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Title:
Modular bootstrap at High energy and Beurling-Selberg Extremization
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Abstract:
We consider the universality of existence and saturation of asymptotic bounds in various quantities in 2D conformal field theory (CFT) . In particular, we obtain the lower and upper bounds on the number of operators within a high energy interval [Δ−δ,Δ+δ], of a modular invariant 2D CFT with a positive spectral density. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval [Δ−δ,Δ+δ] and have Fourier transforms of finite support. The optimization of the bounds over this choice turns out to be exactly the Beurling-Selberg extremization problem, widely known in analytic number theory. We review solutions of this problem and present the corresponding bounds on the number of operators. When 2δ ∈ Z≥0 the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in c > 1 theories. If time permits, I will briefly sketch over how the analysis can be generalized for asymptotics of OPE coefficient, possibly with some additional numerical input and for CFTs with global symmetry.
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