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In a previous paper, we introduced quasi-BPS categories for moduli stacks of semistable Higgs bundles. Under a certain condition on the rank, Euler characteristic, and weight, the quasi-BPS categories (called BPS in this case) are non-commutative analogues of Hitchin integrable systems. We proposed a conjectural equivalence between BPS categories which swaps Euler characteristics and weights. The conjecture is inspired by the Dolbeault Geometric Langlands equivalence of Donagi--Pantev, by the Hausel--Thaddeus mirror symmetry, and by the $\chi$-independence phenomenon for BPS invariants of curves on Calabi-Yau threefolds. In this paper, we show that the above conjecture holds at the level of topological K-theories. When the rank and the Euler characteristic are coprime, such an isomorphism was proved by Groechenig--Shen. Along the way, we show that the topological K-theory of BPS categories is isomorphic to the BPS cohomology of the moduli of semistable Higgs bundles., Comment: 59 pages

We introduce quasi-BPS categories for twisted Higgs bundles, which are building blocks of the derived category of coherent sheaves on the moduli stack of semistable twisted Higgs bundles over a smooth projective curve. Under some condition (called BPS condition), the quasi-BPS categories are non-commutative analogues of Hitchin integrable systems. We begin the study of these quasi-BPS categories by focusing on a conjectural symmetry which swaps the Euler characteristic and the weight. Our conjecture is inspired by the Dolbeault Geometric Langlands equivalence of Donagi--Pantev, by the Hausel--Thaddeus mirror symmetry, and by the $\chi$-independence phenomenon for BPS invariants of curves on Calabi-Yau threefolds. We prove our conjecture in the case of rank two and genus zero. In higher genus, we prove a derived equivalence of rank two stable twisted Higgs moduli spaces as a special case of our conjecture. In a separate paper, we prove a version of our conjecture for the topological K-theory of quasi-BPS categories and we discuss the relation between quasi-BPS categories and BPS invariants of the corresponding local Calabi-Yau threefold., Comment: 64 pages

We introduce and begin the study of quasi-BPS categories for K3 surfaces, which are a categorical version of the BPS cohomologies for K3 surfaces. We construct semiorthogonal decompositions of derived categories of coherent sheaves on moduli stacks of semistable objects on K3 surfaces, where each summand is a categorical Hall product of quasi-BPS categories. We also prove the wall-crossing equivalence of quasi-BPS categories, which generalizes Halpern-Leistner's wall-crossing equivalence of moduli spaces of stable objects for primitive Mukai vectors on K3 surfaces. We also introduce and study a reduced quasi-BPS category. When the weight is coprime to the Mukai vector, the reduced quasi-BPS category is proper, smooth, and its Serre functor is trivial \'{e}tale locally on the good moduli space. Moreover we prove that its topological K-theory recovers the BPS invariants of K3 surfaces, which are known to be equal to the Euler characteristics of Hilbert schemes of points on K3 surfaces. We regard reduced quasi-BPS categories as noncommutative hyperk\"ahler varieties which are categorical versions of crepant resolutions of singular symplectic moduli spaces of semistable objects on K3 surfaces., Comment: 66 pages

In previous works, we introduced and studied certain categories called quasi-BPS categories associated to symmetric quivers with potential, preprojective algebras, and local surfaces. They have properties reminiscent of BPS invariants/ cohomologies in enumerative geometry, for example they play important roles in categorical wall-crossing formulas. In this paper, we make the connections between quasi-BPS categories and BPS cohomologies more precise via the cycle map for topological K-theory. We show the existence of filtrations on topological K-theory of quasi-BPS categories whose associated graded are isomorphic to the monodromy invariant BPS cohomologies. Along the way, we also compute the topological K-theory of categories of matrix factorizations in terms of the monodromy invariant vanishing cycles (a version of this comparison was already known by work of Blanc-Robalo-To\"en-Vezzosi), prove a Grothendieck-Riemann-Roch theorem for matrix factorizations, and prove the compatibility between the Koszul equivalence in K-theory and dimensional reduction in cohomology. In a separate paper, we use the results from this paper to show that the quasi-BPS categories of K3 surfaces recover the BPS invariants of the corresponding local surface, which are Euler characteristics of Hilbert schemes of points on K3 surfaces., Comment: 80 pages, the main theorems apply now to all quivers

