From wikipedia:

In dynamics, all continuous time dynamical systems, with and without noise, are Witten-type TQFTs and the phenomenon of the spontaneous breakdown of the corresponding topological supersymmetry encompasses such well-established concepts as chaos, turbulence, 1/f and crackling noises, self-organized criticality etc.

These are all very "deep" phenomena which are studied in continuous-time dynamics, and its somewhat surprising that TQFT has such a wide range of applications. Unfortunately, that wikipedia article doesn't include inline citations, so it isn't clear which source that information comes from. Could someone point to a textbook or review on the applications of TQFT to continuous-time dynamical systems?

EDIT: Is the "topological supersymmetry" bascially the structural stability of the system? That would make sense, except structural stability doesn't imply topological invariance, just invariance under a certain restricted set of diffeomorphisms (i.e. perturbations).

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    $\begingroup$ I never trust statements of the form “all A are B”, because too often treating an A as a B involves throwing away too much information. Second thought: I am reminded, perhaps wrongly, of the work of Diego Hofman and collaborators. $\endgroup$ May 21, 2019 at 23:08
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    $\begingroup$ further down on the page there is a link to which has more information en.wikipedia.org/wiki/… $\endgroup$ May 22, 2019 at 0:36
  • $\begingroup$ Crossposted at: math.stackexchange.com/q/3233856/195021 $\endgroup$ May 28, 2019 at 17:29

1 Answer 1


The Wikipedia entry cited in the OP is paraphrased from Topological field theory of dynamical systems by Igor Ovchinnikov (2012):

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.

Ovchinnikov discusses the notion of topological supersymmetry in a dynamical system, and the breaking of that symmetry at the transition to chaotic dynamics, in Introduction to Supersymmetric Theory of Stochastics (2016). A more colloquial description is given in Chaos or Order (2017).

I should add that (judging from the citation trail) this line of research does not seem to have been taken up by other groups, and it remains to be seen how productive the connection identified by Ovchinnikov will turn out to be.

  • $\begingroup$ Have you read Ovchinnikov's papers? I would be curious to hear a more senior mathematician/physicist's opinion on the merits of his work. As OP said, these are deep phenomena and Ovchinnikov seems to be undertaking an ambitious project. But the cynical part of me wonders why his papers remain somewhat obscure... $\endgroup$
    – user144699
    Apr 24, 2020 at 19:49

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