Topological Field Theory and Stochastic Dynamics

Dear Colleagues,

More than forty years ago, Parisi and Sourlas established that a special class of stochastic differential equations (SDEs) with gradient flows possess a hidden supersymmetry. Further work by mathematical physicists on models of this type exposed the topological nature of this supersymmetry and led to the formulation of cohomological field theories (ChTF), a class of models on the border of algebraic topology and high-energy physics. More recently, the results obtained in the context of pseudo-unitary quantum theory opened up the possibility to extend the Parisi–Sourlas approach to SDEs of general form. This extension showed that the topological supersymmetry actually exists in all stochastic models. Moreover, its spontaneous breakdown generalizes dynamical systems’ nonintegrability, also known as chaos, turbulence, and the butterfly effect, from deterministic to stochastic dynamics, and encompasses such physical concepts as 1/f noise, self-organization, and complex dynamics.

Due to the unmatched applicability of the general form SDEs in science, the so-emerging supersymmetric theory of stochastic dynamics offers unprecedented possibility to apply, at least partially, the machinery of ChTF to various natural and engineered dynamical systems, including bio-chemo-electric dynamics in brain, stockmarkets, etc., thus bringing the corresponding disciplines to a new level of mathematical rigor and predictive power. In return, mathematical physics may acquire the widest experimental testbed for fundamental theoretical concepts that were previously available only on paper. This promises a fruitful crossfertilization between ChTF, dynamical systems theory, and other related disciplines. The goal of this Special Issue is to foster this crossfertilization.

Dr. Igor V. Ovchinnikov
Guest Editor

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