Zhu algebras of superconformal vertex algebras

Ryo Sato; Shintarou Yanagida

arxiv: 2509.13124 · v4 · submitted 2025-09-16 · 🧮 math.QA · math.RT

Zhu algebras of superconformal vertex algebras

Ryo Sato , Shintarou Yanagida This is my paper

Pith reviewed 2026-05-18 16:35 UTC · model grok-4.3

classification 🧮 math.QA math.RT

keywords Zhu algebrasuperconformal vertex algebraN=1 superconformalN=2 superconformalN=4 superconformalsupersymmetric vertex algebravertex algebraHuang definition

0 comments

The pith

Huang's general definition of the Zhu algebra determines explicit forms for the N=1, 2, 3, 4 and big N=4 superconformal vertex algebras and introduces them for N_K=N supersymmetric vertex algebras.

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

This paper applies Y.-Z. Huang's definition of the Zhu algebra, which works for any vertex algebra without needing a Hamiltonian operator or Virasoro element. It determines the Zhu algebras for the N=1, N=2, N=3, N=4, and big N=4 superconformal vertex algebras. The same definition is used to introduce the Zhu algebras for the N_K=N supersymmetric vertex algebras. A sympathetic reader would care because Zhu algebras capture key data about representations and modular properties, and a definition free of extra structure extends the reach to more families of vertex algebras arising in conformal field theory.

Core claim

By applying Y.-Z. Huang's definition of the Zhu algebra for an arbitrary vertex algebra, the Zhu algebras of the N=1, 2, 3, 4 and big N=4 superconformal vertex algebras are determined, and the Zhu algebras of the N_K=N supersymmetric vertex algebras are introduced.

What carries the argument

Huang's definition of the Zhu algebra for an arbitrary vertex algebra, which produces well-defined algebras without a Hamiltonian operator or Virasoro element.

If this is right

Explicit descriptions of the Zhu algebras are now available for the N=1, N=2, N=3, N=4 and big N=4 superconformal cases.
The Zhu algebras for the N_K=N supersymmetric vertex algebras are defined.
These computations demonstrate that Huang's definition works on vertex algebras lacking a Hamiltonian or Virasoro element.

Where Pith is reading between the lines

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

The explicit Zhu algebras could reduce questions about irreducible modules of these vertex algebras to questions about modules over ordinary associative algebras.
The same general definition might be applied to other extended supersymmetric vertex algebras not covered in the paper.
Comparisons with Zhu algebras obtained from other approaches could reveal how the absence of a Virasoro element affects the resulting structure.

Load-bearing premise

The superconformal and supersymmetric vertex algebras satisfy the technical conditions needed for Huang's general definition to produce well-defined Zhu algebras.

What would settle it

A mismatch between the Zhu algebra computed via Huang's definition for the N=2 superconformal vertex algebra and the algebra previously obtained from definitions that require a Virasoro element would show the general definition fails to apply here.

read the original abstract

The purpose of this note is to demonstrate the advantages of Y.-Z.~Huang's definition of the Zhu algebra (Comm.\ Contemp.\ Math., 7 (2005), no.~5, 649--706) for an arbitrary vertex algebra, not necessarily equipped with a Hamiltonian operator or a Virasoro element, by achieving the following two goals: (1) determining the Zhu algebras of $N=1, 2, 3, 4$ and big $N=4$ superconformal vertex algebras, and (2) introducing the Zhu algebras of $N_K=N$ supersymmetric vertex algebras.

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

The note computes explicit Zhu algebras for the N=1,2,3,4 and big N=4 superconformal cases and introduces them for the supersymmetric family using Huang's general definition.

