QM Research Seminar: Donaldson-Thomas theory for local systems on the 3-Torus

Speaker: Sarunas Kaubrys (University of Edinburgh)


Abstract:

In this talk I will start with an overview of Cohomological Donaldson-Thomas (CoDT) theory, which we view as a way to count objects or extract invariants from the moduli space of objects of a 3-Calabi-Yau category. Although the origins of the theory come from the algebraic geometry of coherent sheaves on 3-Calabi-Yau varieties, we will focus on the topological example of closed oriented real 3-manifolds. The category of local systems or fundamental group representations of such a 3-manifold has a canonical 3-Calabi-Yau structure. CoDT invariants for local systems on a 3-manifold can be defined for any connected reductive group and are conjecturally connected to invariants in low dimensional topology called Skein modules. I will present my work on the computation of CoDT invariants of the 3 -torus. The main structural result used in the computation is called cohomological integrality which I will introduce. One of the consequences of such a integrality theorem is a Langlands duality between SLn CoDT invariants and PGLn CoDT invariants for prime n.