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arxiv: 2605.19708 · v1 · pith:MM46RTJLnew · submitted 2026-05-19 · 🧮 math.QA · math-ph· math.MP· math.RT

Reduction and inverse-reduction functors I: standard mathsf{V^k}(mathfrak{sl}₂)-modules

Justine Fasquel , Ethan Fursman , David Ridout This is my paper

Pith reviewed 2026-05-20 01:37 UTC · model grok-4.3

classification 🧮 math.QA math-phmath.MPmath.RT
keywords quantum hamiltonian reductioninverse-reduction functorsaffine vertex operator algebrasstandard modulesspectral sequencessl_2representation theory
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The pith

Composing reduction and inverse-reduction functors computes the action of reduction on standard modules of the affine sl_2 vertex-operator algebra.

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

The paper presents a general formalism for composing quantum hamiltonian reduction and inverse-reduction functors. It then applies the composition to the standard modules of the affine vertex-operator algebra V^k(sl_2). This computes how reduction acts on modules that include fully relaxed highest-weight modules and their spectral flows. A sympathetic reader would care because this extends the traditional use of reduction, which was limited to highest-weight modules, to a wider class of representations in vertex algebra theory.

Core claim

By composing the reduction and inverse-reduction functors, the paper computes the action of reduction on the standard modules of V^k(sl_2). A general formalism for the composition is developed and exemplified in this case, with the appearance of unbounded spectral sequences noted as potentially of independent interest.

What carries the argument

The composition of the reduction functor and the inverse-reduction functor, which determines the effect of reduction on fully relaxed modules and spectral flows.

If this is right

  • Reduction maps standard modules to modules whose structure can be read off from the composed functors.
  • Spectral flows of relaxed modules are handled by inserting the inverse-reduction step before reduction.
  • Unbounded spectral sequences arise naturally in the composition for these modules.
  • The formalism works at least for the affine sl_2 case and supplies a template for other algebras.

Where Pith is reading between the lines

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

  • The same composition technique could be tested on affine algebras of higher rank to see whether the spectral sequences remain unbounded.
  • Characters or fusion rules for relaxed modules might be computable by chasing the functors through the composition.
  • Links to logarithmic conformal field theory could be probed by applying the method to modules at rational levels.

Load-bearing premise

The general formalism for composing the functors applies directly to the standard modules of V^k(sl_2) without additional restrictions on the level k or the module parameters.

What would settle it

An explicit calculation for a chosen level k and a concrete standard module showing that the result of the composed functors differs from direct reduction would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.19708 by David Ridout, Ethan Fursman, Justine Fasquel.

Figure 1
Figure 1. Figure 1: The zeroth page 𝐸 Li 0 of Li’s spectral sequence for 𝐶0 . Here the arrows indicate the action of the zeroth differential DLi 0 . We have also shaded green the part of the 𝑝𝑞-plane in which the bigraded subspaces (𝐸 Li 0 ) 𝑝,𝑞 are non-zero. From (3.15), it follows that the modes of 𝐶0 are graded by Li’s filtration as follows: (3.27) GrLi 𝑑e −𝑛 = 𝑛 − 1, GrLi 𝜑 ∗ −𝑛 = 𝑛 − 1. (Of course, we assign zero grade t… view at source ↗
Figure 2
Figure 2. Figure 2: The initial page of the filtered Cartan spectral sequence with arrows indicating the action of the differential D0 . To compute the cohomology, we introduce an increasing filtration 𝐹 • 𝐵 in which 𝐹 𝑝 𝐵 𝑛 is spanned by states of total ghost number 𝑛 with at most 𝑝 modes acting on the vacuum |0⟩. This gives a filtered spectral sequence whose initial page is (B.17) 𝐾 𝑝,𝑞 0 = 𝐹 𝑝 𝐵 𝑝+𝑞 𝐹 𝑝−1 𝐵 𝑝+𝑞 = Gr𝑝 𝐵 𝑝+𝑞… view at source ↗
read the original abstract

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.

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.

Referee Report

1 major / 2 minor

Summary. The paper develops a general formalism for composing quantum Hamiltonian reduction and inverse-reduction functors on modules over vertex operator algebras. It exemplifies the formalism by computing the explicit action of the reduction functor on the standard (relaxed highest-weight) modules of the affine VOA V^k(sl_2), with the composition realized via unbounded spectral sequences in the derived category.

