arXiv subjects

Yaoxiong Wen

Publications and source records attributed to Yaoxiong Wen.

14 recordsLinked to original sources

arXiv2026

Spaltenstein Varieties Associated with Pseudo-Polarizations

We introduce minimal Richardson orbits and pseudo-polarizations for nilpotent orbits in classical Lie algebras of types B, C, and D. For any nilpotent orbit, we classify all minimal Richardson orbits containing it and thereby determine the associated pseudo-polarizations. We prove that the corresponding Spaltenstein varieties are smooth and pure dimensional, with iterated orthogonal/isotropic Grassmannian fibrations. As an application, we extend the seesaw property and duality of Fu-Ruan-Wen from Richardson orbits to all special orbits in types B and C.

math.AG↗

arXiv2026

Minimal reduction type in classical cases

We prove Yun's minimal reduction conjecture for all classical groups. More precisely, for any topologically nilpotent regular semisimple element $γ$, we show that the associated minimal reduction set $\mathrm{RT}_{\mathrm{min}}(γ)$ consists of a single nilpotent orbit. This result confirms and extends Yun's earlier work in types A and C, and resolves the remaining cases in types B and D. Moreover, we provide an explicit and effective procedure for determining $\mathrm{RT}_{\mathrm{min}}(γ)$.

math.AG↗

arXiv2025

Springer correspondence and mirror symmetries for parabolic Hitchin systems

We prove the Strominger--Yau--Zaslow and topological mirror symmetries for parabolic Hitchin systems of types B and C. In contrast to type A, a geometric reinterpretation of Springer duality is necessary. Furthermore, unlike Hitchin's construction in the non-parabolic case, the map between generic fibers in type B and C needs more analysis due to the change of partitions of Springer dual nilpotent orbits, which is the main difficulty in this article. To tackle this challenge, we first construct and study the geometry of the generic Hitchin fibers of moduli spaces of Higgs bundles associated to the nilpotent orbit closures. Then we study their relation with the generic Hitchin fibers of parabolic Hitchin systems. Along this way, we establish intriguing connections between Springer duality, Kazhdan--Lusztig maps, and singularities of spectral curves, and uncover a new geometric interpretation of Lusztig's canonical quotient.

math.AG↗

arXiv2025

On the generic fibers and true base of parabolic $\mathrm{SO}_{2n}$-Hitchin systems

In this paper, we confirm a physical conjecture regarding the parabolic $\mathrm{SO}_{2n}$-Hitchin system, showing that Hitchin map factors through a finite cover of the Hitchin base that is isomorphic to an affine space. We first show that the generic Hitchin fiber is disconnected, with the number of components determined by the degree of the generalized Springer map, and then construct the cover explicitly. To this end, we introduce and study a new class of moduli spaces, termed \emph{residually nilpotent Hitchin systems}, and analyze their generic Hitchin fibers. Furthermore, we uncover an interesting connection between self-duality of the generic Hitchin fiber and special nilpotent orbits.

math.AG↗

arXiv2025

Wall Crossing and the Fourier-Mukai Transform for Grassmann Flops

We prove the crepant transformation conjecture for relative Grassmann flops over a smooth base $B$. We show that the $I$-functions of the respective GIT quotients are related by analytic continuation and a symplectic transformation. We verify that the symplectic transformation is compatible with Iritani's integral structure, that is, that it is induced by a Fourier-Mukai transform in $K$-theory.

math.AG↗

arXiv2024

A correspondence between the quantum K theory and quantum cohomology of Grassmannians

We utilize physics arguments, and the nonabelian/abelian correspondence, to relate the Givental and Lee's quantum K theory ring of Grassmannians to a twisted variant of the quantum cohomology ring. Furthermore, the quantum K pairing is related to correlators arising from supersymmetric localization. We state some mathematical conjectures, which we illustrate in several examples.

hep-th↗

arXiv2024

Seiberg-like duality for resolutions of determinantal varieties

We study the genus-zero Gromov-Witten theory of two natural resolutions of determinantal varieties, termed the PAX and PAXY models. We realize each resolution as lying in a quiver bundle, and show that the respective quiver bundles are related by a quiver mutation. We prove that generating functions of genus-zero Gromov-Witten invariants for the two resolutions are related by a specific cluster change of variables. Along the way, we obtain a quantum Thom-Porteous formula for determinantal varieties and prove a Seiberg-like duality statement for certain quiver bundles.

