Given a brane tiling, that is a bipartite graph on a torus, we can associate with it a quiver potential and a quiver potential algebra. Under certain consistency conditions on a brane tiling, we prove a formula for the Donaldson-Thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra. We relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling. We prove that the same consistency conditions imply that the quiver potential algebra is a 3-Calabi-Yau algebra. We also formulate a rationality conjecture for the generating functions of the Donaldson-Thomas type invariants.
We derive a combinatorial formula for quantized Donaldson-Thomas invariants of the m-loop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver.
We derive some combinatorial consequences from the positivity of Donaldson-Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.
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.
We study motivic Donaldson-Thomas invariants for a class of quivers with potentials using the strategy of Behrend, Bryan, and Szendroi. This class includes quivers with potentials arising from consistent brane tilings and quivers with zero potential. Our construction is an alternative to the constructions of Kontsevich and Soibelman. We construct an integration map from the equivariant Hall algebra to the quantum torus and show that our motivic Donaldson-Thomas invariants are images of the natural elements in the equivariant Hall algebra. We show that the inegration map is an algebra homomorph…
A system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert scheme-type) framed versions of quiver moduli is derived. This is applied to wall-crossing formulas for the Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman, in particular confirming their integrality.
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.
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 study the relative Donaldson-Thomas theory of A_n x P^1, where A_n is the surface resolution of a type A_n singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra \hat{gl}(n+1) on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov-Witten theory of A_n x P^1 and the quantum cohomology of the Hilbert scheme of points on A_n.
Let $G\subset SL_2(C)\subset SL_3(C)$ be a finite group. We compute motivic Pandharipande-Thomas and Donaldson-Thomas invariants of the crepant resolution $Hilb^G(C^3)$ of $C^3/G$ generalizing results of Gholampour and Jiang who computed numerical DT/PT invariants using localization techniques. Our formulas rely on the computation of motivic Donaldson-Thomas invariants for a special class of quivers with potentials. We show that these motivic Donaldson-Thomas invariants are closely related to the polynomials counting absolutely indecomposable quiver representations over finite fields introduce…
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…
Notes from the report at the Fields institute in Toronto. We introduce the Donaldson-Thomas invariants and describe the wall-crossing formulas for numerical Donaldson-Thomas invariants.
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.
We study higher rank Donaldson-Thomas invariants of a Calabi-Yau 3-fold using Joyce-Song's wall-crossing formula. We construct quivers whose counting invariants coincide with the Donaldson-Thomas invariants. As a corollary, we prove the integrality and a certain symmetry for the higher rank invariants.
We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.
The local Donaldson-Thomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating Gromov-Witten theory, Donaldson-Thomas theory, and the quantum cohomology of the Hilbert scheme of points of the plane. The quantum differential equation of the Hilbert scheme of points of the plane has a natural interpretation in the local Donaldson-Thomas theory of curves. The solution determines the 1-legged equivariant vertex.
We study motivic Donaldson-Thomas invariants in the sense of Behrend-Bryan-Szendroi. A wall-crossing formula under a mutation is proved for a certain class of quivers with potentials.
We study the equivariant Gromov-Witten and Donaldson-Thomas theories of $\mathbf{P}^2$-bundles over curves. We show the equivariant GW/DT correspondence holds to first order for certain curve classes
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 solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson-Thomas correspondence for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for the quintic threefold in degrees one and two.
The present paper is an extension of a previous paper written in collaboration with Markus Reineke dealing with quiver representations. The aim of the paper is to generalize the theory and to provide a comprehensive theory of Donaldson-Thomas invariants for abelian categories of homological dimension one (without potential) satisfying some technical conditions. The theory will apply for instance to representations of quivers, coherent sheaves on smooth projective curves, and some coherent sheaves on smooth projective surfaces. We show that the (motivic) Donaldson-Thomas invariants satisfy the …
Using the degeneration formula for Donaldson-Thomas invariants, we proved a formula for the change of Donaldson-Thomas invariants of local surfaces under blowing up along points.
We study the Gromov-Witten and Donaldson-Thomas correspondence conjectured in [MNOP1, MNOP2], for trivial elliptic fibrations. In particular, we verify the Gromov-Witten and Donaldson-Thomas correspondence for primary fields when the threefold is $E \times S$ where $E$ is a smooth elliptic curve and $S$ is a smooth surface with numerically trivial canonical class.
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 study the Donaldson-Thomas type invariants for the Calabi-Yau threefold Deligne-Mumford stacks under flops. A crepant birational morphism between two smooth Calabi-Yau threefold Deligne-Mumford stacks is called an orbifold flop if the flopping locus is the quotient of weighted projective lines by a cyclic group action. We prove that the Donaldson-Thomas invariants are preserved under orbifold flops.
