Comments on the holographic description of Narain theories

TL;DR

The paper investigates the holographic description of Narain CFTs and proposes that, in the large central charge limit, their primary-state density is bounded below by the density of $U(1)$ gravity, leading to a maximal spectral gap $\Delta_1 \le c/(2\pi e)$. It tests this conjecture using chiral lattice CFTs and quantum stabilizer-code CFTs, deriving a new bound on quantum codes and showing that the density variance is exponentially small in $c$ for both code and chiral ensembles, indicating strong self-averaging. The authors adapt the Hartman–Keller–Stoica sparseness framework to Narain theories, identifying a Narain-sparseness regime with $\varrho(\Delta) \le e^{2\pi\Delta}$ for $0<\Delta \le c/(4\pi)$, while averaged Narain theories (the $U(1)$ gravity case) remain sparse and Cardy behavior sets in beyond $\Delta \sim c/(2\pi)$. If correct, this scenario supports a bulk dual where each Narain CFT maps to a quasi-classical bulk sector plus light matter and suggests a deep link between holography, ensemble averages, and discrete geometry via sphere packing and coding theory.

Abstract

We discuss the holographic description of Narain $U(1)^c\times U(1)^c$ conformal field theories, and their potential similarity to conventional weakly coupled gravity 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 $Δ_1=c/(2πe)$. 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.

Figures (5)

• Figure 1: Number of points of a cubic "lattice" of size $\sqrt{2}$ within a ball of radius $2\alpha c$. Blue line: $\lambda(\alpha,0)$. Orange line: $\lambda(\alpha,2)$. Green line: $\ln(\pi e\alpha)$. For $\alpha<p/4$, formally $\lambda(\alpha,p)=-\infty$, which means no lattice vectors lie within the sphere of radius $2\alpha c < p c/2$. Inset: $\lambda(\alpha,0)$, blue line, vs sparseness condition $2\pi \alpha$, black dashed line, see section \ref{['sec:HKS']}.
• Figure 2: Density of primary states of averaged code theory. Dashed blue and orange lines: ${\overline \lambda}_1$ and ${\overline \lambda}_2$ correspondingly. Green line: ${\overline \lambda}=\max({\overline \lambda}_1,{\overline \lambda}_2)$. Brown line: Cardy formula $\lambda_C=\ln(2\pi\, e\, \alpha)$.
• Figure 3: Full density of states of the averaged Narain theory -- $U(1)$ gravity. Blue lines: $S_1/c$ and $S_2/c$ as functions of $e=E/c$, where the full density of states (entropy) of $U(1)$ gravity is $S=\max(S_1,S_2)$. $S_1$ dominates for $E<0$ and $S_2$ for $E>0$. Brown line: the sparseness condition $2\pi(e+1/12)$. $S_1(e)/c$ violates sparseness condition at small $e+1/12$. Red line: Cardy formula $\ln(\varrho)/c=\pi\sqrt{4e/3}$\ref{['Cardyrho']}.
• Figure 4: Sparseness condition $2\pi \alpha$ (blue line) vs. density of states of averaged chiral theory, $\lambda_{\it ch}(\alpha)$ (orange) and $\lambda_C(\alpha)$ (green).
• Figure 5: Leading (exponential) contribution to the square root of the variance of the density of states (blue line) vs. mean density of states (orange line).