math.AG↗

arXiv2022

Mirror symmetry for special nilpotent orbit closures

Motivated by geometric Langlands, we initiate a program to study the mirror symmetry between nilpotent orbit closures of a semisimple Lie algebra and those of its Langlands dual. The most interesting case is $B_n$ via $C_n$. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symmetry we seek may coincide with Springer duality in the context of special orbits. Unfortunately, such a naive statement fails. To remedy the situation, we propose a conjecture which asserts the mirror symmetry for certain parabolic/induced covers of special orbits. Then, we prove the conjecture for Richardson orbits and obtain certain partial results in general. In the process, we reveal some very interesting and yet subtle structures of these finite covers, which are related to Lusztig's canonical quotients of special nilpotent orbits. For example, there is a mysterious asymmetry in the footprint or range of degrees of these finite covers. Finally, we provide two examples to show that the mirror symmetry fails outside the footprint.

math.AG↗

arXiv2022

Difference Equation for Quintic 3-Fold

In this paper, we use the Mellin-Barnes-Watson method to relate solutions of a certain type of $q$-difference equations at $Q=0$ and $Q=\infty$. We consider two special cases; the first is the $q$-difference equation of $K$-theoretic $I$-function of the quintic, which is degree 25; we use Adams' method to find the extra 20 solutions at $Q=0$. The second special case is a fuchsian case, which is confluent to the differential equation of the cohomological $I$-function of the quintic. We compute the connection matrix and study the confluence of the $q$-difference structure.

math.AG↗

arXiv2021

Quantum $K$-theory and $q$-Difference equations

This is a set of lecture notes for the first author's lectures on the difference equations in 2019 at the Institute of Advanced Study for Mathematics at Zhejiang University. We focus on explicit computations and examples. The convergence of local solutions is discussed.

math.AG↗

arXiv2020

Nonabelian mirrors and Gromov-Witten invariants

We propose Picard-Fuchs equations for periods of nonabelian mirrors in this paper. The number of parameters in our Picard-Fuchs equations is the rank of the gauge group of the nonabelian GLSM, which is eventually reduced to the actual number of Kähler parameters. These Picard-Fuchs equations are concise and novel. We justify our proposal by reproducing existing mathematical results, namely Picard-Fuchs equations of Grassmannians and Calabi-Yau manifolds as complete intersections in Grassmannians. Furthermore, our approach can be applied to other nonabelian GLSMs, so we compute Picard-Fuchs equations of some other Fano-spaces, which were not calculated in the literature before. Finally, the cohomology-valued generating functions of mirrors can be read off from our Picard-Fuchs equations. Using these generating functions, we compute Gromov-Witten invariants of various Calabi-Yau manifolds, including complete intersection Calabi-Yau manifolds in Grassmannians and non-complete intersection Calabi-Yau examples such as Pfaffian Calabi-Yau threefold and Gulliksen-Negård Calabi-Yau threefold, and find agreement with existing results in the literature. The generating functions we propose for non-complete intersection Calabi-Yau manifolds are genuinely new.

hep-th↗

arXiv2020

Quantum $K$-theory of toric varieties, level structures, and 3d mirror symmetry

We introduce a new version of 3d mirror symmetry for toric stacks, inspired by a 3d $\mathcal{N} = 2$ abelian mirror symmetry construction in physics. Given some toric data, we introduce the $K$-theoretic $I$-function with effective level structure for the associated toric stack. When a particular stability condition is chosen, it restricts to the $I$-function for the particular toric GIT quotient. The mirror of a toric stack is defined by the Gale dual of the original toric data. We then proved the mirror conjecture that the $I$-functions of a mirror pair coincide, under the mirror map, which switches Kähler and equivariant parameters, and maps $q\mapsto q^{-1}$.

math.AG↗

arXiv2020

Level correspondence of $K$-theoretic $I$-function in Grassmann duality

In this paper, we prove a class of nontrivial q-Pochhammer symbol identities with extra parameters by iterated residue method. Then we use these identities to find relations of the quasi-map $K$-theoretical $I$-functions with level structure between Grassmannian and its dual Grassmannian. Here we find an interval of levels within which two $I$-functions are the same, and on the boundary of that interval, two $I$-functions are intertwining with each other. We call this phenomenon level correspondence in Grassmann duality.

math.AG↗

arXiv2019

K-Theoretic $I$-function of $V//_θ \mathbf{G}$ and Application

In this paper, we compute K-theoretic $I$-function with level structure (defined by quasi-map theory) of GIT-quotient of a vector space via abelian and non-abelian correspondence. As a consequence, we generalize Givental-Lee's result to find the analogous "Toda operators" for the $I$-function with nontrivial level structures in the case of complete flag variety.

math.AG↗