  To a given algebraic curve we assign an infinite family of quantum curves
(Schr\"odinger equations), which are in one-to-one correspondence with, and
have the structure of, Virasoro singular vectors. For a spectral curve of a
matrix model we build such quantum curves out of an appropriate representation
of the Virasoro algebra, encoded in the structure of the
$\alpha/\beta$-deformed matrix integral and its loop equation. We generalize
this construction to a large class of algebraic curves by means of a refined
topological recursion. We also specialize this construction to various specific
matrix models with polynomial and logarithmic potentials, and among other
results, show that various ingredients familiar in the study of conformal field
theory (Ward identities, correlation functions and a representation of Virasoro
operators acting thereon, BPZ equations) arise upon specialization of our
formalism to the multi-Penner matrix model.
