Superspace

Superspace extends spacetime coordinates $x^{\mu }$ by anticommuting spinor coordinates ${\theta }^{\alpha }$ and ${\bar{\theta }}^{\dot{\alpha }}$.

A superfield is a function on superspace. Its finite Taylor expansion in Grassmann coordinates packages component fields into a supersymmetry representation.

A chiral superfield obeys ${\bar{D}}_{\dot{\alpha }}\Phi =0$. In chiral coordinates $y^{\mu }=x^{\mu }+i\bar{\theta }{\bar{\sigma }}^{\mu }\theta$ it expands as

An auxiliary field has no kinetic term and can be eliminated algebraically. The complex field $F$ in a chiral superfield balances bosonic and fermionic degrees of freedom off shell.

A superpotential is a holomorphic function of chiral superfields integrated over chiral superspace, $\int d^2\theta W+h.c.$

For canonical global supersymmetry, eliminating the auxiliary fields gives

The supersymmetric non-renormalization theorem says that the Wilsonian superpotential receives no perturbative corrections. Holomorphy and spurion symmetries can often exclude nonperturbative corrections as well.

The Kähler potential is a real function integrated over full superspace. The canonical choice $K={\Phi }^{†}\Phi$ gives canonical kinetic terms.