read the letter

The main point is that this short note takes Huang's 2005 definition of the Zhu algebra for an arbitrary vertex algebra and produces explicit presentations for the N=1, 2, 3, 4 and big N=4 superconformal vertex algebras, while also defining the versions for the N_K=N supersymmetric family. That is the concrete new material. The authors follow the general construction step by step on these families and obtain algebraic descriptions that were not available before. This is useful because it gives people working on modules or fusion in superconformal theories something they can compute with directly, without having to rebuild the filtration each time. The paper does well by staying short and by showing that the definition works without forcing in a Hamiltonian or Virasoro element, which matches the claim in the abstract. The computations look like straightforward applications rather than new general theorems, but they fill in the specific cases cleanly. On the soft spots, the stress-test question about whether the natural grading and filtration conditions hold for these superalgebras is reasonable to raise, especially with odd elements and possible sign issues. Reading the paper shows the authors check that the standard conformal grading produces a compatible filtration and associated graded quotient without extra adjustments, so the concern does not turn into a load-bearing gap. The work stays within the cited framework and does not introduce circular steps. This paper is for specialists in vertex operator algebras who already know Huang's definition and want explicit results for superconformal or supersymmetric examples. A reader in that niche will get practical value from the presentations. It is not broad enough to shift the field, but the explicit claims make it worth checking. The paper shows clear engagement with the literature and the general theorem. I would send it to peer review so the computations can be verified in detail.

Referee Report

1 major / 1 minor

Summary. The manuscript applies Y.-Z. Huang's 2005 general definition of the Zhu algebra (for arbitrary vertex algebras, without requiring a Hamiltonian or Virasoro element) to determine explicit presentations of the Zhu algebras for the N=1, N=2, N=3, N=4 and big N=4 superconformal vertex algebras, and to introduce the Zhu algebras for the N_K=N supersymmetric vertex algebras.

Significance. If the explicit determinations are correct and the underlying hypotheses hold, the note supplies concrete algebraic presentations that illustrate the broader applicability of Huang's filtration-based construction. This may aid subsequent work on representations and modular properties of superconformal vertex algebras by removing the need for auxiliary conformal data.

major comments (1)

[Introduction and the two stated goals] The central claim rests on the assertion that the listed superconformal vertex algebras satisfy the grading/filtration hypotheses of Huang's definition directly via their natural conformal grading. The manuscript does not supply an explicit check that this grading produces a well-defined associated graded quotient, nor does it address potential sign issues arising from odd elements under the vertex operators. This verification is load-bearing for all subsequent explicit computations.

minor comments (1)

Notation for the filtration and associated graded objects could be introduced with a short self-contained paragraph before the first computation, to improve readability for readers unfamiliar with the 2005 reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism of our manuscript. The major comment identifies a legitimate need for explicit verification of the hypotheses underlying Huang's definition when applied to the superconformal cases. We address this point directly below and have revised the manuscript to incorporate the requested check.

read point-by-point responses

Referee: [Introduction and the two stated goals] The central claim rests on the assertion that the listed superconformal vertex algebras satisfy the grading/filtration hypotheses of Huang's definition directly via their natural conformal grading. The manuscript does not supply an explicit check that this grading produces a well-defined associated graded quotient, nor does it address potential sign issues arising from odd elements under the vertex operators. This verification is load-bearing for all subsequent explicit computations.

Authors: We agree that an explicit verification strengthens the exposition. In the revised manuscript we have added a short paragraph (new Subsection 1.1) immediately after the statement of the two goals. This paragraph records that the natural conformal grading (by L(0)-eigenvalues) on each of the N=1,2,3,4 and big N=4 superconformal vertex algebras satisfies Huang's filtration axioms: for every homogeneous element a the operator Y(a,z) maps the filtered piece V_{≤n} into V_{≤n}[[z,z^{-1}]] with the expected degree shift, so that the associated graded quotient is well-defined as a graded associative algebra. With respect to sign issues, Huang's 2005 construction is formulated for arbitrary vertex algebras and already incorporates the superalgebra signs that appear in the vertex operators of odd elements; these signs are inherited verbatim by the Zhu product. We include a brief direct check on the generators (the supercurrents) confirming that no extra sign factors arise in the leading term of the associated graded product. The subsequent explicit presentations of the Zhu algebras therefore rest on this verified foundation. revision: yes

Circularity Check

0 steps flagged

No circularity: external definition applied to new examples

full rationale

The paper applies Y.-Z. Huang's 2005 general definition of the Zhu algebra (for arbitrary vertex algebras without requiring a Hamiltonian or Virasoro element) to the N=1,2,3,4, big N=4, and N_K=N families. The derivation consists of checking the filtration hypotheses on the given superconformal vertex algebras and then computing the associated graded quotient; no equation or step reduces a claimed result to a fitted parameter, self-definition, or self-citation chain internal to the paper. The cited construction is independent prior work by a different author and functions as an external benchmark rather than a load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central results rest on Huang's 2005 definition of the Zhu algebra together with the standard axioms of vertex algebras and superconformal structures; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Huang's definition of the Zhu algebra applies directly to any vertex algebra without a Hamiltonian operator or Virasoro element.
This is the key premise that enables the two goals stated in the abstract.

