N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

3.3. The Lagrangian 45
7) 0 = ∂ ¯ β − β
¯Z − Z
9) 0 = ∂L ¯ δ 10) 0 = ∂ ¯ δ .
8) 0 = ∂β − 2∂Lδ − β
¯Z − Z
From eqs. 3), 4) and 9), 10) it follows that α = α(Z) and δ = δ(Z), respectively. Since
α does not depend on L, we can integrate eq. 6),
β = 2¯αL
Z − ¯ Z + ˆ β(Z, ¯ Z) .
Next we insert this into eq. 8); the L-dependent terms cancel and the Z-dependence of
ˆβ is fixed,
∂ ˆ β = − ˆ β
Z − ¯ Z ⇒ ˆ β = ¯ h( ¯ Z)
Z − ¯ Z .
Now we can determine α from eq. 5),
∂α =
α + ¯α
Z − ¯ Z
⇒ α = i(κZ + ϱ) , κ, ϱ ∈ R . (3.48)
We use this in eq. 7) to derive an analogous condition on ¯ h( ¯ Z),
¯∂ ¯ h = − h + ¯ h
Z − ¯ Z ⇒ ¯ h = −2i (ν ¯ Z + µ) , ν, µ ∈ R .
With α and β known, eq. 1) may be integrated. The reality of γ requires ν = 0, and
we find
γ = − i 2ϱ + κ(Z +
2
¯ Z)
Z − ¯ L
Z
2 − iµ
Z − ¯ Z L + σ(Z, ¯ Z) , (3.49)
with σ real. It remains to solve eq. 2). When β and γ are inserted, the L-dependent
terms drop out and we are left with δ = ∂σ. Eq. 10) then implies
σ(Z, ¯ Z) = f(Z) + ¯ f( ¯ Z) , (3.50)
where f is an arbitrary function of Z. We have thus found the most general linear
multiplet one can build from the linear vector-tensor multiplet with gauged central
charge.
When the coefficients are inserted into the Ansatz (2.58), it comes as no surprise that
several terms group together to form the expression iD (i D j) L, as this is evidently a
linear superfield by itself (cf. the remark in section 2.3.1). The complete pre-Lagrangian
finally reads
where
L ij
L ij = L ij
linVT + Lij cc , (3.51)
linVT = iϱ D i L D j L − ¯ D i L ¯ D j L + L D (i D j) L + iµ D (i D j) L
2L
+ iκ ¯Z (i j)
¯Z
D Z D L + Z ¯ (i
D Z¯ ¯ j) 1 D L + 4 − Z
L (Z + ¯ Z) D i D j Z
+ Z D i L D j L − ¯ Z ¯ D i L ¯ D j
L , ϱ, µ, κ ∈ R ,
(3.52)