We study certain categories associated to symmetric quivers with potential, called quasi-BPS categories. We construct semiorthogonal decompositions of the categories of matrix factorizations for moduli stacks of representations of (framed or unframed) symmetric quivers with potential, where the summands are categorical Hall products of quasi-BPS categories. These results generalize our previous results about the three loop quiver. We prove several properties of quasi-BPS categories: wall-crossing equivalence, strong generation, and categorical support lemma in the case of tripled quivers with potential. We also introduce reduced quasi-BPS categories for preprojective algebras, which have trivial relative Serre functor and are indecomposable when the weight is coprime with the total dimension. In this case, we regard the reduced quasi-BPS categories as noncommutative local hyperk\"ahler varieties, and as (twisted) categorical versions of crepant resolutions of singularities of good moduli spaces of representations of preprojective algebras. The studied categories include the local models of quasi-BPS categories of K3 surfaces. In a follow-up paper, we establish analogous properties for quasi-BPS categories of K3 surfaces., Comment: 55 pages

We construct semiorthogonal decompositions of Donaldson-Thomas (DT) categories for reduced curve classes on local surfaces into products of quasi-BPS categories and Pandharipande-Thomas (PT) categories, giving a categorical analogue of the numerical DT/PT correspondence for Calabi-Yau 3-folds. The main ingredient is a categorical wall-crossing formula for DT/PT quivers (which appear as Ext-quivers in the DT/PT wall-crossing) proved in our previous paper. We also study quasi-BPS categories of points on local surfaces and propose conjectural computations of their K-theory analogous to formulas already known for the three dimensional affine space., Comment: 42 pages

In this paper, we prove the categorical wall-crossing formula for certain quivers containing the three loop quiver, which we call DT/PT quivers. These quivers appear as Ext-quivers for the wall-crossing of DT/PT moduli spaces on Calabi-Yau 3-folds. The resulting formula is a semiorthogonal decomposition which involves quasi-BPS categories studied in our previous papers, and we regard it as a categorical analogue of the numerical DT/PT correspondence. As an application, we prove a categorical DT/PT correspondence for sheaves supported on reduced plane curves in the affine three dimensional space., Comment: 35 pages

Quasi-BPS categories appear as summands in semiorthogonal decompositions of DT categories for Hilbert schemes of points in the three dimensional affine space and in the categorical Hall algebra of the two dimensional affine space. In this paper, we prove several properties of quasi-BPS categories analogous to BPS sheaves in cohomological DT theory. We first prove a categorical analogue of Davison's support lemma, namely that complexes in the quasi-BPS categories for coprime length and weight are supported over the small diagonal in the symmetric product of the three dimensional affine space. The categorical support lemma is used to determine the torsion-free generator of the torus equivariant K-theory of the quasi-BPS category of coprime length and weight. We next construct a bialgebra structure on the torsion free equivariant K-theory of quasi-BPS categories for a fixed ratio of length and weight. We define the K-theoretic BPS space as the space of primitive elements with respect to the coproduct. We show that all localized equivariant K-theoretic BPS spaces are one dimensional, which is a K-theoretic analogue of the computation of (numerical) BPS invariants of the three dimensional affine space., Comment: 45 pages, to appear in Forum Math. Sigma

