On the Topological Nature of the Butterfly Effect

Abstract

Non-integrability in the sense of dynamical systems, also known as dynamical chaos, is a strongly nonlinear qualitative phenomenon. Its most promising theoretical descriptions are likely to emerge from non-perturbative approaches, with symmetry-based methods being particularly reliable. One such symmetry-based framework is supersymmetric theory of stochastic dynamics (STS). STS reformulates a general form stochastic (partial) differential equations (SDE) as a cohomological topological field theory (TFT) and identifies the order associated with the spontaneous breakdown of the corresponding topological supersymmetry (TS) as the stochastic generalization of chaos. The Faddeev-Popov ghosts of STS act as a systematic bookkeeping tool for the dynamic differentials from the definition of the butterfly effect (BE): the infinitely long dynamical memory unique to chaos. 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.