Cohomological Donaldson–Thomas theory of a quiver with potential and quantum enveloping algebras

Abstract

This paper concerns the cohomological aspects of Donaldson-Thomas theory for Jacobi algebras and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman. We prove the Hodge-theoretic categorification of the integrality conjecture and the wall crossing formula, and furthermore realise the isomorphism in both of these theorems as Poincaré-Birkhoff-Witt isomorphisms for the associated cohomological Hall algebra.We do this by defining a perverse filtration on the cohomological Hall algebra, a result of the "hidden properness" of the semisimplification map from the moduli stack of semistable representations of the Jacobi algebra to the coarse moduli space of polystable representations.This enables us to construct a degeneration of the cohomological Hall algebra, for generic stability condition and fixed slope, to a free supercommutative algebra generated by a mixed Hodge structure categorifying the BPS invariants. As a corollary of this construction we furthermore obtain a Lie algebra structure on this mixed Hodge structure -the Lie algebra of BPS invariants -for which the entire cohomological Hall algebra can be seen as the positive part of a Yangian-type quantum group.

Positivity for quantum cluster algebras

“…This theorem turns out to be an easy consequence of the cohomological wall crossing and integrality theorems of [DM16]. In fact, it follows from a very special case since we do not have to consider potentials (equivalently, we set W = 0).…”

“…Proof. Theorem A in [DM16] upgrades (84) to the category of mixed Hodge structures. In particular, there is an inclusion BPS ζ v,Hdg (Q) ⊗ L 1/2 ⊂ A ζ v,Hdg (Q).…”

“…Purity for cohomological DT invariants of quivers (possibly containing oriented cycles) without potential is in fact a direct consequence of the main theorems of [DM16]; we briefly explain how.…”

“…We introduce the background notation for talking about either flavour of Donaldson-Thomas theory for quivers. We will work with right modules throughout, and so our sign conventions will agree with [Dav18] and differ slightly from [DM16]. A quiver Q is determined by two sets Q 1 and Q 0 , the arrows and vertices respectively, along with a pair of maps s, t : Q 1 → Q 0 taking an arrow to its source and target, respectively.…”

“…However, the quantum setup presents new challenges, particularly in the proof of positivity. The present paper uses ideas from Donaldson-Thomas (DT) theory [KS,Kel11,Nag13,DM16,Bri17,MR19] in concert with scattering diagram technology [KS06, GS11, CPS, FS15, KS14, GHK15b, GHKK18, Mana] to prove positivity in the quantum setting.…”