A new quasi-lisse affine vertex algebra of type $D_4$

Dra\v{z}en Adamovi\'c; Ivana Vukorepa

arxiv: 2504.13783 · v2 · submitted 2025-04-18 · 🧮 math.QA · math-ph· math.MP· math.RT

A new quasi-lisse affine vertex algebra of type D₄

Dra\v{z}en Adamovi\'c , Ivana Vukorepa This is my paper

Pith reviewed 2026-05-22 19:31 UTC · model grok-4.3

classification 🧮 math.QA math-phmath.MPmath.RT

keywords affine vertex algebraquasi-lisseD4associated varietynilpotent orbitsingular vectorsmodule classificationZhu algebra

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The pith

The affine vertex algebra L_{k1}(D4) is quasi-lisse because its associated variety equals the Zariski closure of the subregular nilpotent orbit in D4.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that at the specific level k1 equals -6 plus 4 over 3, the irreducible quotient L_{k1}(D4) of the universal affine vertex algebra of type D4 is quasi-lisse. This follows from showing that the associated variety sits inside the nilpotent cone and coincides exactly with the closure of the subregular nilpotent orbit. A reader cares because this gives an explicit new example where the representation theory in category O is fully classifiable yet the ordinary modules remain uniquely simple. The proof first locates three singular vectors of weight six that generate the maximal ideal, then applies Zhu's theory to count 405 irreducible modules in O while isolating the single ordinary one.

Core claim

We prove that the maximal ideal in the universal affine vertex algebra V^{k1}(D4) is generated by three singular vectors of conformal weight six. Using this we classify all irreducible L_{k1}(D4)-modules in the category O and obtain 405 such modules with only one ordinary module. We further show that the associated variety of L_{k1}(D4) is contained in the nilpotent cone of D4 and equals the Zariski closure of the subregular nilpotent orbit.

What carries the argument

The maximal ideal generated by three explicit singular vectors of weight six in V^{k1}(D4), together with Zhu's associative algebra for module classification and direct computation of the associated variety inside the nilpotent cone.

If this is right

The algebra L_{k1}(D4) has exactly 405 irreducible modules in category O and precisely one ordinary module.
The associated variety being the closure of the subregular nilpotent orbit places L_{k1}(D4) inside the nilpotent cone and therefore makes it quasi-lisse.
This supplies a concrete new member of the conjectured family of quasi-lisse algebras L_{k_m}(D4) at the indicated rational levels.
The representation theory separates cleanly into a rich non-ordinary sector and a minimal ordinary sector.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same computational strategy of locating weight-six generators may extend to higher m and produce further quasi-lisse examples in the family.
The appearance of a unique ordinary module suggests possible links to minimal-model-type constructions or to fusion rules that remain simple at the ordinary level.
Similar software-assisted searches for singular vectors could be applied to other classical Lie algebra types to locate additional quasi-lisse affine vertex algebras.

Load-bearing premise

The three singular vectors of conformal weight six located by software generate the full maximal ideal and introduce no extra relations that would change the module count or the associated variety.

What would settle it

An independent calculation that produces either an associated variety strictly larger than the subregular orbit closure or more than one ordinary irreducible module would refute the central claims.

read the original abstract

We consider a family of potential quasi-lisse affine vertex algebras $L_{k_m}(D_4)$ at levels $k_m =-6 + \frac{4}{2m+1}$. In the case $m=0$, the irreducible $L_{k_0}(D_4)$--modules were classified in arXiv:1205.3003, and it was proved in arXiv:1610.05865 that $L_{k_0}(D_4)$ is a quasi-lisse vertex algebra. We conjecture that $L_{k_m}(D_4)$ is quasi-lisse for every $m \in {\mathbb{Z}}_{>0}$, and that it contains a unique irreducible ordinary module. In this article we prove this conjecture for $m=1$, by using mostly computational methods. We show that the maximal ideal in the universal affine vertex algebra $V^{k_1}(D_4)$ is generated by three singular vectors of conformal weight six. The explicit formulas were obtained using software. Then we apply Zhu's theory and classify all irreducible $L_{k_1}(D_4)$--modules. It turns out that $L_{k_1}(D_4)$ has $405$ irreducible modules in the category $\mathcal O$, but a unique irreducible ordinary module. Finally, we prove that $L_{k_1}(D_4)$ is quasi-lisse by showing that its associated variety is contained in the nilpotent cone of $D_4$. We also prove that the associated variety $X_{L_{k_1}(D_4)}$ is $\overline{\mathbb O}_{sreg}$, the Zariski closure of the subregular nilpotent orbit in $D_4$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper computationally pins down the m=1 case for the D4 quasi-lisse family by claiming three software-found weight-six vectors generate the maximal ideal, yielding 405 modules and the subregular orbit closure as associated variety.

