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.
Highlights
Donaldson–Thomas invariants, first introduced in [46], already have an extensive literature
In this paper we describe and prove the natural categorification of the integrality conjecture in the context of the cohomological Hall algebra for a quiver with potential, and explain how this gives rise to a mathematical definition of the Lie algebra of BPS states, for which the cohomological Hall algebra can be thought of as the positive half of an associated quantum group
This algebra categorifies Donaldson–Thomas theory in the sense that Donaldson– Thomas invariants for the category of ζ-semistable modules in S are obtained by considering the class in the Grothendieck group of the Λζμ-graded mixed Hodge structures of the underlying mixed Hodge structure of (1)
Summary
For the Donaldson–Thomas theory and cohomological Hall algebras associated to quivers, a vital role is always played by the grading by NQ0, the semigroup of dimension vectors, which should be seen as a weight space decomposition in the language of Lie algebras and quantum groups. Xidi i∈Q0 is a formal Laurent power series in q1/2, given by the characteristic function of the dth graded piece of the right hand side of (2)—for q1/2 records the cohomological degree. These power series may have finite order poles at q1/2 = 0, due to the shifts appearing in (2). Because of the second H on the right hand side of (5), this isomorphism would not follow directly from the usual six functor formalism for monodromic mixed Hodge modules on stacks, but instead relies essentially on the “hidden properness” of the morphism pμζ. Song [24], and later the proof of the integrality conjecture for path algebras in the absence of a potential by the second author and Markus Reineke [34], which utilised Efimov’s theorem [15] on free supercommutativity for cohomological Hall algebras without potentials to understand the local behaviour of the morphism pμζ
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