N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

42 Chapter 3. The Linear Case
Again the terms in parentheses will turn out to be the covariant derivative of the
potential Vµ. For Z = i the equation reduces to Gµν = Vµν.
We are not done yet, as the equations for W µ and Gµν are still coupled. To simplify
the following calculations let us introduce the abbreviations
H µ ≡ 1
Vµν ≡ I
|Z| 2
2εµνρσ∂νBρσ + Λ µ
Vµν + ˜ Σµν
Then we first solve eq. (3.31) for Gµν,
and insert this into eq. (3.30),
(3.32)
R
+
|Z| 2
˜Vµν − Σµν . (3.33)
Gµν = Vµν − 2I
|Z| 2 A[µWν] − R
|Z| 2 εµνρσA ρ W σ , (3.34)
IW µ = H µ + V µν Aν − 2I
|Z| 2 A[µ W ν] Aν . (3.35)
Collecting the terms with W µ , this can be written as
where the field dependent matrix K µν is given by
IK µν Wν = |Z| 2 (H µ + V µν Aν) , (3.36)
K µν = η µν E + A µ A ν , (3.37)
E being the expression (1.55) that we have already encountered in the central charge
transformation of the spinors ψ i . To solve for W µ , we need to invert K µν . It can be
easily checked that
(K −1 )µν = 1
E
ηµν − |Z| −2
AµAν . (3.38)
Due to the appearance of E in the denominator, this expression is nonpolynomial in
the gauge field Aµ.
Having determined W µ , we obtain Gµν from eq. (3.34). The final solution to the Bianchi
identities then reads
W µ = |Z|2 µ µν
H + V Aν − |Z|
IE
−2 A µ AνH ν
(3.39)
Gµν = Vµν − 2
E A[µ
Hν] + Vν]ρA ρ − R
IE εµνρσA ρ H σ + V σλ
Aλ . (3.40)
The fundamental fields of course are the gauge potentials Vµ and Bµν rather than
W µ and Gµν, so we now have to determine their supersymmetry and central charge
transformations as well. These can be obtained most easily from eqs. (3.30) and (3.31).
Applying ∆ z to the former we have
∆ z (C) (IW µ − Λ µ ) = 1
2εµνρσ z µν
∂ν ∆ (C) Bρσ − G ∂νC + CAν δzG µν .