Beurling-Selberg extremization and modular bootstrap at high energies

<jats:p>We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions <jats:inline-formula><jats:alternatives><jats:tex-math>[\Delta - \delta,\Delta + \delta]</jats:tex-math><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></jats:alternatives></jats:inline-formula> at asymptotically large <jats:inline-formula><jats:alternatives><jats:tex-math>\Delta</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>Δ</mml:mi></mml:math></jats:alternatives></jats:inline-formula> in 2d unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval <jats:inline-formula><jats:alternatives><jats:tex-math>[\Delta - \delta,\Delta + \delta]</jats:tex-math><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></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:tex-math>\delta \geq 0</jats:tex-math><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></jats:alternatives></jats:inline-formula>. When <jats:inline-formula><jats:alternatives><jats:tex-math>2\delta \in \mathbb Z_{\geq 0}</jats:tex-math><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></jats:alternatives></jats:inline-formula> the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in <jats:inline-formula><jats:alternatives><jats:tex-math>c>1</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>c</mml:mi><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula> theories.</jats:p>