N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

2.2. Consistent Deformations 23
Using eqs. (2.26) and (2.28), this gives
0 = iε kj (iεβγ ¯ Z δzψ i α + D i αGβγ) + iεβγ D i αM jk − εαβε ij Z¯ δzψ k γ + i α ↔ j
β .
Symmetrizing in ijk, we obtain our first consistency condition (the second equation
being the complex conjugate of the first),
D (i
α M jk) = 0 , ¯ D (i
˙α ¯ M jk) = 0 . (C.1)
Only such deformations M ij that obey this condition can be taken into account. In
the following we will assume M ij to satisfy eq. (C.1).
If we symmetrize in the spinor indices αβγ, we find that the spin-3/2 part of D i αGβγ
vanishes. This must be so as the maximum helicity in nongravitational theories is ±1.
The remaining components all involve D αi Gαβ. Thus we can express the action of D i α
on the self-dual part of Gµν through δzψ i and M ij . We find
Now we consider
D i αGβγ = −2 εα(β
{D i α , ¯ D j
˙α } ψk β
iZ¯ δzψ i − 1DjM
ij
3 γ) . (2.29)
!
= iε ij Dα ˙αψ k β .
With the supersymmetry transformations of ψ i as above and using [ D i α , Dβ ˙α ]L =
−iεαβ ¯ λ i ˙αU, this can be written as
0 = iε kj (Dβ ˙αψ i α + εαβ ¯ λ i ˙αU − D i αWβ ˙α) − iε ik ¯ D j
˙α Gαβ − iε ij Dα ˙αψ k β
+ iεαβ ε ik ( ¯ λ j
˙α U + i ¯ Z δz ¯ ψ j
˙α ) + Di αN jk
β ˙α + iεαβ ¯ D j
˙α M ik .
We decompose the equation into parts which are symmetric and antisymmetric in the
indices αβ, respectively. Let us consider the former: symmetrized in ijk it provides us
with a second consistency condition,
D (i jk)
(βN α) ˙α = 0 , ¯ D (i
( ˙ β
jk)
N ˙α)α = 0 . (C.2)
The remaining components determine the action of ¯ D i ˙α on Gαβ, of which we give the
complex conjugate expression,
D i α ¯ G ˙α β ˙ = 2 D ¯i α( ˙α ψ ˙β) − 2
3i ¯ Dj( β˙ N ij
˙α)α , (2.30)
and supply the relation D i (α W β) ˙α = 1
2 ¯ D i ˙αGαβ. From the part antisymmetric in αβ follows
first of all a relation between M ij and N ij
α ˙α
, which is a third consistency condition,
¯D (i
˙α M jk) = i
2 Dα(iN jk)
α ˙α , D(i α ¯ M jk) = i
2 ¯ D ˙α(i N jk)
α ˙α . (C.3)
Moreover, we obtain the central charge transformation of ¯ ψi and thus of ψ i ,
Zδzψ i α = i Dα ˙α ¯ ψ ˙αi − iλ i αU + i
3 Dαj ¯ M ij − 1
3 ¯ D ˙α j N ij
α ˙α
¯Zδz ¯ ψ i ˙α = iDα ˙αψ αi + i ¯ λ i ˙αU − i
3 ¯ D ˙αjM ij + 1
3 Dα j N ij
α ˙α ,
(2.31)