Averaging over codes and an SU(2) modular bootstrap

Error-correcting codes are known to define chiral 2d lattice CFTs where all the U(1) symmetries are enhanced to SU(2). In this paper, we extend this construction to a broader class of length-n codes which define full (non-chiral) CFTs with SU(2)(n) symmetry, where n = c + (c) over bar. We show that codes give a natural discrete ensemble of 2d theories in which one can compute averaged observables. The partition functions obtained from averaging over all codes weighted equally is found to be given by the sum over modular images of the vacuum character of the full extended symmetry group, and in this case the number of modular images is finite. This averaged partition function has a large gap, scaling linearly with n, in primaries of the full SU(2)(n) symmetry group. Using the sum over modular images, we conjecture the form of the genus-2 partition function. This exhibits the connected contributions to disconnected boundaries characteristic of wormhole solutions in a bulk dual.

Automated summary

This paper investigates how error-correcting codes define conformal field theories and extends their applicability. By calculating averaged observables of error-correcting codes, the partition function is obtained and shown to have special properties. The findings contribute to the understanding of the relationship between conformal field theories and error-correcting codes.