[Paper Review] Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras

Ben Davison, Sven Meinhardt|Edinburgh Research Explorer|Jan 11, 2016

Algebraic structures and combinatorial models31 references17 citations

TL;DR

This paper establishes a cohomological categorification of the integrality conjecture and wall-crossing formula in Donaldson-Thomas theory for quivers with potential, using a perverse filtration on the cohomological Hall algebra. It realizes these isomorphisms as Poincaré-Birkhoff-Witt type isomorphisms and constructs a Lie algebra of BPS invariants, showing the cohomological Hall algebra is the positive part of a quantum group.

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.

Motivation & Objective

• To categorify the integrality conjecture and wall-crossing formula in cohomological Donaldson-Thomas theory for quivers with potential.
• To define a perverse filtration on the cohomological Hall algebra arising from the semisimplification map of moduli stacks.
• To realize the integrality and wall-crossing isomorphisms as Poincaré-Birkhoff-Witt isomorphisms for the cohomological Hall algebra.
• To construct a Lie algebra structure on the mixed Hodge structure of BPS invariants, identifying the cohomological Hall algebra as the positive part of a quantum group.
• To provide a mathematical foundation for the Lie algebra of BPS states in the context of cohomological Hall algebras.

Proposed method

• Define a perverse filtration on the cohomological Hall algebra using the 'hidden properness' of the semisimplification map from the moduli stack of semistable representations to the coarse moduli space of polystable representations.
• Construct a degeneration of the cohomological Hall algebra for generic stability conditions and fixed slope to a free supercommutative algebra generated by a mixed Hodge structure categorifying BPS invariants.
• Use monodromic mixed Hodge modules and equivariant vanishing cycles to analyze the cohomological structure of the Hall algebra.
• Apply the Harder-Narasimhan filtration to filter the cohomological Hall algebra and show that the associated graded map is an isomorphism of pure mixed Hodge modules.
• Establish a relative cohomological Hall algebra structure via correspondences on framed moduli spaces of quiver representations.
• Prove that the cohomological Hall algebra is isomorphic to the twisted symmetric algebra of the BPS Lie algebra via a Poincaré-Birkhoff-Witt type isomorphism.

Experimental results

Research questions

• RQ1 How can the integrality conjecture in Donaldson-Thomas theory be categorified at the cohomological level for quivers with potential?
• RQ2 What is the role of the perverse filtration in organizing the cohomological Hall algebra structure and its degeneration?
• RQ3 How does the wall-crossing formula manifest in the cohomological Hall algebra, and can it be realized as a Poincaré-Birkhoff-Witt isomorphism?
• RQ4 What Lie algebra structure emerges from the BPS invariants in this cohomological framework?
• RQ5 How does the cohomological Hall algebra relate to a quantum group, and what is the significance of the positive part construction?

Key findings

• The integrality conjecture is categorified via a Poincaré-Birkhoff-Witt isomorphism for the cohomological Hall algebra.
• The wall-crossing formula is realized as a Poincaré-Birkhoff-Witt isomorphism in the cohomological Hall algebra setting.
• The cohomological Hall algebra degenerates to a free supercommutative algebra generated by a mixed Hodge structure categorifying BPS invariants.
• A Lie algebra structure is constructed on the mixed Hodge structure of BPS invariants, termed the BPS Lie algebra.
• The cohomological Hall algebra is identified as the positive part of a Yangian-type quantum group.
• The isomorphism between the cohomological Hall algebra and the symmetric algebra of the BPS Lie algebra is established via a twisted symmetric algebra construction, confirming the quantum group structure.