论文标题
在高能量时，贝尔林 - 塞尔伯格极端和模块化自举
Beurling-Selberg Extremization and Modular Bootstrap at High Energies
论文作者
论文摘要
我们考虑在缩放尺寸的窗口中的运算符数量上的上限和下限，$ [δ-δ，δ+δ] $在2D单位模块化不变性CFT中均非$δ$。这些界限取决于函数的选择，该功能将间隔$ [δ-δ，δ+δ] $的特征功能进行了主要化和缩小，并具有有限支持的傅立叶变换。在此选择中，边界的优化完全是贝尔林 - 塞尔伯格极端化问题，在分析数理论中广为人知。我们回顾了此问题的解决方案，并在任何$δ\ geq 0 $的运算符数量上介绍了相应的界限。当$2δ\ in \ Mathbb z _ {\ geq 0} $中，边界通过具有整数间距的已知分区函数饱和。类似的结果适用于固定旋转和Virasoro初选的运营商，以$ C> 1 $理论。
We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions $[Δ- δ,Δ+ δ]$ at asymptotically large $Δ$ in 2d unitary modular invariant CFTs. 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 for any $δ\geq 0$. When $2δ\in \mathbb Z_{\geq 0}$ 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.
