Topological field theory of dynamical systems

Area of Science:

• Theoretical Physics
• Dynamical Systems Theory
• Statistical Mechanics

Background:

• Path-integral representations are fundamental for analyzing dynamical systems.
• Topological field theories offer a unique lens for understanding system symmetries.
• Non-renormalization theorems ensure the stability of certain symmetries.

Purpose of the Study:

• To demonstrate that path-integral representations of continuous-time dynamical models are topological field theories.
• To categorize dynamical models based on their topological field theory properties and Q-symmetry.
• To explore the implications of Q-symmetry breaking in various dynamical regimes.

Main Methods:

• Formulating dynamical models as cohomological or Witten-type topological field theories.
• Analyzing the stability of global topological supersymmetry (Q-symmetry) using non-renormalization theorems.
• Classifying models into Markovian, chaotic, and self-organized critical (SOC) categories based on Q-symmetry behavior.

Main Results:

• All continuous-time dynamical models can be viewed as topological field theories with global topological supersymmetry (Q-symmetry).
• Dynamical models are categorized into three types: Markovian (unbroken Q-symmetry), chaotic (spontaneously broken Q-symmetry), and SOC (dynamically broken Q-symmetry).
• Self-organized criticality (SOC) acts as a phase separating Markovian and chaotic dynamics, collapsing to the 'edge of chaos' in the deterministic limit.

Conclusions:

• The topological field theory framework provides a unified understanding of diverse dynamical systems.
• Q-symmetry breaking in chaotic and SOC systems explains phenomena like 1/f noise and sensitivity to initial conditions.
• Further investigation into Q-symmetry breaking in non-equilibrium systems like quenches and the Barkhausen effect is warranted.