# Chiral multiplet : Fundamental and adjoint representation and its Lagrangian

In supersymmetry theory, consider $4d$ $N=1$ theory, we know that chiral superfield (In fundamental representation $\Phi \rightarrow e^{i\alpha} \Phi$) $\Phi$ and its lagrangian is given as \begin{align} \Phi^{\dagger}e^{-2gV}\Phi \end{align}

1. I want to know the lagrangian for chiral superfield in adjoint representation.

2. Further, i know for $4d$ $N=1$ theory, fundamental representation of chiral superfield has $(\phi, \psi, F)$ How about the case for adjoint representation?

• You can put a field in whatever representation you like. The structure is the same. I don't understand the question. – Prahar Oct 22 '15 at 15:53
• @Prahar, See for example, Gutaksov-Schwimmer duality, Here we added a one adjoint matter, (in terms of superpotential $Tr(X^l)$) I want to know how it represented in the Lagrangian – phy_math Oct 24 '15 at 6:49