, the volume complexity of an interval for the BTZ [40,44] is:…”

“…and it is non-trivially independent on temperature. Subregion C V at equilibrium is a topologically protected quantity: for multiple intervals, the authors of [44] found the following result using the the Gauss-Bonnet theorem where l tot is the total length of all the segments and κ is the finite part, that depends on topology…”

“…In the CV conjecture, we can use eqs. (3.33) and (3.34) from [44] to determine mutual complexity. Considering the case of a double segment, we find that for d < d 0 the mutual complexity is constant:…”