Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

By M. Kontsevich, Y. Soibelman. 2010

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We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0ab94685f08167cde6585228a9e277a4b332f25d

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Citations in the corpus

Homological algebra of knots and BPS states (2011, background): “Z(γ2), i.e. it deﬁnes a homomorphism Z ∈ Hom(Γ,C). 10 SERGEI GUKOV AND MARKO STOSIˇ C´ In the present case, the relevant algebras include the motivic Hall algebra [17], the cohomological Hall algebra [31], and its various ramiﬁcations, e.g. cluster algebras. Therefore, the problem can be approached ”
Cluster algebras and derived categories (2012, background): “In other words, for each vertex t of the n-regular tree and each 1 ≤ j ≤ n, the cluster variable xj(t) belongs to the ring Z[x±11 , . . . , x ±1 n , xn+1, . . . , x2n].”
A classical model for derived critical loci (2013, background): “ sheaves, D-modules and mixed Hodge modules || ✲ ✲✲ ✲✲ D–T invariants of Calabi–Yau 3- folds [17,18,33] motivic D–T invariants [18] categoriﬁed D–T theory [9] and cohomological Hall algebras[19] ‘Fukaya categories’ of complex and algebraic symplectic manifolds Figure 1.1: Relations between diﬀeren”
n-angulated categories (2010, background): “However, as we will see, this apparently dry subject matter is linked to some exciting developments in algebraic geometry and string theory [25] [24] [8] [9].”
Arithmetic harmonic analysis on character and quiver varieties II (2008, methodology): “v(q). A more recent work by Mozgovoy[26] proves Conjecture 3.1.2 for any dimension vector for quivers with at least one loop at each vertex. His proof is accomplished via work of Kontsevich-Soibelman [18] and Eﬁmov [4] on motivic Donaldson-Thomas invariants associated to quivers. By Kac[16], there e”
Symmetries and stabilization for sheaves of vanishing cycles (2012, background): “ﬁcation of Donaldson– Thomas invariants of Calabi–Yau 3-folds in algebraic geometry [6,7], which was inspiredbyearlierworkofBehrendand Fantechi[4], Dimca andSzendr˝oi[9], and Kontsevich and Soibelman [15]. Let Zbe a Calabi–Yau 3-fold over C, and Ma propermoduli C-schemeof stable coherent sheaveson Z”
Quiver W-algebras (2015, background): “The relation to cohomological Hall algebra of quiver [21] remains to be clarified.”
A Darboux theorem for derived schemes with shifted symplectic structure (2013, neutral): “Related results are important in Donaldson–Thomas theory [18–20].”
The integrality conjecture and the cohomology of preprojective stacks (2016, methodology): “om the above formulae) and using Hua’s formula [11] to rewrite (34) and the ﬁrst term of (35) as rational functions in q, we recover Theorem 5 of [10]. 7. Degenerations ofCohomologicalHallalgebras In [13], a method was given for associating a Cohomological Hall algebra (CoHA for short) to the data o”