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Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras
Ben Davison,Sven Meinhardt +1 more
TLDR
The cohomological aspects of Donaldson-Thomas theory for Jacobi algebras and the associated cohomologically Hall algebra, introduced by Kontsevich and Soibelman, were studied in this paper.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 Poincare-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.read more
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Cohomological Hall algebras, vertex algebras and instantons
TL;DR: In this paper, an action of the double of Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov is defined.
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On cohomological Hall algebras of quivers: Generators
Olivier Schiffmann,Eric Vasserot +1 more
TL;DR: The cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties were studied in this paper.
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On cohomological Hall algebras of quivers : Yangians
Olivier Schiffmann,Eric Vasserot +1 more
TL;DR: The cohomological Hall algebra Y of a Lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q and its actions on the cohomology of quiver varieties was studied in this paper.
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Moduli stacks of semistable sheaves and representations of Ext–quivers
TL;DR: In this paper, it was shown that the moduli stacks of semistable sheaves on smooth projective varieties are analytic locally on their coarse moduli spaces described in terms of representations of the associated Ext-quivers with convergent relations.
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Cohomological Hall algebra of Higgs sheaves on a curve
TL;DR: The cohomological Hall algebra of the Calabi-Yau category of coherent Higgs sheaves on a smooth projective curve has been studied in this article in the context of Borel-Moore homology.
References
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Geometric Invariant Theory
TL;DR: Geometric invariant theory for moduli spaces has been studied extensively in the mathematical community as mentioned in this paper, with a large number of applications to the moduli space construction problem, see, for instance, the work of Mumford and Fogarty.
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The Yang-Mills equations over Riemann surfaces
Michael Atiyah,Raoul Bott +1 more
TL;DR: In this article, the Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory, and the main result is that this is a perfect 9 functional provided due account is taken of its gauge symmetry.
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Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
Maxim Kontsevich,Yan Soibelman +1 more
TL;DR: In this article, the authors define new invariants of 3d Calabi-Yau categories endowed with a stability structure, which are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field.
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Moduli of representations of finite dimensional algebras
TL;DR: In this paper, a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k is presented, where the problem of classifying A -modules with a fixed class in the Grothendieck group K0(mod-A), represented by a 'dimension vector' a, is converted into one of classification orbits for the action of a reductive algebraic group GL(a) on a subvariety VA(a), of the representation space 9t{Q, a) of the quiver.
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Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
Maxim Kontsevich,Yan Soibelman +1 more
Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
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