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Zachary Fehily

Bio: Zachary Fehily is an academic researcher from University of Melbourne. The author has contributed to research in topics: Mathematics & Physics. The author has an hindex of 2, co-authored 4 publications receiving 14 citations.

Papers
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TL;DR: In this article, Adamovic and Kontrec showed that the simple Bershadsky-polyakov algebras with admissible non-integral weights are always rational in the category of highest-weight modules.
Abstract: The Bershadsky–Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $$\mathfrak {sl}_{3}$$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications (Arakawa in Commun Math Phys 323:627–633, 2013, Adamovic and Kontrec in Classification of irreducible modules for Bershadsky–Polyakov algebra at certain levels). In particular, we prove that the simple Bershadsky–Polyakov algebras with admissible nonintegral $$\mathsf {k}$$ are always rational in category $$\mathscr {O}$$ , whilst they always admit nonsemisimple relaxed highest-weight modules unless $$\mathsf {k}+\frac{3}{2} \in \mathbb {Z}_{\geqslant 0}$$ .

18 citations

Posted Content
TL;DR: In this article, the modular properties of the Bershadsky-polyakov characters and the associated Grothendieck fusion rules were investigated for affine vertex operator algebras.
Abstract: The Bershadsky-Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with $\mathfrak{sl}_3$ by quantum hamiltonian reduction. In [arXiv:2007.03917], we explored the representation theories of the simple quotients of these algebras when the level $\mathsf{k}$ is nondegenerate-admissible. Here, we combine these explorations with Adamovi\'{c}'s inverse quantum hamiltonian reduction functors to study the modular properties of Bershadsky-Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with $\mathfrak{sl}_2$, except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov's $\mathsf{W}_3$ algebras.

3 citations

Journal ArticleDOI
TL;DR: In this article, the simple relaxed highest-weight modules with finite-dimensional weight spaces were classified for all admissible but non-integral levels, significantly generalising the known highest weight classifications.
Abstract: The Bershadsky-Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $\mathfrak{sl}_3$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications [arXiv:1005.0185, arXiv:1910.13781]. In particular, we prove that the simple Bershadsky-Polyakov algebras with admissible nonintegral $\mathsf{k}$ are always rational in category $\mathscr{O}$, whilst they always admit nonsemisimple relaxed highest-weight modules unless $\mathsf{k}+\frac{3}{2} \in \mathbb{Z}_{\ge0}$.

2 citations

Posted Content
TL;DR: In this article, it was shown that the representation theory of subregular W-algebras can be realized in terms of the minimal model vertex algebra and the half lattice vertex algebra.
Abstract: Subregular W-algebras are an interesting and increasingly important class of quantum hamiltonian reductions of affine vertex algebras. Here, we show that the $\mathfrak{sl}_{n+1}$ subregular W-algebra can be realised in terms of the $\mathfrak{sl}_{n+1}$ regular W-algebra and the half lattice vertex algebra $\Pi$. This generalises the realisations found for $n=1$ and $2$ in [arXiv:1711.11342, arXiv:2007.00396] and can be interpreted as an inverse quantum hamiltonian reduction in the sense of Adamovi\'c. We use this realisation to explore the representation theory of $\mathfrak{sl}_{n+1}$ subregular W-algebras. Much of the structure encountered for $\mathfrak{sl}_{2}$ and $\mathfrak{sl}_{3}$ is also present for $\mathfrak{sl}_{n+1}$. Particularly, the simple $\mathfrak{sl}_{n+1}$ subregular W-algebra at nondegenerate admissible levels can be realised purely in terms of the $\mathsf{W}_{n+1}$ minimal model vertex algebra and $\Pi$.