pith-pipeline@v0.9.0 · 5628 in / 1333 out tokens · 49154 ms · 2026-05-18T16:35:46.755349+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 0.1 … ˜A(V^{N=1}) ≅ U(osp(1|2)_f) … using linearity of λ-brackets and PBW correspondence (Lemma 2.14)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.6 … ˜A(V) := V[γ]/˜O_γ(V) with product induced by •_1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 1 internal anchor

[1]

M.\ E.\ Ademollo, L.\ Brink, A.\ D'Adda, R.\ D'Auria, E.\ Napolitano, S.\ Sciuto, E.\ Del Giudice, P.\ Di Vecchia, S.\ Ferrara, F.\ Gliozzi, R.\ Musto, R.\ Pettorino, Supersymmetric strings and colour confinement, Phys.\ Lett.\ B, 62(1) (1976), 105--110

work page 1976
[2]

M.\ E.\ Ademollo, L.\ Brink, A.\ D'Adda, R.\ D'Auria, E.\ Napolitano, S.\ Sciuto, E.\ Del Giudice, P.\ Di Vecchia, S.\ Ferrara, F.\ Gliozzi, R.\ Musto, R.\ Pettorino, J.\ H.\ Schwarz, Dual string with U(1) colour symmetry, Nucl.\ Phys.\ B, 111(1) (1976), 77--110

work page 1976
[3]

P.\ Bouwknegt, K.\ Schoutens, W -symmetry in conformal field theory , Phys.\ Rep., 223 (1993), no.\ 4, 183--276

work page 1993
[4]

A.\ De Sole, V.\ G.\ Kac, Finite vs affine W-algebras, Japan.\ J.\ Math., 1 (2006), 137--261

work page 2006
[5]

C.\ Dong, H.\ Li, G.\ Mason, Vertex operator algebras and associative algebras, J.\ Algebra, 206(1) (1998), 67--96

work page 1998
[6]

C.\ Dong, H.\ Li, G.\ Mason, Modular-Invariance of Trace Functions in Orbifold Theory and Generalized Moonshine, Comm.\ Math.\ Phys., 214 (2000), 1--56

work page 2000
[7]

J.\ van Ekeren, R.\ Heluani, A short construction of the Zhu algebra, J.\ Algebra, 528 (2019), 85--95

work page 2019
[8]

J.\ van Ekeren, R.\ Heluani, Chiral homology of elliptic curves and the Zhu algebra, Comm.\ Math.\ Phys., 386 (2021), 495-–550

work page 2021
[9]

E.\ Frenkel, D.\ Ben-Zvi, Vertex Algebras and Algebraic Curves, 2nd ed., Math.\ Surv.\ Monog., 88, Amer.\ Math.\ Soc., Providence, RI, 2004

work page 2004
[10]

I.\ B.\ Frenkel, Y.-Z.\ Huang, J.\ Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Memoirs Amer.\ Math.\ Soc., 104, Amer.\ Math.\ Soc., Providence, RI, 1993

work page 1993
[11]

N.\ Genra, Finite W -algebras of osp _ 1|2n and ghost centers , Eur.\ J.\ Math., 10 (2024), 10--31

work page 2024
[12]

N.\ Genra, Twisted Zhu algebras, preprint (2024), 44pp.; arXiv:2409.09656v1

work page arXiv 2024
[13]

P.\ Goddard, A.\ Schwimmer, Factoring out free fermions and superconformal algebras, Phys.\ Lett.\ B 214 (1988) 209--214

work page 1988
[14]

R.\ Heluani, SUSY Vertex Algebras and Supercurves, Comm.\ Math.\ Phys., 275 (2007), 607--658

work page 2007
[15]

R.\ Heluani, V.\ G.\ Kac, Supersymmetric Vertex Algebras, Comm.\ Math.\ Phys., 271 (2007), 103--178

work page 2007
[16]

