Within this definition of the finite-time SEO, the Witten index can be recognized as the sharp trace of the generalized transfer operator. It also links the Witten index to the Lefschetz index, I L = Tr ⁡ ⁡ ( − − 1 ) n ^ ^ M t ′ t ∗ ∗ = ∑ ∑ x ∈ ∈ fix ⁡ ⁡ M t t ′ sign ⁡ ⁡ det ⁡ ⁡ ( δ δ j i − − ∂ ∂ M t t ′ i ( x ) / ∂ ∂ x j ) , a topological constant that equals the Euler characteristic of the (closed) phase space. Namely, W = Tr ⁡ ⁡ ( − − 1 ) n ^ ^ ⟨ ⟨ M t ′ t ∗ ∗ ⟩ ⟩ noise = ⟨ ⟨ Tr ⁡ ⁡ ( − − 1 ) n ^ ^ M t ′ t ∗ ∗ ⟩ ⟩ noise = I L .