String Theory and M-Theory

12.3 The AdS/CFT correspondence 665
idea that is being explored is to formulate dualities for asymptotically AdS
space-times that have large regions within them that are nearly de Sitter.
EXERCISES
EXERCISE 12.5
Explain why the superpotential of an N = 1 supersymmetric theory with
an unbroken R symmetry should have R charge equal to two.
SOLUTION
The four Grassmann coordinates of N = 1 superspace decompose into two
chiral components θα and two antichiral components ¯ θ ˙α. The superfield
formulation of N = 1 gauge theories contains two types of terms, called
D terms and F terms, where D terms are given by integrals over the full
superspace and F terms are given by integrals over chiral superspace. The
Lagrangian density has the general structure
L = LK + LV + LW ,
where the first term is a D term and the last two terms are F terms and
their complex conjugates. For example,
LW = d 2
θ W + d 2 θ¯ W ,
where the superpotential W is a holomorphic function of chiral superfields.
Under an R-symmetry transformation, the different components of a superfield
carry different charges. One can assign uniform R-charge assignment
to the entire superfield if one adopts the rule that θ carries R-charge
+1 and ¯ θ carries R-charge −1. These statements are a consequence of the
commutation relations
[R, Qα] = Qα and [R, Q ˙α] = −Q ˙α.
Integration over θ is like differentiation, and therefore d 2 θ carries R-charge
−2. Therefore, for LW to be R-invariant, it is necessary that the superpotential
W have R-charge equal to 2. This entails finding suitable R-charge
assignments for all the chiral superfields so that each term in the superpotential
has R-charge equal to 2. ✷