  We propose a codimension two holography between a gravitational theory on a
$d+1$ dimensional wedge spacetime and a $d-1$ dimensional CFT which lives on
the corner of the wedge. Formulating this as a generalization of AdS/CFT, we
explain how to compute the free energy, entanglement entropy and correlation
functions of the dual CFTs from gravity. In this wedge holography, the
holographic entanglement entropy is computed by a double minimization
procedure. Especially, for a four dimensional gravity ($d=3$), we obtain a two
dimensional CFT and the holographic entanglement entropy perfectly reproduces
the known result expected from the holographic conformal anomaly. We also
discuss a lower dimensional example ($d=2$) and find that a universal quantity
naturally arises from gravity, which is analogous to the boundary entropy.
Moreover, we consider a gravity on a wedge region in Lorentzian AdS, which is
expected to be dual to a CFT with a space-like boundary. We formulate this new
holography and compute the holographic entanglement entropy via a Wick rotation
of the AdS/BCFT construction. Via a conformal map, this wedge spacetime is
mapped into a geometry where a bubble-of-nothing expands under time evolution.
We reproduce the holographic entanglement entropy for this gravity dual via CFT
calculations.
