Number Theory Seminar

Fedosov and, independently, de Wilde and Lecomte proved that every smooth symplectic manifold admits a distinguished deformation quantization, well defined up to noncanonical isomorphism. Bezrukavnikov and Kaledin later established an analogous result for affine symplectic varieties over a field of characteristic zero. Moreover, the assignment sending an affine symplectic variety to the category of modules over its quantized algebra extends to a functor from the groupoid of all symplectic varieties and symplectic isomorphisms to the 2-groupoid of categories. 

I will review some of these results and then explain an analogous picture in positive characteristic, developed in work of Bezrukavnikov–Kaledin and Bogdanova–Kubrak–Travkin–Vologodsky. A new feature in characteristic p is that the categorical quantization is defined over a mod-p analogue of the twistor line rather than merely over a formal disk. If time permits, I will also explain some (partially conjectural) results describing the topological Hochschild homology of the quantized category and their implications for the image of the period map.