We show how the traversable wormhole induced by a double-trace
deformation of the thermofield double state can be understood as a
modular inclusion of the algebras of exterior operators. The effect of
this deformation is the creation of a new region of spacetime deep in
the bulk, corresponding to a non-trivial center between the left and
right algebras. This set-up provides a precise framework for
investigating how black hole interiors are encoded in the CFT. In
particular, we use modular theory to demonstrate that state dependence
is an inevitable feature of any attempt to represent operators behind
the horizon. Building on this geometrical structure, we propose that
modular inclusions may provide a more precise means of investigating the
nascent relationship between entanglement and geometry in the context of
the emergent spacetime paradigm.
Emergent spacetime and black hole probes from automorphic forms