Effective action, spectrum and first law of wedge holography

In this paper, we study the effective action, the mass spectrum and the first law of entanglement entropy for a novel doubly holographic model called wedge holography. We work out 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 effective action yields the correct Weyl anomaly. Interestingly, 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. We also analyze the mass spectrum of 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. 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.