Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers

We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomasinvariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich-Soibelman. This is achieved by computing, via an arithmetic Fourier transform, the dimensions of the isoytpical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincaré polynomials is an extension of Hua's formula for Kac polynomials of quivers involving Hall-Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties. The main resultsLet Γ = (I, Ω) be a quiver: that is, an oriented graph on a finite set I = {1, . . . , r} with Ω a finite multiset of oriented edges. In his study of the representation theory of quivers, Kac [17] introduced A v (q), the number of isomorphism classes of absolutely indecomposable representations of Γ over the finite field F q of dimension v = (v 1 , . . . , v r ) and showed they are polynomials in q. We call A v (q) the Kac polynomial for Γ and v. Following ideas of Kac [17], Hua [16] proved the following generating function identity:where P denotes the set of partitions of all positive integers, Log is the plethystic logarithm (see [14, §2.3.3]), , is the pairing on partitions defined bywith m j (λ) the multiplicity of the part j in the partition λ, which implies positivity for certain refined DT-invariants of symmetric quivers with no potential.The goal of Kontsevich-Soibelman's theory is to attach refined (or motivic, or quantum) DonaldsonThomas invariants (or DT-invariants for short) to Calabi-Yau 3-folds X. The invariants should only depend on the derived category of coherent sheaves on X and some extra data; this raises the possibility of defining DTinvariants for certain Calabi-Yau 3-categories which share the formal properties of the geometric situation, but are algebraically easier to study. The simplest of such examples are the Calabi-Yau 3-categories attached to quivers (symmetric or not) with no potential (c.f. [12]).Denote by Γ = (I, Ω) the double quiver, that is Ω = Ω Ω opp , where Ω opp is obtained by reversing all edges in Ω. The refined DT-invariants of Γ (a slight renormalization of those introduced by Kontsevich and Soibelman [19]) are defined by the following combinatorial construction.In fact, as a consequence of Efimov's proof [10] of [20, Conjecture 1], DT v (q) actually has non-negative coefficients. We will give an alternative proof of this in (1.10) by interpreting its coefficients as dimensions of cohomology groups of an associated quiver variety. Remark 1.2. We should stress that we have restricted to double quivers for the benefit of exposition; our results extend easily to any symmetric quiver. We outline how to treat the general case in §3.2. The technical starting point in this paper is a common generalization of (1.3) and Hua'...

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“…Up to a power of q the invariants DT n (q) are those considered by Reineke [30]. Here is a list of the first few values.…”

Section: Examplesmentioning

confidence: 99%

Positivity for Kac polynomials and DT-invariants of quivers

2013

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“…The technical difficulties occurring in their approach disappear in the special situation of representations of quivers (with zero potential). This case has been intensively studied by the second author in a series of papers [30], [31], [32]. Despite some computations of motivic or even numerical Donaldson-Thomas invariants for quivers with or without potential (see [1], [8], [7], [28]), the true nature of Donaldson-Thomas invariants still remains mysterious.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, we will give, in Theorem 4.7, an explicit formula for the intersection Betti numbers of the classical spaces of matrix invariants (that is, the quotient of tuples of linear operators by simultaneous conjugation), using the explicit formula for motivic DT invariants for loop quivers in [32].…”

Section: Introductionmentioning

confidence: 99%

Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

Meinhardt

Reineke

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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100

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Abstract. The main result of this paper is the statement that the Hodge theoretic Donaldson-Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson-Thomas "function" to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson-Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

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“…Remark 3.6. A slightly weaker statement of Conjecture 1 for the quivers with one vertex and several loops was recently proved by Markus Reineke [13]. The complete proof of [7, Conjecture 1] and thus of Conjecture 1 was recently obtained by Efimov [2].…”

Section: Donaldson-thomas Invariantsmentioning

confidence: 96%

“…It follows from the conjecture of Kontsevich and Soibelman [7, Conjecture 1] on the properties of the cohomological Hall algebra of a symmetric quiver, that the Donaldson-Thomas invariants are polynomials with non-negative coefficients. An interesting combinatorial interpretation of the Donaldson-Thomas invariants for the quiver with one vertex and several loops was given by Reineke [13]. The full conjecture [7, Conjecture 1] was recently proved by Efimov [2].…”

Section: Introductionmentioning

confidence: 99%