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A Finite Boson–Fermion Pair in a Superconformal-Index Trace

Physical scope

Finite two-state graded Hilbert-space example of the index trace's boson–fermion cancellation. It is not a representation of the full four-dimensional N=4 superconformal algebra and asserts no continuum SCFT or empirical result.

Example

Let $H = C ∣ b ⟩ \oplus C ∣ f ⟩$ be a two-dimensional Hilbert space with orthonormal even state $∣ b ⟩$ and odd state $∣ f ⟩$ . For a fixed $λ > 0$ , define an odd operator by $Q ∣ b ⟩ = \sqrt{λ / 2} ∣ f ⟩$ and $Q ∣ f ⟩ = 0$ , let $S = Q^{†}$ , and set $U = I$ . With $δ = 2 (S Q + Q S)$ , both states have eigenvalue $λ$ , so $\ker δ = {0}$ . The graded trace from the SCFT-index definition is

$I (β) = {Tr}_{H} ((- 1)^{F} e^{- β δ}) = e^{- β λ} - e^{- β λ} = 0$

for every $β > 0$ , equal to the trace on $\ker δ$ . This is a finite model of one supercharge pair only; it is not claimed to realize the full superconformal algebra or a four-dimensional gauge theory.

Example references

Kinney, Maldacena, Minwalla, and Raju, §3, equations (3.1)–(3.3), printed pp. 12–13

David Tong, Supersymmetric Field Theory, §§2.2–2.3 and 6.4.2

Facts & Assumptions

Given: The finite graded Hilbert space, operator $Q$ , its adjoint $S$ , and $λ > 0$ specified in the example.

[F1]

Four-Dimensional SCFT and Protected Observable Data: the index trace is $Tr ((- 1)^{F} e^{- β δ} U)$ with $δ = 2 (S Q + Q S)$ and $U$ commuting with both supercharges.

[A1]

The states $∣ b ⟩, ∣ f ⟩$ are an orthonormal basis, $U = I$ , and $λ$ is a fixed positive real number.

Verification

1.1givenF1A1algebra

From $Q ∣ b ⟩ = \sqrt{λ / 2} ∣ f ⟩$ , $Q ∣ f ⟩ = 0$ , and $S = Q^{†}$ , we have $S Q ∣ b ⟩ = (λ / 2) ∣ b ⟩$ , $Q S ∣ b ⟩ = 0$ , $S Q ∣ f ⟩ = 0$ , and $Q S ∣ f ⟩ = (λ / 2) ∣ f ⟩$ . Hence $δ ∣ b ⟩ = λ ∣ b ⟩$ and $δ ∣ f ⟩ = λ ∣ f ⟩$ , so the kernel is zero.

2.1givenF1A1step 1.1algebra∎

The even state contributes $e^{- β λ}$ to the graded trace and the odd state contributes $- e^{- β λ}$ . Their sum is zero for every $β > 0$ , and it equals the zero trace on the zero-dimensional kernel.

Depends on

Four-Dimensional SCFT and Protected Observable Data