Nonabelian Hodge theory for stacks and BPS cohomology | BIMSA AG Seminar

Abstract

If C is a smooth projective complex curve, the nonabelian Hodge correspondence gives adiffeomorphism between the coarse moduli space of degree d rank r semistable Higgs bundles onC, and r-dimensional d-twisted representations of the fundamental group of the underlying Riemannsurface of C. lf r and d are not coprime, there are strictly semistables with nontrivial stabilizers, andit perhaps makes more sense to study the respective stacks, instead of coarse moduli spaces. ltseems to be too much to ask that there is any kind of isomorphism between these stacks. but whatwe can show is that the Borel-Moore homology of the two stacks are naturally isomorphic. Theproof uses the classical nonabelian Hodge correspondence, but also a lot of new cohomological DTtheory, and a version of the cohomological integrality conjecture for 2-Calabi-Yau categories. This isjoint work with Hennecart and Schlegel Mejia.