Abstract
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/(2πe). 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.
Highlights
If the sparseness condition (1.1) is satisfied, the density of states for ∆ > c/6 at leading order in 1/c is given by the Cardy formula
Which matches the density of states of the BTZ black hole, suggesting that the gravity dual is a theory of quasiclassical gravity plus o(c) additional matter fields
The conventional picture is that sparse 2d CFTs with large central charge are expected to be dual to quasiclassical gravity with some additional fields
Summary
A Narain CFT describes c ∈ N free scalar fields compactified on a c-dimensional torus. This is the analog of the Cardy formula for Narain theories, as can be deduced directly from modular invariance [20] It applies universally for ∆ c but as we will see below, in certain cases its validity extends to much smaller values of ∆. For ∆ 1 the summation over can be replaced by an integration, yielding (2.4) in the c 1 limit For this U(1) gravity theory, the Cardy formula (2.6) is valid for all values of ∆ > c/(2πe), i.e. when the number of states is exponential. 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 has a holographic description, similar to the one of the Narain case. By extending the conjecture (1.5) to chiral theories, we can predicts the spectral gap in large c limit to be bounded by ∆1 ≤ c/(4πe)
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