N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

50 Chapter 3. The Linear Case
and it is easily verified that these satisfy the consistency conditions (C.1–4) for any
function F that is chiral, ¯ D ˙αiF = 0, and invariant under central charge and gauge
transformations. In particular, F may be a gauge invariant combination of vector
superfields φ I , with appropriately extended spinor and covariant derivatives, i.e.
Dµ = ∂µ + Aµδz + A I µδI , etc. ,
where L and Z transform trivially under the δI. This provides a means of coupling
the vector-tensor multiplet to additional (even nonabelian) vector multiplets, as long
as the Bianchi identities admit such a coupling. We shall see that this is the case only
for a very specific function F(Z, φ).
To determine the Bianchi identities, we follow the steps in section 3.2. With the benefit
of hindsight we take F to depend only on the φ I , which simplifies the calculations
considerably. The deformations then read
with M ij
1 as in eq. (3.20) and
N ij
α ˙α = 0 , M ij = − ¯ M ij = M ij
1 + M ij
2 , (3.68)
M ij
2 = 1
χiIχ I
jJ FIJ − ¯χ iI ¯χ jJ FIJ
¯ + 2D ijI (FI − ¯ FI) , (3.69)
where a subscript on F denotes a differentiation with respect to φ and similar for ¯ F,
FI1...In ≡ ∂ ∂
. . . I1 ∂φ ∂φIn F , ¯ FI1...In ≡ ∂
∂ ¯ φI1 . . . ∂
∂ ¯ φ In
¯F . (3.70)
The derivatives of F are not independent of each other, for gauge invariance implies
0 = δIF = δIφ K FK = −fIJ K φ J FK , (3.71)
and differentiating once more with respect to φ we obtain another identity,
0 = fIJ K FK + fIL K φ L FJK . (3.72)
We observe that, modulo the prefactor 1/I, the expression M ij
2 is precisely the linear
superfield from which one constructs the super Yang-Mills Lagrangian, cf. section 1.2.
Applying a supersymmetry generator to M ij yields
DαjM ij = 3
D
ijIχ
I
J j FIJ + F I µνσ µνχiJ FIJ − iDµ( ¯ FIσ µ ¯χ iI ) + iFIσ µ Dµ ¯χiI
− 1
2 χiI ¯ φ J fIJ K FK + 1
3 (χiIχjJ ) χK j FIJK + i
4
ij
λjM2 + . . . ,
α
(3.73)
where only contributions from M ij
2 have been written explicitly, while the dots denote
the terms already given in eq. (3.21) (where now M ij = M ij
1 ). Next we apply ¯ D ˙αi.
Making frequent use of the above identities for the derivatives of F, we arrive at
¯D ˙αiDαjM ij = − 3
I iσµ
α ˙α Dν 2(FI − ¯ FI)F I µν − 2i (FI + ¯ FI) ˜ F I µν + FIJ χiI σµν χJ i
− ¯ FIJ ¯χI
i ¯σµν ¯χiJ + . . . .
(3.74)