Abstract
We study the influence of angular momentum on quantum complexity for CFT states holographically dual to rotating black holes. Using the holographic complexity=action (CA) and complexity=volume (CV) proposals, we study the full time dependence of complexity and the complexity of formation for two dimensional states dual to rotating BTZ. The obtained results and their dependence on angular momentum turn out to be analogous to those of charged states dual to Reissner-Nordström AdS black holes. For CA, our computation carefully accounts for the counterterm in the gravity action, which was not included in previous analysis in the literature. This affects the complexity early time dependence and its effect becomes negligible close to extremality. In the grand canonical ensemble, the CA and CV complexity of formation are linear in the temperature, and diverge with the same structure in the speed of light angular velocity limit. For CA the inclusion of the counterterm is crucial for both effects. We also address the problem of studying holographic complexity for higher dimensional rotating black holes, focusing on the four dimensional Kerr-AdS case. Carefully taking into account all ingredients, we show that the late time limit of the CA growth rate saturates the expected bound, and find the CV complexity of formation of large black holes diverges in the critical angular velocity limit. Our holographic analysis is complemented by the study of circuit complexity in a two dimensional free scalar model for a thermofield double (TFD) state with angular momentum. We show how this can be given a description in terms of non-rotating TFD states introducing mode-by-mode effective temperatures and times. We comment on the similarities and differences of the holographic and QFT complexity results.
Highlights
Complexity=volume (CV) [4, 5] and complexity=action (CA) [6, 7]
In this work we studied various aspects of holographic complexity for states with rotation dual to AdS black holes, and extended the QFT complexity analysis to a rotating thermofield double state of a 2d free boson
For the case of rotating BTZ black holes, we carried out a thorough study, refining existing results and analysing in detail the role of the counterterm action in CA [9] and the full time dependence of the holographic complexity proposals
Summary
We extend part of the above holographic analysis to four-dimensional KerrAdS black holes. The axial, rather than spherical, symmetry of the solution complicates. In the left and center panels: J = 0.1 (purple solid), J = 1 (green dashed) and J = 1.999 (blue dotted). As we comment below, even the null hypersurfaces foliation of Kerr-AdS spacetimes needed to construct the WDW patch was only worked out recently and is only known in implicit form [107].
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