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Ben Davison

Publications and source records attributed to Ben Davison.

At least 19 recordsLinked to original sources
arXiv2026

Hecke operators on symplectic surfaces and $\chi$-independence

We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise to a global BPS Lie algebra attached to the stack of coherent sheaves on S. The link between this structure and the applications to chi-independence and tautological generation is provided by Hecke operators on BPS cohomology, which modify one-dimensional sheaves by zero-dimensional quotients. To make this construction work, we prove that there is an identification between the affinized BPS cohomology of the semistable locus and the primitive part of the coproduct on the entire moduli stack

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arXiv2025

The generic Markov CoHA is not spherically generated

Let $Q$ be the Markov quiver, and let $W$ be an infinitely mutable potential for $Q$. We calculate some low degree refined BPS invariants for the resulting Jacobi algebra, and use them to show that the critical cohomological Hall algebra $\mathcal{H}_{Q,W}$ is not necessarily spherically generated, and is not independent of the choice of infinitely mutable potential $W$. This leads to a counterexample to a conjecture of Gaiotto, Grygoryev and Li \cite[\S 2.1]{GGL}, but also suggestions for how to modify it. In the case of generic cubic $W$, we discuss a way to modify the conjecture, by excluding the non-spherical part via the decomposition of $\mathcal{H}_{Q,W}$ according to the characters of a discrete symmetry group.

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arXiv2025

Cohomology of symmetric stacks

We construct decompositions of: (1) the cohomology of smooth stacks, (2) the Borel--Moore homology of $0$-shifted symplectic stacks, and (3) the vanishing cycle cohomology of $(-1)$-shifted symplectic stacks, assuming a good moduli space exists and the tangent space has a pointwise orthogonal structure. These conditions are satisfied by many stacks of interest, including moduli stacks of semistable $G$-bundles and (twisted) $G$-Higgs bundles on curves, $G$-character stacks of oriented closed 2-manifolds and various 3-manifolds, and moduli stacks of semistable coherent sheaves on Calabi--Yau threefolds and K3 surfaces with generic polarization. As a special case, we prove a PBW-type theorem for cohomological Hall algebras of $3$-Calabi--Yau categories with commutative orientation data, a strong form of the cohomological integrality conjecture for such categories. We define the BPS cohomology as the primary summand of the decomposition. When the stack is smooth, the BPS cohomology coincides with the intersection cohomology of the good moduli space, generalizing a theorem of Meinhardt--Reineke. Using the BPS cohomology for singular spaces, we propose a formulation of the topological mirror symmetry conjecture for the stack of $G$-Higgs bundles generalizing the work of Hausel and Thaddeus for type A groups, and a version of Langlands duality for character stacks of compact oriented 3-manifolds, following Ben-Zvi--Gunningham--Jordan--Safronov.

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arXiv2024

Degree two Gopakumar-Vafa invariants of local curves

We investigate the Gopakumar-Vafa (GV) theory of local curves, namely, the total spaces of rank two vector bundles with canonical determinant on smooth projective curves. Under a certain genericity condition on the rank two bundles, we propose a general mechanism to compute the degree two GV invariants of local curves. In particular, we determine all the degree two GV invariants when the base curve has genus two. Combined with previous work by Bryan and Pandharipande, we obtain the GV/GW correspondence in this case. When the base curve has genus greater than two, we calculate GV invariants for some extremal genera, providing evidence for the GV/GW conjecture for curves of higher genus.

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arXiv2023

Okounkov's conjecture via BPS Lie algebras

Let $Q$ be an arbitrary finite quiver. We use nonabelian stable envelopes to relate representations of the Maulik-Okounkov Lie algebra $\mathfrak{g}^{MO}_Q$ to representations of the BPS Lie algebra associated to the tripled quiver $\tilde Q$ with its canonical potential. We use this comparison to provide an isomorphism between the Maulik-Okounkov Lie algebra and the BPS Lie algebra. Via this isomorphism we prove Okounkov's conjecture, equating the graded dimensions of the Lie algebra $\mathfrak{g}^{MO}_Q$ with the coefficients of Kac polynomials. Via general results regarding cohomological Hall algebras in dimensions two and three we furthermore give a complete description of $\mathfrak{g}^{MO}_Q$ as a generalised Kac-Moody Lie algebra with Cartan datum given by intersection cohomology of singular Nakajima quiver varieties, and prove a conjecture of Maulik and Okounkov, stating that their Lie algebra is obtained from a Lie algebra defined over the rationals, by extension of scalars. Finally, we explain how our results suggest the correct definition of critical stable envelopes in vanishing cycle cohomology.

