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Scalar Chiral-Primary Unitarity Bound

Physical scope

Conditional representation-theoretic norm bound for scalar superconformal primaries in a unitary representation of the selected N=1 subalgebra of the source-scoped four-dimensional N=4 SU(2) SCFT data. No existence or continuum construction is asserted.

Statement

Let $H$ be a Hilbert-space representation of the selected $N = 1$ superconformal subalgebra, with positive inner product and common invariant domain on which the radial adjoint relations in the definition hold. Let $∣ O ⟩ \neq 0$ be a scalar superconformal primary with dilation eigenvalue $d_{O}$ and $U (1)_{R}$ eigenvalue $r_{O}$ . Then

$d_{O} \geq \frac{3}{2} ∣ r_{O} ∣ .$

If $r_{O} \geq 0$ , equality holds exactly when the selected dotted chiral-branch supercharge $Q_{c h} = {\overset{ˉ}{Q}}_{\dot{-}}^{s t d} / \sqrt{2}$ annihilates $∣ O ⟩$ . If $r_{O} \leq 0$ , equality holds exactly when the undotted anti-chiral component $Q_{a c} = Q_{-}^{s t d} / \sqrt{2}$ annihilates $∣ O ⟩$ . This is a conditional algebraic statement about a state already present in the assumed unitary representation; it does not construct that representation or a continuum SCFT.

Statement references

David Tong, Supersymmetric Field Theory, §6.4.2

Kinney, Maldacena, Minwalla, and Raju, An Index for 4 Dimensional Super Conformal Theories, §3

Facts & Assumptions

Given: A positive Hilbert-space representation of the selected superconformal subalgebra, a nonzero scalar superconformal primary, and the common operator domains stated above.

[F1]

Four-Dimensional SCFT and Protected Observable Data: on a scalar primary, the two adjoint supercharge anticommutators reduce to $2 {S_{c h}, Q_{c h}} = D - \frac{3}{2} R$ and $2 {S_{a c}, Q_{a c}} = D + \frac{3}{2} R$ , with $S_{c h} = Q_{c h}^{†}$ and $S_{a c} = Q_{a c}^{†}$ .

[A1]

The state $∣ O ⟩$ is nonzero, lies in the common invariant domain of all four operators, and is annihilated by the special superconformal charges $S_{c h}$ and $S_{a c}$ because it is a superconformal primary.

Proof

1.1givenF1A1algebra

Using $S_{c h} ∣ O ⟩ = 0$ and the Hilbert-adjoint relation, [F1] gives $(d_{O} - \frac{3}{2} r_{O}) ∥ O ∥^{2} = ⟨ O ∣ 2 {S_{c h}, Q_{c h}} ∣ O ⟩ = 2 ∥ Q_{c h} O ∥^{2} \geq 0$ . Thus $d_{O} \geq \frac{3}{2} r_{O}$ , with equality exactly when $Q_{c h} ∣ O ⟩ = 0$ .

2.1givenF1A1step 1.1algebra∎

The conjugate anticommutator similarly gives $(d_{O} + \frac{3}{2} r_{O}) ∥ O ∥^{2} = ⟨ O ∣ 2 {S_{a c}, Q_{a c}} ∣ O ⟩ = 2 ∥ Q_{a c} O ∥^{2} \geq 0$ . Combining the two inequalities gives $d_{O} \geq \max (\frac{3}{2} r_{O}, - \frac{3}{2} r_{O}) = \frac{3}{2} ∣ r_{O} ∣$ ; the active equality condition is the first norm when $r_{O} \geq 0$ and the conjugate norm when $r_{O} \leq 0$ .

Depends on

Four-Dimensional SCFT and Protected Observable Data