Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

In this talk we consider moduli spaces of flat Lie algebroid connections on a Riemann surface. These types of moduli spaces constitute a simultaneous generalization of several classes of moduli spaces which are broadly used in differential geometry, algebraic geometry, and mathematical physics, such as moduli spaces of Higgs bundles, twisted Higgs bundles, flat connections, and logarithmic or meromorphic connections.

For a general choice of a Lie algebroid, the geometry of theses spaces is mostly unknown, but, for rank 1 Lie algebroids, we were able to obtain interesting properties about their motives and their topology. We obtain some mild conditions which guarantee that these moduli spaces are smooth nonempty irreducible algebraic varieties, we compute various higher homotopy groups and we prove that there exist correspondences between the motives of different moduli spaces of twisted Higgs bundles and of flat Lie algebroid connections which allow us to compute explicitly the motives and E-polynomials (and, thus, Betti numbers) of these moduli spaces in rank 2 and 3. In particular, we prove that the motive of the moduli space of Lie algebroid connections is invariant with respect to changes in the Lie algebroid structure and in the underlying twisting line bundle as long as the degree of the twisting bundle remains constant. Along the way, we were able to verify computationally in low rank and genus a conjectural formula by Mozgovoy on the motives of moduli spaces of twisted Higgs bundles. Joint work with André Oliveira.