Donaldson-Thomas theory and cohomological Hall algebras of character stacks
Given a smooth finitely generated algebra with a potential one can study the refined Donaldson-Thomas theory of its moduli stack of representations via motivic or cohomological methods. In this thesis we focus on fundamental group algebras whose stacks of representations are known as character varieties or character stacks. These arise naturally in the realm of algebraic geometry and Donaldson-Thomas theory via the non-abelian Hodge correspondence which relates the study of Higgs bundles to character varieties. In the first part of this thesis we consider approaches to studying the motivic Donaldson-Thomas invariants of fundamental group algebras over mapping tori of Riemann surfaces by constructing an isomorphism between the fundamental group algebra and the Jacobi algebra of a so-called brane tiling on the Riemann surface. Using the critical locus structure of a Jacobi algebra this presents us with a natural way to study the motivic Donaldson-Thomas invariants of the character varieties of mapping tori and we present ideas on how this can be accomplished. In the second part of this thesis we focus on the cohomological Donaldson-Thomas theory of fundamental group algebras over Riemann surfaces. Again utilising brane tilings we prove that the cohomological Hall algebra of the character variety of a Riemann surface has a natural 2 Calabi-Yau structure arising from a 2D Jacobi algebra, and hence can be obtained by dimensional reduction of the corresponding 3D cohomological Hall algebra of the 3D Jacobi algebra.
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