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Gauge-invariant bilinears in one SU(2) hypermultiplet

Physical scope

A local algebraic check of the mass and cubic superpotential terms in the classical N=2 SU(2) model for one fundamental hypermultiplet; it is not a quantum or empirical claim.

Example

Use the defining representation $R = C^{2}$ with group homomorphism $ρ_{f u n d} : S U (2) \to U (2)$ given by $ρ_{f u n d} (u) = u$ acting by matrix multiplication. In each superspace fiber let $Q$ be a column in $R$ , let $\tilde{Q}$ be a row in $R^{*}$ , and let $Φ$ act by $ρ_{f u n d *} (Φ)$ . Under any chiral supergauge transformation $U_{R} = e^{i Λ_{R}}$ , set $Q^{'} = U_{R} Q$ , ${\tilde{Q}}^{'} = \tilde{Q} U_{R}^{- 1}$ , and $Φ^{'} = U_{R} Φ U_{R}^{- 1}$ . Then the mass bilinear $\tilde{Q} Q$ and cubic bilinear $\tilde{Q} Φ Q$ in the adopted $N = 2$ superpotential are gauge invariant. This checks gauge invariance of these two terms only; it does not prove the extended supersymmetry variation.

Example references

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model

Four-dimensional $N = 1$ non-Abelian vector superfield in connection normalization

Facts & Assumptions

Given: The fundamental representation and chiral supergauge transformations in the Example, under the adopted $N = 2$ classical model.

[A1]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: one hypermultiplet has chiral superfields $Q$ in $R$ and $\tilde{Q}$ in $R^{*}$ , with superpotential $\sqrt{2} \tilde{Q} Φ Q + m \tilde{Q} Q$ .

[F1]

Four-dimensional $N = 1$ non-Abelian vector superfield in connection normalization: in a representation $R$ , a chiral matter superfield transforms as $Φ_{R} \mapsto e^{i Λ_{R}} Φ_{R}$ under an admissible chiral supergauge parameter.

Verification

Given: [A1], [F1], and the fundamental representation $U = e^{i Λ_{R}}$ with its dual action on $R^{*}$ .

1.1givenA1F1algebra

The dual action is right multiplication by $U^{- 1}$ , so ${\tilde{Q}}^{'} Q^{'} = \tilde{Q} U^{- 1} U Q = \tilde{Q} Q$ . Thus the mass bilinear is invariant.

2.1givenA1F1step 1.1algebra∎

The adjoint action gives ${\tilde{Q}}^{'} Φ^{'} Q^{'} = \tilde{Q} U^{- 1} (U Φ U^{- 1}) U Q = \tilde{Q} Φ Q$ . Hence the cubic bilinear, and therefore the stated linear combination of mass and cubic terms, is invariant.

Gauge-invariant bilinears in one SU(2) hypermultiplet

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels