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SU(2) and SO(3) Line-Spectrum Choices in the $N = 4$ Duality Family

Physical scope

Finite center-charge and line-spectrum calculation for the su(2) N=4 duality family. T is the theta-angle shift and can permute the line-spectrum choices; quantum S-equivalence is reported as conjectural. This item proves the finite charge arithmetic and identifies the T parameter shift; it asserts no confinement law, continuum construction, or proof of quantum S-duality.

Example

Use the $s u (2)$ center-charge group $C = Z_{2} \times Z_{2}$ with pairing $⟨ (e, m), (e^{'}, m^{'}) ⟩ = e m^{'} - m e^{'} (m o d 2)$ . The source-reported line-spectrum subgroups associated with this gauge algebra are

$L_{S U (2)} = {(0, 0), (1, 0)}, L_{S O (3)_{+}} = {(0, 0), (0, 1)}, L_{S O (3)_{-}} = {(0, 0), (1, 1)} .$

Each is maximal among subgroups on which the pairing vanishes; such subgroups are called isotropic. Under the source-reported duality maps, $S : (e, m) \mapsto (m, - e)$ sends $L_{S U (2)}$ to $L_{S O (3)_{+}}$ , while $T : (e, m) \mapsto (e + m, m)$ fixes $L_{S U (2)}$ and exchanges $L_{S O (3)_{+}}$ with $L_{S O (3)_{-}}$ . With dimensionless $g > 0$ and $θ \in R$ , set $τ = θ / (2 π) + 4 π i / g^{2}$ . The accompanying coupling maps are $S : τ \mapsto - 1 / τ$ and $T : τ \mapsto τ + 1$ . Here $T$ is the shift $θ \mapsto θ + 2 π$ at fixed $g$ , identified by Kapustin and Witten as a classical symmetry on $R^{4}$ ; their additional quantum $S$ -equivalence is conjectural. The $T$ operation can exchange the two $S O (3)$ line-spectrum choices, so it need not fix the complete theory data. This verifies the finite line-charge permutation and identifies the $T$ parameter shift; it does not prove quantum $S$ -duality.

Example references

Aharony, Seiberg, and Tachikawa, §§1.1–1.2 and 2.4

Kapustin and Witten, §2.2

Facts & Assumptions

Given: The finite group $C = Z_{2} \times Z_{2}$ , its alternating pairing, and the source-reported $S$ and $T$ maps in the statement.

[F1]

Four-Dimensional SCFT and Protected Observable Data: the center-charge classes are $(e, m) \in Z_{2} \times Z_{2}$ , the mutual-locality pairing is $e m^{'} - m e^{'} (m o d 2)$ , and the source-reported maps are $S (e, m) = (m, - e)$ and $T (e, m) = (e + m, m)$ . The dimensionless coupling is $τ = θ / (2 π) + 4 π i / g^{2}$ for $g > 0$ and real $θ$ .

[F2]

Aharony, Seiberg, and Tachikawa, §§1.2 and 2.4: for gauge algebra $s u (2)$ the source assigns the three maximal line classes to $S U (2)$ , $S O (3)_{+}$ , and $S O (3)_{-}$ and gives the source-reported $S$ action on line charges; its $T$ action follows from the stated $θ \mapsto θ + 2 π$ transformation, which exchanges the two $S O (3)$ choices. Kapustin and Witten, §2.2 identify $T$ as the classical theta-shift symmetry on $R^{4}$ and explicitly classify the additional quantum $S$ -duality as a conjecture.

[A1]

All coordinates and pairings in this example are computed in the field $Z_{2}$ ; in particular $- 1 = 1$ there.

Verification

1.1givenF1F2A1algebra

The three nonzero elements of $C$ are $(1, 0)$ , $(0, 1)$ , and $(1, 1)$ . Each generates an order-two subgroup with the origin. The pairing is alternating, so each such subgroup is isotropic. Any subgroup containing two distinct nonzero elements contains two linearly independent columns in $Z_{2}^{2}$ ; the determinant $e m^{'} - m e^{'}$ of two independent columns is nonzero and therefore equals $1$ in $Z_{2}$ . Hence such a subgroup is not isotropic. Therefore these are exactly the three maximal isotropic subgroups, and [F2] gives their source-reported gauge-group labels.

2.1givenF1F2A1step 1.1algebra∎

Modulo $2$ , the map $S$ sends $(1, 0)$ to $(0, 1)$ , $(0, 1)$ to $(1, 0)$ , and $(1, 1)$ to $(1, 1)$ . The map $T$ sends $(1, 0)$ to $(1, 0)$ , $(0, 1)$ to $(1, 1)$ , and $(1, 1)$ to $(0, 1)$ . These invertible linear maps therefore send the displayed subgroups exactly as stated. At fixed $g$ , replacing $θ$ by $θ + 2 π$ adds $1$ to $τ$ by [F1]. Source [F2] identifies this $T$ operation as a classical symmetry on $R^{4}$ , with the stated permutation of line-spectrum choices, and describes the additional quantum $S$ -equivalence as conjectural. That $S$ -equivalence is not established by the finite arithmetic.

Remarks

The diagram uses horizontal coordinate $e$ and vertical coordinate $m$ . Only the four marked lattice points are elements of $C$ ; the drawn segments join the two elements of each subgroup as a visual aid. The algebraic proof above does not rely on the drawing.

Depends on

Four-Dimensional SCFT and Protected Observable Data

SU(2) and SO(3) Line-Spectrum Choices in the $N = 4$ Duality Family

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

SU(2) and SO(3) Line-Spectrum Choices in the N=4N=4 Duality Family

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

SU(2) and SO(3) Line-Spectrum Choices in the $N = 4$ Duality Family