Renormalized holographic subregion complexity under relevant perturbations

Abstract

We construct renormalized holographic entanglement entropy (HEE) and subregion complexity (HSC) in the CV conjecture for asymptotically AdS 4 and AdS 5 geometries under relevant perturbations. Using the holographic renormalization method developed in the gauge/gravity duality, we obtain counter terms which are invariant under coordinate choices. We explicitly define different forms of renormalized HEE and HSC, according to conformal dimensions of relevant operators in the d = 3 and d = 4 dual field theories. We use a general embedding for arbitrary entangling subregions and showed that any choice of the coordinate system gives the same form of the counter terms, since they are written in terms of curvature invariants and scalar fields on the boundaries. We show an explicit example of our general procedure. Intriguingly, we find that a divergent term of the HSC in the asymptotically AdS 5 geometry under relevant perturbations with operators of conformal dimensions in the range 0 < ∆ < 1 2 and 7 2 < ∆ < 4 cannot be cancelled out by adding any coordinate invariant counter term. This implies that the HSCs in these ranges of the conformal dimensions are not renormalizable covariantly. We also write the plot of the renormalization procedure in the case of asymptotically AdS d+1 geometries, with d > 4.

Islands and complexity of eternal black hole and radiation subsystems for a doubly holographic model

“…For details see[73][74][75][76][77][78][79] and references therein 3. It is worth noting that in their paper, the authors argued that for a gravitating bath, one does not get a Page curve and instead end up with a constant entanglement curve.…”

Islands and complexity of eternal black hole and radiation subsystems for a doubly holographic model

“…For details see[69][70][71][72][73][74][75] and references therein 7. It is worth noting that in their paper, the authors argued that for a gravitating bath, one does not get a Page curve and instead end up with a constant entanglement curve.…”

Generalised proofs of the first law of entanglement entropy

“…To include the latter, one would simply add in the bulk field corresponding to the irrelevant operator, and compute renormalized quantities perturbatively in the irrelevant deformation. Other analyses where our methodology would be useful to extend the class of theories/states under consideration include discussions of subregion complexity and the first law of complexity [30,31] as well as analyses of the relation of holographic entanglement entropy to inverse mean curvature flow [32].…”