On the cohomological Hall algebra of a character variety

TL;DR

This paper explores the structure of the cohomological Hall algebra (CoHA) associated with character varieties, comparing different multiplication constructions and generalizing previous work on commuting matrices.

Contribution

It generalizes the multiplication on the 2D CoHA to character varieties and compares different presentations of the algebra.

Findings

Comparison of multiplication structures on the character variety CoHA

Extension of CoHA construction to zero-locus of functions on smooth varieties

Insights into algebraic structures from different presentations

Abstract

A multiplication on the 2D cohomological Hall algebra (CoHA) of the variety of commuting matrices was described by Schiffman and Vasserot. This construction can be generalised to other varieties that exist as the zero-locus of a function on a smooth ambient variety. On the 2D CoHA of the character variety of the fundamental group of a genus $g$ Riemann surface, we compare the multiplication induced by the standard presentation and that of a brane tiling presentation.