arXiv subjects

Wan Keng Cheong

Publications and source records attributed to Wan Keng Cheong.

8 recordsLinked to original sources

arXiv2026

Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras

We prove an equivalence between the actions of the Gaudin algebras with irregular singularities for $\mathfrak{gl}_d$ and $\mathfrak{gl}_{p+m|q+n}$ on the Fock space of $d(p+m)$ bosonic and $d(q+n)$ fermionic oscillators. This establishes a duality of $(\mathfrak{gl}_d, \mathfrak{gl}_{p+m|q+n})$ for Gaudin models. As an application, we show that the Gaudin algebra with irregular singularities for $\mathfrak{gl}_{p+m|q+n}$ acts cyclically on each weight space of a certain class of infinite-dimensional modules over a direct sum of Takiff superalgebras over $\mathfrak{gl}_{p+m|q+n}$ and that the action is diagonalizable with a simple spectrum under a generic condition. We also study the classical versions of Gaudin algebras with irregular singularities and demonstrate a duality of $(\mathfrak{gl}_d, \mathfrak{gl}_{p+m|q+n})$ for classical Gaudin models.

math.RT↗

arXiv2025

Bethe algebras for unitarizable modules over classical Lie (super)algebras and a duality

Let $\mathfrak{g}$ denote the classical Lie algebra $\mathfrak{gl}_d$, $\mathfrak{sp}_{2d}$, or $\mathfrak{so}_{2d}$ with a fixed $*$-structure $σ$. Let $M_1, \ldots, M_\ell$ be unitarizable $\mathfrak{g}$-modules (with respect to $σ$), and let ${\bf z}=(z_1, \ldots, z_\ell) \in \mathbb{C}^\ell$. We investigate the action of the Bethe algebra $\mathcal{B}_{\mathfrak{g}}^μ$ for $\mathfrak{g}$ with respect to $μ\in \mathfrak{g}^*$ on the tensor product $\underline{M}({\bf z}):=M_1(z_1) \otimes \cdots \otimes M_\ell(z_\ell)$ of evaluation $\mathfrak{g}[t]$-modules. We show that if $μ\circ σ$ equals the complex conjugation of $μ$, then $\mathcal{B}_{\mathfrak{g}}^μ$ is diagonalizable on any finite-dimensional $\mathcal{B}_{\mathfrak{g}}^μ$-submodule of $\underline{M}({\bf z})$ for ${\bf z} \in \mathbb{R}^\ell$. This, together with the result derived from the duality of Bethe algebras (see below), suggests that a simple spectrum conjecture for $\mathcal{B}_{\mathfrak{g}}^μ$ should hold. We establish a duality of Bethe algebras for the general linear Lie (super)algebras $\mathfrak{gl}_d$ and $\mathfrak{gl}_{p+m|q+n}$. As an application, we show that under a generic condition, the Bethe algebra for $\mathfrak{gl}_{p+m|q+n}$ with respect to ${\bf z} \in \mathbb{C}^{p+q+m+n}$ is diagonalizable with a simple spectrum on any weight space of $L_1(w_1) \otimes \cdots \otimes L_d(w_d)$, where the $L_i$ are (infinite-dimensional) unitarizable highest weight $\mathfrak{gl}_{p+m|q+n}$-modules corresponding to generalized partitions of depth 1, and $w_1, \ldots, w_d \in \mathbb{C}$. We also obtain the corresponding result for $\mathfrak{gl}_{p+m}$ by setting $q=n=0$.

math.RT↗

arXiv2025

Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras

We show that under a generic condition, the quadratic Gaudin Hamiltonians associated to $\mathfrak{gl}(p+m|q+n)$ are diagonalizable on any singular weight space in any tensor product of unitarizable highest weight $\mathfrak{gl}(p+m|q+n)$-modules. Moreover, every joint eigenbasis of the Hamiltonians can be obtained from some joint eigenbasis of the quadratic Gaudin Hamiltonians for the general linear Lie algebra $\mathfrak{gl}(r+k)$ on the corresponding singular weight space in the tensor product of some finite-dimensional irreducible $\mathfrak{gl}(r+ k)$-modules for $r$ and $k$ sufficiently large. After specializing to $p=q=0$, we show that similar results hold as well for the cubic Gaudin Hamiltonians associated to $\mathfrak{gl}(m|n)$. We also relate the set of singular solutions of the (super) Knizhnik-Zamolodchikov equations for $\mathfrak{gl}(p+m|q+n)$ to the set of singular solutions of the Knizhnik-Zamolodchikov equations for $\mathfrak{gl}(r+k)$ for $r$ and $k$ sufficiently large.

math.RT↗

arXiv2024

The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz

Let $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$ be the Gaudin algebra of the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ with respect to a sequence $\underline{\boldsymbol{z}} \in \mathbb{C}^\ell$ of pairwise distinct complex numbers, and let $M$ be any $\ell$-fold tensor product of irreducible polynomial modules over $\mathfrak{gl}_{m|n}$. We show that the singular space $M^{\rm sing}$ of $M$ is a cyclic $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$-module and the Gaudin algebra $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$ of $M^{\rm sing}$ is a Frobenius algebra. We also show that $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$ is diagonalizable with a simple spectrum for a generic $\underline{\boldsymbol{z}}$ and give a description of an eigenbasis and its corresponding eigenvalues in terms of the Fuchsian differential operators with polynomial kernels. This may be interpreted as the completeness of a reformulation of the Bethe ansatz for $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$.

math.RT↗

arXiv2024

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Let $M$ be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra $\mathcal{G}$, where $\mathcal{G}$ is $\mathfrak{gl}(m|n)$, $\mathfrak{osp}(2m|2n)$ or $\mathfrak{spo}(2m|2n)$. We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for $\mathcal{G}$ on $M$ can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for $\mathcal{G}$ are diagonalizable on the space spanned by singular vectors of $M$ and hence on $M$. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra $\mathfrak{so}(2n)$ and the symplectic Lie algebra $\mathfrak{sp}(2n)$, on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.

math-ph↗

arXiv2013

Strengthening the Cohomological Crepant Resolution Conjecture for Hilbert-Chow morphisms

Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold invariants of [Sym^n(S)].

math.AG↗

arXiv2009

Connected Gromov-Witten invariants of [Sym^n(A_r)]

We explore the theory of connected Gromov-Witten invariants of the symmetric product stack [Sym^n(A_r)]. We derive closed-form expressions for all equivariant invariants with two insertions and reveal a natural correspondence between the theory and the relative Gromov-Witten theory of the threefold A_r x P^1. When n is less than or equal to 3, we determine 3-point (usual) Gromov-Witten invariants of [Sym^n(A_1)].

math.AG↗

arXiv2009

Orbifold quantum cohomology of the symmetric product of A_r

Let A_r be the minimal resolution of the cyclic quotient singularity C^2/Z_{r+1}. We study the equivariant quantum cohomology ring of the n-fold symmetric product stack [Sym^n(A_r)] of A_r. We calculate the operators of quantum multiplication by divisor classes. Under the assumption of the nonderogatory conjecture, these operators completely determine the ring structure, which provides an affirmative answer to the Crepant Resolution Conjecture on [Sym^n(A_r)] and Hilb^n(A_r). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Sym^n(A_r)]/Hilb^n(A_r) and the relative Gromov-Witten/Donaldson-Thomas theories of A_r x P^1.

math.AG↗