We study the integrability from the spectral form factor in the Chern–Simons formulation. The effective action in the higher spin sector was not derived so far. Therefore, we begin from the SL(3) Chern–Simons higher spin theory. Then the dimensional reduction in this Chern–Simons theory gives the SL(3) reparametrization invariant Schwarzian theory, which is the boundary theory of an interacting theory between the spin-2 and spin-3 fields at the infrared or massless limit. We show that the Lorentzian SL(3) Schwarzian theory is dual to the integrable model, SL(3) open Toda chain theory. Finally, we demonstrate the application of open Toda chain theory from the SL(2) case. The numerical result shows that the spectral form factor loses the dip-ramp-plateau behavior, consistent with integrability. The spectrum is not a Gaussian random matrix spectrum. We also give an exact solution of the spectral form factor for the SL(3) theory. This solution provides a similar form to the SL(2) case for [Formula: see text]. Hence the SL(3) theory should also do not have a Gaussian random matrix spectrum.

中文翻译：

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Chern-Simons 公式中的可积性和光谱形状因子

我们从 Chern-Simons 公式中的光谱形状因子研究可积性。到目前为止，还没有在更高的旋转领域产生有效的作用。因此，我们从 SL(3) Chern-Simons 高自旋理论开始。然后这个 Chern-Simons 理论的降维给出了 SL(3) 重参数化不变施瓦兹理论，它是红外或无质量极限处自旋 2 和自旋 3 场之间相互作用理论的边界理论。我们表明洛伦兹 SL(3) Schwarzian 理论与可积模型 SL(3) 开户田链理论是对偶的。最后，我们从 SL(2) 案例中展示了开放 Toda 链理论的应用。数值结果表明，光谱形状因子失去了倾斜-斜坡-平台行为，与可积性一致。该谱不是高斯随机矩阵谱。我们还给出了 SL(3) 理论的光谱形状因子的精确解。该解决方案为 [公式：参见文本] 提供了与 SL(2) 案例类似的形式。因此，SL(3) 理论也不应该有高斯随机矩阵谱。