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arXiv2023

BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories

We determine the structure of the BPS algebra of 2-Calabi-Yau Abelian categories for which the stack of objects admits a good moduli space. We prove that this algebra is isomorphic to the positive part of the enveloping algebra of a generalised Kac-Moody Lie algebra generated by the intersection cohomology of certain connected components (corresponding to roots) of the good moduli space. Some major examples include the BPS algebras of (1) the category of semistable coherent sheaves of given slope on a K3 surface or, more generally, quasiprojective symplectic surface, (2) semistable Higgs bundles on a smooth projective curve, (3) preprojective algebras of quivers, (4) multiplicative preprojective algebras and (5) fundamental groups of (quiver) Riemann surfaces. We define the BPS Lie algebras of 2-Calabi-Yau categories and prove that they coincide with the ones obtained by dimensional reduction from the critical cohomological Hall algebra in the case in which the 2-Calabi-Yau category is the category of representations of a preprojective algebra. Consequences include (1) A proof in full generality of the Bozec-Schiffmann positivity conjecture for absolutely cuspidal polynomials, a strengthening of the Kac positivity conjecture (2) A proof of the cohomological integrality conjecture for the category of semistable coherent sheaves on local K3 surfaces (3) A description of the cohomology (in all degrees) of Nakajima quiver varieties as direct sums of irreducible lowest weight representations over the BPS Lie algebra.

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arXiv2022

BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks

We define and study a sheaf-theoretic cohomological Hall algebra for suitably geometric Abelian categories $\mathcal{A}$ of homological dimension at most two, and a sheaf-theoretic BPS algebra under the conditions that $\mathcal{A}$ is 2-Calabi-Yau and has a good moduli space. We show that the BPS algebra for the preprojective algebra $\Pi_Q$ of a totally negative quiver is the free algebra generated by the intersection cohomology of the closure of the locus parameterising simple $\Pi_Q$-modules in the coarse moduli space. We define and study the BPS Lie algebra of arbitrary 2-Calabi-Yau categories $\mathcal{A}$ for which the Euler form is negative on all pairs of non-zero objects, which recovers the BPS algebra as its universal enveloping algebra for such "totally negative" 2CY categories. We show that for totally negative 2CY categories the BPS algebra is freely generated by intersection complexes of certain coarse moduli spaces as above, and the Borel-Moore homology of the stack of objects in such $\mathcal{A}$ satisfies a Yangian-type PBW theorem for the BPS Lie algebra. In this way we prove the cohomological integrality theorem for these categories. We use our results to prove that for $C$ a smooth projective curve, and for $r$ and $d$ not necessarily coprime, there is a nonabelian Hodge isomorphism between the Borel-Moore homologies of the stack of rank $r$ and degree $d$ Higgs bundles, and the appropriate stack of twisted representations of the fundamental group of $C$. In addition we prove the Bozec-Schiffmann positivity conjecture for totally negative quivers; we prove that their polynomials counting cuspidal functions in the constructible Hall algebra for $Q$ have positive coefficients, strengthening the positivity theorem for the Kac polynomials of such quivers.

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arXiv2022

Affine BPS algebras, W algebras, and the cohomological Hall algebra of $\mathbb{A}^2$

We introduce affinizations and deformations of the BPS Lie algebra associated to a tripled quiver with potential, and use them to precisely determine the $T$-equivariant cohomological Hall algebra $\mathcal{H}_{\mathbb{A}^2}^T$ of compactly supported coherent sheaves on $\mathbb{A}^2$, acted on by a torus $T$. In particular we show that this algebra is spherically generated for all $T$.

