蝴蝶效应的拓扑本质

动力系统意义上的非可积性，亦即动力混沌，是一种强非线性的定性现象。其最有前景的理论描述可能来自非微扰方法，其中尤其可靠的是基于对称性的方法。一种这样的基于对称性的框架是随机动力学的超对称理论 (STS)。STS将一般形式的随机(偏)微分方程 (SDE) 重构为cohomological拓扑场论 (TFT)，并将相应拓扑超对称性 (TS) 自发破缺相关联的秩识别为混沌的随机推广。STS的Faddeev-Popov鬼魂作为系统化的记账工具，用于从蝴蝶效应 (BE) 的定义中提取动力学微分：混沌特有的无限长动力学记忆。 accordingly, the effective field theory (EFT) of the TS breaking is essentially a field theory of the BE in the long-wavelength limit. Building on this perspective, here we demonstrate that one way to build such EFTs is the background field method with the external $\mathfrak{gl}(1|1)$ supergauge field coupled to N=2 supercurrents of STS, the fermion number conservation, and translations in time. By the Goldstone theorem, the resulting EFTs are conformal field theories (CFTs) and the operator product expansion provides an explanation for 1/f noise -- the experimental signature of chaos in the form of dynamical power-law correlations. Moreover, the generating functional of the background field possesses its own TS, revealing the topological nature of the BE. Particularly, when the Anti-de Sitter(AdS)/CFT correspondence is an acceptable approximation, the holographic dual of EFT is a cohomological TFT on AdS in which the associated TS and the isometry of the basespace underlie the BE and 1/f noise, respectively.