The Abel--Jacobi map over the twistor-${\mathbb{P}}^1$ and real local class field theory

We study the Abel–Jacobi map over the twistor-${\mathbb{P}}^1$ in the context of Scholze’s geometrisation of the real local Langlands correspondence. In a similar spirit to a result of Fargues over the Fargues–Fontaine curve, we prove that pullback along the Abel–Jacobi map induces an equivalence on Picard groupoids and use this to recover local class field theory for archimedean local fields.