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arXiv2021

Nonabelian Hodge theory for stacks and a stacky P=W conjecture

We introduce a version of the P=W conjecture relating the Borel-Moore homology of the stack of representations of the fundamental group of a genus g Riemann surface with the Borel-Moore homology of the stack of degree zero semistable Higgs bundles on a smooth projective complex curve of genus $g$. In order to state the conjecture we propose a construction of a canonical isomorphism between these Borel-Moore homology groups. We relate the stacky P=W conjecture with the original P=W conjecture concerning the cohomology of smooth moduli spaces of twisted objects, and the PI=WI conjecture concerning the intersection cohomology groups of singular moduli spaces of untwisted objects. In genus zero and one, we prove the conjectures that we introduce in this paper.

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arXiv2021

A boson-fermion correspondence in cohomological Donaldson-Thomas theory

We introduce and study a fermionization procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson--Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel-Moore homology of the stack of representations of the $\mu$-deformed preprojective algebra introduced by Crawley-Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras, and my earlier results on the Borel-Moore homology of the stack of representations of the undeformed preprojective algebra.

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arXiv2021

Purity and 2-Calabi-Yau categories

For various 2-Calabi-Yau categories $\mathscr{C}$ for which the stack of objects $\mathfrak{M}$ has a good moduli space $p\colon\mathfrak{M}\rightarrow \mathcal{M}$, we establish purity of the mixed Hodge module complex $p_{!}\underline{\mathbb{Q}}_{\mathfrak{M}}$. We do this by using formality in 2CY categories, along with \'etale neighbourhood theorems for stacks, to prove that the morphism $p$ is modelled \'etale-locally by the semisimplification morphism from the stack of modules of a preprojective algebra. Then via the integrality theorem in cohomological Donaldson-Thomas theory we prove purity of $p_{!}\underline{\mathbb{Q}}_{\mathfrak{M}}$. It follows that the Beilinson-Bernstein-Deligne-Gabber decomposition theorem for the constant sheaf holds for the morphism $p$, despite the possibly very singular and stacky nature of $\mathfrak{M}$. We use this to define cuspidal cohomology for $\mathfrak{M}$, which is conjecturally a complete space of generators for the BPS algebra associated to $\mathscr{C}$. We prove purity of the Borel-Moore homology of the moduli stack $\mathfrak{M}$, provided its good moduli space $\mathcal{M}$ is projective, or admits a suitable contracting $\mathbb{C}^*$-action. In particular, when $\mathfrak{M}$ is the moduli stack of Gieseker-semistable sheaves on a K3 surface, this proves a conjecture of Halpern-Leistner. We use these results to moreover prove purity for several stacks of coherent sheaves that do not admit a good moduli space. Without the usual assumption that $r$ and $d$ are coprime, we prove that the Borel-Moore homology of the stack of semistable degree $d$ rank $r$ Higgs sheaves is pure and carries a perverse filtration with respect to the Hitchin base, generalising the usual perverse filtration for the Hitchin system to the case of singular stacks of Higgs sheaves.

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arXiv2020

BPS Lie algebras and the less perverse filtration on the preprojective CoHA

The affinization morphism for the stack $\mathfrak{M}(\Pi_Q)$ of representations of a preprojective algebra $\Pi_Q$ is a local model for the morphism from the stack of objects in a general 2-Calabi-Yau category to the good moduli space. We show that the derived direct image of the dualizing complex along this morphism is pure, and admits a decomposition in the sense of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem. We introduce a new perverse filtration on the Borel-Moore homology of $\mathfrak{M}(\Pi_Q)$, using this decomposition. We show that the zeroth piece of the resulting filtration on the cohomological Hall algebra built out of the Borel-Moore homology of $\mathfrak{M}(\Pi_Q)$ is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra $\mathfrak{g}_{\Pi_Q}$. This Lie algebra is defined via the Kontsevich-Soibelman theory of critical cohomological Hall algebras for 3-Calabi-Yau categories. We then lift this Lie algebra to a Lie algebra object in the category of perverse sheaves on the coarse moduli space of $\Pi_Q$-modules, and use this algebra structure to prove results about the summands appearing in the above decomposition theorem. In particular, we prove that the intersection cohomology of singular spaces of semistable $\Pi_Q$-modules provide "cuspidal cohomology" - a conjecturally complete subspace of canonical generators for $\mathfrak{g}_{\Pi_Q}$.

