  Quantum chaotic systems are often defined via the assertion that their
spectral statistics coincides with, or is well approximated by, random matrix
theory. In this paper we explain how the universal content of random matrix
theory emerges as the consequence of a simple symmetry-breaking principle and
its associated Goldstone modes. This allows us to write down an effective-field
theory (EFT) description of quantum chaotic systems, which is able to control
the level statistics up to an accuracy ${\cal O} \left(e^{-S} \right)$ with $S$
the entropy. We explain how the EFT description emerges from explicit
ensembles, using the example of a matrix model with arbitrary invariant
potential, but also when and how it applies to individual quantum systems,
without reference to an ensemble. Within AdS/CFT this gives a general framework
to express correlations between "different universes" and we explicitly
demonstrate the bulk realization of the EFT in minimal string theory where the
Goldstone modes are bound states of strings stretching between bulk spectral
branes. We discuss the construction of the EFT of quantum chaos also in higher
dimensional field theories, as applicable for example for higher-dimensional
AdS/CFT dual pairs.
