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A finite supersymmetric pair with Witten index one

Physical scope

A finite-dimensional supersymmetric quantum-mechanical Hilbert-space example with exact discrete trace. It is not a continuum QFT, an observed system, or a claim about a naturally realized particle.

Example

Assume ${A C}_{ω}$ and use natural units $c = ℏ = 1$ . The model has time $t \in R$ , with time units, and spatial domain the positively oriented singleton $X = {*}$ with its unique zero-dimensional metric. Its spacetime is $M = R \times X$ , with flat metric $d t^{2}$ and orientation $d t$ ; neither domain has a boundary. Let $E_{*} > 0$ be an energy and let $H = C^{3}$ have orthonormal basis $e_{0}, e_{+}, e_{-}$ . The state bundle is the trivial bundle $M \times C^{3} \to M$ ; the internal/gauge group is ${1}$ , acting as the identity on each fibre. There are no spatial field bundles or additional matter representations. State histories are $C^{1}$ curves $ψ : R \to H$ , and the adopted dynamical postulate is $i d ψ / d t = K ψ$ , with $K$ specified below. The Hilbert space and all matrices are supplied directly, with no continuum regulator, functional measure, or path-integral claim. Give $H$ the grading $H^{+} = span {e_{0}, e_{+}}$ and $H^{-} = span {e_{-}}$ , with $Γ = + I$ on $H^{+}$ and $- I$ on $H^{-}$ . On the full domain $D (Q) = H$ , define the self-adjoint odd supercharge by $Q e_{0} = 0$ , $Q e_{+} = \sqrt{E_{*}} e_{-}$ , and $Q e_{-} = \sqrt{E_{*}} e_{+}$ . Set $K = Q^{2}$ . The Witten index is $1$ for every $β > 0$ .

Example references

Tong, §1.2.2

Facts & Assumptions

Given: The finite-dimensional graded Hilbert-space model and the operators in the Example; $β > 0$ has inverse-energy units.

[A1]

Assume ${A C}_{ω}$ , as used by the exact Hilbert and trace definitions. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Physical Witten index and selected BPS sector: for a pure-point graded physical index datum with trace-class heat operators, the Witten index is the absolutely convergent even-minus-odd diagonal sum in a supplied compatible basis.

Verification

Given: [A1], [F1], and the finite-dimensional model stated in the Example.

1.1A1givenalgebra

The matrix of $Q$ in $(e_{0}, e_{+}, e_{-})$ is $(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & \sqrt{E_{*}} \\ 0 & \sqrt{E_{*}} & 0 \end{array})$ , which is self-adjoint. It sends each parity subspace into the other and annihilates $e_{0}$ , so $Γ Q = - Q Γ$ . Squaring gives $K e_{0} = 0$ and $K e_{+} = E_{*} e_{+}$ , $K e_{-} = E_{*} e_{-}$ .

2.1A1F1step 1.1algebra

Thus the heat operator is diagonal with eigenvalues $1, e^{- β E_{*}}, e^{- β E_{*}}$ on $(e_{0}, e_{+}, e_{-})$ ; it is finite rank and trace class, and this basis is a complete compatible basis for the pure-point datum under the trace conventions using [A1].

3.1F1step 2.1algebra∎

By [F1], $W_{E} (β) = ⟨ e_{0}, e_{0} ⟩ + e^{- β E_{*}} ⟨ e_{+}, e_{+} ⟩ - e^{- β E_{*}} ⟨ e_{-}, e_{-} ⟩ = 1 + e^{- β E_{*}} - e^{- β E_{*}} = 1$ . The single even zero mode supplies the net index; the positive-energy even/odd pair cancels for every $β > 0$ .

Depends on

Dependency tree · 2 levels

Sources

David Tong, Supersymmetric Quantum Mechanics