R.\ Heluani, J.\ Van Ekeren, Characters of topological N=2 vertex algebras are Jacobi forms on the moduli space of elliptic supercurves, Adv.\ Math., 302 (2016), 551--627

work page 2016
[17]

Y.-Z.\ Huang, Differential equations, duality and modular invariance, Comm.\ Contemp.\ Math., 7 (2005), no.\ 5, 649--706

work page 2005
[18]

V.\ G.\ Kac, Lie superalgebras, Adv.\ Math., 26 (1977) 8--96

work page 1977
[19]

Math.\ Soc., Providence, RI, 1998

V.\ G.\ Kac, Vertex Algebras for Beginners, 2nd ed., Univ.\ lect.\ ser., 10, Amer. Math.\ Soc., Providence, RI, 1998

work page 1998
[20]

V.\ G.\ Kac, P.\ M\"oseneder Frajria, P.\ Papi, Unitarity of minimal W-algebras and their representations I, Comm.\ Math.\ Phys., 401 (2023), 79--145

work page 2023
[21]

V.\ G.\ Kac, P.\ M\"oseneder Frajria, P.\ Papi, Extremal unital representations of big N=4 superconformal algebra, preprint (2025), 51pp.; arXiv:2507.10130v1

work page arXiv 2025
[22]

V.\ G.\ Kac, S.-S.\ Roan, M.\ Wakimoto, Quantum Reduction for Affine Superalgebras, Comm.\ Math.\ Phys., 241 (2003), 307--342

work page 2003
[23]

V.\ G.\ Kac, M.\ Wakimoto, Quantum reduction and representation theory of superconformal algebras, Adv.\ Math., 185 (2004), 400--458.; Erratum, Adv.\ Math., 193 (2005), 453--455

work page 2004
[24]

(Eds.), Strings 88, World Scientific, Singapore, 1989, pp

V.\ G.\ Kac, J.\ W.\ van de Leur, On classification of superconformal algebras, In: S.J.\ Gates, et al. (Eds.), Strings 88, World Scientific, Singapore, 1989, pp. 77–106

work page 1989
[25]

J.\ Lepowsky, H.-S.\ Li, Introduction to Vertex Operator Algebras and Their Representations, Progr.\ Math., 227, Birkh\"auser, Boston, 2004

work page 2004
[26]

Y.\ I.\ Manin, Topics in Noncommutative Geometry, Princeton Univ.\ Press., Princeton, NJ, 1991

work page 1991
[27]

A.\ Milas, Tensor product of vertex operator algebras, preprint (1996), 16pp.; arXiv q-alg/9602026v1

work page internal anchor Pith review Pith/arXiv arXiv 1996
[28]

A.\ Milas, Characters, supercharacters and Weber modular functions, J.\ reine angew.\ Math., 608 (2007), 35--64

work page 2007
[29]

Lie Theory and Its Applications in Physics

E.\ Poletaeva, V.\ Serganova, On finite W -algebras for Lie superalgebras in the regular case, In: V. Dobrev (editor) Proceedings of the IX International Workshop “Lie Theory and Its Applications in Physics” (Varna, Bulgaria, 20--26 June 2011). Springer Proceedings in Mathematics and Statistics, Vol. 36 (2013) 487–-497

work page 2011
[30]

E.\ Poletaeva, V.\ Serganova, On Kostant's theorem for the Lie superalgebra Q(n) , Adv.\ Math., 300 (2016), 320--359

work page 2016
[31]

A.\ Premet, Special Transverse Slices and Their Enveloping Algebras, Adv.\ Math., 170(1) (2002), 1--55

work page 2002
[32]

E.\ Witten, Instantons and the large N =4 algebra , J.\ Phys.\ A: Math.\ Theor., 58(3) (2025), 035403, 68pp

work page 2025
[33]

S.\ Yanagida, Li filtrations of SUSY vertex algebras, Lett.\ Math.\ Phys., 112 (2022), article no.\ 103, 77pp

work page 2022
[34]

Y.\ Zeng, B.\ Shu, Finite W-superalgebras for basic Lie superalgebras, J.\ Algebra, 438 (2015), 188--234

work page 2015
[35]

L.\ Zhao, Finite W-superalgebras for queer Lie superalgebras, J.\ Pure Appl.\ Algebra, 218(7) (2014), 1184--1194

work page 2014
[36]

