Motivic characteristic classes in cohomological Hall algebras

TL;DR

This paper explores the computation of characteristic classes for Dynkin quiver orbits within cohomological Hall algebras, revealing identities akin to Donaldson-Thomas quantum dilogarithm identities and their invariance under stability conditions.

Contribution

It introduces a reduction method for computing CSM and MC classes of quiver orbits and proves a stability-independent identity in CoHA and KHA related to these classes.

Findings

Reduction of class computation to basic classes $c^o_eta$, $C^o_eta$

Proved stability-independent identities in CoHA and KHA

Expressed classes as commutators via wall-crossing arguments

Abstract

The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are important characteristic classes of singular varieties in cohomology and K theory---and their theory overlaps with the theory of Okounkov's stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes $c^o_\beta$, $C^o_\beta$ parameterized by positive roots $\beta$. We prove an identity in a deformed version of Kontsevich-Soibelman's Cohomological (and K-theoretic) Hall Algebra (CoHA, KHA), namely, that a product of exponentials of $c^o_\beta$ (or $C^o_\beta$) classes formally depending on a stability function Z, does not depend on Z. This identity---which encodes infinitely many identities among rational functions in growing number of variables---has the structure of Donaldson-Thomas type quantum dilogarithm identities. Using a wall-crossing argument we present the $c^o_\beta$, $C^o_\beta$ classes as certain commutators in the CoHA, KHA.