# Knizhnik-Zamolodchikov equation is a connection on “affine slice”

The question is - what is the precise meaning of the phrase in the title? I heard it from Andrey Okounkov during one of his lectures.

The problem is that he didn't really specified which slice is meant, if it is about the Slodowi one, then it is too mysterious.

If it will help: the topic of his talk was about equivariant cohomology and intersection theory.

So, it turns out that we should take affine transversal slice to closure of an $$G(\mathcal{O})$$ orbit in affine grassmanian $$G(K) / G(\mathcal{O})$$, then there is "some" $$X$$ of smallest dimension in this slice, which is of our interest. $$X$$ turns out to be a singular variety (e.g. $$\mathbb{C}^{2} / (\mathbb{Z} / n\mathbb{Z})$$). Let $$\widetilde{X} \to X$$ be it's minimal resolution. We take quantum cohomology of $$\widetilde{X}$$ and see how basis forms of $$H^{2}_{q}(X)$$ act on the whole $$H^{*}_{q}$$ and these guys give rise to a connection which is Trigonometric Knizhnik-Zamolodchikov connection.