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Conditional Dimension of the SU(2) Quadratic Chiral Primary

Physical scope

Conditional scaling-dimension calculation for a nonzero scalar chiral primary in a unitary realization of the selected four-dimensional N=4 SU(2) model. No continuum SCFT construction or general protection-under-deformation theorem is asserted.

Example

Use the selected $N = 4$ SU(2) field data and natural units $c = ℏ = 1$ . Let $Z$ be the complex scalar component of the first canonical adjoint chiral superfield; its engineering mass dimension is one and its selected $N = 1$ $U (1)_{R}$ charge is $2 / 3$ . Suppose a proposed unitary SCFT realization supplies a nonzero scalar superconformal primary state $∣ O ⟩$ associated with the renormalized composite $O = {Tr}_{2} (Z^{2})$ in a prescription that preserves the exact selected $U (1)_{R}$ symmetry, and suppose the selected chiral-branch supercharge $Q_{c h} = {\overset{ˉ}{Q}}_{\dot{-}}^{s t d} / \sqrt{2}$ from the SCFT data satisfies $Q_{c h} ∣ O ⟩ = 0$ . Then $O$ has $R$ charge $4 / 3$ and scaling dimension exactly $2$ . This is conditional shortening data for this specified operator state; it does not assert that the classical action constructs a continuum SCFT or that every scalar observable is protected.

Example references

David Tong, Supersymmetric Field Theory, §§4.4.2 and 6.4.2

Kinney, Maldacena, Minwalla, and Raju, An Index for 4 Dimensional Super Conformal Theories, §4.1

Facts & Assumptions

Given: The selected classical $N = 4$ SU(2) field data, a proposed unitary SCFT realization supplying the named primary state, a renormalization prescription preserving its exact selected U(1)_R symmetry, annihilation by the selected positive-branch supercharge, and the local scalar-primary unitarity theorem.

[F1]

Four-dimensional $N = 4$ SU(2) super-Yang-Mills model: the chosen global group is SU(2), the chiral superfields are adjoint-valued, and the canonical fields are obtained from the connection-normalized fields by division by $g$ .

[F2]

Four-Dimensional SCFT and Protected Observable Data: the selected $N = 1$ $U (1)_{R}$ assigns charge $2 / 3$ to each adjoint chiral superfield.

[F3]

Scalar Chiral-Primary Unitarity Bound: if a scalar superconformal primary has $r \geq 0$ and is annihilated by the selected chiral-branch supercharge $Q_{c h}$ , then its scaling dimension is $d = 3 r / 2$ .

[A1]

In the assumed quantum realization, $O = {Tr}_{2} (Z^{2})$ in the chosen renormalization prescription corresponds to a nonzero scalar superconformal primary state and $Q_{c h} ∣ O ⟩ = 0$ for the selected chiral-branch supercharge; the prescription preserves the exact selected $U (1)_{R}$ symmetry. This is a hypothesis of the example, not a consequence of the classical postulate.

Verification

1.1givenF1F2A1algebra

For a gauge transformation $U \in S U (2)$ , $(U Z U^{- 1})^{2} = U Z^{2} U^{- 1}$ . For finite matrices, ${Tr}_{2} (A B) = \sum_{i, j} A_{i j} B_{j i} = \sum_{i, j} B_{j i} A_{i j} = {Tr}_{2} (B A)$ , so ${Tr}_{2} (U Z^{2} U^{- 1}) = {Tr}_{2} (Z^{2} U^{- 1} U) = {Tr}_{2} (Z^{2})$ . Under an $R$ rotation by $α$ , $Z \mapsto e^{2 i α / 3} Z$ , hence $Z^{2} \mapsto e^{4 i α / 3} Z^{2}$ and the linear trace has charge $4 / 3$ .

2.1givenF3A1step 1.1algebra∎

By step 1.1, the state has $r_{O} = 4 / 3 > 0$ . The positive-branch hypothesis in [F3] therefore applies to the nonzero primary in [A1], giving $d_{O} = \frac{3}{2} \cdot \frac{4}{3} = 2$ . This is a conditional representation-theoretic deduction; the engineering dimension of the classical composite is not being used as a substitute for the scaling-dimension proof.

Conditional Dimension of the SU(2) Quadratic Chiral Primary

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources