A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and $c>1$. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin $j \neq 0$. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.

中文翻译：

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Rademacher 展开和 2d CFT 的频谱

Rademacher 的解析数论的经典结果给出了非正权模形式的傅立叶系数的精确公式。我们应用类似的技术来研究二维酉共形场理论的谱，没有扩展手征代数和 $c>1$。通过利用分配函数的全模约束，我们根据理论的光谱提出了光谱密度的表达式。该表达式根据 Rademacher 展开给出，它收敛于自旋 $j \neq 0$。对于有限数量的光算子，该表达式与 Maloney、Witten 和 Keller 开发的 Poincare 构造的变体一致。有了这个框架，我们研究了对纯引力对偶的配分函数中负态密度的存在，