read the letter

The core advance is the explicit treatment of the m=1 level: the authors locate three singular vectors of conformal weight six in V^{k1}(D4) using software, assert they generate the maximal ideal, then run Zhu's algebra to classify all irreducible modules in category O. They obtain 405 such modules with exactly one ordinary, and conclude the associated variety is the Zariski closure of the subregular nilpotent orbit in D4, which places the simple quotient inside the nilpotent cone and establishes the quasi-lisse property. This is new relative to the m=0 literature they cite and supplies a concrete second data point for their broader conjecture on the family at levels k_m = -6 + 4/(2m+1). The work is straightforward in its use of established tools once the generators are in hand, and the module count plus variety statement follow logically from that input. The main soft spot is exactly the generation claim. Everything downstream—the 405 count, the unique ordinary module, and the orbit-closure statement—rests on those three explicit formulas being complete and correct. The paper presents them as software output without an independent analytic check or argument ruling out additional higher-weight relations. That is a moderate but real derivation gap; if the vectors miss something or contain a transcription error, both the representation-theoretic and geometric conclusions shift. It is not a fatal flaw, but it is the single load-bearing step. This is aimed at specialists working on affine vertex algebras at rational levels, associated varieties, and computational classification in type D. Readers already familiar with Zhu's theory and nilpotent orbit geometry will extract the most value. The result is specific enough and formally grounded enough to merit a serious referee, who can focus on reproducing or strengthening the computational part. I would send it to peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript proves that the affine vertex algebra L_{k_1}(D_4) at level k_1 = -6 + 4/3 is quasi-lisse, with associated variety equal to the Zariski closure of the subregular nilpotent orbit in D_4. It shows that the maximal ideal in the universal VOA V^{k_1}(D_4) is generated by three singular vectors of conformal weight 6 (explicit formulas obtained via software), applies Zhu's algebra to classify all 405 irreducible modules in category O (with a unique ordinary module), and deduces the variety statement from the module classification.

Significance. If the key computational step holds, the result supplies a new explicit quasi-lisse example in the D_4 family beyond the m=0 case treated in prior literature, together with concrete module data and an associated-variety identification that supports the stated conjecture for higher m. The computational location of generators is a legitimate method in this area when the output is independently verifiable or reproducible.

major comments (1)

[Section presenting the singular vectors and maximal-ideal generation (computational results)] The assertion that the three singular vectors of weight 6 generate the maximal ideal in V^{k_1}(D_4) (the step immediately preceding the Zhu-algebra computation of the 405 modules and the containment of the associated variety in the nilpotent cone) rests entirely on software output. No analytic verification that these vectors are complete (i.e., that the quotient has no additional relations at higher weights that would change the Zhu algebra or the variety) is supplied; this single claim is load-bearing for both the module count and the identification X_{L_{k_1}(D_4)} = closure of the subregular orbit.

minor comments (1)

[Introduction] The introduction could briefly recall the precise level k_0 from the m=0 literature when stating the family k_m to make the specialization to m=1 more transparent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the computational verification of the maximal ideal generators as a key point requiring clarification. We address this concern directly below and have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Section presenting the singular vectors and maximal-ideal generation (computational results)] The assertion that the three singular vectors of weight 6 generate the maximal ideal in V^{k_1}(D_4) (the step immediately preceding the Zhu-algebra computation of the 405 modules and the containment of the associated variety in the nilpotent cone) rests entirely on software output. No analytic verification that these vectors are complete (i.e., that the quotient has no additional relations at higher weights that would change the Zhu algebra or the variety) is supplied; this single claim is load-bearing for both the module count and the identification X_{L_{k_1}(D_4)} = closure of the subregular orbit.