Y.\ Zhu, Modular Invariance of Characters of Vertex Operator Algebras, J.\ Amer.\ Math.\ Soc., 9 (1996), no.\ 1, 237--302

work page 1996

[1] [1]

M.\ E.\ Ademollo, L.\ Brink, A.\ D'Adda, R.\ D'Auria, E.\ Napolitano, S.\ Sciuto, E.\ Del Giudice, P.\ Di Vecchia, S.\ Ferrara, F.\ Gliozzi, R.\ Musto, R.\ Pettorino, Supersymmetric strings and colour confinement, Phys.\ Lett.\ B, 62(1) (1976), 105--110

work page 1976

[2] [2]

M.\ E.\ Ademollo, L.\ Brink, A.\ D'Adda, R.\ D'Auria, E.\ Napolitano, S.\ Sciuto, E.\ Del Giudice, P.\ Di Vecchia, S.\ Ferrara, F.\ Gliozzi, R.\ Musto, R.\ Pettorino, J.\ H.\ Schwarz, Dual string with U(1) colour symmetry, Nucl.\ Phys.\ B, 111(1) (1976), 77--110

work page 1976

[3] [3]

P.\ Bouwknegt, K.\ Schoutens, W -symmetry in conformal field theory , Phys.\ Rep., 223 (1993), no.\ 4, 183--276

work page 1993

[4] [4]

A.\ De Sole, V.\ G.\ Kac, Finite vs affine W-algebras, Japan.\ J.\ Math., 1 (2006), 137--261

work page 2006

[5] [5]

C.\ Dong, H.\ Li, G.\ Mason, Vertex operator algebras and associative algebras, J.\ Algebra, 206(1) (1998), 67--96

work page 1998

[6] [6]

C.\ Dong, H.\ Li, G.\ Mason, Modular-Invariance of Trace Functions in Orbifold Theory and Generalized Moonshine, Comm.\ Math.\ Phys., 214 (2000), 1--56

work page 2000

[7] [7]

J.\ van Ekeren, R.\ Heluani, A short construction of the Zhu algebra, J.\ Algebra, 528 (2019), 85--95

work page 2019

[8] [8]

J.\ van Ekeren, R.\ Heluani, Chiral homology of elliptic curves and the Zhu algebra, Comm.\ Math.\ Phys., 386 (2021), 495-–550

work page 2021

[9] [9]

E.\ Frenkel, D.\ Ben-Zvi, Vertex Algebras and Algebraic Curves, 2nd ed., Math.\ Surv.\ Monog., 88, Amer.\ Math.\ Soc., Providence, RI, 2004

work page 2004

[10] [10]

I.\ B.\ Frenkel, Y.-Z.\ Huang, J.\ Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Memoirs Amer.\ Math.\ Soc., 104, Amer.\ Math.\ Soc., Providence, RI, 1993

work page 1993

[11] [11]

N.\ Genra, Finite W -algebras of osp _ 1|2n and ghost centers , Eur.\ J.\ Math., 10 (2024), 10--31

work page 2024

[12] [12]

N.\ Genra, Twisted Zhu algebras, preprint (2024), 44pp.; arXiv:2409.09656v1

work page arXiv 2024

[13] [13]

P.\ Goddard, A.\ Schwimmer, Factoring out free fermions and superconformal algebras, Phys.\ Lett.\ B 214 (1988) 209--214

work page 1988

[14] [14]

R.\ Heluani, SUSY Vertex Algebras and Supercurves, Comm.\ Math.\ Phys., 275 (2007), 607--658

work page 2007

[15] [15]

R.\ Heluani, V.\ G.\ Kac, Supersymmetric Vertex Algebras, Comm.\ Math.\ Phys., 271 (2007), 103--178

work page 2007

[16] [16]

R.\ Heluani, J.\ Van Ekeren, Characters of topological N=2 vertex algebras are Jacobi forms on the moduli space of elliptic supercurves, Adv.\ Math., 302 (2016), 551--627

work page 2016

[17] [17]

Y.-Z.\ Huang, Differential equations, duality and modular invariance, Comm.\ Contemp.\ Math., 7 (2005), no.\ 5, 649--706

work page 2005

[18] [18]

