We investigate the time-dependent entanglement entropy in the AdS space with a dS boundary which represents an expanding spacetime. On this time-dependent spacetime, we show that the Ryu–Takayanagi formula, which is usually valid in the static spacetime, provides a leading contribution to the time-dependent entanglement entropy. We also study the leading behavior of the entanglement entropy between the visible and invisible universes in an inflationary cosmology. The result shows that the quantum entanglement monotonically decreases with time and finally saturates a constant value inversely proportional to the square of the Hubble constant. Intriguingly, we find that even in the expanding universes, the time-dependent entanglement entropy still satisfies the area law determined by the physical distance.

We holographically investigate the renormalization group flow in a two-dimensional conformal field theory deformed by a relevant operator. If the relevant operator allows another fixed point, the UV conformal field theory smoothly flows to a new IR conformal field theory. From the holographic point of view, such a renormalization group flow can be realized as a dual geometry interpolating two different AdS boundaries. On this interpolating geometry, we investigate how the c-function of the entanglement entropy behaves along the RG flow analytically and numerically, which reproduces the expected central charges of UV and IR. We also show that the c-function monotonically decreases from UV to IR without any phase transition. *

We study the holographic entanglement entropy under small deformations of AdS, including timedependence. It is found through perturbative analysis that the divergent terms are not affected and the change appears only in the finite terms. We also consider the entanglement thermodynamic first law, and calculate the entanglement temperature and confirm that it is inversely proportional to the size of the entangling region.

Motivated by the recent progress on gravity duals of supersymmetric Chern-Simons matter theories, we consider classical membrane solutions in AdS 4 × M 1,1,1 . In particular, we present several types of exact solutions rotating in the Sasaki-Einstein 7-manifold whose isometry is SU (3)× SU (2) × U (1). We analyze the limiting behavior of macroscopic membranes and discuss how one can identify the dual operators and the implications of our result on their conformal dimensions.Recently we have seen great progress on our understanding of field theory duals for various AdS 4 backgrounds in string or M-theory. Most notable is the discovery of N = 6 supersymmetric Chern-Simons-matter theories which provide conformal field theory duals for AdS 4 ×S 7 /Z k [1]. This three-dimensional field theory has U (N )×U (N ) gauge symmetry with Chern-Simons (CS) kinetic terms of quantized level (k, −k). The interaction is also described by a quartic superpotential. In fact, as a quiver guage theory the data is exactly the same as the well-known conifold theory in four-dimensions [2], apart from the extra information on CS levels.It is an important issue how to generalize this duality to other backgrounds AdS 4 × Y 7 .In this paper, we are particularly interested in the examples which preserve N = 2, or eight supercharges. Mathematically Y 7 is then required to be Sasaki-Einstein, and we choose the so-called M 1,1,1 space [3]. It is constructed as U (1)-fibration over a six-dimensional Kähler-Einstein manifold CP 2 × CP 1 . The dual field theory is thus expected to enjoy SU (3) × SU (2) × U (1) global symmetry, where the U (1) part is the usual R-symmetry.In order to establish the duality relation, we need the Kaluza-Klein spectrum of 11dimensional supergravity on M 1,1,1 . It is computed in [4], and a three-dimensional quiver gauge theory was proposed as the dual of AdS 4 ×M 1,1,1 in [5], but a consistent superpotential could not be written down. Now with the new insight of Chern-Simons theories without the usual second-order Maxwell-type kinetic terms, we have a more reliable candidate. The CS duals have been given for a general class of the so-called Y p,q (CP 2 ) metrics [6,7]. For our interest here the relevant one has gauge symmetry U (N ) 3 with CS levels (k, k, −2k). The dual geometry is conjectured to be orbifolds AdS 4 × M 1,1,1 /Z k . A cubic superpotential, with conformal dimension two, is written in terms of the nine bifundamental chiral multipelts.The AdS/CFT duality relation implies that the M-theory spectrum in AdS 4 × M 1,1,1 /Z k gives the space of gauge singlet operators on the quiver gauge theory side. Instead of trying to quantize the supermembrane theory in a curved background, one can study classical membrane solutions and compare the result to field theory operators with large conformal dimensions. This is the strategy advocated first in [8] for AdS 5 × S 5 , and has been extensively used in the study of AdS/CFT relations. Quantitative results for non-supersymmetric solutions lead to very non-trivial checks of...