N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

3.2. Transformations and Bianchi Identities 39
This finishes the solution to the consistency conditions (C.1–4) subject to the restriction
(2.71) and F = 0. The complete set of coefficient functions reads
A = B = 2
¯Z − Z
D = ¯ Z∂g
¯Z − Z , E = ¯ Z ¯ ∂¯g
¯Z − Z
1
, C =
¯Z
L + Zg ¯ + Z¯g ¯
− Z
, a = b = c = F = G = 0 ,
(3.16)
with some arbitrary function g(Z). When inserted into the Ansatz (2.41), the gdependent
terms can be written as
¯Z i j
¯Z
D (g D Z) + ¯ i
D (¯g ¯ j
D Z) ¯ , (3.17)
− Z
and if there is a function f(Z) with ∂f = g, they simplify to
¯Z i j
¯Z
D D f(Z) + ¯ i
D ¯ j
D f( ¯ Z) ¯ . (3.18)
− Z
We shall first consider the simplest case g = 0, which corresponds to the constraints
D (i
α ¯ D j)
L = 0
˙α
D (i D j) L = 2
¯Z − Z
D (i Z D j) L + ¯ D (i ¯ Z ¯ D j) L + 1
2 L Di D j Z .
(3.19)
In the limit Z = i they reduce to the free constraints (2.3). We return to g = 0 in
section 3.4.
3.2 Transformations and Bianchi Identities
By construction, the constraints (3.19) satisfy the necessary consistency conditions
(C.1–4). The task now is to solve, if possible, the Bianchi identities (BI.1) and (BI.2).
Then we would have shown the constraints to be consistent and could proceed to
determine the invariant action. To do this, we need to compute DαjM ij , DiDjM ij and
¯D ˙αi ¯ DαjM ij , which suffices as in the case at hand M ij is imaginary and N ij
α ˙α = 0, cf.
section 2.2.
In terms of component fields the deformation M ij reads
M ij = − ¯ M ij = 1 (i j)
λ ψ − ¯(i λ ¯j) ij
ψ + iLY
I
. (3.20)
Applying a supersymmetry generator Dαj and summing over j, we obtain
DαjM ij = 3
Fµνσ
2I
µν ψ i + i
2Gµνσ µν λ i + i
2Wµσ µ¯i λ − i∂µ ¯ Zσ µ ψ¯ i ij
+ Y ψj
− Lσ µ ∂µ ¯ λ i − 1
2 DµL σ µ¯ λ i − i
2 ¯ ZUλ i + i λjM
ij
2 α ,
(3.21)