Derived categories of moduli of vector bundles on curves

The moduli space of vector bundles of fixed rank and determinant of coprime degree is a smooth projective Fano variety, and as such its derived category is expected to have an interesting semiorthogonal decomposition. Recently, Narasimhan and Fonarev--Kuznetsov have (independently) shown that for rank 2 there exists an admissible embedding of the derived category of the curve into that of the moduli space, given by the universal Poincaré bundle. In a joint work with Swarnava Mukhopadhyay we have generalised this to arbitrary rank (and determinant of degree 1), and exhibited a second copy of the derived category of the curve. More generally we can look for a decomposition into indecomposable pieces. At least for rank 2 we can state a precise conjecture, and in a joint work with Sergey Galkin and Swarnava Mukhopadyay we give evidence for this using mirror symmetry for Fano varieties. If time permits I will discuss what is known about derived categories of symmetric products of curves.