Can anyone describe (or give a reference for) the 2d superspace formulation of N=(2,2) SUSY in Euclidean signature?


I'm reading Hori's excellent introduction to QFT in the book 'Mirror symmetry', and my question is basically Ex. 12.1.1. page 273. What I imagine the answer is is a super version of the usual story of differential forms on complex manifolds, i.e. we complexify, find square roots of the $\partial\_z$ and $\partial\_{\bar{z}}$ operators, then find a subalgebra of 'chiral' fields analogous to the subalgebra of holomorphic forms.



N=(2,2) supersymmetry in two-dimensional Euclidean space
You imagine well. Hori is talking about $\mathbb{R}^{2|2}$, which is arguably the simplest *super Riemann surface*.


There is lots on this subject, mostly in the Physics literature, which I'm hesitant to recommend. In the Mathematics literature, you might wish to read Deligne and Freed's [Supersolutions](http://arxiv.org/abs/hep-th/9901094) and in particular section 2.6.


