arXiv subjects

Sabin Cautis

Publications and source records attributed to Sabin Cautis.

At least 19 recordsLinked to original sources

arXiv2025

Abelian Hall categories

To a quiver we associate a finite length monoidal abelian category which categorifies the corresponding preprojective K-theoretic Hall algebra of Varagnolo-Vasserot. The simples in this category provide a (dual) canonical basis of the Hall algebra. In particular, if the quiver is affine, this provides a basis for the positive half of the corresponding quantum toroidal algebra. We also show that this abelian category is naturally endowed with renormalized r-matrices.

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arXiv2025

On the center of the generic affine Hecke algebra

We identify the center of the generic affine Hecke algebra $H_q$ corresponding to some root datum with the semigroup algebra $\mathbb C[q][\check X^+]$ of the dominant chamber of its coweight lattice. This is done by first identifying a maximal commutative subalgebra $A_q \subset H_q$ with the Rees algebra associated to the semigroup algebra of the coweight lattice for the filtration induced by the length function. We explain how this subalgebra $A_q$ can also be identified with (quantum) cohomology of the toric variety given by the fan corresponding to the coroot lattice (a Hessenberg variety).

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arXiv2024

Rigid dualizing complexes of affine Hecke algebras

We identify the rigid dualizing complex of the (generic) affine Hecke algebra $H_q$ attached to a reduced root system and deduce some structural properties as a consequence. For example, we show that the classical Hecke algebra $H_{q^\pm}$ as well as $H_q/q$ are, under a certain condition on the root system, Frobenius over their centers with Nakayama automorphism given by an explicit involution.

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arXiv2024

Ind-geometric stacks

We develop the theory of ind-geometric stacks, in particular their coherent and ind-coherent sheaf theory. This provides a convenient framework for working with equivariant sheaves on ind-schemes, especially in derived settings. Motivating examples include the coherent Satake category, the double affine Hecke category, and related categories in the theory of Coulomb branches.

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arXiv2024

Tamely presented morphisms and coherent pullback

We study two classes of morphisms in infinite type: tamely presented morphisms and morphisms with coherent pullback. These are generalizations of finitely presented morphisms and morphisms of finite Tor-dimension, respectively. The class of tamely presented schemes and stacks is restricted enough to retain the key features of finite-type schemes from the point of view of coherent sheaf theory, but wide enough to encompass many infinite-type examples of interest in geometric representation theory. The condition that a diagonal has coherent pullback is a natural generalization of smoothness to the tamely presented setting, and we show such objects retain many good cohomological properties of smooth varieties. Our results are motivated by the study of convolution products in the double affine Hecke category and related categories in the theory of Coulomb branches.

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arXiv2023

Canonical bases for Coulomb branches of 4d $\mathcal{N}=2$ gauge theories

We construct and study a nonstandard t-structure on the derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima space of triples $\mathcal{R}_{G,N}$, where $N$ is a representation of a reductive group $G$. Its heart $\mathcal{KP}_{G,N}$ is a finite-length, rigid, monoidal abelian category with renormalized $r$-matrices. We refer to objects of $\mathcal{KP}_{G,N}$ as Koszul-perverse coherent sheaves. Simple objects of $\mathcal{KP}_{G,N}$ define a canonical basis in the quantized $K$-theoretic Coulomb branch of the associated gauge theory. These simples possess various characteristic properties of Wilson-'t Hooft lines, and we interpret our construction as an algebro-geometric definition of the category of half-BPS line defects in a 4d $\mathcal{N}=2$ gauge theory of cotangent type.

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arXiv2021

Associated graded of Hodge modules and categorical sl_2 actions

One of the most mysterious aspects of Saito's theory of Hodge modules are the Hodge and weight filtrations that accompany the pushforward of a Hodge module under an open embedding. In this paper we consider the open embedding in a product of complementary Grassmannians given by pairs of transverse subspaces. The push-forward of the structure sheaf under this open embedding is an important Hodge module from the viewpoint of geometric representation theory and homological knot invariants. We compute the associated graded of this push-forward with respect to the induced Hodge filtration as well as the resulting weight filtration. The main tool is a categorical $\sl_2$ action on the category of $\D_h$-modules on Grassmannians. Along the way we also clarify the interaction of kernels for $\D_h$-modules with the associated graded functor. Both of these results may be of independent interest.

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arXiv2019

Curved Rickard complexes and link homologies

Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson-Seed arXiv:1303.6240 and Gorsky-Hogancamp arXiv:1712.03938 to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in arXiv:1410.7156 to categorified quantum groups (this was the original motivation for this paper).

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arXiv2018

Exotic t-structures and actions of quantum affine algebras

We explain how quantum affine algebra actions can be used to systematically construct "exotic" t-structures. The main idea, roughly speaking, is to take advantage of the two different descriptions of quantum affine algebras, the Drinfeld--Jimbo and the Kac--Moody realizations. Our main application is to obtain exotic t-structures on certain convolution varieties defined using the Beilinson--Drinfeld and affine Grassmannians. These varieties play an important role in the geometric Langlands program, knot homology constructions, K-theoretic geometric Satake and the coherent Satake category. As a special case we also recover the exotic t-structures of Bezrukavnikov--Mirkovic on the (Grothendieck--)Springer resolution in type A.

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arXiv2018

Cluster theory of the coherent Satake category

We study the category of G(O)-equivariant perverse coherent sheaves on the affine Grassmannian of G. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible Satake category. Instead, we use the Beilinson-Drinfeld Grassmannian to construct renormalized r-matrices. These are canonical nonzero maps between convolution products which satisfy axioms weaker than those of a braiding. We also show that the coherent Satake category is rigid, and that together these results strongly constrain its convolution structure. In particular, they can be used to deduce the existence of (categorified) cluster structures. We study the case G = GL_n in detail and prove that the loop rotation equivariant coherent Satake category of GL_n is a monoidal categorification of an explicit quantum cluster algebra. More generally, we construct renormalized r-matrices in any monoidal category whose product is compatible with an auxiliary chiral category, and explain how the appearance of cluster algebras in 4d N=2 field theory may be understood from this point of view.

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arXiv2017

Categorical geometric symmetric Howe duality

We provide a natural geometric setting for symmetric Howe duality. This is realized as a (loop) sl(n) action on derived categories of coherent sheaves on certain varieties arising in the geometry of the Beilinson-Drinfeld Grassmannian. The main construction parallels our earlier work on categorical sl(n) actions and skew Howe duality. In that case the varieties involved arose in the geometry of the affine Grassmannian. We discuss some relationships between the two actions.

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arXiv2017

Remarks on coloured triply graded link invariants

We explain how existing results (such as categorical sl(n) actions, associated braid group actions and infinite twists) can be used to define a triply graded link invariant which categorifies the HOMFLY polynomial of links coloured by arbitrary partitions. The construction uses a categorified HOMFLY clasp defined via cabling and infinite twists. We briefly discuss differentials and speculate on related structures.

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arXiv2017

Quantum K-theoretic geometric Satake

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $ G $ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group. Bezrukavnikov-Finkelberg developed a derived version of this equivalence which relates the derived category of $ G^\vee$-equivariant constructible sheaves on $ Gr $ with the category of $G$-equivariant ${\mathcal O}(\mathfrak g)$-modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group $ U_q \mathfrak g $. We define a convolution category $ KConv(Gr) $ whose morphism spaces are given by the $ G^\vee \times \mathbb C^\times $-equivariant algebraic K-theory of certain fibre products. We conjecture that $KConv(Gr)$ is equivalent to a full subcategory of the category of $ U_q \mathfrak g $-equivariant $ \mathcal O_q(G) $-modules. We prove this conjecture when $G = SL_n$. A key tool in our proof is the $SL_n$ spider, which is a combinatorial description of the category of $U_q \mathfrak{sl}_n$ representations. By applying horizontal trace, we show that the annular $SL_n$ spider describes the category of $ U_q \mathfrak{sl}_n $-equivariant $ \mathcal O_q(SL_n) $-modules. Then we use quantum loop algebras to relate the annular $SL_n $ spider to $ KConv(Gr) $. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.

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arXiv2016

The Elliptic Hall algebra and the deformed Khovanov Heisenberg category

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann, specialized at $σ= \barσ^{-1} = q$. A key step in the proof may be of independent interest: we show that the sum (over $n$) of the Hochschild homologies of the positive affine Hecke algebras $\mathrm{AH}_n^+$ is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the $q$-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.

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arXiv2015

On a categorical Boson-Fermion correspondence

We propose a categorical version of the Boson-Fermion correspondence and its twisted version. One can view it as a relative of the Frenkel-Kac-Segal construction of quantum affine algebras.

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arXiv2015

W-algebras from Heisenberg categories

The trace (or zeroth Hochschild homology) of Khovanov's Heisenberg category is identified with a quotient of the algebra W_{1+\infty}. This induces an action of W_{1+\infty} on symmetric functions.

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arXiv2014

Knot homology via derived categories of coherent sheaves IV, coloured links

We define a deformation of our earlier link homologies for fundamental representations of sl_m. The deformed homology of a link is isomorphic to the deformed homology of the disjoint union of its components. Moreover, there exists a spectral sequence starting with the old homology and converging to this deformed homology.

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arXiv2014

Derived Reid's recipe for abelian subgroups of SL3(C)

For any finite subgroup G in SL3(C), work of Bridgeland-King-Reid constructs an equivalence between the G-equivariant derived category of C^3 and the derived category of the crepant resolution Y = G-Hilb(C^3) of C^3/G. When G is abelian we show that this equivalence gives a natural correspondence between irreducible representations of G and certain sheaves on exceptional subvarieties of Y, thereby extending the McKay correspondence from two to three dimensions. This categorifies Reid's recipe and extends earlier work from [CL09] and [Log10] which dealt only with the case when C^3/G has one isolated singularity.

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