Category $\mathcal{O}$ for truncated shifted Yangians and the bi-infinite Bott-Samelson variety

In this paper, we study the category $\mathcal{O}$ of representations of shifted Yangians associated to a simply-laced simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. In particular, we prove that the (complexified) Grothendieck ring of this category is isomorphic to the Cox ring of the open bi-infinite Bott-Samelson variety, which is a pro-variety we construct from Bott-Samelson varieties for alternating heaps. Using work of Francone-Leclerc, we prove a conjecture of Hernandez-Zhang by identifying the above Grothendieck ring with a cluster algebra defined by Geiss-Hernandez-Leclerc. Our methods also yield an action of the Langlands dual group $G^{\vee}$ on this Grothendieck ring, and show that the shifted coproducts defined in work of the first and fifth authors with collaborators give rise to coproducts for truncated shifted Yangians. This machinery then allows us to prove further conjectures of Frenkel-Hernandez and Geiss-Hernandez-Leclerc on extended $QQ$-systems, and to obtain a generalization of a duality defined by Hernandez-Leclerc.