Significance. If the central computation is valid, the work supplies a concrete bridge between highest-weight and relaxed modules via functor composition, which is useful for representation theory of affine VOAs and applications in conformal field theory. The explicit sl_2 example and the general formalism are strengths; the handling of unbounded spectral sequences, if rigorously justified, adds technical interest.

major comments (1)
  1. [General formalism and sl_2 example computation] The central claim that the functor composition computes the action of reduction on standard modules (as stated in the abstract and exemplified in the sl_2 case) depends on the unbounded spectral sequences converging in the appropriate derived category without extra vanishing hypotheses. The manuscript provides no explicit convergence argument or check for generic level k or relaxed highest-weight parameters; this is load-bearing because failure of convergence would mean the computed action does not match the direct reduction functor.
minor comments (2)
  1. [Introduction] The introduction could briefly recall the definition of standard modules and the precise domain of the functors to make the setup self-contained for readers outside the immediate subfield.
  2. [Notation and setup] Notation for the reduction and inverse-reduction functors is introduced but could be summarized in a single table or diagram for quick reference when reading the composition statements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying the need for a more explicit treatment of convergence for the unbounded spectral sequences. This is a substantive point that strengthens the paper, and we address it directly below.

read point-by-point responses

  1. Referee: The central claim that the functor composition computes the action of reduction on standard modules (as stated in the abstract and exemplified in the sl_2 case) depends on the unbounded spectral sequences converging in the appropriate derived category without extra vanishing hypotheses. The manuscript provides no explicit convergence argument or check for generic level k or relaxed highest-weight parameters; this is load-bearing because failure of convergence would mean the computed action does not match the direct reduction functor.

    Authors: We agree that the manuscript would benefit from an explicit convergence argument, particularly for generic k and generic relaxed highest-weight parameters, as this underpins the identification of the composed functor with the direct reduction functor. In the revised version we will add a dedicated subsection (likely in Section 4) that establishes convergence in the derived category. The argument proceeds by showing that the spectral sequence is bounded below in each degree and that, for generic parameters, the only non-vanishing contributions occur in a finite range of filtration degrees; this is verified directly for the standard modules of V^k(sl_2) by using the explicit description of the inverse-reduction functor and the fact that the higher derived functors of reduction vanish outside a controlled range. We will also include a brief remark on the non-generic cases where additional vanishing hypotheses may be needed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper develops a general formalism for composing reduction and inverse-reduction functors in the derived category, then applies this formalism to compute the action on standard modules of V^k(sl_2). No step reduces a claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation by construction. The central computation is presented as following from the new general setup applied to the specific case, with the appearance of unbounded spectral sequences noted as potentially independent. The derivation chain therefore contains independent mathematical content and does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on background results from quantum Hamiltonian reduction and vertex algebra theory; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Standard properties of quantum Hamiltonian reduction functors and their inverses hold in the category of modules over affine vertex operator algebras.
    Invoked when the general formalism is applied to V^k(sl_2).

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Reference graph

Works this paper leans on

77 extracted references · 77 canonical work pages · 26 internal anchors

  1. [1]

    Realizations of simple affine vertex algebras and their modules: the cases $\widehat{sl(2)}$ and $\widehat{osp(1,2)}$

    D Adamovi ´c. Realizations of simple affine vertex algebras and their modules: the cases š𝑠𝑙(2)and œ𝑜𝑠𝑝(1,2).Comm. Math. Phys., 366:1025–1067, 2019.arXiv:1711.11342 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  2. [2]

    Relaxed and logarithmic modules of c𝔰𝔩3.Math

    D Adamovi ´c, T Creutzig and N Genra. Relaxed and logarithmic modules of c𝔰𝔩3.Math. Ann., 389:281–324, 2023.arXiv:2110.15203 [math.RT]. REDUCTION AND INVERSE-REDUCTION FUNCTORS I: STANDARDV k (𝔰𝔩 2 )-MODULES 29

    work page arXiv 2023

  3. [3]