We discuss the GW/DT correspondence for 3-folds in both the absolute and relative cases. Descendents in Gromov-Witten theory are conjectured to be equivalent to Chern characters of the universal sheaf in Donaldson-Thomas theory. Relative constraints in Gromov-Witten theory are conjectured to correspond in Donaldson-Thomas theory to cohomology classes of the Hilbert scheme of points of the relative divisor. Independent of the conjectural framework, we prove degree 0 formulas for the absolute and relative Donaldson-Thomas theories of toric varieties.
Using a homotopy approach, we prove in this paper a conjecture of Maulik, Nekrasov, Okounkov and Pandharipande on the dimension zero Donaldson-Thomas invariants of all smooth complex threefolds.
We compute the Donaldson-Thomas invariants for two types of Calabi-Yau 3-folds. These invariants are associated to the moduli spaces of rank-2 Gieseker semistable sheaves. None of the sheaves are locally free, and their double duals are locally free stable sheaves investigated earlier by Donaldson and Thomas, Li and Qin respectively. We show that these Gieseker moduli spaces are isomorphic to some Quot-schemes. We prove a formula for Behrend's functions when torus actions present with positive dimensional fixed point sets, and use it to obtain the generating series of the relevant Donaldson-Th…
We modify the standard perfect symmetric obstruction theory for moduli spaces of simple perfect complexes, to the situation of complexes on abelian threefolds with fixed determinant and Fourier-Mukai determinant. As outcome we attach nontrivial Donaldson-Thomas invariants to moduli spaces for complexes modulo twist and translation.
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.
We show that the generating series of generalized Donaldson-Thomas invariants on the local projective plane with any positive rank is described in terms of modular forms and theta type series for indefinite lattices. In particular it absolutely converges to give a holomorphic function on the upper half plane.
We present a construction of Donaldson-Thomas invariants for three-dimensional projective Calabi-Yau Deligne-Mumford stacks. We also study the structure of these invariants for etale gerbes over such stacks.
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 determine the structure of certain moduli spaces of ideal sheaves by generalizing an earlier result of the first author. As applications, we compute the (virtual) Hodge polynomials of these moduli space, and calculate the Donaldson-Thomas invariants of certain 3-folds with trivial canonical classes.
We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Marino, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.
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.
We conjecture an equivalence between the Gromov-Witten theory of 3-folds and the holomorphic Chern-Simons theory of Donaldson-Thomas. For Calabi-Yau 3-folds, the equivalence is defined by the change of variables, exp(iu)=-q, where u is the genus parameter of GW theory and q is charge parameter of DT theory. The conjecture is proven for local Calabi-Yau toric surfaces.
Given a complex 4-fold $X$ with an (Calabi-Yau 3-fold) anti-canonical divisor $Y$, we study relative Donaldson-Thomas invariants for this pair, which are elements in the Donaldson-Thomas cohomologies of $Y$. We also discuss gluing formulas which relate relative invariants and $DT_{4}$ invariants for Calabi-Yau 4-folds.
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.
Let $X$ be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kähler form $ω$ and a holomorphic four-form $Ω$. Under certain assumptions, we define Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifold. We also study sheaves counting on local Calabi-Yau four-folds. We relate the sheaves countings over $K_{Y}$ with the Donaldson-Thomas invariants for the associated compact three-fold $Y$. In some very special cases, we prove the DT/GW correspondence for $X$. Finally, we comput…
We show that there is a natural perverse sheaf on the moduli space of semistable sheaves on a smooth projective Calabi-Yau 3-fold which is locally the perverse sheaf of vanishing cycles for a local Chern-Simons functional. This gives us a cohomology theory whose Euler numbers are Donaldson-Thomas invariants.
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…
Motivated by the counting of BPS states in string theory with orientifolds, we study moduli spaces of self-dual representations of a quiver with contravariant involution. We develop Hall module techniques to compute the number of points over finite fields of moduli stacks of semistable self-dual representations. Wall-crossing formulas relating these counts for different choices of stability parameters recover the wall-crossing of orientifold BPS/Donaldson-Thomas invariants predicted in the physics literature. In finite type examples the wall-crossing formulas can be reformulated in terms of id…
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.
Donaldson-Thomas invariant is expressed as the weighted Euler characteristic of the so-called Behrend (constructible) function. In \cite{Behrend} Behrend introduced a DT-type invariant for a morphism. Motivated by this invariant, we extend the motivic Hirzebruch class to naive Donaldson-Thomas type analogues. We also discuss a categorification of the DT-type invariant for a morphism from a bivariant-theoretic viewpoint, and we finally pose some related questions for further investigations.
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.
