In this mini lecture course, I am going to explain some results of the joint work https://arxiv.org/abs/2211.17261 with Bogdanova-Travkin-Vologodsky (and beyond). The rough goal is to generalize to characteristic p the Simpson's construction of "twistor space" M_X --> P^1 of lambda-connections on a complex variety X. This is a G_m-equivariant fibration of analytic spaces with the feature that variations of Hodge structures on X naturally give special sections P^1 --> M_X; and as a whole M_X ---> P^1 can be viewed as (the moduli space of objects of) a certain "compactified quantization" of the symplectic variety T^*X (where sheaves on T^*X = aka Higgs bundles are identified with the fiber of M_X over 0 in P^1; they deform to D-modules on X sitting as the fiber over {1}; finally this is compactified by the fiber at infinity which is given by Higgs bundles on the complex conjugate variety \bar{X}).
A new feature of the de Rham complex of a smooth variety X in char p is that it now has two filtrations: Hodge and conjugate. These filtrations are Koszul-dual to two filtrations on the sheaf D_X of differential operators on X, which produce a sheaf of algebras D_{P^1,X} on X x P^1 via the Rees construction. The category of D_{P^1,X}-modules considered as a quasi-coherent sheaf of categories over P^1 bears a formal resemblance to M_X --> P^1; namely its fiber over 0 is given by Higgs bundles on X, fiber over 1 is the category of D-modules on X, while the fiber over infinity is Morita-equivalent to Higgs bundles on the Frobenius twist X^(1) (which naturally replaces the complex conjugate). Moreover, F-gauges on X correspond to globally defined objects of D_{P^1,X}-mod together with an identification of fibers at 0 and infinity via Frobenius pull-back.
The main result of our work (which I hope to explain during the course) is that the sheaf of categories D_{P^1,X} in fact makes sense for any restricted symplectic variety in characteristic p. Namely, given such (Y,omega) we show that there is a natural sheaf of categories QCoh_{P^1}(Y) over P^1 with analogous properties which gives back D_{P^1,X} in the case Y=T^*X with its canonical restricted symplectic structure. Moreover, its sections over the formal disk around 0 recover modules over the Frobenius-constant quantization of Bezrukavnikov-Kaledin (whenever the latter exists); even though these quantized algebras typically only exist over the formal disk, our result shows that the corresponding deformation of the category of quasi-coherent sheaves extends to A^1 and can even be naturally compactified at infinity. This also allows for a natural definition of "quantum F-gauges" in this more general context.
In the course I plan to cover some elementary theory of D-modules and restricted symplectic/Poisson geometry in char p, technique of formal geometry (in the sense of Gelfand-Kazhdan) and quasi-coherent sheaves of categories on stacks, and then finally sketch the construction of the sheaf QCoh_{P^1}(Y).
