Quantum Error-Correcting Codes and CFTs

There is a Narain CFT naturally associated to any self-dual quantum stabilizer code.  For these "code CFTs", the constraints of modular invariance reduce to simple algebraic equations satisfied by the enumerator polynomial of the code, which can be solved à la bootstrap to provide many examples of physically distinct CFTs with identical spectra.  The spectral gap of a code CFT corresponds to the error-correcting capacity of its associated code, which can be bounded from above and below.   I'll conclude with a discussion of the relation between the ensemble of large-c code CFTs and U(1) gravity.