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arXiv2020

Deformed dimensional reduction

Since its first use by Behrend, Bryan, and Szendr\H{o}i in the computation of motivic Donaldson-Thomas (DT) invariants of $\mathbb{A}_{\mathbb{C}}^3$, dimensional reduction has proved to be an important tool in motivic and cohomological DT theory. Inspired by a conjecture of Cazzaniga, Morrison, Pym, and Szendr\H{o}i on motivic DT invariants, work of Dobrovolska, Ginzburg, and Travkin on exponential sums, and work of Orlov and Hirano on equivalences of categories of singularities, we generalize the dimensional reduction theorem in motivic and cohomological DT theory and use it to prove versions of the Cazzaniga-Morrison-Pym-Szendr\H{o}i conjecture in these settings.

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arXiv2019

Strong positivity for quantum theta bases of quantum cluster algebras

We construct "quantum theta bases," extending the set of quantum cluster monomials, for various versions of skew-symmetric quantum cluster algebras. These bases consist precisely of the indecomposable universally positive elements of the algebras they generate, and the structure constants for their multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the quantum strong cluster positivity conjecture for these algebras. The classical limits recover the theta bases considered by Gross-Hacking-Keel-Kontsevich. Our approach combines the scattering diagram techniques used in loc. cit. with the Donaldson-Thomas theory of quivers.

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arXiv2019

The local motivic DT/PT correspondence

We show that the Quot scheme $Q_L^n = \textrm{Quot}_{\mathbb A^3}(\mathscr I_L,n)$ parameterising length $n$ quotients of the ideal sheaf of a line in $\mathbb{A}^3$ is a global critical locus, and calculate the resulting motivic partition function (varying $n$), in the ring of relative motives over the configuration space of points in $\mathbb{A}^3$. As in the work of Behrend-Bryan-Szendr\H{o}i this enables us to define a virtual motive for the Quot scheme of $n$ points of the ideal sheaf $\mathscr I_C\subset \mathscr O_Y$, where $C\subset Y$ is a smooth curve embedded in a smooth 3-fold $Y$, and we compute the associated motivic partition function. The result fits into a motivic wall-crossing type formula, refining the relation between Behrend's virtual Euler characteristic of $\textrm{Quot}_Y(\mathscr I_C,n)$ and of the symmetric product $\textrm{Sym}^nC$. Our "relative" analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert-Chow map $Q_L^n \rightarrow \textrm{Sym}^n(\mathbb{A}^3)$, and connections with cohomological Hall algebra representations.

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arXiv2019

Refined invariants of finite-dimensional Jacobi algebras

We define and study refined Gopakumar-Vafa invariants of contractible curves in complex algebraic 3-folds, alongside the cohomological Donaldson--Thomas theory of finite-dimensional Jacobi algebras. These Gopakumar-Vafa invariants can be constructed one of two ways: as cohomological BPS invariants of contraction algebras controlling the deformation theory of these curves, as defined by Donovan and Wemyss, or by feeding the moduli spaces that Katz used to define genus zero Gopakumar-Vafa invariants into the machinery developed by Joyce et al. The conjecture that the two definitions give isomorphic results is a special case of a kind of categorified version of the strong rationality conjecture due to Pandharipande and Thomas, that we discuss and propose a means of proving. We prove the positivity of the cohomological/refined BPS invariants of all finite-dimensional Jacobi algebras. This result supports this strengthening of the strong rationality conjecture, as well as the conjecture of Brown and Wemyss stating that all finite-dimensional Jacobi algebras for appropriate symmetric quivers are isomorphic to contraction algebras.

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arXiv2018

Enumerating coloured partitions in 2 and 3 dimensions

We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson-Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption.

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