Deletion-contraction triangles for Hausel–Proudfoot varieties

To a graph, Hausel and Proudfoot associate two complex manifolds, \mathfrak{B} and \mathfrak{D} , which behave, respectively, like moduli of local systems on a Riemann surface and moduli of Higgs bundles. For instance, \mathfrak{B} is a moduli space of microlocal sheaves, which generalize local systems, and \mathfrak{D} carries the structure of a complex integrable system. We show the Euler characteristics of these varieties count spanning subtrees of the graph, and the point-count over a finite field for \mathfrak{B} is a generating polynomial for spanning subgraphs. This polynomial satisfies a deletion-contraction relation, which we lift to a deletion-contraction exact triangle for the cohomology of \mathfrak{B} . There is a corresponding triangle for \mathfrak{D} . Finally, we prove that \mathfrak{B} and \mathfrak{D} are diffeomorphic, the diffeomorphism carries the weight filtration on the cohomology of \mathfrak{B} to the perverse Leray filtration on the cohomology of \mathfrak{D} , and all these structures are compatible with the deletion-contraction triangles.