Recently established connection between additive codes and Narain CFTs provides a new tool to construct theories with special properties and solve modular bootstrap constraints by reducing them to algebraic identities. We generalize previous constructions to include many new theories, in particular we show that all known optimal Narain theories, i.e. those maximizing the value of spectral gap, can be constructed from codes. For asymptotically large central charge
Non-rational Narain CFTs from codes over F4
A bstract We construct a map between a class of codes over F 4 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.
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- Award ID(s):
- 2013812
- NSF-PAR ID:
- 10348655
- Date Published:
- Journal Name:
- Journal of High Energy Physics
- Volume:
- 2021
- Issue:
- 11
- ISSN:
- 1029-8479
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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A<sc>bstract</sc> c we show there are code theories with the spectral gap growing linearly withc , with the coefficient saturating the conjectural upper bound. We therefore conjecture that optimal Narain theories for any value ofc can be obtained from codes. -
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A<sc>bstract</sc> We give a general construction relating Narain rational conformal field theories (RCFTs) and associated 3d Chern-Simons (CS) theories to quantum stabilizer codes. Starting from an abelian CS theory with a fusion group consisting of
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null (Ed.)A bstract There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E 8 theory, which is based on the root lattice of E 8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.more » « less
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Abstract We continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $ . We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω -stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].more » « less
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A bstract We discuss the holographic description of Narain U(1) c × U(1) c conformal field theories, and their potential similarity to conventional weakly coupled gravitational theories in the bulk, in the sense that the effective IR bulk description includes “U(1) gravity” amended with additional light degrees of freedom. Starting from this picture, we formulate the hypothesis that in the large central charge limit the density of states of any Narain theory is bounded by below by the density of states of U(1) gravity. This immediately implies that the maximal value of the spectral gap for primary fields is ∆ 1 = c /(2 πe ). To test the self-consistency of this proposal, we study its implications using chiral lattice CFTs and CFTs based on quantum stabilizer codes. First we notice that the conjecture yields a new bound on quantum stabilizer codes, which is compatible with previously known bounds in the literature. We proceed to discuss the variance of the density of states, which for consistency must be vanishingly small in the large- c limit. We consider ensembles of code and chiral theories and show that in both cases the density variance is exponentially small in the central charge.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/JHEP11(2021)016
