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Yukinobu Toda

Publications and source records attributed to Yukinobu Toda.

At least 19 recordsLinked to original sources
arXiv2026

The Dolbeault geometric Langlands conjecture via limit categories

We introduce limit categories for cotangent stacks of smooth stacks as an effective version of classical limits of categories of D-modules on them. We develop their general theory and pursue their relation with categories of D-modules. In particular, we establish the functorial properties of limit categories such as the smooth pull-back and projective push-forward. Using the notion of limit categories, we propose a precise formulation of the Dolbeault geometric Langlands conjecture, proposed by Donagi-Pantev as the classical limit of the de Rham geometric Langlands equivalence. It states an equivalence between the derived categories of moduli stacks of semistable Higgs bundles and limit categories of moduli stacks of all Higgs bundles. We prove the existence of a semiorthogonal decomposition of the limit category into quasi-BPS categories, which are categorical versions of BPS invariants on a non-compact Calabi-Yau 3-fold. This semiorthogonal decomposition is interpreted as a Langlands dual to the semiorthogonal decomposition constructed in our previous work on the category of coherent sheaves on the moduli stack of semistable Higgs bundles. We also construct Hecke operators on limit categories for Higgs bundles. They are expected to be compatible with Wilson operators under our formulation of Dolbeault geometric Langlands conjecture. The conjectured equivalence implies an equivalence between BPS categories for semistable Higgs bundles, which we expect to be a categorical version of the topological mirror symmetry conjecture for Higgs bundles by Hausel-Thaddeus.

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arXiv2026

The Dolbeault geometric Langlands correspondence for type A groups beyond the elliptic locus

In this paper, we prove a Dolbeault geometric Langlands equivalence for $\GL_r$ and for the Langlands dual pair $\SL_r/\PGL_r$ over an open locus of the Hitchin base which strictly contains the elliptic locus. This open locus contains the points corresponding to spectral curves with at worst type $A$ singularities, without any restriction on the number of irreducible components. The Dolbeault geometric Langlands equivalence considered here is the one formulated in our previous work with Tudor Pădurariu, which links categorical Donaldson--Thomas theory with the geometric Langlands correspondence. It relates coherent sheaves on moduli stacks of semistable Higgs bundles to the limit category associated with the full moduli stack of Higgs bundles. The use of limit categories is essential beyond the elliptic locus, where the full Higgs moduli stack is no longer quasi-compact and contains infinitely many Harder--Narasimhan strata. The key step is to prove the Whittaker normalization conjecture over the locus of spectral curves with type $A$ singularities, following and extending the strategy developed in the author's proof of the $\GL_2$ case over the reduced spectral curve locus. As a consequence, we also obtain the Dolbeault geometric Langlands conjecture for $\SL_2/\PGL_2$ over the reduced spectral curve locus.

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arXiv2026

A proof of Dolbeault geometric Langlands for $\mathrm{GL}_2$ with reduced spectral curves

In our previous paper with Tudor Pădurariu, we introduced the notion of limit categories for moduli stacks of Higgs bundles and formulated the Dolbeault geometric Langlands correspondence. These limit categories are expected to provide an effective ``classical limit'' of the categories of D-modules on the moduli stack of bundles, and our formulation links categorical Donaldson-Thomas theory with the geometric Langlands correspondence. In this paper, we prove the above Dolbeault geometric Langlands correspondence for $\mathrm{GL}_2$ over the locus in the Hitchin base where the spectral curves are reduced. This is the first non-trivial case in which the relevant moduli stacks are not quasi-compact, and the use of limit categories is essential to the formulation and proof of the correspondence. Our approach also outlines a strategy for proving the correspondence in greater generality and explains the current obstructions to such an extension.

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arXiv2026

Semiorthogonal decompositions for stacks

We give a systematic construction of semiorthogonal decompositions of derived categories of coherent sheaves on quasi-smooth derived algebraic stacks over $\mathbb{C}$, where the summands are subcategories defined by weight conditions, and the inclusion functors are given by parabolic induction. The summands are indexed by the component lattice of the stack, a central combinatorial structure in intrinsic Donaldson-Thomas theory. As examples, we obtain semiorthogonal decompositions for moduli stacks of semistable $G$-bundles or $G$-Higgs bundles on a curve, and moduli stacks of de Rham or Betti $G$-local systems on a curve, for reductive groups $G$ not necessarily of type A.

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arXiv2026

Topological K-theory of quasi-BPS categories of symmetric quivers with potential

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-Toën-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.

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arXiv2025

Quasi-BPS categories for symmetric quivers with potential

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 hyperkähler 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.

