Narain CFTs and quantum codes at higher genus

Abstract

Code CFTs are 2d conformal field theories defined by error-correcting codes. Recently, Dymarsky and Shapere generalized the construction of code CFTs to include quantum error-correcting codes. In this paper, we explore this connection at higher genus. We prove that the higher-genus partition functions take the form of polynomials of higher-weight theta functions, and that the higher-genus modular group acts as simple linear transformations on these polynomials. We explain how to solve the modular constraints explicitly, which we do for genus 2. The result is that modular invariance at genus 1 and genus 2 is much more constraining than genus 1 alone. This allows us to drastically reduce the space of possible code CFTs. We also consider a number of examples of “isospectral theories” — CFTs with the same genus 1 partition function — and we find that they have different genus 2 partition functions. Finally, we make connection to some 2d CFTs known from the modular bootstrap. The n = 4 theory conjectured to have the largest possible gap in Virasoro characters, the SO(8) WZW model, is a code CFT, allowing us to give an expression for its genus 2 partition function. We also find some other known CFTs which are not code theories but whose partition functions satisfy the same simple polynomial ansatz as the code theories. This leads us to speculate about the usefulness of the code polynomial form beyond the study of code CFTs.

Averaging over codes and an SU(2) modular bootstrap

“…Beyond genus 1, we conjecture a formula for the average genus-2 partition function (3.14), again using the sum over modular images starting from the (genus-2) character of the identity multiplet, recently discussed in [20,21]. This allows us to compute the expectation value of the product of the partition function on two disconnected tori.…”

“…Once we require that all character degeneracies are positive integers, we shall find a finite number of solutions, and furthermore we shall find that imposing higher-genus modular invariance reduces this number of possible solutions. 14 The idea of bootstrapping code theories by considering invariant polynomials was introduced in [6] and higher-genus constraints were incorporated in [20,21].…”

“…As a result, for a given code of length n there are a finite number of possible partition functions which are consistent with both modular invariance and positive SU(2) character degeneracies. Imposing higher-genus modular invariance [20,21] leads to an even smaller set of possible partition functions. In section 2, we apply this approach to our new construction of non-chiral SU(2) n theories, and compare to the number of theories obtained directly from the list of codes [22].…”

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

“…Another work on this code CFT is [KY23, HM22, DS21], and it would be interesting to apply it to these applications. There are other interesting works [ADK+21,HKM23] where the notion of Narain CFTs is extended.…”

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.…”