Beurling-Selberg extremization and modular bootstrap at high energies

Abstract

We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions [\Delta - \delta,\Delta + \delta] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">[</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>−</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>δ</mml:mi> <mml:mo stretchy="false" form="postfix">]</mml:mo> </mml:mrow> </mml:math> at asymptotically large \Delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Δ</mml:mi> </mml:math> in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval [\Delta - \delta,\Delta + \delta] <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">[</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>−</mml:mo> <mml:mi>δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>δ</mml:mi> <mml:mo stretchy="false" form="postfix">]</mml:mo> </mml:mrow> </mml:math> 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 for any \delta \geq 0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . When 2\delta \in \mathbb Z_{\geq 0} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>δ</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> 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&gt;1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> theories.