On the Topological Nature of the Butterfly Effect

TL;DR

The paper develops a symmetry-based framework for chaos through the supersymmetric theory of stochastic dynamics (STS), identifying the butterfly effect with spontaneous breaking of topological supersymmetry in stochastic systems. By constructing an effective field theory via a background GL(1|1) supergauge field, it derives that BE corresponds to a conformal, and often non-Hermitian, field theory whose generating functional retains a topological supersymmetry. The operator product expansion explains 1/f noise as a consequence of conformal structure, while an AdS/CFT perspective casts the EFT as a cohomological topological field theory with a holographic dual, where Wilson loops and localization encode the long-range memory of chaotic dynamics. Together, these results provide a unified, topologically grounded account of BE and its experimental signatures, with a holographic interpretation offering new avenues to characterize scale-invariant chaos.

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.

Figures (1)

• Figure 1: (left) In the overdamped sine-Gordon equation OVCHINNIKOV2024114611, the ordinary chaotic phase is preceded by the phase of the noise-induced chaos, where the TS is spontaneously broken by noise-induced antiinstantons ($\bar{I}$) and instantons ($I$) matching them. These are the processes of annihilation and creation of the pairs of solitons which are the left and right moving kinks. The solitons move at a fixed velocity, making the effective field theory (EFT) a Lorentian conformal field theory (CFT), despite the original SDE being Galilean. (right) The AdS/CFT correspondence is an approximation where a CFT is described by a dual theory in the bulk of the AdS space with the conformal boundary being the spacetime of the original theory such as the one in the left. If the AdS/CFT correspondence is a reasonable approximation for the EFT of the BE, the dual theory is a cohomological TFT of AdS, making certain matrix elements, such as Wilson loops, natural objects of interest.