Topological aspects of the Parisi–Sourlas construction can be briefly outlined in the following manner. The delta-functional, i.e., the collection of the infinite number of delta-functions, ensures that only solutions of the Langevin SDE contribute to W . In the context of BRST procedure, these solutions can be viewed as Gribov copies. Each solution contributes either positive or negative unity: W = ⟨ I N ( ξ ξ ) ⟩ noise with I N ( ξ ξ ) = ∑ ∑ solutions sign ⁡ ⁡ J being the index of the so-called Nicolai map, ξ ξ = ( x ˙ ˙ + ∂ ∂ U ) / ( 2 Θ Θ ) 1 / 2 , which in this case is the map from the space of closed paths in X to the space of noise configurations, a map that provides a noise configuration at which a given closed path is a solution of the Langevin SDE. I N ( ξ ξ ) can be viewed as a realization of Poincaré–Hopf theorem on the infinite-dimensional space of close paths with the Langevin SDE playing the role of the vector field and with the solutions of Langevin SDE playing the role of the critical points with index sign ⁡ ⁡ J . I N ( ξ ξ ) is independent of the noise configuration because it is of topological character. The same it true for its stochastic average, W , which is not the partition function of the model but, instead, its Witten index.