  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.
