N=2 Supersymmetric Gauge Theories with Nonpolynomial Interactions

2.3. The Ansatz 29
18) ′
19) ′
20) ′
21) ′
22) ′
0 = ∂(A − 2a) − 1
2 (A − 2a)2 + 2D(F − c) − 2ĒG 0 = ∂L(A − 2a) − ¯ BG + 2a(F − c) (2.51)
0 = ¯ ∂(A − 2a) − 1
2 B ¯ B + 2bF
0 = ¯ ∂G + 1
2
ĀG + 1
2 B(2c − F − ¯ F )
0 = ¯ ∂B − 2E(F − c) − 1
2 B(Ā − 2ā) + 2 ¯ DG .
Before we are trying to solve these equations, it is important to note that the solutions to
the consistency conditions decompose into mutually disjoint equivalence classes, where
two sets of constraints are deemed equivalent if they are related by a local superfield
redefinition
L → L = L( ˆ L, Z, ¯ Z) . (2.52)
Each representative of such a class effectively describes the same physics. We may
employ this to choose representatives which simplify the subsequent calculations as
much as possible. Using
D i αL = L ′ D i α ˆ L + ∂L D i αZ (2.53)
and similar for ¯ D i ˙αL, where L ′ ≡ ∂ ˆ L L = 0, we rewrite the Ansatz (2.40), (2.41) in terms
of ˆ L. The coefficients as functions of ˆ L are then given by
â = a + c ∂L − ∂L ′ /L ′
ˆ b = (b + a ¯ ∂L + ā ∂L + c ∂L ¯ ∂L − ∂ ¯ ∂L)/L ′
ĉ = c L ′ − L ′′ /L ′
 = A + 2F ∂L − 2∂L ′ /L ′
ˆB = B + 2G ¯ ∂L (2.54)
Ĉ = (C − ∂L)/L ′
ˆD = (D + A ∂L + F ∂L ∂L − ∂ 2 L)/L ′
Ê = (E + B ¯ ∂L + G ¯ ∂L ¯ ∂L)/L ′
ˆF = F L ′ − L ′′ /L ′
ˆG = GL ′ .
The differential equations (2.46), (2.48) and (2.50) are invariant under the above transformations
in the sense that if A, B, etc. are solutions to the consistency conditions,
then Â, ˆ B, etc. satisfy the same equations with ∂L replaced by ∂ ˆ L . Consider for instance
eq. 1) in (2.46),
∂ ˆ F − 1
2 ∂ ˆ L Â = ∂ (F L ′ − L ′′ /L ′ ) − 1
2 ∂ ˆ L (A + 2F ∂L − 2∂L ′ /L ′ )
= ∂F L ′ + ∂L ∂LF L ′ + F ∂L ′ − ∂L ′′ /L ′ + L ′′ ∂L ′ /L ′2
− 1
2 (L′ ∂LA + 2L ′ ∂LF ∂L + 2F ∂L ′ − 2∂L ′′ /L ′ + 2L ′′ ∂L ′ /L ′2 )
= L ′ (∂F − 1
2 ∂LA) = 0 .