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$N = 4$ action as a massless adjoint-hypermultiplet specialization

Physical scope

A conditional equality of the N=1 superspace action terms and on-shell field content for the N=4 SU(2) model and the massless-adjoint-hyper specialization of the adopted N=2 model; it does not prove sixteen-supercharge invariance.

Example

Set the $N = 2$ hypermultiplet representation to $R = a d$ on $H_{R} = s u (2)_{C}$ , use $h_{a d} (X, Y) = κ_{C} (\overline{X}, Y)$ and the induced dual Hermitian form, and set its mass to $m = 0$ . The map $ȷ (X) (Y) = κ_{C} (X, Y)$ is an equivariant isometry from the adjoint representation to its dual, so it identifies $\tilde{Q}$ with an adjoint field. If $Q$ and $\tilde{Q}$ are the canonically normalized chiral fields, define connection-normalized adjoint chiral fields $Φ_{2} = g Q$ and $Φ_{3} = g \tilde{Q}$ ; let $Φ_{1}$ be the connection-normalized $N = 2$ vector chiral field. Then the $N = 2$ hypermultiplet kinetic terms and cubic superpotential become exactly the two additional adjoint-chiral terms in the adopted $N = 4$ $N = 1$ action, and the on-shell component counts agree: one vector, three complex adjoint scalars, and four adjoint Weyl fermions. The calculation does not prove the $N = 4$ supersymmetry variation.

Example references

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

Four-dimensional $N = 4$ SU(2) super-Yang-Mills model

Facts & Assumptions

Given: The $N = 2$ action with one hypermultiplet, the $N = 4$ $N = 1$ action, and the stated connection/canonical field rescaling.

[A1]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: the hyper kinetic terms are canonical and its superpotential is $\sqrt{2} \tilde{Q} Φ_{1} Q + m \tilde{Q} Q$ .

[A2]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: the $N = 2$ vector action has gauge and adjoint-chiral terms with coefficient $1 / g^{2}$ .

[A3]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: for $R = a d$ , $ȷ (X) = κ_{C} (X, \cdot)$ is an equivariant isometric identification of the adjoint with its dual, using $h_{a d}$ and the induced dual Hermitian form.

[A4]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: $κ (X, Y) = - 2 {Tr}_{2} (X Y)$ .

[A5]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: $κ_{C}$ is the complex-bilinear extension of $κ$ .

[A6]

Four-dimensional $N = 4$ SU(2) super-Yang-Mills model: the connection-normalized $N = 4$ action has coefficient $1 / g^{2}$ and cubic term $\sqrt{2} κ_{C} (Φ_{1}, [Φ_{2}, Φ_{3}])$ .

[A7]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: the connection-normalized vector multiplet has one complex adjoint scalar and two adjoint Weyl fermions.

[A8]

Four-dimensional $N = 2$ SU(2) vector and hypermultiplet model: each chiral $N = 1$ hypermultiplet field has one complex scalar and one Weyl fermion.

[A9]

Four-dimensional $N = 4$ SU(2) super-Yang-Mills model: on shell, the $N = 4$ content is one gauge connection, six real adjoint scalars, and four adjoint Weyl fermions.

Verification

Given: [A1]–[A9], $R = a d$ , $m = 0$ , and $g > 0$ .

1.1givenA1A2A3A6algebra

The map $ȷ (X) (Y) = κ_{C} (X, Y)$ intertwines the adjoint and dual actions, since $κ_{C} (X, {Ad}_{u^{- 1}} Y) = κ_{C} ({Ad}_{u} X, Y)$ for every $u \in S U (2)$ ; in the $κ$ -orthonormal real basis $T_{a}$ , it preserves the Hermitian form because both forms have coordinate expression $\sum_{a} \overline{X^{a}} Y^{a}$ . Differentiating the group intertwining identity and complex-linearly extending gives intertwining for the represented complexified Lie algebra; the matrix exponential power series then gives the same identity for the superfield exponentials in the D-terms. Substituting $Q = Φ_{2} / g$ and $\tilde{Q} = Φ_{3} / g$ into the two hypermultiplet D-terms in [A1] therefore gives $g^{- 2}$ times the D-terms of $Φ_{2}$ and $Φ_{3}$ in [A6]. The $N = 2$ vector term already equals the $N = 4$ vector and $Φ_{1}$ terms by [A2] and [A6].

1.2givenA3algebra

By [A3], under the invariant-form identification of the dual adjoint with the adjoint, $\tilde{Q} Φ_{1} Q = κ_{C} (Φ_{3}, [Φ_{1}, Φ_{2}]) / g^{2}$ .

2.1givenA4A5step 1.2algebra

By [A4]–[A5], $κ_{C} (X, Y) = - 2 {Tr}_{2} (X Y)$ on the complexified adjoint. Cyclicity gives ${Tr}_{2} (Φ_{3} Φ_{1} Φ_{2}) = {Tr}_{2} (Φ_{1} Φ_{2} Φ_{3})$ and ${Tr}_{2} (Φ_{3} Φ_{2} Φ_{1}) = {Tr}_{2} (Φ_{1} Φ_{3} Φ_{2})$ . Therefore $κ_{C} (Φ_{3}, [Φ_{1}, Φ_{2}]) = κ_{C} (Φ_{1}, [Φ_{2}, Φ_{3}])$ .

3.1givenA1A6step 2.1algebra

The cubic term in [A1], after the substitution from [given] and equality in step 2.1, is the $N = 4$ cubic term in [A6]; the mass term vanishes because $m = 0$ .

4.1givenA3A7A8A9step 3.1algebra∎

By [A3], [A7], and [A8], the $N = 2$ vector and adjoint hyper together have one gauge connection, three complex adjoint scalars, and four adjoint Weyl fermions. The three complex scalars are six real scalars, so these counts equal the on-shell $N = 4$ content in [A9].

$N = 4$ action as a massless adjoint-hypermultiplet specialization

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

N=4N=4 action as a massless adjoint-hypermultiplet specialization

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

$N = 4$ action as a massless adjoint-hypermultiplet specialization