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arXiv2025

$K$-theoretic pullbacks for Lagrangians on derived critical loci

Given a regular function $ϕ$ on a smooth stack, and a $(-1)$-shifted Lagrangian $M$ on the derived critical locus of $ϕ$, under fairly general hypotheses, we construct a pullback map from the Grothendieck group of coherent matrix factorizations of $ϕ$ to that of coherent sheaves on $M$. This map satisfies a functoriality property with respect to the composition of Lagrangian correspondences, as well as the usual bivariance and base-change properties. We provide three applications of the construction, one in the definition of quantum $K$-theory of critical loci (Landau-Ginzburg models), paving the way to generalize works of Okounkov school from Nakajima quiver varieties to quivers with potentials, one in establishing a degeneration formula for $K$-theoretic Donaldson-Thomas theory of local Calabi-Yau 4-folds, the other in confirming a $K$-theoretic version of Joyce-Safronov conjecture.

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arXiv2024

Topological K-theory of quasi-BPS categories for Higgs bundles

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 $χ$-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.

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arXiv2024

Quasi-BPS categories for Higgs bundles

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 $χ$-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.

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arXiv2023

Quasi-BPS categories for K3 surfaces

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 étale 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 hyperkähler varieties which are categorical versions of crepant resolutions of singular symplectic moduli spaces of semistable objects on K3 surfaces.

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arXiv2023

Categorical and K-theoretic Donaldson-Thomas theory of $\mathbb{C}^3$ (part II)

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.

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arXiv2023

Categorical and K-theoretic Donaldson-Thomas theory of $\mathbb{C}^3$ (part I)

We begin the study of categorifications of Donaldson-Thomas invariants associated with Hilbert schemes of points on the three-dimensional affine space, which we call DT categories. The DT category is defined to be the category of matrix factorizations on the non-commutative Hilbert scheme with a super-potential whose critical locus is the Hilbert scheme of points. The first main result in this paper is the construction of semiorthogonal decompositions of DT categories, which can be regarded as categorical wall-crossing formulae of the framed triple loop quiver. Each summand is given by the categorical Hall product of some subcategories of matrix factorizations, called quasi-BPS categories. They are categories of matrix factorizations on twisted versions of noncommutative resolutions of singularities considered by Špenko-Van den Bergh, and were used by the first author to prove a PBW theorem for K-theoretic Hall algebras. We next construct explicit objects of quasi-BPS categories via Koszul duality equivalences, and show that they form a basis in the torus localized K-theory. These computations may be regarded as a numerical K-theoretic analogue in dimension three of the McKay correspondence for Hilbert schemes of points. In particular, the torus localized K-theory of DT categories has a basis whose cardinality is the number of plane partitions, giving a K-theoretic analogue of MacMahon's formula.

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arXiv2022

The local categorical DT/PT correspondence

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.

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arXiv2022

The categorical DT/PT correspondence and quasi-BPS categories for local surfaces

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.

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arXiv2022

Derived categories of Quot schemes of zero-dimensional quotients on curves

We prove the existence of semiorthogonal decompositions of derived categories of Quot schemes of zero-dimensional quotients on curves in terms of derived categories of symmetric products of curves. The above result is a categorical analogue of a similar formula for the class of Quot schemes in the Grothendieck ring of varieties by Bagnarol-Fantechi-Perroni. It is a special case of a more general Quot formula of relative dimension one, which is regarded as a Bosonic counterpart of the Quot formula conjectured by Jiang and proved by the author. The proof involves categorical wall-crossing formula for framed one loop quiver, which itself is motivated and has applications to categorical wall-crossing formula of Donaldson-Thomas invariants.

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arXiv2022

Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds

As an analogy to Gopakumar-Vafa conjecture on Calabi-Yau 3-folds, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold $X$ using Gromov-Witten theory. When $X$ is holomorphic symplectic, Gromov-Witten invariants vanish and one can consider the corresponding reduced theory. In a companion work, we propose a definition of Gopakumar-Vafa type invariants for such a reduced theory. In this paper, we give them a sheaf theoretic interpretation via moduli spaces of stable pairs.

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arXiv2022

Curve counting via stable objects in derived categories of Calabi-Yau 4-folds

In our previous paper with Maulik, we proposed a conjectural Gopakumar-Vafa (GV) type formula for the generating series of stable pair invariants on Calabi-Yau (CY) 4-folds. The purpose of this paper is to give an interpretation of the above GV type formula in terms of wall-crossing phenomena in the derived category. We introduce invariants counting LePotier's stable pairs on CY 4-folds, and show that they count certain stable objects in D0-D2-D8 bound states in the derived category. We propose a conjectural wall-crossing formula for the generating series of our invariants, which recovers the conjectural GV type formula. Examples are computed for both compact and toric cases to support our conjecture.

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