BRST gauge fixing procedure goes along the same lines as in case of Langevin SDEs. The topological interpretation of the BRST procedure is just the same and the path integral representation of the Witten index is defined by the gauge fermion, Ψ Ψ , given by the same expression but with the generalized version of d ¯ ¯ = ı ı F − − Θ Θ ı ı e a ( Q , ı ı e a ) . There is one important subtlety, however, that appears on the way to the operator representation of the model. Unlike for Langevin SDEs, classical mechanics, and other SDEs with additive noises, the path integral representation of the finite-time SEO is an ambiguous object. This ambiguity originates from non-commutativity of momenta and position operators, e.g., B ^ ^ x ^ ^ ≠ ≠ x ^ ^ B ^ ^ . As a result, B ( τ τ ) x ( τ τ ) in the path integral representation has a whole one-parameter family of possible interpretations in the operator representation, ( α α B ^ ^ x ^ ^ + ( 1 − − α α ) x ^ ^ B ^ ^ ) | ψ ψ ( τ τ ) ⟩ ⟩ , where | ψ ψ ( τ τ ) ⟩ ⟩ denotes an arbitrary wavefunction. Accordingly, there is a whole α α -family of infinitesimal SEOs, H ^ ^ α α = L ^ ^ F − − Θ Θ L ^ ^ e a L ^ ^ e a = [ d ^ ^ , d ¯ ¯ ^ ^ α α ] , with d ¯ ¯ ^ ^ α α = ı ı ^ ^ F α α − − Θ Θ ı ı ^ ^ e a L ^ ^ e a , ı ı ^ ^ F α α being the interior multiplication by the subscript vector field, and the "shifted" flow vector field being F α α = F − − Θ Θ ( 2 α α − − 1 ) ( e a ⋅ ⋅ ∂ ∂ ) e a . Noteworthy, unlike in Langevin SDEs, d ¯ ¯ ^ ^ α α is not a supercharge and STS cannot be identified as a N=2 supersymmetric theory in the general case.