The operator content of a unitary, compact, two-dimensional Conformal Field Theory (CFT) on the circle or on the line is encoded by a spectral density of primary states, which is a sum of delta functions supported on a finite set of primary conformal dimensions. Modular invariance of the torus partition function can then be stated as the fact that this spectral density should equal its Fourier transform. In the mathematics literature, ‘’crystalline measures’’ are a special class of distributions of finite support, whose Fourier transform is also finitely supported. Relatedly, recent results in the sphere packing problem have pointed toward a renewed interest in the study of the so-called ‘’Fourier interpolation formulas’’, which allow one to reconstruct a function from some minimum information about its Fourier transform. In this informal talk, we will give a small introduction to these math concepts and try to draw inspiration for the modular bootstrap problem in 2d CFTs.

IPhT