A local ${\mathfrak{g}\mathfrak{l}}_{1\mathrm{\mid }1}$-action on odd Khovanov homology

We show that odd Khovanov homology carries an action of the super Lie algebra ${\mathfrak{g}\mathfrak{l}}_{1\mathrm{\mid }1}$, given extra choice of markings on the link. Moreover, we show that this action arises from an action on super ${\mathfrak{g}\mathfrak{l}}_2$-foams, in the extended-TQFT framework developed by the second author and Vaz; in particular, it extends to tangles. Finally, we relate the action to torsion $\mathbb{Z}\mathrm{/}n\mathbb{Z}$ in pretzel links $P(n,n,-n)$. In particular, this shows that all torsion can appear in odd Khovanov homology.