Topological field theory of dynamical systems

TL;DR

This paper demonstrates that dynamical systems can be described as topological field theories with supersymmetry, classifying models into categories based on Q-symmetry breaking, and linking these to phenomena like chaos and self-organized criticality.

Contribution

It introduces a topological field theory framework for dynamical systems, revealing a classification based on Q-symmetry breaking and connecting it to physical phenomena such as chaos and SOC.

Findings

Dynamical models form a topological field theory with supersymmetry.

Classification of models into unbroken, broken, and dynamically broken Q-symmetry.

Self-organized criticality corresponds to a phase with dynamically broken Q-symmetry.

Abstract

Here, it is shown that the path-integral representation of any stochastic or deterministic continuous-time dynamical model is a cohomological or Witten-type topological field theory, i.e., a model with global topological supersymmetry (Q-symmetry). As many other supersymmetries, Q-symmetry must be perturbatively stable due to what is generically known as non-renormalization theorems. As a result, all (equilibrium) dynamical models are divided into three major categories: Markovian models with unbroken Q-symmetry, chaotic models with Q-symmetry spontaneously broken on the mean-field level by, e.g., fractal invariant sets (e.g., strange attractors), and intermittent or self-organized critical (SOC) models with Q-symmetry dynamically broken by the condensation of instanton-antiinstanton configurations (earthquakes, avalanches etc.) SOC is a full-dimensional phase separating chaos and Markovian dynamics. In the deterministic limit, however, antiinstantons disappear and SOC collapses into the "edge of chaos". Goldstone theorem stands behind spatio-temporal self-similarity of Q-broken phases known under such names as algebraic statistics of avalanches, 1/f noise, sensitivity to initial conditions etc. Other fundamental differences of Q-broken phases is that they can be effectively viewed as quantum dynamics and that they must also have time-reversal symmetry spontaneously broken. Q-symmetry breaking in non-equilibrium situations (quenches, Barkhausen effect, etc.) is also briefly discussed.