A complexity/fidelity susceptibility g-theorem for AdS3_3/BCFT2_2

We use a recently proposed holographic Kondo model as a well-understood example of AdS/boundary CFT (BCFT) duality, and show explicitly that in this model the bulk volume decreases along the RG flow. We then obtain a proof that this volume loss is indeed a generic feature of AdS/BCFT models of the type proposed by Takayanagi in 2011. According to recent proposals holographically relating bulk volume to such quantities as complexity or fidelity susceptibility in the dual field theory, this suggests the existence of a complexity or fidelity susceptibility analogue of the Affleck-Ludwig g-theorem, which famously states the decrease of boundary entropy along the RG flow of a BCFT. We comment on this possibility.Comment: 24 pages, 4 figures v2: added citations and minor clarification

WdW-patches in AdS3_{3} and complexity change under conformal transformations II

We study the null-boundaries of Wheeler-de Witt (WdW) patches in three dimensional Poincare-AdS, when the selected boundary timeslice is an arbitrary (non-constant) function, presenting some useful analytic statements about them. Special attention will be given to the piecewise smooth nature of the null-boundaries, due to the emergence of caustics and null-null joint curves. This is then applied, in the spirit of our previous paper arXiv:1806.08376, to the problem of how complexity of the CFT$_2$ groundstate changes under a small local conformal transformation according to the action (CA) proposal. In stark contrast to the volume (CV) proposal, where this change is only proportional to the second order in the infinitesimal expansion parameter $\sigma$, we show that in the CA case we obtain terms of order $\sigma$ and even $\sigma\log(\sigma)$. This has strong implications for the possible field-theory duals of the CA proposal, ruling out an entire class of them.Comment: 31 pages + appendices, 9 figures v2: minor improvements, matches published versio

Discrete scale invariance in holography and an argument against the complexity = action proposal

The AdS/CFT correspondence often motivates research on questions in gravitational physics whose relevance might not be immediately clear from a purely GR perspective, but which are nevertheless interesting. In these proceedings, we summarise two such results recently obtained by the author. One concerns, broadly speaking, the possible isometry groups of a spacetime sourced by physical matter. The other one provides a possible argument against the recently proposed complexity = action conjecture

Discrete scale invariance in holography and an argument against the complexity=action proposal

The AdS/CFT correspondence often motivates research on questions in gravitational physics whose relevance might not be immediately clear from a purely GR-perspective, but which are nevertheless interesting. In these proceedings, we summarise two such results recently obtained by the author. One concerns, broadly speaking, the possible isometry-groups of a spacetime sourced by physical matter. The other one provides a possible argument against the recently proposed complexity=action conjecture.Comment: 5 pages, 1 figure. Slightly extended version of proceedings for the 6th Conference of the Polish Society on Relativit

Entanglement and defect entropies in gauge/gravity duality

In this thesis we investigate and use geometrical prescriptions for the calculation of entanglement entropy in field theories that have a gravity dual according to gauge/gravity duality. The main results of this work will arise from the application of our findings to the study of entanglement and defect entropies in a holographic model of the Kondo effect. Gauge/gravity duality is an important tool for the study of strongly coupled systems. We give a short review over the related idea of the holographic principle and the realisation of the AdS/CFT correspondence in string theory. We also introduce the concept of entanglement entropy and review the methods of holographically calculating it. We then apply recent prescriptions for calculating holographic entanglement entropy in gravitational theories with higher curvature terms to specific example spacetimes, such as stationary black holes, and obtain analytical solutions for extremal surfaces defining entanglement entropy that wrap around the black holes. We argue that these surfaces are unphysical by discussing how they violate certain well motivated causality constraints. We then investigate the geometrical properties of certain models of dualities between AdS spaces and boundary CFTs, with a special interest in a recently proposed holographic model of the Kondo effect. Understanding the impact of energy conditions on the allowed bulk geometries will be one of the main results of this thesis. We then apply the knowledge gained from these studies to the specific Kondo model, and numerically calculate entanglement and impurity entropies. These quantities can be interpreted in terms of the RG flow that the Kondo model undergoes. It will also be discussed in detail to which extend the holographic model reproduces field theory expectations, and how it can be improved. Furthermore, we investigate recent proposals of defining holographic measures of complexity. This is a quantity in quantum information theory. We end with an outlook on possible future research directions

Bending branes for DCFT in two dimensions

We consider a holographic dual model for defect conformal field theories (DCFT) in which we include the backreaction of the defect on the dual geometry. In particular, we consider a dual gravity system in which a two-dimensional hypersurface with matter fields, the brane, is embedded into a three-dimensional asymptotically Anti-de Sitter spacetime. Motivated by recent proposals for holographic duals of boundary conformal field theories (BCFT), we assume the geometry of the brane to be determined by Israel junction conditions. We show that these conditions are intimately related to the energy conditions for the brane matter fields, and explain how these energy conditions constrain the possible geometries. This has implications for the holographic entanglement entropy in particular. Moreover, we give exact analytical solutions for the case where the matter content of the brane is a perfect fluid, which in a particular case corresponds to a free massless scalar field. Finally, we describe how our results may be particularly useful for extending a recent proposal for a holographic Kondo model.Comment: 35 pages + appendices, 12 figures, v2: added references and a paragraph on negative tension solutions, v3: updated reference

Complexity change under conformal transformations in AdS3_{3}/CFT2_{2}

Using the volume proposal, we compute the change of complexity of holographic states caused by a small conformal transformation in AdS$_{3}$/CFT$_{2}$. 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.Comment: v2: 23 pages, 5 figures, added references and one entirely new section about a comparison to a field theory proposal v3: 27 pages, 5 figures, minor improvements. Matches published versio

Conformal field theory complexity from Euler-Arnold equations

Defining complexity in quantum field theory is a difficult task, and the main challenge concerns going beyond free models and associated Gaussian states and operations. One take on this issue is to consider conformal field theories in 1+1 dimensions and our work is a comprehensive study of state and operator complexity in the universal sector of their energy-momentum tensor. The unifying conceptual ideas are Euler-Arnold equations and their integro-differential generalization, which guarantee well-posedness of the optimization problem between two generic states or transformations of interest. The present work provides an in-depth discussion of the results reported in arXiv:2005.02415 and techniques used in their derivation. Among the most important topics we cover are usage of differential regularization, solution of the integro-differential equation describing Fubini-Study state complexity and probing the underlying geometry.Comment: 31 pages + appendicies, 2 figures, extended version of arXiv:2005.02415 v2: added references and minor improvement