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Supersymmetric Quantum Mechanics, Cohomology, and the Witten Index

Supersymmetric quantum mechanics connects spectral pairing, cohomology, and index theory in a calculable setting. The algebra constrains positive-energy states, while domains, normalizability, and boundary behavior decide which zero-energy states actually exist and whether a protected index is defined.

Enter this chapter

Section titled “Enter this chapter”

Begin with Supercharges, Partner Hamiltonians, and Positive Energy to construct the Hamiltonian from a supercharge and its Hilbert-space adjoint.

Helpful background. Self-Adjointness, Extensions, and Unitary Evolution; Chains, Homology, Cohomology, and Exact Sequences; Fredholm and Dirac Index Theorems and Zero-Mode Counting provide the operator domains, cohomology, and Fredholm index conditions used here.

Topics

Section titled “Topics”
  1. Supercharges, Partner Hamiltonians, and Positive Energy — Derives nonnegative energy with compatible domains.
  2. Spectral Pairing, Ground States, and Supersymmetry Breaking — Pairs positive-energy states and tests the ground state.
  3. Q-Cohomology, Hodge Decomposition, and Zero-Energy States — Relates harmonic representatives to supercharge cohomology.
  4. Morse Deformation, Gradient Flow, and Semiclassical Tunneling — Exposes a semiclassical complex through Morse deformation.
  5. The Witten Index, Vacuum Counting, and Its Failure Modes — Distinguishes a signed index from an ordinary state count.
  6. Continuum Spectra, Boundaries, Wall Crossing, and Ground-State Bundles — Tracks continuum, boundary, wall-crossing, and Berry-transport effects.

How the ideas fit together

Section titled “How the ideas fit together”

Positive-energy states cancel in a well-defined graded trace because supersymmetry pairs their opposite parities. Unpaired zero-energy states remain, making the index a signed count. A nonzero index then implies an unbroken supersymmetric ground state, but a zero index permits either no such states or cancellation between even and odd ground states. Continuous spectra or states escaping to infinity can invalidate the trace or deformation argument, so algebra alone does not guarantee index invariance. Cooper, Khare, and Sukhatme 1995, § 2.

Where to go next

Section titled “Where to go next”

Continue to Supermultiplets, Superspace, and Off-Shell Closure to check transformations and closure for an explicit set of component fields.

References

Section titled “References”