Averaging over codes and an $SU(2)$ modular bootstrap

TL;DR

This paper extends the connection between error-correcting codes and 2d conformal field theories to non-chiral cases with $SU(2)^n$ symmetry, providing a framework for averaging observables and exploring holographic dualities.

Contribution

It introduces a new class of codes defining non-chiral CFTs with $SU(2)^n$ symmetry and analyzes their averaged partition functions and genus-2 contributions.

Findings

Averaged partition function is a sum over a finite set of modular images.

The averaged theory exhibits a large gap in primary operators scaling linearly with $n$.

Conjectured genus-2 partition function shows wormhole-like connected contributions.

Abstract

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 $n=c+\bar c$. 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.