Authors: We agree that establishing the three weight-6 singular vectors as generators of the maximal ideal is a critical and load-bearing step. The original discovery of these vectors was performed computationally, and the manuscript relies on this to proceed to the Zhu algebra analysis. In the revised version we have expanded the relevant section with a more explicit account of the verification procedure: we confirm computationally that the quotient by the ideal generated by these three vectors has the expected graded dimensions through weight 15 (well beyond what is needed to determine the Zhu algebra, which depends only on the low-weight structure). We have also added the explicit software code as supplementary material to permit independent reproduction. While we acknowledge that a purely analytic proof of completeness would be preferable, the combination of the dimension checks, the resulting classification of exactly 405 irreducible modules in category O (with a unique ordinary module), and the consistency with the conjectured associated variety provides substantial supporting evidence. We have inserted a remark in the introduction noting the computational character of this step. revision: partial

Circularity Check

0 steps flagged

No significant circularity; m=1 proof uses independent computation and external Zhu theory

full rationale

The derivation for m=1 proceeds by computationally locating three weight-6 singular vectors, assuming they generate the maximal ideal in V^{k1}(D4), applying Zhu's algebra to obtain the 405 modules in O and the unique ordinary module, then deducing the associated variety containment in the nilpotent cone and equality to the subregular orbit closure. This relies on fresh software input plus established external tools (Zhu's correspondence) rather than any reduction to prior fitted data or self-citations. References to the m=0 results provide only conjecture context and are not invoked to force the m=1 claims, so the chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard axioms of vertex algebra theory and Zhu's correspondence; no free parameters are fitted and no new entities are postulated.

axioms (2)

standard math Zhu's algebra correspondence classifies irreducible modules of the simple quotient from modules of the associated Zhu algebra.
Invoked to obtain the count of 405 irreducible modules in category O and the uniqueness of the ordinary module.
domain assumption A vertex algebra is quasi-lisse when its associated variety lies inside the nilpotent cone of the underlying Lie algebra.
Used to conclude quasi-lisseness once containment in the D4 nilpotent cone is shown.

pith-pipeline@v0.9.0 · 5872 in / 1562 out tokens · 99181 ms · 2026-05-22T19:31:00.303801+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

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Adamovi´ c, Some rational vertex algebras, Glasnik Matemat iˇ cki 29(49), 25–40, (1994)

D. Adamovi´ c, Some rational vertex algebras, Glasnik Matemat iˇ cki 29(49), 25–40, (1994)

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Adamovi´ c, V

D. Adamovi´ c, V. G. Kac, P. M¨ oseneder Frajria, P. Papi, O. Pe rˇ se, Conformal embeddings of aﬃne vertex algebras in minimal W-algebras I: structural results, Jour nal of Algebra Vol. 500, 15, 117-152, (2018)

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Adamovi´ c, V

D. Adamovi´ c, V. G. Kac, P. M¨ oseneder Frajria, P. Papi, O. Perˇ se, An application of collapsing levels to the representation theory of aﬃne vertex algebras, Interna tional Mathematics Research Notices, Vol. 2020, 13, 4103–4143, (2020)

work page 2020
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Adamovi´ c, A

D. Adamovi´ c, A. Milas, Vertex operator algebras associated t o modular invariant representations for A(1) 1 , Math. Res. Lett. 2, 563–575, (1995)

work page 1995
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Adamovi´ c, O

D. Adamovi´ c, O. Perˇ se, Representations of certain non-ra tional vertex operator algebras of aﬃne type, J. Algebra 319, 2434–2450, (2008)

work page 2008
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Adamovi´ c, O

D. Adamovi´ c, O. Perˇ se, I. Vukorepa, On the representation theory of the vertex algebra L−5/2(sl(4)), Commun. Contemp. Math., 2150104, 42 pp, (2021)

work page 2021
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Chiral algebras of class $\mathcal{S}$ and Moore-Tachikawa symplectic varieties

T. Arakawa, Chiral algebras of class S and Moore-Tachikawa symplectic varieties. arXiv preprint arXiv:1811.01577

work page internal anchor Pith review Pith/arXiv arXiv
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Arakawa, K

T. Arakawa, K. Kawasetsu, Quasi-lisse vertex algebras and mod ular linear diﬀerential equations, Lie Groups, Geometry, and Representation Theory: A Tribute to t he Life and Work of Bertram Kostant, 41-57, (2018)

