We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we give an alternative proof of Fomin-Zelevinsky's conjectures on $F$-polynomials and $g$-vectors.
We compute motivic Donaldson-Thomas invariants for crepant resolutions of quotients of affine three-space by even dihedral groups in terms of an affine type D root system, using double dimensional reduction and the representation theory of affine type D quivers.
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 s…
We develop theoretical aspects of refined Donaldson-Thomas theory for threefold flops, and use these to determine all DT invariants for a doubly infinite family of length 2 flopping contractions. Our results show that a refined version of the strong-rationality conjecture of Pandharipande-Thomas holds in this setting, and also that refined DT invariants do not classify flops. Our main innovation is the application of tilting theory to better understand the stability conditions and cyclic A-infinity-deformation theory of these spaces. Where possible we work in the motivic setting, but we also c…
The semi-stable ChowHa of a quiver with stability is defined as an analog of the Cohomological Hall algebra of Kontsevich and Soibelman via convolution in equivariant Chow groups of semi-stable loci in representation varieties of quivers. We prove several structural results on the semi-stable ChowHa, namely isomorphism of the cycle map, a tensor product decomposition, and a tautological presentation. For symmetric quivers, this leads to an identification of their quantized Donaldson-Thomas invariants with the Chow-Betti numbers of moduli spaces.
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 rigorous proofs of all standard results concerning the integration map, wall-crossing, PT-DT correspondence, etc. following Kontsevich and Soibelman. We also show the equivalence of their approach and the one given by Joyce and Song. Secondly, we relate Donaldson-Thomas functions for such a category with arbitrary potential to those with zero potential under some mild conditions. As a result of this,…
We establish a geometric interpretation of orientifold Donaldson-Thomas invariants of $σ$-symmetric quivers with involution. More precisely, we prove that the cohomological orientifold Donaldson-Thomas invariant is isomorphic to the rational Chow group of the moduli space of $σ$-stable self-dual quiver representations. As an application we prove that the Chow Betti numbers of moduli spaces of stable $m$-tuples in classical Lie algebras can be computed numerically. We also prove a cohomological wall-crossing formula relating semistable Hall modules for different stabilities.
We prove a family of factorization formulas for the combinatorial Donaldson--Thomas invariant for an acyclic quiver. A quantum dilogarithm identity due to Reineke, later interpreted by Rimanyi by counting codimensions of quiver loci, gives two extremal cases of our formulation in the Dynkin case. We establish our interpolating factorizations explicitly with a dimension counting argument by defining certain stratifications of the space of representations for the quiver and calculating Betti numbers in the corresponding equivariant cohomology algebras.
Kontsevich and Soibelman defined the notion of Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. A family of examples of such categories can be constructed from an arbitrary cluster variety. The corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety, known as Donaldson-Thomas transformation. Fix two integers $m$ and $n$ with $1<m<m+1<n$. It is known that the configuration space $\mathrm{Conf}_n(\mathbb{P}^{m-1})$, closely related to Grassmannian $\mathrm{Gr}_m(n)$, is a cluster Poisson variety. In this pap…
The aim of the paper is to provide a rather gentle introduction into Donaldson-Thomas theory using quivers with potential. The reader should be familiar with some basic knowledge in algebraic or complex geometry. The text contains many examples and exercises to support the process of understanding the main concepts and ideas.
Let $A$ be an abelian variety. We introduce $A$-equivariant Grothendieck rings and $A$-equivariant motivic Hall algebras, and endow them with natural integration maps to the ring of dual numbers. The construction allows a systematic treatment of reduced Donaldson-Thomas invariants by Hall algebra techniques. We calculate reduced Donaldson-Thomas invariants for $\mathrm{K3} \times E$ and abelian threefolds for several imprimitive curve classes. This verifies (in special cases) multiple cover formulas conjectured by Oberdieck-Pandharipande and Bryan-Oberdieck-Pandharipande-Yin.
In this paper we calculate the motivic Donaldson-Thomas invariants for (-2)-curves arising from 3-fold flopping contractions in the minimal model programme. We translate this geometric situation into the machinery developed by Kontsevich and Soibelman, and using the results and framework developed previously by the authors we describe the monodromy on these invariants. In particular, in contrast to all existing known Donaldson-Thomas invariants for small resolutions of Gorenstein singularities these monodromy actions are nontrivial.
Kontsevich and Soibelman defined the Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety can produce an example of such a category, whose corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety, known as the Donaldson-Thomas transformation. In this paper we prove a conjecture of Goncharov and Shen in the case of $\mathrm{GL}_n$, which describes the Donaldson-Thomas transformation of the double quotient of the double Bruhat cells $H \backslash \mathrm{GL}_n^{u,v}/H$ where $H$ is a maximal t…
We generalize the notion of expanded degenerations and pairs for a simple degeneration or smooth pair to the case of smooth Deligne-Mumford stacks. We then define stable quotients on the classifying stacks of expanded degenerations and pairs and prove the properness of their moduli's. On 3-dimensional smooth projective DM stacks this leads to a definition of relative Donaldson-Thomas invariants and the associated degeneration formula.
This paper is a continuation of author's previous work arXiv:1911.07949, where we defined Donaldson-Thomas invariants of quantum Fermat threefolds. In this paper, we study the generic quantum Fermat threefold. We give explicit local models for Hilbert schemes of points as quivers with potential, and compute degree zero Donaldson-Thomas invariants. The result is expressed in terms of certain colored plane partitions.
We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $σ$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson-Thomas theory of quivers which allows the quiver representations to have orthogonal and symplectic structure groups. The associated invariants are call…
Let $X$ and $X'$ be nonsingular projective $3$-folds related by a flop of a disjoint union of $(-2)$-curves. We prove a flop formula relating the Donaldson-Thomas invariants of $X$ to those of $X'$, which implies some simple relations among BPS state counts. As an application, we show that if $X$ satisfies the GW/DT correspondence for primary insertions and descendants of the point class, then so does $X'$. We also propose a conjectural flop formula for general flops.
We further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied in [Bryan'19]. These are smooth proper Calabi-Yau threefolds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a "banana configuration". In [Bryan'19] the Donaldson-Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations…
We prove the flow tree formula conjectured by Alexandrov and Pioline which computes Donaldson-Thomas invariants of quivers with potentials in terms of a smaller set of attractor invariants. This result is obtained as a particular case of a more general flow tree formula reconstructing a consistent scattering diagram from its initial walls.
We study a class of Riemann-Hilbert problems arising naturally in Donaldson-Thomas theory. In certain special cases we show that these problems have unique solutions which can be written explicitly as products of gamma functions. We briefly explain connections with Gromov-Witten theory and exact WKB analysis.
Motives of Brauer-Severi schemes of Cayley-smooth algebras associated to homogeneous superpotentials are used to compute inductively the motivic Donaldson-Thomas invariants of the corresponding Jacobian algebras. This approach can be used to test the conjectural exponential expressions for these invariants, proposed in arXiv:1510.08116. As an example we confirm the second term of the conjectured expression for the motivic series of the homogenised Weyl algebra.
