Comments on the holographic description of Narain theories

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 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 ∆ 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.

Adding Flavor to the Narain Ensemble

“…It is easy to see that this occurs when D > 2 by considering the explicit behavior of the lattice sum; see [2] for details. 4 To see that this is an eigenfunction of the Laplacian we note that τ…”

“…At first sight, constructing a random conformal field theory seems quite difficult, as it would involve an ensemble average over the space of conformal field theories, a space which is itself quite poorly understood. For this reason, recent work in this direction [2,3] has focused on CFTs with enhanced symmetry algebras where the space of CFTs can be understood precisely (related works in this direction include [4][5][6][7][8]).…”

A note on ensemble holography for rational tori

“…In this section we discuss the tentative bulk dual interpretation of the ensemble average over A k × A k rational torus CFTs encountered in the previous sections. In analogy with Narain duality [11][12][13][14][15][16][17][18][19][20] and earlier examples of rational ensemble holography [24][25][26], the bulk dual would be an exotic gravity theory described by a Chern-Simons action supplemented by a prescription to sum over certain topologies in the path integral. In the case of interest, we have two compact U (1) Chern-Simons theories at levels k and −k, and we will review some of the standard dictionary with rational torus CFT on the boundary [22,23].…”

“…However, the prescription specifying which topologies are to be included is far from clear. Various extensions of this example have been studied since then [13][14][15][16][17][18][19][20].…”

Averaging over moduli in deformed WZW models

“…It would be useful to have more examples of averaged CFTs in order to explore these issues further. The original Narain duality has been extended in several directions recently, to include chemical potentials [26], orbifolds of free bosons [27], and more general quadratic forms [28] (see also [29][30][31][32][33] for related progress). In this paper, we generalize the Narain duality to an ensemble of CFTs defined by the moduli space of the SU (N + 1) k WZW model deformed by exactly marginal current-current operators.…”