Abstract
We construct a map between a class of codes over F4 and a family of non-rational Narain CFTs. This construction is complementary to a recently introduced relation between quantum stabilizer codes and a class of rational Narain theories. From the modular bootstrap point of view we formulate a polynomial ansatz for the partition function which reduces modular invariance to a handful of algebraic easy-to-solve constraints. For certain small values of central charge our construction yields optimal theories, i.e. those with the largest value of the spectral gap.
Highlights
Narain theories, two-dimensional conformal theories of free fields compactified on a multidimensional torus, have enjoyed renewed attention recently
Two years ago spinless bootstrap constraints for theories with U(1)n ×U(1)n symmetry were shown in [9] to reduce to linear programming bounds of Cohen and Elkies on the density of sphere packings [11]
A certain family of Narain theories was found to be related to quantum stabilizer codes [12, 13]. These developments are conceptually similar. In both cases a subset of modular bootstrap constraints reduces to a well-known problem, the linear programming bounds of [11] in the case of sphere packings and those of Calderbank et al in the case of quantum codes [14]
Summary
Two-dimensional conformal theories of free fields compactified on a multidimensional torus, have enjoyed renewed attention recently. The problems of maximizing the CFT spectral gap, sphere packing density, and code Hamming (or other appropriate) distance are qualitatively similar, which can be utilized e.g. to shed light on holographic properties of Narain theories [14]. For Narain CFTs, we propose to call a theory optimal if it maximizes the spectral gap for the given central charge c and extremal if it saturates (any subset) of the modular bootstrap constraints.
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