1 citations

26 Jun 2023
TL;DR: In this paper , it was shown that an inverse reduction embedding between the affine vertex operator algebra and the minimal W-algebra exists, which generalises the regular-to-subregular inverse reduction of [arXiv:2111.05536], and similarly uses free-field realisations and their associated screening operators.
Abstract: Originating in the work of A.M. Semikhatov and D. Adamovi\'c, inverse reductions are embeddings involving W-algebras corresponding to the same Lie algebra but different nilpotent orbits. Here, we show that an inverse reduction embedding between the affine $\mathfrak{sl}_{n+1}$ vertex operator algebra and the minimal $\mathfrak{sl}_{n+1}$ W-algebra exists. This generalises the realisations for $n=1,2$ in [arXiv:1711.11342, arXiv:2110.15203]. A similar argument is then used to show that inverse reduction embeddings exists between all hook-type $\mathfrak{sl}_{n+1}$ W-algebras, which includes the principal/regular, subregular, minimal $\mathfrak{sl}_{n+1}$ W-algebras, and the affine $\mathfrak{sl}_{n+1}$ vertex operator algebra. This generalises the regular-to-subregular inverse reduction of [arXiv:2111.05536], and similarly uses free-field realisations and their associated screening operators.

Cited by
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TL;DR: In this paper, a realisation of the universal/simple Bershadsky-Polyakov vertex algebra as subalgebras of the tensor product of the Zamolodchikov vertex algebra and an isotropic lattice vertex algebra is presented.
Abstract: We present a realisation of the universal/simple Bershadsky–Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to $$\mathfrak {sl}_{2}$$ and $$\mathfrak {osp} (1 \vert 2)$$ given in Adamovic (Commun Math Phys 366:1025–1067, 2019). Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of Kawasetsu and Ridout (Commun Math Phys 368:627–663, 2019).

18 citations

Journal ArticleDOI
TL;DR: In this paper, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl n+1) in the minimal nilpotent orbit of sl n + 1 has been constructed in terms of Gelfand-Tsetlin tableaux.
Abstract: We explicitly construct, in terms of Gelfand--Tsetlin tableaux, a new family of simple positive energy representations for the simple affine vertex algebra V_k(sl_{n+1}) in the minimal nilpotent orbit of sl_{n+1}. These representations are quotients of induced modules over the affine Kac-Moody algebra of sl_n+1 and include in particular all admissible simple highest weight modules and all simple modules induced from sl_2. Any such simple module in the minimal nilpotent orbit has bounded weight multiplicities.

11 citations

Journal ArticleDOI
TL;DR: In this paper, a realisation of the universal/simple Bershadsky-Polyakov vertex algebra as subalgebras of the tensor product of the Zamolodchikov vertex algebra and an isotropic lattice vertex algebra is presented.
Abstract: We present a realisation of the universal/simple Bershadsky--Polyakov vertex algebras as subalgebras of the tensor product of the universal/simple Zamolodchikov vertex algebras and an isotropic lattice vertex algebra. This generalises the realisation of the universal/simple affine vertex algebras associated to $\mathfrak{sl}_2$ and $\mathfrak{osp}(1|2)$ given in arXiv:1711.11342. Relaxed highest-weight modules are likewise constructed, conditions for their irreducibility are established, and their characters are explicitly computed, generalising the character formulae of arXiv:1803.01989.

10 citations

Journal ArticleDOI
TL;DR: In this article , Adamović et al. studied the modular properties of Bershadsky-Polyakov characters and deduced the associated Grothendieck fusion rules.
Abstract: Abstract The Bershadsky–Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with $$\mathfrak {sl}_3$$ sl 3 by quantum Hamiltonian reduction. In Fehily et al. (Comm Math Phys 385:859–904, 2021), we explored the representation theories of the simple quotients of these algebras when the level $$\mathsf {k}$$ k is nondegenerate-admissible. Here, we combine these explorations with Adamović’s inverse quantum Hamiltonian reduction functors to study the modular properties of Bershadsky–Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with $$\mathfrak {sl}_2$$ sl 2 , except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov’s $$\mathsf {W}_3$$ W 3 algebras.

4 citations

Posted Content
TL;DR: In this article, the authors study the simple Bershadsky-polyakov algebra W_k = \mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their irreducible modules.
Abstract: We study the simple Bershadsky-Polyakov algebra $\mathcal W_k = \mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their irreducible modules. In this way we confirm the conjecture from arXiv:1910.13781. Next, we study the case $k=1$. We discover that this vertex algebra has a Kazama-Suzuki-type dual isomorphic to the simple afine vertex superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. Using the free-field realization of $L_{k'} (osp(1 \vert 2))$ from arXiv:1711.11342, we get a free-field realization of $\mathcal W_k$ and their highest weight modules. In a sequel, we plan to study fusion rules for $\mathcal W_k$.

4 citations