Integral trace conjecture for categorified quantum $\mathfrak{sl}_n$

Let $\mathfrak{g}=\mathfrak{sl}_n$. Write $\mathcal{U}_{\mathbb{Z}}$ for the integral form of the categorified quantum group, $\mathcal{U}^{\ast}_{\mathbb{Z}}$ for its version whose 2-hom spaces allow homogeneous morphisms of arbitrary degree, and $\dot{\mathcal{U}}_{\mathbb{Z}}$ and $\dot{\mathcal{U}}$ for the corresponding integral and rational idempotent forms. Let $\operatorname{Tr}$ denote the trace of a 2-category, and let $\mathbf{U}(\mathfrak{sl}_n[t])$ be the idempotented current algebra defined in the source.

Integral trace conjecture. We have

$$K_0(\dot{\mathcal{U}}_{\mathbb{Z}})=\operatorname{Tr}(\mathcal{U}_{\mathbb{Z}})=K_0(\dot{\mathcal{U}})$$

and $\operatorname{Tr}(\mathcal{U}^{\ast}_{\mathbb{Z}})$ coincides with the integral idempotented version of the current algebra $\mathbf{U}(\mathfrak{sl}_n[t])$.

The conjecture proposes integral trace and Grothendieck-group identifications for the categorified quantum group of type $\mathfrak{sl}_n$, together with a current-algebra description of the trace after the grading is killed. The paper describes evidence for this claim and expects analogous results for finite-type simply-laced Kac–Moody algebras.