Abstract
We establish a precise relation between the modular bootstrap, used to con- strain 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 repro- duce 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 {Delta}_0underset{sim }{<}c/mathrm{8.503.}
Highlights
Charting the space of quantum field theories is a central task of theoretical physics, which has received renewed impetus with the modern resurgence of the conformal bootstrap program
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 show that the analytic functionals developed earlier for the correlator conformal bootstrap can be adapted to this context
Summary
Charting the space of quantum field theories is a central task of theoretical physics, which has received renewed impetus with the modern resurgence of the conformal bootstrap program. The weak gravity conjecture [3], motivated by properties of extremal black holes, translates into constraints on the spectrum of charged states in large-N CFTs, with no known origin in quantum field theory. The c → ∞ limit is especially interesting because large-c CFTs with sparse spectrum are dual to AdS3 quantum gravity In this limit, we are able to find an improved (though still suboptimal) functional that leads to the Virasoro analytic bound ∆0(c) c/8.503. Like the Hellerman bound, our bound constrains the spectrum of black holes in 3D quantum gravity It is related (though somewhat indirectly due to the distinction between Virasoro and U(1)c) to constraints on the density of sphere packing in high dimensions. Two appendices contain basic reference material on modular forms and further technical details on analytic functionals
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