Effective Action, Spectrum and First Law of Wedge Holography

Abstract

In this paper, we study the mass spectrum, the effective action and the first law of entanglement entropy for a novel doubly holographic model called wedge holography. Remarkably, there is always a massless mode of gravitons on the end-of-the-world branes in wedge holography. This happens because one imposes Neumann boundary condition on both branes. On the other hand, the massless mode disappears if one imposes Dirichlet boundary condition on one of the branes as in brane world theory and AdS/BCFT. We also study the effective action of quantum gravity on the branes. In the perturbative formulation, it is given by an infinite sum of Pauli-Fierz actions. In the non-perturbative formulation, the effective action is composed of a higher derivative gravity and a matter action. Usually, a higher derivative gravity can be renormalizable but suffers the ghost problem. For our case, since the effective theory on the brane is equivalent to Einstein gravity in the bulk, it must be ghost-free. We notice that the matter action plays an important role in eliminating the ghost. We also provide evidences that the higher derivative gravity on the brane is equivalent to a ghost-free multi-gravity. Besides, we prove that the non-perturbative effective action yields the correct Weyl anomaly. Remarkably, although the effective action on the brane is an infinite tower of higher derivative gravity, the holographic Weyl anomaly is exactly the same as that of Einstein gravity. Finally, we verify the first law of entanglement entropy for wedge holography. Interestingly, the massive fluctuations are irrelevant to the first order perturbation of the holographic entanglement entropy. Thus, in many aspects, the effective theory on the brane behaves like massless Einstein gravity.

Entanglement Island and Page Curve in Wedge Holography

“…which yields a massless gravitational mode on the brane [46]. On the other hand, the gravity becomes massive if one imposes Dirichlet boundary condition (DBC) [44] or conformal boundary condition (CBC) [45] on one or two of the branes.…”

“…Technically, that is because one imposes Neumann boundary condition (NBC) on the brane Q while imposing Dirichlet boundary condition (DBC) on the AdS boundary M . However, according to [46], only if one sets both NBCs on the two boundaries Q and M does the massless gravity appear. Besides, the massless mode is non-normalizable since the AdS boundary M locates at infinity.…”

“…Thus, it is also called codim-2 holography. In wedge holography, the effective theory on the branes is a CFT plus a ghost-free higher derivative gravity, or an equivalent multi-metric gravity, which behaves like Einstein gravity in many aspects [46]. For example, they yield the same holographic Weyl anomaly and the first law of entanglement entropy.…”

Multiverse in Karch-Randall Braneworld

“…In this setup, Q 2 contains a black hole whose Hawking radiation is collected by Q 1 . Literature on wedge holography can be found in [47][48][49][50].…”

AdS/BCFT and Island for curvature-squared gravity

“…Recently, a novel doubly holographic model called wedge holography has been proposed [46]. Remarkably, the effective theory of wedge holography on the brane is a ghost-free higher derivative gravity, which includes a massless mode and behaves like Einstein gravity in many aspects [47]. Wedge holography can be regarded as a holographic dual of the edge mode living on the boundary (codim-1 defect) [48].…”