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, x ˙ ˙ = F , that lead from, say, less stable fixed point of F 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, a and b , with a being more stable than b , ∫ ∫ … … ∫ ∫ x ( ± ± ∞ ∞ ) = a , b ( ∏ ∏ i O i ( t i ) ) e ( Q , Ψ Ψ ) D Φ Φ | Θ Θ → → 0 = ⟨ ⟨ a | T ( ∏ ∏ i O ^ ^ i ( t i ) ) | b ⟩ ⟩ . Here ⟨ ⟨ a | and | b ⟩ ⟩ 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; T denotes chronological ordering; O 's are observables that are the Poincare duals of some closed submanifolds in X ; O ^ ^ ( t ) = M ^ ^ t 0 , t O ^ ^ M ^ ^ t , t 0 are the observables in the Heisenberg representation with t 0 being an unimportant reference time moment. The critical points have different indexes of stability so that the states | a ⟩ ⟩ and | b ⟩ ⟩ are topologically inequivalent as they represent unstable manifolds of different dimensionalities. The above matrix elements are independent of t i ′ s as they actually represent the intersection number of O -manifolds on the instanton as exemplified in the figure.