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Justine Fasquel

Publications and source records attributed to Justine Fasquel.

12 recordsLinked to original sources
arXiv2026

Reduction and inverse-reduction functors I: standard $\mathsf{V^k}(\mathfrak{sl}_2)$-modules

Quantum hamiltonian reduction is a fundamental tool of conformal field theory and vertex algebra representation theory. It has traditionally been applied to study highest-weight modules. On the other hand, inverse quantum hamiltonian reduction lends itself to the study of fully relaxed highest-weight modules and their spectral flows, sometimes called the standard modules. This is the first of several papers that study the composition of reduction and inverse-reduction functors. A general formalism is presented and exemplified with the simplest example, thereby computing the action of reduction on the standard modules of the affine vertex-operator algebra associated with $\mathfrak{sl}_2$. The appearence of unbounded spectral sequences in this formalism may be of independent interest.

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arXiv2026

On mode transition algebras for $\mathbb{Z}$-graded vertex algebras and applications to bosonic ghosts

We study the mode transition algebras and Zhu algebras in the setting of $\mathbb{Z}$-graded vertex algebras, with particular focus on the Weyl vertex algebra at central charge 2 (also known as bosonic ghosts or the $\beta\gamma$-system). We show that the mode transition algebras of the Weyl vertex algebra at central charge 2 admit unity elements that form a family of strong unities in the sense of Damiolini-Gibney-Krashen. The existence of unities for the mode transition algebra of the Weyl vertex algebra at central charge 2 allows us to explicitly construct all higher level Zhu algebras of the Weyl vertex algebra at central charge 2. We further analyze weak modules of the Weyl vertex algebra at central charge 2 induced from Zhu algebras, proving that every such module is already induced from the level-zero Zhu algebra. We then prove that all indecomposable reducible weight modules induced from a Zhu algebra are not weakly interlocked, and hence not strongly interlocked in the sense of Barron-Batistelli-Orosz Hunziker-Yamskulna. More generally, we show that the property of being weakly interlocked is preserved under the action of an invertible Li's $\mathbf{\Delta}$ operator. As an application, we prove that all indecomposable reducible weight modules of the Weyl vertex algebra at central charge 2 obtained via spectral flow of Zhu-induced modules are likewise not weakly interlocked. These results clarify the role of being weakly interlocked in the modularity properties of bosonic ghost modules previously studied by Ridout-Wood and Allen-Wood.

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arXiv2025

Quantum Hamiltonian reductions for W-algebras

In this paper, we establish a general criterion for good pairs, namely pairs consisting of a nilpotent orbit and an even good grading in a simple Lie algebra, which guarantees the existence of a quantum Hamiltonian reduction between associated affine W-algebras. In particular, we show that for type A, any two affine W-algebras associated with two adjacent nilpotent orbits are related by quantum Hamiltonian reductions in full generality.

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arXiv2025

Minimal W-algebras of $\mathfrak{so}_N$ at level minus one

For $N \in\mathbb Z_{\geq 7}$ we show that the simple minimal $\mathcal{W}$-algebra of $\mathfrak{so}_N$ at level minus one is isomorphic to the even subalgebra of the tensor product of the simple affine vertex superalgebra of $\mathfrak{osp}_{1|2}$ at level $\frac{N-6}{2}$ with $N-4$ free fermions. In particular when $N$ is even this minimal $\mathcal{W}$-algebra is strongly rational as conjectured by Arakawa-Moreau.

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arXiv2024

Connecting affine $\mathcal{W}$-algebras: A case study on $\mathfrak{sl}_4$

We introduce the partial reductions and inverse Hamiltonian reductions between affine $\mathcal{W}$-algebras along the closure relations of associated nilpotent orbits in the case of $\mathfrak{sl}_4$, fulfilling all the missing constructions in the literature. We also apply the partial reductions to modules in the Kazhdan-Lusztig category and show compatibility with the usual reductions of Weyl modules.