Let $Y$ be a smooth projective threefold and let $f:Y\to X$ be a birational map with $Rf_*\mathcal{O}_Y=\mathcal{O}_X$. When $Y$ is Calabi-Yau, Bryan-Steinberg defined enumerative invariants associated to such maps called $f$-relative stable (or Bryan-Steinberg) invariants. When $X$ has Gorenstein singularities and $f$ has relative dimension one, they compared these invariants to the Donaldson-Thomas, or equivalently the Pandharipande-Thomas invariants of $Y$. We define Bryan-Steinberg invariants for maps $f$ as above without assuming that $Y$ is Calabi-Yau. For $X$ with Gorenstein and rational singularities, $f$ of relative dimension one, and for insertions from $X$ and arbitrary descendant levels, we conjecture a relation between the generating functions of Bryan-Steinberg and Pandharipande-Thomas invariants of $Y$. We check the conjecture for the contraction $f: Y\to X$ of a rational curve $C$ with normal bundle $N_{C/Y}\cong \mathcal{O}_C(-1)^{\oplus 2}$ using degeneration and localization techniques to reduce to a Calabi-Yau situation, which we then treat using Joyce's motivic Hall algebra., Comment: 38 pages

K-theoretic Hall algebras (KHAs) of quivers with potential $(Q,W)$ are a generalization of preprojective KHAs of quivers, which are conjecturally positive parts of the Okounkov-Smironov quantum affine algebras. In particular, preprojective KHAs are expected to satisfy a Poincar\'e-Birkhoff-Witt theorem. We construct semi-orthogonal decompositions of categorical Hall algebras using techniques developed by Halpern-Leistner, Ballard-Favero-Katzarkov, and \v{S}penko-Van den Bergh. For a quotient of $\text{KHA}(Q,W)_{\mathbb{Q}}$, we refine these decompositions and prove a PBW-type theorem for it. The spaces of generators of $\text{KHA}(Q,0)_{\mathbb{Q}}$ are given by (a version of) intersection K-theory of coarse moduli spaces of representations of $Q$., Comment: This article is a revised version of Section 7 from arXiv:1911.05526. 33 pages, to appear in Sel. Math. New Ser

Given a quiver with potential $(Q,W)$, Kontsevich-Soibelman constructed a Hall algebra on the critical cohomology of the stack of representations of $(Q,W)$. Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann-Vasserot, Maulik-Okounkov, Yang-Zhao etc. about geometric constructions of Yangians and their representations; indeed, given a quiver $Q$, there exists an associated pair $\left(\widetilde{Q},\widetilde{W}\right)$ whose CoHA is conjecturally the positive half of the Maulik-Okounkov Yangian $Y_{\text{MO}}(\mathfrak{g}_Q)$. For a quiver with potential $(Q,W)$, we follow a suggestion of Kontsevich-Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers and we prove a wall-crossing theorem for KHAs. We expect the KHA for $\left(\widetilde{Q},\widetilde{W}\right)$ to recover the positive part of quantum affine algebra $U_q\left(\widehat{\mathfrak{g}_Q}\right)$ defined by Okounkov-Smirnov., Comment: 31 pages, submitted. The article is a revised version of Sections 2, 3, 6, 9, and 10 in arXiv:1911.05526

For a smooth surface $S$, Porta-Sala defined a categorical Hall algebra generalizing previous work in K-theory of Zhao and Kapranov-Vasserot. We construct semi-orthogonal decompositions for categorical Hall algebras of points on $S$. We refine these decompositions in K-theory for a topological K-theoretic Hall algebra., Comment: 23 pages

For $(Q,W)$ a symmetric quiver with potential satisfying a K\"unneth-type condition, we construct (positive and negative) extensions of its K-theoretic Hall algebra which are bialgebras. In particular, there are bialgebra extensions of preprojective KHAs of a quiver $Q$ and one can construct their Drinfeld double algebra., Comment: 21 pages

