9 research outputs found
Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras
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 Poincar\'e-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.Comment: v5 final version, 64 pages, to appear in Invent. Math. Many thanks to
the anonymous referee for helpful suggestion