N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

3.4. Chern-Simons Couplings 51
Here the covariant derivative actually reduces to the partial derivative since the terms
in square brackets are gauge invariant, and a similar remark as above applies to the
dots. With the result from section 3.2, the second Bianchi identity (BI.2) takes the
form
where ˆ Σµν is given by
Dν IG ˜µν µν
+ RG + Σ ˆ µν = − ˜ F µν Wν , (3.75)
ˆΣµν = Σµν − i 2(FI − ¯ FI)F I µν − 2i (FI + ¯ FI) ˜ F I µν
+ FIJ χiI σµν χJ i − ¯ FIJ ¯χI
i ¯σµν ¯χiJ .
(3.76)
Since ( ˆ Σµν − Σµν) is δz-invariant, we can replace Σµν with the extended expression in
eq. (3.28) and thus in the solution (3.31). Hence, the second Bianchi identity does not
restrict the φ-dependence of the function F. It is the first Bianchi identity for W µ ,
however, that imposes a constraint on F. It now reads
µ µ
Dµ IW − Λ = 1
4 FµνG µν + i
ij
IDiDjM 2 −
12 1
12 (2Yij + ¯ λi ¯ Dj)M ij
2 + c.c. ,
where the last term originates from DiDjM ij
1 . So let us apply a generator D α i to eq.
(3.73); the contribution from M ij
2 is
DiDjM ij
2 = 12
FIJD
I
µ φ I Dµ ¯ φ J − 1
2FIJ(Fµν − i ˜ Fµν) I F µνJ − iFIJ χiIσ µ Dµ ¯χJ
i
+ 1
2FIJD I ijD ijJ + 1
2FIJ χiIχK i ¯ φ L fKL J − 1FIJ
¯χI
2 i ¯χiKφ L fKL J
− 1
4 FIJ(φ K ¯ φ J fKL I )(φ M ¯ φ N fMN L ) + ∂µ
+ 1
2 FIJKD ijIχ J i χK j + 1
+ 1
12 FIJKL χiIχ jJ χ K i χL j
( ¯ FI − FI)D µ ¯ φ I
2FIJK F I µν χJ i σ
µνχiK i
+
I (Yij + λiDj)M ij
2 .
(3.77)
Clearly, the imaginary part 2 of the expression inside the square brackets can combine
into a total derivative only if FIJ is constant and real. For a compact gauge group this
fixes F modulo a normalization,
F(φ) = e
2 δIJφ I φ J , (3.78)
e being a coupling constant of mass dimension −1. Then the first Bianchi identity
reduces to
µ
Dµ IW − Λ ˆ µ = 1
2FµνG µν , (3.79)
where
ˆΛ µ = Λ µ − e 2ε µνρσ (A I ν∂ρA I σ − 1
3AIνA J ρ A K σ fJK I )
+ i(φ − ¯ φ) I D µ (φ + ¯ φ) I − χiI σ µ ¯χ I (3.80)
i
2 The real part is exactly the super Yang-Mills Lagrangian (1.45) for an arbitrary prepotential F.