  Synthetic dimensions alter one of the most fundamental properties in nature,
the dimension of space. They allow, for example, a real three-dimensional
system to act as effectively four-dimensional. Driven by such possibilities,
synthetic dimensions have been engineered in ongoing experiments with ultracold
matter. We show that rotational states of ultracold molecules can be used as
synthetic dimensions extending to many - potentially hundreds of - synthetic
lattice sites. Microwaves coupling rotational states drive fully controllable
synthetic inter-site tunnelings, enabling, for example, topological band
structures. Interactions leads to even richer behavior: when molecules are
frozen in a real space lattice with uniform synthetic tunnelings, dipole
interactions cause the molecules to aggregate to a narrow strip in the
synthetic direction beyond a critical interaction strength, resulting in a
quantum string or a membrane, with an emergent condensate that lives on this
string or membrane. All these phases can be detected using measurements of
rotational state populations.
