Using the volume proposal, we compute the change of complexity of holographic states caused by a small conformal transformation in AdS3/CF T2. This computation is done perturbatively to second order. We give a general result and discuss some of its properties. As operators generating such conformal transformations can be explicitly constructed in CFT terms, these results allow for a comparison between holographic methods of defining and computing computational complexity and purely field-theoretic proposals. A comparison of our results to one such proposal is given.
Within gauge/gravity duality, we consider the AdS-Schwarzschild metric in arbitrary dimensions. We obtain analytical closed-form results for the two-point function, Wilson loop and entanglement entropy for strip geometries in the finite-temperature fieldtheory dual. According to the duality, these are given by the area of minimal surfaces of different dimension in the gravity background. Our analytical results involve generalised hypergeometric functions. We show that they reproduce known numerical results to great accuracy. Our results allow to identify new physical behaviour: for instance, we consider the entanglement density, i.e. the difference of entanglement entropies at finite and vanishing temperature divided by the volume of the entangling region. For field theories of dimension seven or higher, we find that the entanglement density displays non-monotonic behaviour as function of · T , with the strip width and T the temperature. This implies that the area theorem, proven for RG flows in general dimensions, does not apply here. This may signal the emergence of new degrees of freedom for AdS Schwarzschild black holes in eight or more dimensions.
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