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In many notes on supersymmetry one writes an expansion for the chiral superfields as: $$ \phi(y,\theta) = z(x) + \sqrt{2} \theta \psi (x) + i \theta \sigma^{\mu} \bar{\theta} \partial_{\mu} z(x)- \theta \theta f (x) - \frac{i}{\sqrt{2}} \theta \theta \sigma^{\mu} \partial_{\mu} \psi(x) \bar{\theta} -\frac{1}{4} \theta \theta \bar{\theta \theta} \partial^2 z(x) .$$

This chiral fields contains one scalar and one Weyl fermion. Is this expansion valid for extended supersymmetry too? If not, what is the expansion (or fields that appear in chiral field) for $\mathcal{N}=2$ gauge theory?

Are these chiral fields the same fields that appear in quiver diagrams for quiver gauge theories?

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In $N=1$ case there is one set of spinor coordinates: $\theta$, $\bar{\theta}$, as in your expansion. In extended $N=n$ supersymmetry there are $n$ sets of spinor coordinates: $\theta_i$ and $\bar{\theta}_i$, where $i=1,2,..,n$.

Because of that it becomes a lot more difficult to use the natural framework of $N=n$ superspaces. For example in $N=2$ supersymmetry (super Yang-Mills) there is $N=2$ vector multiplet (which is also chiral) with the superfield strength $\Psi$ satisfying \begin{equation}\label{eq1} D_iD_j\Psi=\bar{D}_i\bar{D}_j\bar{\Psi},\tag{1} \end{equation} where $D_i$ and $\bar{D}_i$ are spinorial covariant derivatives of $N=2$ SUSY algebra (spinor indices supressed). Since \eqref{eq1} is very difficult to solve (I believe it's not known analytically) we can use another way to determine its field content: $N=1$ formalism, in terms of which the $N=2$ vector multiplet is constructed (on-shell) from $N=1$ vector and chiral multiplets (call them $V$ and $\Phi$ respectively), all in the adjoint representation of a given gauge group. We may even expand $\Psi$ in terms of, say $\theta_1$, using $V=V(x,\theta_2,\bar{\theta}_2)$ and $\Phi=\Phi(x,\theta_2)$: \begin{equation} \Psi=\Phi+\sqrt{2}\theta_1W+\theta_1^2 X,\tag{2} \end{equation} where $W$ is the chiral superfield strength of $V$, and $X$ is a chiral function of $\Phi$ and $V$.

A short addition: Matter fields in $N=2$ case are represented by hypermultiplets which contain two $N=1$ multiplets: chiral and anti-chiral.

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