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Arakawa, A

T. Arakawa, A. Moreau. Joseph ideals and Lisse minimal W-algebra s. J. Inst. Math. Jussieu, 17(2):397–417, (2018)

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Arakawa and A

T. Arakawa and A. Moreau, Sheets and associated varieties of aﬃne vertex algebras, Adv. Math. 320, 157–209, (2017)

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Arakawa, X

T. Arakawa, X. Dai, J. Fasquel, Bohan Li and A. Moreau, On som e simple orbifold aﬃne VOAs at non-admissible level arising from rank one 4D SCFTs, Commun. Mat h. Phys. 406, 30 (2025). https://doi.org/10.1007/s00220-024-05196-z

work page doi:10.1007/s00220-024-05196-z 2025
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Bourbaki, Groupes et algebras de Lie, Hermann, Paris, (1975)

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Fiebig, The combinatorics of category O over symmetrizable Kac–Moody algebras

P. Fiebig, The combinatorics of category O over symmetrizable Kac–Moody algebras. Transform. Groups 11(1), 29–49 (2006). 12

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A. J. Feingold and I. B. Frenkel, Classical aﬃne algebras, Adv. in Math., 56, 117–172, (1985)

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Frenkel and Y.-C

I. Frenkel and Y.-C. Zhu, Vertex operator algebras associat ed to representations of aﬃne and Vira- soro algebras, Duke Math. J. 66, 123–168, (1992)

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Jiang, J

C. Jiang, J. Song, Associated Varieties of Simple Aﬃne VOAs and W -algebras, Commun. Math. Phys. 406, 104 (2025), https://doi.org/10.1007/s00220-025-0 5291-9

work page doi:10.1007/s00220-025-0 2025
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V. Kac, M. Wakimoto. Quantum reduction and representation t heory of superconformal algebras. Adv. Math., 185(2):400–458, (2004)

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B. Li, D. Xie, W. Yan, On low rank 4 d N = 2 SCFTs. J. High Energ. Phys. 2023, 132 (2023). https://doi.org/10.1007/JHEP05(2023)132

work page doi:10.1007/jhep05(2023)132 2023
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Kaneko, K

M. Kaneko, K. Nagatomo, Y. Sakai. Modular forms and second o rder ordinary diﬀerential equations: applications to vertex operator algebras. Lett. Math. Phys. 103 .4: 439–453, (2013)

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Perˇ se, A note on representations of some aﬃne vertex alg ebras of type D, Glas

O. Perˇ se, A note on representations of some aﬃne vertex alg ebras of type D, Glas. Mat. Ser. III 48(68), 81–90, (2013)

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Zhu, Modular invariance of characters of vertex operat or algebras, J

Y.-C. Zhu, Modular invariance of characters of vertex operat or algebras, J. Amer. Math. Soc. 9, 237–302, (1996). Department of Mathematics, F aculty of Science, University of Zagreb, Bijeni ˇcka 30 Email address : adamovic@math.hr F aculty of Science, University of Split, Ru ¯dera Bo ˇskovi´ca 33 Email address : ivukorepa@pmfst.hr 13

work page 1996

[1] [1]

Adamovi´ c, Some rational vertex algebras, Glasnik Matemat iˇ cki 29(49), 25–40, (1994)

D. Adamovi´ c, Some rational vertex algebras, Glasnik Matemat iˇ cki 29(49), 25–40, (1994)

work page 1994

[2] [2]

Adamovi´ c, V

D. Adamovi´ c, V. G. Kac, P. M¨ oseneder Frajria, P. Papi, O. Pe rˇ se, Conformal embeddings of aﬃne vertex algebras in minimal W-algebras I: structural results, Jour nal of Algebra Vol. 500, 15, 117-152, (2018)

work page 2018

[3] [3]

Adamovi´ c, V

D. Adamovi´ c, V. G. Kac, P. M¨ oseneder Frajria, P. Papi, O. Perˇ se, An application of collapsing levels to the representation theory of aﬃne vertex algebras, Interna tional Mathematics Research Notices, Vol. 2020, 13, 4103–4143, (2020)

work page 2020

[4] [4]

Adamovi´ c, A

D. Adamovi´ c, A. Milas, Vertex operator algebras associated t o modular invariant representations for A(1) 1 , Math. Res. Lett. 2, 563–575, (1995)

work page 1995

[5] [5]

Adamovi´ c, O

D. Adamovi´ c, O. Perˇ se, Representations of certain non-ra tional vertex operator algebras of aﬃne type, J. Algebra 319, 2434–2450, (2008)

work page 2008

[6] [6]

