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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.

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    $\begingroup$ I'm not sure if I understand your question, but have you looked at the article by Deligne and Morgan in the "Quantum Fields and Strings" book? $\endgroup$
    – Kevin H. Lin
    Commented Dec 9, 2009 at 17:18

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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 and in particular section 2.6.

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