N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

2.3. The Ansatz 33
unknown coefficients in the Ansatz. This is quite obvious as the number of equations
is not sufficient to fix the dependence of the coefficients on all three variables L, Z, ¯ Z.
Our goal is to generalize the solutions found in the previous section. To this end we
simplify the Ansatz by setting
a = b = c = 0 ⇒ N ij
α ˙α = 0 , (2.71)
which can be justified a posteriori by showing that the resulting constraints do indeed
yield the linear and nonlinear vector-tensor multiplet with gauged central charge. Note
that we are still allowed to redefine
L = κ ˆ L + f(Z) + ¯ f( ¯ Z) , κ ∈ R ∗ . (2.72)
This is the general solution to the differential equations
L ′′ = ∂L ′ = ¯ ∂∂L = 0 ,
which according to eq. (2.54) guarantee the preservation of Ansatz (2.71).
The simplification is motivated by the fact that now condition (C.4) may easily be
evaluated. With the reduced Ansatz put in, it reads
0 = δz ( 1
2L + ZC) DiD j Z + (1 + ZA) D (i Z D j) L + (1 + ZB) ¯ D (i Z¯ D¯ j)
L
+ ZD D i Z D j Z + ZE ¯ D i
Z¯ D¯ j
Z ¯ i j
+ ZF D L D L + ZG D¯ i
L D¯ j
L (2.73)
+ c.c.
from which we infer
as well as
23) 0 = 2 + ZA + ¯ Z ¯ B 24) 0 = ZF + ¯ Z ¯ G , (2.74)
25) u(Z, ¯ Z) = L + ZC + ¯ Z ¯ C 26) v(Z, ¯ Z) = ZD + ¯ ZĒ , (2.75)
u and v being arbitrary L-independent functions that contribute only to δz-invariant
terms inside the square brackets in eq. (2.73). Equations 23 − 26) allow to eliminate
B, D and G in the eqs. (2.46), (2.50) and (2.51), leaving the four unknown functions
A, C, E and F , whereas u and v may be removed using the gauge freedom (2.72). To
show this, let us first consider eq. 16) in (2.50). Replacing B and D according to the
relations above, we obtain
0 = ¯ ∂v − ¯ Z( ¯ ∂
1 Ē − ĀĒ) . 2
Eq. 7) then implies ¯ ∂v = 0. Next we consider eq. 15). Using eqs. 25), 3) and 26), we
find
0 = ∂u − Z(∂C − 1
2AC) − ¯ ZĒ = ∂u − v .