page_title: Supersymmetric theory of stochastic dynamics
section_title: STS as a topological field theory
hierarchical_section_title: Supersymmetric theory of stochastic dynamics / STS as a topological field theory
caption_attribution_description: English: Figure representing instanton matrix element in STS
context_page_description: Supersymmetric theory of stochastic dynamics or stochastics is an exact theory of stochastic differential equations, the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.
context_section_description: The topological sector of STS can be recognized as a member of the Witten-type topological field theories. In other words, some objects in STS are of topological character with the Witten index being the most famous example. There are other classes of topological objects. One class of objects is related to instantons, i.e., transient dynamics. Crumpling paper, protein folding, and many other nonlinear dynamical processes in response to quenches, i.e., to external (sudden) changes of parameters, can be recognized as instantonic dynamics. From the mathematical point of view, instantons are families of solutions of deterministic equations of motion, , that lead from, say, less stable fixed point of  to a more stable fixed point. Certain matrix elements calculated on instantons are of topological nature. An example of such matrix elements can be defined for a pair of critical points,  and , with  being more stable than ,
  Here  and  are the bra and ket of the corresponding perturbative supersymmetric ground states, or vacua, which are the Poincare duals of the local stable and unstable manifolds of the corresponding critical point;  denotes chronological ordering; 's are observables that are the Poincare duals of some closed submanifolds in ;  are the observables in the Heisenberg representation with  being an unimportant reference time moment. The critical points have different indexes of stability so that the states  and  are topologically inequivalent as they represent unstable manifolds of different dimensionalities. The above matrix elements are independent of  as they actually represent the intersection number of -manifolds on the instanton as exemplified in the figure.
  The above instantonic matrix elements are exact only in the deterministic limit. In the general stochastic case, one can consider global supersymmetric states, 's, from the De Rham cohomology classes of  and observables, , that are Poincare duals of closed manifolds non-trivial in homology of . The following matrix elements,  are topological invariants representative of the structure of De Rham cohomology ring of .
caption: The square acbd represents an instanton, i.e, the family of trajectories of deterministic flow (dotted arrowed curves) leading from one critical point (b) to another (a). The bra/ket (
    
      
        
          ⟨
          a
          
            |
          
        
      
      {\displaystyle \langle a|}
    
  /
    
      
        
          
            |
          
          b
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      {\displaystyle |b\rangle }
    
  ) of the locally supersymmetric ground states (vacua) corresponding to these critical points are the Poincare duals of the local unstable/stable manifolds. Operators 
    
      
        
          
            
              
                
                  O
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              1
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              2
            
          
        
      
      {\displaystyle {\hat {O}}_{1,2}}
    
   are Poincare duals of vertical/horizontal lines. The value of the instantonic matrix element is the intersection number (the difference in the numbers of positive (black) and negative (white) intersections). It is invariant under the temporal evolution due to the flow, 
    
      
        
          
            
              
                
                  O
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              1
            
          
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          0
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          →
          
            
              
                
                  O
                  ^
                
              
            
            
              1
            
          
          (
          t
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      {\displaystyle {\hat {O}}_{1}(0)\to {\hat {O}}_{1}(t)}
    
   .
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