Folding conjecture for equivariant foam categories and non-simply laced webs

Let $\mathfrak{g}$ be a simply laced simple Lie algebra and let $\tau$ be a diagram automorphism. Let $\mathfrak{g}_\tau$ be the non-simply laced Lie algebra Langlands dual to the fixed-point subalgebra $\mathfrak{g}^\tau$; in the cases $\mathfrak{sl}_{2n}$, $\mathfrak{so}_{2n}$, $\mathfrak{e}_6$, and $\mathfrak{so}_8$ with automorphisms of orders $2$, $2$, $2$, and $3$, respectively, this gives $\mathfrak{so}_{2n+1}$, $\mathfrak{sp}_{2n-2}$, $\mathfrak{f}_4$, and $\mathfrak{g}_2$. A graded, linear, monoidal bicategory $\mathbf{Foam}(\mathfrak{g})$ and a full monoidal sub-bicategory $\mathbf{B}(\mathfrak{g})\subset\mathbf{Foam}(\mathfrak{g})$ are as in the claim. Folding conjecture. The following decategorifications and identifications should hold:

$${\mathbb C}(q)\otimes_{{\mathbb Z}[q^{\pm}]}K_0(\mathbf{Foam}(\mathfrak{g}))\cong\mathbf{Web}(\mathfrak{g})\cong\mathbf{FundRep}(U_q(\mathfrak{g})),$$

and, for an automorphism $\tau$ of the same order as the diagram automorphism,

$${\mathbb C}(q)\otimes_{{\mathbb Z}[q^{\pm}]}K_0^\tau(\mathbf{B}(\mathfrak{g})^\tau)\cong\mathbf{Web}(\mathfrak{g}_\tau)\cong\mathbf{FundRep}(U_q(\mathfrak{g}_\tau)).$$

This framework aims to explain webs and representation categories in non-simply laced type by folding simply laced foam categorifications. The paper presents it as a proposed general framework, and the general assertions remain open.