Tilting bundles and 3-dimensional field theory

The category of coherent sheaves on an interesting variety X has an extremely annoying property: does not have enough projectives, so it cannot be equivalent to the category of modules over an algebra. However, if you pass to the derived category, this defect can be fixed in many interesting cases, by finding a tilting generator: that is, a vector bundle T such that any coherent sheaf can be resolved by a complex consisting of sums of copies of T, and Ext^i(T,T)=0 for all i>0. If A=End(T), then we obtain an equivalence of derived categories between finitely generated A-modules and Coh(X). This provides a bridge between representation theory and algebraic geometry. This bridge was largely built for us by Bezrukavnikov and Kaledin, but they did not make it very easy for the rest of us to cross it; in particular, while both ends of the bridge are in characteristic 0, the middle is in characteristic p. I’ll talk about how recent input from work in 3d field theory by Braverman, Finkelberg and Nakajima allows us in certain cases to reroute this bridge and avoid characteristic p, while getting a more concrete understanding of the algebra A on the other side.