Donaldson-Thomas theory and cohomological Hall algebras of character stacks
View/ Open
Date
18/01/2023Author
Mistry, Vivek
Metadata
Abstract
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.
Collections
Related items
Showing items related by title, author, creator and subject.
-
Blanchfield and Seifert algebra in high-dimensional boundary link theory I. Algebraic K-theory
Ranicki, Andrew; Sheiham, D. (2006-11-01)The classification of high-dimensional μ–component boundary links motivates decomposition theorems for the algebraic K–groups of the group ring A[Fμ] and the noncommutative Cohn localization Σ-1A[Fμ], for any μ≥1 and an ... -
Gradings on the Brauer algebras and double affine BMW algebras
Mkrtchyan, Anna (The University of Edinburgh, 2021-07-31)In this thesis we study algebras that appear in di erent generalisations of the well-known Schur-Weyl duality. This classical result gives a remarkable connection between the irreducible finite-dimensional representations ... -
1. On the uniformisation of algebraic curves; 2. On the solution of linear differential equations by definite integrals
Ahmed, Mohamed Mursi (The University of Edinburgh, 1931)