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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
Consistency conditions for brane tilings
Abstract Given a brane tiling on a torus, we provide a new way to prove and generalise the recent results of Szendrői, Mozgovoy and Reineke regarding the Donaldson–Thomas theory of the moduli space
The integrality conjecture and the cohomology of preprojective stacks
By importing the compactly supported cohomology of various stacks of representations of the preprojective algebra $\Pi_Q$, for $Q$ an arbitrary quiver, into the theory of cohomological
Positivity for quantum cluster algebras
Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first
Purity for graded potentials and quantum cluster positivity
Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-equivariant with respect
Superpotential algebras and manifolds
We study a special class of Calabi–Yau algebras (in the sense of Ginzburg): those arising as the fundamental group algebras of acyclic manifolds. Motivated partly by the usefulness of ‘superpotential
Donaldson-Thomas theory for categories of homological dimension one with potential
The aim of the paper is twofold. Firstly, we give an axiomatic presentation of Donaldson-Thomas theory for categories of homological dimension at most one with potential. In particular, we provide
The critical CoHA of a quiver with potential
Pursuing the similarity between the Kontsevich--Soibelman construction of the cohomological Hall algebra of BPS states and Lusztig's construction of canonical bases for quantum enveloping algebras,
The motivic Donaldson–Thomas invariants of ($-$2)-curves
We calculate the motivic Donaldson–Thomas invariants for (−2)-curves arising from 3-fold flopping contractions in the minimal model program. We translate this geometric situation into the machinery
The critical CoHA of a self dual quiver with potential
In this paper we provide an explanation for the many beautiful infinite product formulas for generating functions of refined DT invariants for symmetric quivers with potential: they are