Topological Field Theory and Stochastic Dynamics

In the late 1980s, Baulieu and Grossman demonstrated that the supersymmetric formulation of Langevin stochastic differential equations (SDEs), proposed earlier by Parisi and Sourlas, belongs to the family of Witten-type topological field theories (TFTs). From a certain angle, this finding may appear puzzling: TFTs have no local degrees of freedom, whereas SDEs do exhibit local fluctuations. In this paper, we address this apparent contradiction in the context of the supersymmetric theory of stochastic dynamics, a generalization of the Parisi–Sourlas-Baulieu-Grossman approach to SDEs of arbitrary form. The resolution lies in recognizing that, as a mathematical construct, each SDE encodes two subsystems: the continuous-time dynamical system (DS) and its noise. The Parisi–Sourlas construction employs a gauge-fixing procedure to rewrite the partition function of the noise (PFN) in terms of the variables of the DS, resulting in the Witten index. Being a topological invariant, this representative of the PFN is independent of the duration of the evolution, reflecting the absence of dynamics in the noise, as expected since the noise is a static probabilistic entity that experiences no backaction from the DS. The partition function of the DS itself is obtained by twisting, i.e., by imposing anti-periodic boundary conditions (APBC) on the Faddeev–Popov ghosts, thereby rendering them physical – a step that can be justified by interpreting the ghosts, e.g., as differentials used in numerical experiments to track Lyapunov exponents. Since APBC are not compatible with the topological character of the Witten index, one might be tempted to conclude that the twist destroys the topological/BRST supersymmetry (TS) of the model altogether. In the context of stochastic dynamics, however, this conclusion is incorrect. The TS remains intact because it is a property of the stochastic evolution operator (SEO), the fundamental construct corresponding to open boundary conditions and known in DS theory as the generalized transfer operator. The Witten index and the partition function do not define different models; rather, they are different objects of the same model defined by the SEO. What APBC do change, however, is that the partition function exhibits exponential growth in time when TS is spontaneously broken – a situation that, as it turns out, is equivalent to the definition of chaos in random DSs introduced by Ruelle. Within the resulting picture of chaos, 1/f noise emerges as a consequence of the Goldstone theorem, while the butterfly effect is described by an effective field theory that may possess a hidden topological structure, as we speculate on the basis of the Ginzburg-Landau approach and AdS/CFT duality.