For a large class of good moduli spaces $X$ of symmetric stacks $\mathcal{X}$, we define noncommutative motives $\mathbb{D}^{\text{nc}}(X)$ which can be regarded as categorifications of the intersection cohomology of $X$. These motives are summands of noncommutative resolutions of singularities $\mathbb{D}(X)\subset D^b(\mathcal{X})$ of $X$. The category $\mathbb{D}(X)$ is a global analogue of the noncommutative resolutions of singularities of $V//G$ for $V$ a symmetric representation of a reductive group $G$ constructed by \v{S}penko-Van den Bergh., Comment: 34 pages. Submitted

For a proper map $f:X\to S$ between varieties over $\mathbb{C}$ with $X$ smooth, we introduce increasing filtrations $\textbf{P}^{\leq \cdot}_f\subset P^{\leq \cdot}_f$ on $\text{gr}^\cdot K_\cdot(X)$, the associated graded on $K$-theory with respect to the codimension filtration, both sent by the cycle map to the perverse filtration on cohomology ${}^pH^{\leq \cdot}_f(X)$. The filtrations $P^{\leq \cdot}_f$ and $\textbf{P}^{\leq \cdot}_f$ are functorial with respect to proper pushforward; $P^{\leq \cdot}_f$ is functorial with respect to pullback. We use the above filtrations to propose two definitions of (graded) intersection $K$-theory $\text{gr}^\cdot IK_\cdot(S)$ and $\text{gr}^\cdot \textbf{I}K_\cdot(S)$. Both have cycle maps to intersection cohomology $IH^{\cdot}(S)$. We conjecture a version of the decomposition theorem for semismall surjective maps and prove it in some particular cases., Comment: 35 pages, minor revisions

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., Comment: 40 pages

Given a quiver with potential $(Q,W)$, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of $(Q,W)$. As shown by Davison-Meinhardt, this algebra comes with a filtration whose associated graded algebra is supercommutative. A special case of this construction is related to work of Nakajima, Varagnolo, Maulik-Okounkov etc. about geometric constructions of Yangians and their representations; indeed, given a quiver $Q$, there exists an associated pair $(\widetilde{Q},\widetilde{W})$ for which the CoHA is conjecturally the positive half of the Yangian $Y_{\text{MO}}(\mathfrak{g}_Q)$. The goal of this article is to extend these ideas to K-theory. More precisely, we construct a K-theoretic Hall algebra using category of singularities, define a filtration whose associated graded algebra is a deformation of a symmetric algebra, and compare the $\text{KHA}$ and the $\text{CoHA}$ using the Chern character. As before, we expect our construction for the special class of quivers $(\widetilde{Q},\widetilde{W})$ to recover the positive part of quantum affine algebra $U_q(\hat{\mathfrak{g}_Q})$ defined by Okounkov-Smirnov, but for general pairs $(Q,W)$ we expect new phenomena., Comment: 118 pages

A natural place to study the Chow ring of the classifying space $BG$, for $G$ a linear algebraic group, is Voevodsky's triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives $M(BG)$ and $M^c(BG)$, respectively. We show that, for any group $G$ of order $p^3$ over a field of characteristic not $p$ which contains a primitive $p^2$-th root of unity, the motive $M(BG)$ is a mixed Tate motive. We also show that, for a finite group $G$ over a field of characteristic zero, $M(BG)$ is a mixed Tate motive if and only $M^c(BG)$ is a mixed Tate motive., Comment: 17 pages

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation $T$ in the class, the cartesian product $T\times T$ of the transformation with itself is ergodic, but the product $T\times T^{-1}$ of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative., Comment: Added references and revised some arguments; removed old section 6; main results unchanged

For each vector $v$ we define the notion of a $v$-positive type for infinite measure-preserving transformations, a refinement of positive type as introduced by Hajian and Kakutani. We prove that a positive type transformation need not be $(1,2)$-positive type. We study this notion in the context of Markov shifts and multiple recurrence and give several examples.

We study the notions of weak rational ergodicity and rational weak mixing as defined by Jon Aaronson. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and consider their relation to other notions of mixing in infinite measure., Comment: Some typos corrected, a reference added, arguments expanded