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arXiv2024

Modularity of admissible-level $\mathfrak{sl}_{3}$ minimal models with denominator $2$

We use the newly developed technique of inverse quantum hamiltonian reduction to investigate the representation theory of the simple affine vertex algebra $\mathsf{A}_{2}(\mathsf{u},2)$ associated to $\mathfrak{sl}_{3}$ at level $\mathsf{k} = -3+\frac{\mathsf{u}}{2}$, for $\mathsf{u}\ge3$ odd. Starting from the irreducible modules of the corresponding simple Bershadsky-Polyakov vertex operator algebras, we show that inverse reduction constructs all irreducible lower-bounded weight $\mathsf{A}_{2}(\mathsf{u},2)$-modules. This proceeds by first constructing a complete set of coherent families of fully relaxed highest-weight $\mathsf{A}_{2}(\mathsf{u},2)$-modules and then noting that the reducible members of these families degenerate to give all remaining irreducibles. Using this fully relaxed construction and the degenerations, we deduce modular S-transforms for certain natural generalised characters of these irreducibles and their spectral flows. With this modular data in hand, we verify that the (conjectural) standard Verlinde formula predicts Grothendieck fusion rules with nonnegative-integer multiplicities.

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arXiv2024

On the structure of W-algebras in type A

We formulate and prove examples of a conjecture which describes the W-algebras in type A as successive quantum Hamiltonian reductions of affine vertex algebras associated with several hook-type nilpotent orbits. This implies that the affine coset subalgebras of hook-type W-algebras are building blocks of the W-algebras in type A. In the rational case, it turns out that the building blocks for the simple quotients are provided by the minimal series of the regular W-algebras. In contrast, they are provided by singlet-type extensions of W-algebras at collapsing levels which are irrational. In the latter case, several new sporadic isomorphisms between different W-algebras are established.

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arXiv2024

On some simple orbifold affine VOAs at non-admissible level arising from rank one 4D SCFTs

We study the representations of some simple affine vertex algebras at non-admissible level arising from rank one 4D SCFTs. In particular, we classify the irreducible highest weight modules of $L_{-2}(G_2)$ and $L_{-2}(B_3)$. It is known by the works of Adamovi\'{c} and Per\v{s}e that these vertex algebras can be conformally embedded into $L_{-2}(D_4)$. We also compute the associated variety of $L_{-2}(G_2)$, and show that it is the orbifold of the associated variety of $L_{-2}(D_4)$ by the symmetric group of degree 3 which is the Dynkin diagram automorphism group of $D_4$. This provides a new interesting example of associated variety satisfying a number of conjectures in the context of orbifold vertex algebras.

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arXiv2023

Orthosymplectic Feigin-Semikhatov duality

We study the representation theory of the subregular W-algebra $\mathcal{W}^k(\mathfrak{so}_{2n+1},f_{sub})$ of type B and the principal W-superalgebra $\mathcal{W}^\ell(\mathfrak{osp}_{2|2n})$, which are related by an orthosymplectic analogue of Feigin-Semikhatov duality in type A. We establish a block-wise equivalence of weight modules over the W-superalgebras by using the relative semi-infinite cohomology functor and spectral flow twists, which generalizes the result of Feigin-Semikhatov-Tipunin for the N=2 superconformal algebra. In particular, the correspondence of Wakimoto type free field representations is obtained. When the level of the subregular W-algebra is exceptional, we classify the simple modules over the simple quotients $\mathcal{W}_k(\mathfrak{so}_{2n+1},f_{sub})$ and $\mathcal{W}_\ell(\mathfrak{osp}_{2|2n})$ and derive the character formulae.

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arXiv2022

OPEs of rank two W-algebras

In this short note, we provide OPEs for several affine W-algebras associated with Lie algebras of rank two and give some direct applications.

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arXiv2021

Rationality of the exceptional W-algebras $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$ associated with subregular nilpotent elements of $\mathfrak{sp}_4$

We prove the rationality of the exceptional W-algebras associated with the simple Lie algebra $\mathfrak{sp}_4$ and subregular nilpotent elements, proving a new particular case of a conjecture of Kac-Wakimoto. Moreover, we describe the simple $\mathcal{W}_k(\mathfrak{sp}_4,f_{subreg})$-modules and compute their characters. We also explicit the nontrivial action of the component group on the set of these simple modules.

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