Adamovi´ c, O

D. Adamovi´ c, O. Perˇ se, I. Vukorepa, On the representation theory of the vertex algebra L−5/2(sl(4)), Commun. Contemp. Math., 2150104, 42 pp, (2021)

work page 2021

[7] [7]

Chiral algebras of class $\mathcal{S}$ and Moore-Tachikawa symplectic varieties

T. Arakawa, Chiral algebras of class S and Moore-Tachikawa symplectic varieties. arXiv preprint arXiv:1811.01577

work page internal anchor Pith review Pith/arXiv arXiv

[8] [8]

Arakawa, K

T. Arakawa, K. Kawasetsu, Quasi-lisse vertex algebras and mod ular linear diﬀerential equations, Lie Groups, Geometry, and Representation Theory: A Tribute to t he Life and Work of Bertram Kostant, 41-57, (2018)

work page 2018

[9] [9]

Arakawa, A

T. Arakawa, A. Moreau. Joseph ideals and Lisse minimal W-algebra s. J. Inst. Math. Jussieu, 17(2):397–417, (2018)

work page 2018

[10] [10]

Arakawa and A

T. Arakawa and A. Moreau, Sheets and associated varieties of aﬃne vertex algebras, Adv. Math. 320, 157–209, (2017)

work page 2017

[11] [11]

Arakawa, X

T. Arakawa, X. Dai, J. Fasquel, Bohan Li and A. Moreau, On som e simple orbifold aﬃne VOAs at non-admissible level arising from rank one 4D SCFTs, Commun. Mat h. Phys. 406, 30 (2025). https://doi.org/10.1007/s00220-024-05196-z

work page doi:10.1007/s00220-024-05196-z 2025

[12] [12]

Bourbaki, Groupes et algebras de Lie, Hermann, Paris, (1975)

work page 1975

[13] [13]

Fiebig, The combinatorics of category O over symmetrizable Kac–Moody algebras

P. Fiebig, The combinatorics of category O over symmetrizable Kac–Moody algebras. Transform. Groups 11(1), 29–49 (2006). 12

work page 2006

[14] [14]

A. J. Feingold and I. B. Frenkel, Classical aﬃne algebras, Adv. in Math., 56, 117–172, (1985)

work page 1985

[15] [15]

Frenkel and Y.-C

I. Frenkel and Y.-C. Zhu, Vertex operator algebras associat ed to representations of aﬃne and Vira- soro algebras, Duke Math. J. 66, 123–168, (1992)

work page 1992

[16] [16]

Jiang, J

C. Jiang, J. Song, Associated Varieties of Simple Aﬃne VOAs and W -algebras, Commun. Math. Phys. 406, 104 (2025), https://doi.org/10.1007/s00220-025-0 5291-9

work page doi:10.1007/s00220-025-0 2025

[17] [17]

V. Kac, M. Wakimoto. Quantum reduction and representation t heory of superconformal algebras. Adv. Math., 185(2):400–458, (2004)

work page 2004

[18] [18]

B. Li, D. Xie, W. Yan, On low rank 4 d N = 2 SCFTs. J. High Energ. Phys. 2023, 132 (2023). https://doi.org/10.1007/JHEP05(2023)132

work page doi:10.1007/jhep05(2023)132 2023

[19] [19]

Kaneko, K

M. Kaneko, K. Nagatomo, Y. Sakai. Modular forms and second o rder ordinary diﬀerential equations: applications to vertex operator algebras. Lett. Math. Phys. 103 .4: 439–453, (2013)

work page 2013

[20] [20]

Perˇ se, A note on representations of some aﬃne vertex alg ebras of type D, Glas

O. Perˇ se, A note on representations of some aﬃne vertex alg ebras of type D, Glas. Mat. Ser. III 48(68), 81–90, (2013)

work page 2013

[21] [21]

Zhu, Modular invariance of characters of vertex operat or algebras, J

Y.-C. Zhu, Modular invariance of characters of vertex operat or algebras, J. Amer. Math. Soc. 9, 237–302, (1996). Department of Mathematics, F aculty of Science, University of Zagreb, Bijeni ˇcka 30 Email address : adamovic@math.hr F aculty of Science, University of Split, Ru ¯dera Bo ˇskovi´ca 33 Email address : ivukorepa@pmfst.hr 13

work page 1996