I will discuss connections between bounds on the sphere packing density, the Laplacian spectral gaps of hyperbolic manifolds, and conformal field theories. A common theme is the systematic use of harmonic analysis to set up infinite-dimensional linear programs producing bounds on the spectral gap. The resulting bounds are often nearly or exactly sharp. In particular, this allows us to determine the set of spectral gaps attained by hyperbolic $2$-orbifolds. The sharpness of the linear programming bounds has been proven for sphere packing in $8$ and $24$ dimensions and independently for the spectral gap of conformal field theories. I will explain the precise mapping between these two results.