# Chrial multiplet's fundamental and anti-fundamental representation

Here i follow the notation in arXiv 9312104v1 (Witten's Verlinder algebra ~ paper) The usual kinetic energy for a chiral multiplet is given as (In 2 dimensional $N=(2,2)$ supersymmetry theory)

\begin{align} L_{ch} = \frac{1}{4} \int d^2 x d^4 \theta \bar{\Phi} \Phi = d^2 x ( | D_0 \phi|^2 - |D_1 \phi|^2 + |F|^2 + i \bar{\psi}_{+} (D_0 -D_1) \psi_{+} + i \bar{\psi}_{-}(D_0 +D_1 )\psi_{-} + \bar{\phi}D \phi - \bar{\phi} \{ \sigma, \bar{\sigma} \} \phi - \sqrt{2} \bar{\psi}_{+} \bar{\sigma} \psi_{-} - \sqrt{2} \bar{\psi}_{-} \sigma \psi_{+} + i\sqrt{2}\bar{\psi}_{+}\lambda_{-}\phi - i\sqrt{2} \bar{\psi}_{-} \lambda{+}\phi + i \sqrt{2}\bar{\phi}\lambda_{+}\psi_{-} - i\sqrt{2} \bar{\phi} \lambda_{-}\psi_{+}) \end{align} As far as i know, this expression is valid for chiral multiplet in fundamental representation.

I'd like to know what is the Lagrangian for of chiral multiplet in anti-fundamental representation.

• 1. Please link the arXiv abstract of the paper in question. 2. Representation of what? 3. Explain the notation! 4. Why do you think the form of the Lagrangian depends on the particular representation chosen? – ACuriousMind Dec 27 '14 at 13:51

I figure out what is the problem. In both representation, (fundamental or anti-fundamental), the Lagrangian is same. But note in their representation, the generators(basis) is different. It is not problem when we deal the Lagrangian separately, but if we want to deal with system with co-exist of fundamental Q, and antifundamental $\tilde{Q}$, we have to fixed the basis. ($i.e$ either fundamental reps. or antifundamental reps.)
The relation between fundamental and antifundamental generators in $SU(N)$, $T^a = -(T^a)^{*}$ Thus the D-term for fundamental chiral and anti-fundamental chiral multiplets can be given as $D= (\bar{\phi}\phi - \tilde{\phi}\tilde{\bar{\phi}})T^a$