SUSY Quantum Mechanics — Witten Index & Superpotential

Supersymmetric quantum mechanics pairs bosonic and fermionic sectors through supercharges Q, Q†. The Hamiltonians H± = −d²/dx² + W²(x) ∓ W′(x) are isospectral except possibly for zero modes. The Witten index Δ = Tr(−1)^F = n₀⁺ − n₀⁻ counts the difference of zero modes and is topologically robust—it cannot change under continuous deformations that preserve SUSY.

SUSY algebra:
{Q, Q†} = H, [H,Q] = 0

Partner Hamiltonians:
H± = −d² + V±(x)
V±(x) = W(x)² ∓ W′(x)

Witten index:
Δ = Tr(−1)^F = n₀⁺ − n₀⁻
SUSY broken ⟺ Δ = 0
SUSY unbroken ⟺ |Δ| ≥ 1

Zero mode:
ψ₀⁻ ∝ e^{−∫W dx} (if L² norm.)

Isospectrality:
E_n(H+) = E_n(H−) for E>0
Intertwining: A†H− = H+A†