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Register a new userHolographic Complexity Growth Rate in a dual FLRW Universe
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In this paper, taking the large $R$ limit and using the complexity-volume duality, we investigate the holographic complexity growth rate of a field state defined on the universe located at an asymptotical AdS boundary in Gauss-Bonnet gravity and massive gravity, respectively. For the Gauss-Bonnet gravity case, its growth behavior of the state mainly presents three kinds of contributions: one, as a finite term viewed as an interaction term, comes from a conserved charge, the second one is from the spatial volume of the universe and the third one relates the curvature of the horizon in the AdS Gauss-Bonnet black hole, where the Gauss-Bonnet effect plays a vital role on such growth rate. For massive gravity case, except the first divergent term still obeying the growth rate of the spatial volume of the Universe, its results reveal the more interesting novel phenomenons: beside the conserved charge $E$, the graviton mass term also provides its effect to the finite term; and the third divergent term is determined by the spatial curvature of its horizon $k$ and graviton mass effect; furthermore, the graviton mass effect can be completely responsible for the second divergent term as a new additional term saturating an area law.
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It is assumed that the holographic complexities such as the complexity-action (CA) and the complexity-volume (CV) conjecture are dual to complexity in field theory. However, because the definition of the complexity in field theory is still not complete, the confirmation of the holographic duality of the complexity is ambiguous. To improve this situation, we approach the problem from a different angle. We first identify minimal and genuin properties that the filed theory dual of the holographic complexity should satisfy without assuming anything from the circuit complexity or the information theory. Based on these properties, we propose a field theory formula dual to the holographic complexity. Our field theory formula implies that the complexity between certain states in two dimensional CFTs is given by the Liouville action, which is compatible with the path-integral complexity. It gives natural interpretations for both the CA and CV conjectures and identify what their reference states are. When applied to the thermo-field double states, it also gives consistent results with the holographic results in the CA conjecture: both the divergent term and finite term.
We study holographic subregion complexity, and its possible connection to purification complexity suggested recently by Agon et al. In particular, we study the conjecture that subregion complexity is the purification complexity by considering holographic purifications of a holographic mixed state. We argue that these include states with any amount of coarse-graining consistent with being a purification of the mixed state in question, corresponding holographically to different choices of the cutoff surface. We find that within the complexity = volume and complexity = spacetime volume conjectures, the subregion complexity is equal to the holographic purification complexity. For complexity = action, the subregion complexity seems to provide an upper bound on the holographic purification complexity, though we show cases where this bound is not saturated. One such example is provided by black holes with a large genus behind the horizon, which were studied by Fu et al. As such, one must conclude that these offending geometries are not holographic, that CA must be modified, or else that holographic subregion complexity in CA is not dual to the purification complexity of the corresponding reduced state.
Following a methodology similar to cite{Alishahiha:2015rta}, we derive a holographic complexity for two dimensional holographic superconductors (gauge/string superconductors) with backreactions. Applying a perturbation method proposed by Kanno in Ref. cite{kanno}, we study behaviors of the complexity for a dual quantum system near critical points. We show that when a system moves from the normal phase ($T>T_c$) to the superconductor phase ($T<T_c$), the holographic complexity will be divergent.
In this paper, we will propose a universal relation between the holographic complexity (dual to a volume in AdS) and the holographic entanglement entropy (dual to an area in AdS). We will explicitly demonstrate that our conjuncture hold for all a metric asymptotic to AdS$_3$, and then argue that such a relation should hold in general due to the AdS version of the Cavalieri principle. We will demonstrate that it holds for Janus solution, which have been recently been obtained in type IIB string theory. We will also show that this conjecture holds for a circular disk. This conjecture will be used to show that the proposal that the complexity equals action, and the proposal that the complexity equal volume can represent the same physics. Thus, using this conjecture, we will show that the black holes are fastest computers, using the proposal that complexity equals volume.
Quantum complexity of a thermofield double state in a strongly coupled quantum field theory has been argued to be holographically related to the action evaluated on the Wheeler-DeWitt patch. The growth rate of quantum complexity in systems dual to Einstein-Hilbert gravity saturates a bound which follows from the Heisenberg uncertainty principle. We consider corrections to the growth rate in models with flavor degrees of freedom. They are realized by adding a small number of flavor branes to the system. Holographically, such corrections come from the DBI action of the flavor branes evaluated on the Wheeler-DeWitt patch. We relate corrections to the growth of quantum complexity to corrections to the mass of the system, and observe that the bound on the growth rate is never violated.
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