Relation between spectra of Narain CFTs and properties of associated boolean functions

Abstract

Recently, the construction of Narain CFT from a certain class of quantum error correcting codes has been discovered. In particular, the spectral gap of Narain CFT corresponds to the binary distance of the code, not the genuine Hamming distance. In this paper, we show that the binary distance is identical to the so-called EPC distance of the boolean function uniquely associated with the quantum code. Therefore, seeking Narain CFT with large spectral gap can be addressed by getting a boolean function with high EPC distance. Furthermore, this problem can be undertaken by finding lower Peak-to-Average Power ratio (PAR) with respect to the binary truth table of the boolean function. Though this is neither sufficient nor necessary condition for high EPC distance, we construct some examples of relatively high EPC distances referring to the constructions for lower PAR. We also see that codes with high distance are related to induced graphs with low independence numbers.

On the rationality and the code structure of a Narain CFT, and the simple current orbifold

“…In short, it is the B-form of the code, and as will be discussed in detail in section 2.2, such a correspondence provides a codetheoretic translation of the simple current orbifold. An interesting aspect of this finding is its connection with our previous paper [Fur22]. We previously found that the CFT spectral gap is in a sense equivalent to the Extended Periodic Criterion (EPC) distance 2 when translated into code theory.…”

“…In RCFT terminology, this is a map that gives a gluing together of a chiral part and an antichiral part, which is classified using the cohomology of the group. In a previous paper [Fur22], the author introduced the Boolean function f to write the spectral gap ∆ in terms of f and discussed the consideration of f where ∆ is large. Since this KSB can be regarded as a generalization of the Boolean function, we intend to set up a future work to see if in the general RCFT ∆ can be written using KSB.…”

“…What all of these extensions have in common is that there is a correspondence between the properties of the codes and the properties of the lattice. Thus, Narain CFT can be analyzed using codes, and it can also be transformed into discrete mathematical objects [Fur22]. In this article, We will begin by the more general F p code as a code and show how stabilizer codes appear as a special case of the code.…”

Narain CFTs from qudit stabilizer codes

“…There are also other directions of research related to quantum codes and CFTs [62][63][64][65][66][67][68]. It deserves further investigation to see whether our construction is relevant to these recent developments.…”

Optimal Narain CFTs from codes

“…The relation to quantum codes was recently extended and interpreted in terms of CFT Hilbert space in [13]. There are also "bottom-up" generalizations when the connection with codes is perceived as a tool to solve modular bootstrap constraints and construct interesting CFTs [14,15], also see [16,17] for the subsequent developments. In this paper we use this approach and introduce the umbrella construction which generalizes and encompasses the constructions of [14,15].…”