Sphere packing and quantum gravity

Abstract

We establish a precise relation between the modular bootstrap, used to constrain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiral algebra U(1) c maps exactly to the Cohn-Elkies linear programming bound on the sphere packing density in d = 2c dimensions. We also show that the analytic functionals developed earlier for the correlator conformal bootstrap can be adapted to this context. For c = 4 and c = 12, these functionals exactly reproduce the "magic functions" used recently by Viazovska [1] and Cohn et al. [2] to solve the sphere packing problem in dimensions 8 and 24. The same functionals are also applied to general 2D CFTs, with only Virasoro symmetry. In the limit of large central charge, we relate sphere packing to bounds on the black hole spectrum in 3D quantum gravity, and prove analytically that any such theory must have a nontrivial primary state of dimension ∆ 0 c/8.503.

Analytic functional bootstrap for CFTs in d > 1

“…Below we will present complete bases of such functionals, where completeness means that they fully capture the constraints of crossing symmetry on the line. Some of these bases have already appeared in print, while the existence of others was only indicated [22][23][24]29]. In any case, all of them are reviewed and/or constructed (accordingly) in appendix A, to which we refer the reader for further details.…”

“…Note that the full bases of ω − type functionals were previously constructed in [22] based on earlier results [23,24]. The ω + functional bases however have not been fully constructed explicitly before, with the exception of the fermionic β + 0 functional in [29].…”

“…The second class is obtained by tensoring holomorphic and antiholomorphic copies of 1d functionals, some of which have appeared in the literature [22][23][24]29] (including the ones mentioned above), others which we construct in detail. These product functionals exist in 2 Up to some important caveats as we shall see, namely that this functional set is not complete.…”

Correlators of the symmetric product orbifold

“…By the covering space method, this correlator is related to the torus partition function of the Ising CFT. This relation is well-known and was used for instance recently in [40]. Thus, this gives an easy method to derive modular differential equations for rational CFTs; the corresponding modular differential equations are reinterpreted as ordinary differential equations for the four-point function of twist fields.…”

Tauberian-Cardy formula with spin

“…On a more general ground, it would be interesting to see whether Tauberian theorems and/or Modular bootstrap program can say anything about the chaotic, irrational CFTs. An approach borrowing ideas from Tauberian techniques and that of extremal functions appearing in [45][46][47][48] might be useful in this regard. Furthermore, for holographic CFTs, we can only achieve a reduced regime of validity of Cardy formula compared to what is reported in [19].…”