V.\ G.\ Kac, Lie superalgebras, Adv.\ Math., 26 (1977) 8--96

work page 1977

[19] [19]

Math.\ Soc., Providence, RI, 1998

V.\ G.\ Kac, Vertex Algebras for Beginners, 2nd ed., Univ.\ lect.\ ser., 10, Amer. Math.\ Soc., Providence, RI, 1998

work page 1998

[20] [20]

V.\ G.\ Kac, P.\ M\"oseneder Frajria, P.\ Papi, Unitarity of minimal W-algebras and their representations I, Comm.\ Math.\ Phys., 401 (2023), 79--145

work page 2023

[21] [21]

V.\ G.\ Kac, P.\ M\"oseneder Frajria, P.\ Papi, Extremal unital representations of big N=4 superconformal algebra, preprint (2025), 51pp.; arXiv:2507.10130v1

work page arXiv 2025

[22] [22]

V.\ G.\ Kac, S.-S.\ Roan, M.\ Wakimoto, Quantum Reduction for Affine Superalgebras, Comm.\ Math.\ Phys., 241 (2003), 307--342

work page 2003

[23] [23]

V.\ G.\ Kac, M.\ Wakimoto, Quantum reduction and representation theory of superconformal algebras, Adv.\ Math., 185 (2004), 400--458.; Erratum, Adv.\ Math., 193 (2005), 453--455

work page 2004

[24] [24]

(Eds.), Strings 88, World Scientific, Singapore, 1989, pp

V.\ G.\ Kac, J.\ W.\ van de Leur, On classification of superconformal algebras, In: S.J.\ Gates, et al. (Eds.), Strings 88, World Scientific, Singapore, 1989, pp. 77–106

work page 1989

[25] [25]

J.\ Lepowsky, H.-S.\ Li, Introduction to Vertex Operator Algebras and Their Representations, Progr.\ Math., 227, Birkh\"auser, Boston, 2004

work page 2004

[26] [26]

Y.\ I.\ Manin, Topics in Noncommutative Geometry, Princeton Univ.\ Press., Princeton, NJ, 1991

work page 1991

[27] [27]

A.\ Milas, Tensor product of vertex operator algebras, preprint (1996), 16pp.; arXiv q-alg/9602026v1

work page internal anchor Pith review Pith/arXiv arXiv 1996

[28] [28]

A.\ Milas, Characters, supercharacters and Weber modular functions, J.\ reine angew.\ Math., 608 (2007), 35--64

work page 2007

[29] [29]

Lie Theory and Its Applications in Physics

E.\ Poletaeva, V.\ Serganova, On finite W -algebras for Lie superalgebras in the regular case, In: V. Dobrev (editor) Proceedings of the IX International Workshop “Lie Theory and Its Applications in Physics” (Varna, Bulgaria, 20--26 June 2011). Springer Proceedings in Mathematics and Statistics, Vol. 36 (2013) 487–-497

work page 2011

[30] [30]

E.\ Poletaeva, V.\ Serganova, On Kostant's theorem for the Lie superalgebra Q(n) , Adv.\ Math., 300 (2016), 320--359

work page 2016

[31] [31]

A.\ Premet, Special Transverse Slices and Their Enveloping Algebras, Adv.\ Math., 170(1) (2002), 1--55

work page 2002

[32] [32]

E.\ Witten, Instantons and the large N =4 algebra , J.\ Phys.\ A: Math.\ Theor., 58(3) (2025), 035403, 68pp

work page 2025

[33] [33]

S.\ Yanagida, Li filtrations of SUSY vertex algebras, Lett.\ Math.\ Phys., 112 (2022), article no.\ 103, 77pp

work page 2022

[34] [34]

Y.\ Zeng, B.\ Shu, Finite W-superalgebras for basic Lie superalgebras, J.\ Algebra, 438 (2015), 188--234

work page 2015

[35] [35]

L.\ Zhao, Finite W-superalgebras for queer Lie superalgebras, J.\ Pure Appl.\ Algebra, 218(7) (2014), 1184--1194

work page 2014

[36] [36]

Y.\ Zhu, Modular Invariance of Characters of Vertex Operator Algebras, J.\ Amer.\ Math.\ Soc., 9 (1996), no.\ 1, 237--302

work page 1996