  Attached to a weight space in an integrable highest weight representation of
a simply-laced Kac-Moody algebra $\mathfrak{g}$, there are two natural
commutative algebras: the cohomology ring of a quiver variety and the center of
a cyclotomic KLR algebra. In this note, we describe a natural geometric map
between these algebras in terms of quantum coherent sheaves on quiver
varieties.
  The cohomology ring of an algebraic symplectic variety can be interpreted as
the Hochschild cohomology of a quantization of this variety in the sense of
Bezrukavnikov and Kaledin. On the other hand, cyclotomic KLR algebras appear as
Ext-algebras of certain particular sheaves, and thus its center receives a
canonical map from the Hochschild cohomology of the category. We show that this
map is an isomorphism in finite type, and injective in general. We further note
that the Kirwan surjectivity theorem for quivers of finite type is an easy
corollary of these results.
  The most important property of this map is its compatibility with actions of
the current algebra on both the cohomology of quiver varieties and on the
Hochschild cohomology of any category with a categorical action of
$\mathfrak{g}$. The structure of these current algebra actions allow us to show
the desired results.
