Vertex algebras, moduli stacks, cohomological Hall algebras and quantum groups

Thesis

The content of this Thesis comes in three parts. Firstly, it shows a compatibility between two structures on the homology of moduli stacks of certain codimension one categories: Joyce’s vertex algebra structure and the cohomological Hall algebra (CoHA) structure. Our Theorem 3.10.1 can be thought of as saying the homology H rpMq is a vertex analogue of a “quantum group” (i.e. triangular bialgebra).

Secondly, the main technical work of this thesis builds up the machinery to let us compute cohomological Hall algebras using torus localisation. To begin with, we construct a “bivariant” Euler class and use it to get a clean formulation of torus localisation for singular stacks. We then explain how combining this, with stratifications of the stacks under consideration, allows us to compute their CoHA products. We finish by using these techniques to give new formulae for CoHA products, and a new interpretation of existing ones.

Thirdly, we turn to the question of q deforming Joyce’s vertex algebra structure. We interpret the well known (q deformed) Frenkel-Segal-Kac free field realisation in terms of homology of moduli stacks, then make steps to interpreting it as a map of q deformed vertex algebras.

The appendices include the categorical axiomatics needed to talk about vertex analogues of quasitriangular bialgebras and related structures, as well as the construction of the “cohomological” exponential map for algebraic stacks, which is needed to “linearise” closed embeddings by replacing them with the associated normal bundle/complex.

Authors: Latyntsev, A

• DOI: 10.5287/ora-6grjdkrxb
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English