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Finite positive-energy pair cancels from a nonzero Witten index

Example

Fix $ω > 0$ . Let the zero-energy subspace have dimensions $m_{0}^{+} = 2$ and $m_{0}^{-} = 1$ , and let there be one positive-energy pair $H_{ω}^{+} = C ∣ b ⟩$ and $H_{ω}^{-} = C ∣ f ⟩$ . Give the five-dimensional Hilbert space the orthonormal basis $∣ 0_{1} ⟩, ∣ 0_{2} ⟩, ∣ 0_{-} ⟩, ∣ b ⟩, ∣ f ⟩$ and parity $+ 1$ on $∣ 0_{1} ⟩, ∣ 0_{2} ⟩, ∣ b ⟩$ and $- 1$ on $∣ 0_{-} ⟩, ∣ f ⟩$ . Define $Q = 0$ on the three zero-energy states and $Q ∣ b ⟩ = \sqrt{ω} ∣ f ⟩$ , $Q ∣ f ⟩ = \sqrt{ω} ∣ b ⟩$ ; set $H = Q^{2}$ . For each $β > 0$ , supply the diagonal operator $T_{β}$ on the displayed basis by $T_{β} ∣ 0_{1} ⟩ = ∣ 0_{1} ⟩$ , $T_{β} ∣ 0_{2} ⟩ = ∣ 0_{2} ⟩$ , $T_{β} ∣ 0_{-} ⟩ = ∣ 0_{-} ⟩$ , $T_{β} ∣ b ⟩ = e^{- β ω} ∣ b ⟩$ , and $T_{β} ∣ f ⟩ = e^{- β ω} ∣ f ⟩$ . The heat supertrace is $W (β) = 1$ for all $β > 0$ .

Example references

Gates et al., §8.4, mass-squared supertrace formulae.

Tong, §1.2.2, Witten-index definition and positive-energy cancellation.

Pure-point graded trace-class index data: Pure-point graded trace-class index data.

Beta independence of the pure-point supersymmetric heat supertrace: Beta independence of the pure-point supersymmetric heat supertrace.

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ : Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ .

Hilbert space: Hilbert space.

Real and complex inner-product spaces and their induced length: Real and complex inner-product spaces and their induced length.

The Axiom of Countable Choice ( ${A C}_{ω}$ ): The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Trace class operator: Trace class operator.

Facts & Assumptions

Given: The displayed orthogonal parity decomposition, $ω > 0$ , the finite-dimensional Hilbert inner product and the stated $Q, H$ .

[A1]

Assume Countable Choice, as required by the trace-class supplier used to certify the heat operator. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Pure-point graded trace-class index data: for every $β > 0$ , the datum supplies a bounded diagonal $T_{β}$ with eigenvalue $1$ on the zero-energy basis and $e^{- β λ}$ on the parity subspaces of positive energy, and requires $T_{β}$ to be trace class; the heat supertrace is the absolutely convergent diagonal sum with sign $+ 1$ on even basis vectors and $- 1$ on odd basis vectors.

[F2]

Beta independence of the pure-point supersymmetric heat supertrace: for every pure-point graded trace-class index datum, $W (β) = m_{0}^{+} - m_{0}^{-}$ for all $β > 0$ .

[F3]

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ : $ω > 0$ has a positive square root.

[F4]

Trace class operator: every finite-rank operator is compact and hence trace class.

Verification

Given: [A1], [F1]–[F4] and the finite-dimensional data in the Example.

1.1A1F1givenF3F4algebra

The map $Q$ is a self-adjoint matrix on the displayed orthonormal basis: its only nonzero block in the ordered pair $(∣ b ⟩, ∣ f ⟩)$ is $\sqrt{ω} (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , a real symmetric matrix, and it is zero on the orthogonal zero-energy subspace. This block exchanges parity, and its square is $ω I$ ; thus $H$ is $0$ on the three zero states and $ω I$ on the positive pair. For each $β > 0$ , the supplied diagonal $T_{β}$ has eigenvalue $1$ on the zero states and $e^{- β ω}$ on the positive pair, so it is exactly the operator denoted $e^{- β H}$ in [F1]. Its norm is $\max {1, e^{- β ω}} < \infty$ , so it is bounded; its five diagonal entries give finite rank, hence it is trace class by [F4] (equivalently, its singular values have finite sum $3 + 2 e^{- β ω}$ ). Thus the data satisfy the boundedness, trace-class and finite pure-point hypotheses for every $β > 0$ .

2.1A1F1F2step 1.1algebra∎

The three zero modes contribute $2 - 1 = 1$ to the supertrace. The positive pair contributes $e^{- β ω} - e^{- β ω} = 0$ , so direct evaluation gives $W (β) = 1$ . By [F2], this also equals $m_{0}^{+} - m_{0}^{-} = 2 - 1$ . This calculation is finite dimensional and makes no continuum or regularization claim.

Finite positive-energy pair cancels from a nonzero Witten index

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources