arXiv subjects

Gufang Zhao

Publications and source records attributed to Gufang Zhao.

At least 19 recordsLinked to original sources

arXiv2026

Quasimaps to quivers with potentials

This paper is concerned with a non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential, and prove a gluing formula in the formalism of cohomological field theories. The main examples studied in this paper is when the above setting arises from quivers with potentials, where the above construction gives quantum correction to the equivariant Chow homology of the critical locus. Following similar ideas as in quasimaps to Nakajima quiver varieties studied by the Okounkov school, we analyse vertex functions in several examples, including Hilbert schemes of points on $\mathbb{C}^3$, moduli spaces of perverse coherent systems on the resolved conifold, and a quiver which defines higher $\mathfrak{sl}_2$-spin chains. Bethe equations are calculated in these cases. The construction in the present paper is based on the theory of gauged linear sigma models as well as shifted symplectic geometry of Pantev, Toën, Vaquie and Vezzosi, and uses the virtual pullback formalism of symmetric obstruction theory of Park, which arises from the recent development of Donaldson-Thomas theory of Calabi-Yau 4-folds.

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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

A degeneration formula of Donaldson-Thomas theory on local Calabi-Yau 4-folds

We define relative Donaldson-Thomas invariants of local log Calabi-Yau 4-folds, under the assumption that the derived Hilbert scheme on the anti-canonical divisor is a derived critical locus. We prove a degeneration formula and apply it to compute zero dimensional invariants on $\mathbb{C}^4$ and on any local curve. This verifies a conjecture proposed by the first author and Kool.

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arXiv2023

Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds

We study the Drinfeld double of the (equivariant spherical) Cohomological Hall algebra in the sense of Kontsevich and Soibelman, associated to a smooth toric Calabi-Yau 3-fold $X$. By general reasons, the COHA acts on the cohomology of the moduli spaces of certain perverse coherent systems on $X$ via "raising operators". Conjecturally the COHA action extends to an action of the Drinfeld double by adding the "lowering operators". In this paper, we show that the Drinfeld double is a generalization of the notion of the Cartan doubled Yangian defined earlier by Finkelberg and others. We extend this "$3d$ Calabi-Yau perspective" on the Lie theory furthermore by associating a root system to certain families of $X$. We formulate a conjecture that the above-mentioned action of the Drinfeld double factors through a shifted Yangian of the root system. The shift is explicitly determined by the moduli problem and the choice of stability conditions, and is expressed explicitly in terms of an intersection number in $X$. We check the conjectures in several examples, including a special case of an earlier conjecture of Costello.

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arXiv2023

A Langlands duality of elliptic Hecke algebras

Associated to any root datum, there is an elliptic affine Hecke algebra defined by Ginzburg, Kapranov, and Vasserot. In this study, we establish a Fourier-Mukai functor from the representation category of the elliptic affine Hecke algebra to the corresponding category associated with the Langlands dual root datum. To achieve this connection, we employ the elliptic Hecke algebra with dynamical parameters as an intermediary.

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arXiv2023

Elliptic classes via the periodic Hecke module and its Langlands dual

This paper explores a construction of the elliptic classes of the Springer resolution using the periodic Hecke module. The module is established by employing the Poincar\'e line bundle over the product of the abelian variety of elliptic cohomology and its dual. Additionally, we introduce the elliptic twisted group algebra, which acts on the periodic module. The construction of the elliptic twisted group algebra is such that the Demazure-Lusztig (DL) operators with dynamical parameters are rational sections. We define elliptic classes as rational sections of the periodic module, and give explicit formulas of the restriction to fixed points. Our main result shows that a natural assembly of the DL operators defines a rational isomorphism between the periodic module and the one associated to the Langlands dual root system. This isomorphism intertwines the (opposite) elliptic classes with the fixed point basis in the dual system.

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arXiv2022

Categorification via blocks of modular representations II

Bernstein, Frenkel and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak{sl}_2$ using singular blocks of category $\mathcal{O}$ for $\mathfrak{sl}_n$. In earlier work, we construct a positive characteristic analogue using blocks of representations of $\mathfrak{sl}_n$ over a field $\textbf{k}$ of characteristic $p > n$, with zero Frobenius character, and singular Harish-Chandra character. In the present paper, we extend these results and construct a categorical $\mathfrak{sl}_k$-action, following Sussan's approach, by considering more singular blocks of modular representations of $\mathfrak{sl}_n$. We consider both zero and non-zero Frobenius central character. In the former setting, we construct a graded lift of these categorifications which are equivalent to a geometric construction of Cautis, Kamnitzer and Licata. We establish a Koszul duality between two geometric categorificatons constructed in their work, and resolve a conjecture of theirs. For non-zero Frobenius central characters, we show that the geometric approach to categorical symmetric Howe duality by Cautis and Kamnitzer can be used to construct a graded lift of our categorification using singular blocks of modular representations of $\mathfrak{sl}_n$.

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arXiv2022

The cohomological Hall algebras of a preprojective algebra with symmetrizer

This paper aims at a geometric realization of the Yangian of non-simply laced type in terms of quiver with potentials. For every quiver with symmetrizer, there is an extended quiver with superpotential, whose Jacobian algebra is the generalized preprojective algebra of Geiß, Leclerc, and Schröer arXiv:1410.1403. We study the cohomological Hall algebra of Kontsevich and Soibelman associated to this quiver with potential. In particular, we prove a dimensional reduction result, and provide a shuffle formula of this cohomological Hall algebra. In the case when the quiver with symmetrizer comes from a symmetrizable Cartan matrix, we prove that this shuffle algebra satisfies the relations of the Yangian associated to this Cartan matrix.

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arXiv2021

Representations of the elliptic affine Hecke algebras

We prove that irreducible representations of the elliptic affine Hecke algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence with certain nilpotent Higgs bundles on the elliptic curve. The main tool we use is the equivariant elliptic cohomology of the Steinberg variety of the Springer resolution. As a by-product, we study representations at roots of unity in type-$A$. As another by-product, we define a version of elliptic Demazure-Lusztig operators with dynamical parameters that satisfy the braid relations. We discuss speculative indications of this correspondence in 4d $\mathcal N=2$ gauge theory.

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arXiv2021

Wall-crossings and a categorification of $K$-theory stable bases of the Springer resolution

We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure-Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

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arXiv2021

Frobenii on Morava $E$-theoretical quantum groups

In this paper, we study a family of new quantum groups labelled by a prime number $p$ and a natural number $n$ constructed using the Morava $E$-theories. We define the quantum Frobenius homomorphisms among these quantum groups. This is a geometric generalization of Lusztig's quantum Frobenius from the quantum groups at a root of unity to the enveloping algebras. The main ingredient in constructing these Frobenii is the transchromatic character map of Hopkins, Kuhn, Ravenal, and Stapleton. As an application, we prove a Steinberg-type formula for irreducible representations of these quantum groups. Consequently, we prove that, in type $A$ the characters of certain irreducible representations of these quantum groups satisfy the formulas introduced by Lusztig in 2015.

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arXiv2021

Algebraic elliptic cohomology and flops II: $SL$-cobordism

In this paper, we study the algebraic cobordism spectrum $MSL$ in the motivic stable homotopy category of Voevodsky over an arbitrary perfect field $k$. Using the motivic Adams spectral sequence, we compute the geometric part of the $η$-completion of $MSL$ (modulo the maximal subgroup that is $l$-divisble for all primes $l\neq2, char k$). As an application, we study the Krichever's elliptic genus with integral coefficients, restricted to $MSL$. We determine its image, and identify its kernel as the ideal generated by differences of $SL$-flops. This was proved by B. Totaro in the complex analytic setting. In the appendix, we prove some convergence properties of the motivic Adams spectral sequence.

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arXiv2020

Loop Grassmannians of quivers and affine quantum groups

We construct for each choice of a quiver $Q$, a cohomology theory $A$ and a poset $P$ a "loop Grassmannian" $\mathcal{G}^P(Q,A)$. This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a dilation torus $\mathcal{D}\subset\mathbb{G}_m^2$ gives a quantization $\mathcal{G}^P_\mathcal{D}(Q,A)$. The construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program ['Some extensions of the notion of loop Grassmannians', Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardesic issue. No. 532, (2017) 53--74.] and on the construction of affine quantum groups from generalized cohomologies ArXiv: 1708.01418.

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arXiv2020

Categorification via blocks of modular representations for sl(n)

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak{sl}_2$, where they use singular blocks of category $\mathcal{O}$ for $\mathfrak{sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak{sl}_n$ over a field $\textbf{k}$ of characteristic $p$ with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig's conjectures for representations of Lie algebras in positive characteristic.

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arXiv2019

Noncommutative desingularization of orbit closures for some representations of $GL_n$

We describe noncommutative desingularizations of determinantal varieties, determinantal varieties defined by minors of generic symmetric matrices, and pfaffian varieties defined by pfaffians of generic anti-symmetric matrices. For maximal minors of square matrices and symmetric matrices, this gives a non-commutative crepant resolution. Along the way, we describe a method to calculate the quiver with relations for any non-commutative desingularizations coming from exceptional collections over partial flag varieties.

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arXiv2019

Cohomological Hall algebras, vertex algebras and instantons

We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner $\mathcal{W}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds.

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arXiv2018

On two cohomological Hall algebras

We compare two cohomological Hall algebras (CoHA). The first one is the preprojective CoHA introduced by the authors in arXiv:1407.7994 associated to each quiver Q, and each algebraic oriented cohomology theory A. It is defined as the A-homology of the moduli of representations of the preprojective algebra of Q, generalizing the K-theoretic Hall algebra of commuting varieties of Schiffmann-Vasserot. The other one is the critical CoHA defined by Kontsevich-Soibelman associated to each quiver with potentials. It is defined using the equivariant cohomology with compact support with coefficients in the sheaf of vanishing cycles. In the present paper, we show that the critical CoHA, for the quiver with potential of Ginzburg, is isomorphic to the preprojective CoHA as algebras. As applications, we obtain an algebra homomorphism from the positive part of the Yangian to the critical CoHA.

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arXiv2018

The PBW theorem for the affine Yangians

We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.

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