Geometric translations of $(\phi ,\mathrm{\Gamma })$-modules for ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$

We study ``change of weights’’ maps between loci of the stack of $(\phi ,\mathrm{\Gamma })$-modules over the Robba ring with integral Hodge-Tate-Sen weights. We show that in the ${\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$ case these maps can realize translations of $(\phi ,\mathrm{\Gamma })$-modules geometrically. The motivation is to investigate translations of locally analytic representations under the categorical $p$-adic Langlands correspondence.