Solutions of Modular Bootstrap Constraints from Quantum Codes

Abstract

Modular invariance imposes rigid constrains on the partition functions of two-dimensional conformal field theories. Many fundamental results follow strictly from modular invariance, giving rise to the numerical modular bootstrap program. Here we report a way to assign to a particular family of quantum error correcting codes a family of "code CFTs" -a subset in the space of Narain CFTs. This relation reduces modular invariance of the 2d CFT partition function to simple algebraic relations at the level of a multivariate polynomial characterizing the corresponding code. Using this relation we construct many explicit examples of physically distinct isospectral theories, as well as many examples of non-holomorphic functions, which satisfy all basic properties of the 2d CFT partition function, yet are not associated with any known theory.

Comments on the holographic description of Narain theories

“…which can be interpreted as a sum over different handlebodies on the gravity side. This led to a suggestion in [21] that the ensemble of chiral theories also has a holographic description, similar to the one of the Narain case. The Fourier expansion of the Eisenstein series is…”

“…In particular, from this condition follows a bound on the spectral gap for Narain theories, ∆ 1 ≤ c/(2πe). To probe the consistency of this scenario and the hypothesis (1.5), we consider the code CFTs of [21,22] associated with quantum stabilizer codes as well as chiral theories associated with even self-dual lattices. We show that (1.5) leads to a novel bound on quantum codes, which is consistent with previously known results.…”

Chern-Simons invariants from ensemble averages

“…This holographic duality was studied further in e.g. [9][10][11][12][13][14][15]. In this paper we consider the generalization where the associated CFT moduli space is a more general type of Narain moduli space associated with an indefinite quadratic form Q of rank p + q and signature (p, q): In the process of generalizing to an arbitrary integral quadratic form Q, we will encounter many interesting features which were not present in the previous studies.…”

“…where g is a PSL(2, Z) matrix of the form a b c d , and Φ(g) ∈ Z is the Rademacher function. 10 Note that the phase factor exp(2πiσΦ(g)/24) as well as the eta functions η(g • τ ) pη (g • τ ) q depend on the choice of a representative of the quotient Γ ∞ \PSL(2, Z), 10 The modular transformation rule for the eta function is given by…”

“…where the normalization factor M (Q) is known as the mass of the quadratic form Q. Note that holography for chiral theories was discussed in [10], see also [11]. It is far from clear physically, however, why we need to consider such an ensemble average.…”