For compact Kaehler manifolds, Corlette and Simpson have established a nonabelian Hodge correspondence between representations of their fundamental groups and Higgs bundles on them subject to certain semistability conditions. Since the work of Faltings, analogy has been sought for varieties over p-adic fields. Inspired by the recent work of Heuer, we found that for a smooth rigid space over a perfectoid field, there is a natural etale Gm-gerbe on its cotangent bundle, whose coherent sheaf theory realizes a p-adic Simpson correspondence. This shows great similarity to results in prime characteristics discovered by Braverman-Bezrukavnikov and Ogus-Vologodsky. In this talk, I will explain this perspective.

This is joint work in progress with Bhargav Bhatt.

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