    The vertex algebrasR (𝑝) andV (𝑝) .Comm

    D Adamovi ´c, T Creutzig, N Genra and J Yang. The vertex algebrasR (𝑝) andV (𝑝) .Comm. Math. Phys., 383:1207–1241, 2021. arXiv:2001.08048 [math.RT]

    work page arXiv 2021

  4. [4]

    A realisation of the Bershadsky–Polyakov algebras and their relaxed modules.Lett

    D Adamovi ´c, K Kawasetsu and D Ridout. A realisation of the Bershadsky–Polyakov algebras and their relaxed modules.Lett. Math. Phys., 111, 2021.arXiv:2007.00396 [math.QA]

    work page arXiv 2021

  5. [5]

    Weight module classifications for Bershadsky–Polyakov algebras.Comm

    D Adamovi ´c, K Kawasetsu and D Ridout. Weight module classifications for Bershadsky–Polyakov algebras.Comm. Contemp. Math., 26, 2024.arXiv:2303.03713 [math.QA]

    work page arXiv 2024

  6. [6]

    Vertex operator algebras associated to modular invariant representations for $A_1 ^{(1)}$

    D Adamovi ´c and A Milas. Vertex operator algebras associated to modular invariant representations of𝐴 (1) 1 .Math. Res. Lett., 2:563–575, 1995.arXiv:q-alg/9509025

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  7. [7]

    Logarithmic intertwining operators andW(2,2𝑝−1)algebras.J

    D Adamovi ´c and A Milas. Logarithmic intertwining operators andW(2,2𝑝−1)algebras.J. Math. Phys., 48, 2007. arXiv:math.QA/0702081

    work page arXiv 2007

  8. [8]

    Lattice construction of logarithmic modules for certain vertex algebras

    D Adamovi ´c and A Milas. Lattice construction of logarithmic modules for certain vertex algebras.Selecta Math. New Ser., 15, 2009. arXiv:0902.3417 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  9. [9]

    Vanishing of cohomology associated to quantized Drinfeld–Sokolov reduction.Int

    T Arakawa. Vanishing of cohomology associated to quantized Drinfeld–Sokolov reduction.Int. Math. Res. Not., 2004:729–767, 2004. arXiv:math.QA/0303172

    work page arXiv 2004

  10. [10]

    Representation theory of W-algebras.Invent

    T Arakawa. Representation theory of W-algebras.Invent. Math., 169:219–320, 2007.arXiv:0506056 [math.QA]

    work page 2007

  11. [11]

    Associated varieties of modules over Kac-Moody algebras and $C_2$-cofiniteness of W-algebras

    T Arakawa. Associated varieties of modules over Kac–Moody algebras and𝐶 2-cofiniteness of W-algebras.Int. Math. Res. Not., 2015:11605–11666, 2015.arXiv:1004.1554 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  12. [12]

    Weight representations of affine Kac–Moody algebras and small quantum groups.Adv

    T Arakawa, T Creutzig and K Kawasetsu. Weight representations of affine Kac–Moody algebras and small quantum groups.Adv. Math., 477:110365, 2025.arXiv:2311.10233 [math.RT]

    work page arXiv 2025

  13. [13]

    Quantum Langlands duality of representations ofW-algebras.Contemporary Mathematics, 155:2235–2262, 2019.arXiv:1807.01536 [math.QA]

    T Arakawa and E Frenkel. Quantum Langlands duality of representations ofW-algebras.Contemporary Mathematics, 155:2235–2262, 2019.arXiv:1807.01536 [math.QA]

    work page arXiv 2019

  14. [14]

    Arc spaces and vertex algebras

    T Arakawa and A Moreau. Arc spaces and vertex algebras. Available at https://www.imo.universite-paris-saclay.fr/∼anne.moreau/

    work page

  15. [15]

    Modularity of Relatively Rational Vertex Algebras and Fusion Rules of Principal Affine W-Algebras

    T Arakawa and J van Ekeren. Modularity of relatively rational vertex algebras and fusion rules of principal affine𝑊-algebras.Comm. Math. Phys., 370:205–247, 2019.arXiv:1612.09100 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  16. [16]

    Rationality and fusion rules of exceptional W-algebras.J

    T Arakawa and J van Ekeren. Rationality and fusion rules of exceptional W-algebras.J. Eur. Math. Soc., 25:2763–2813, 2023. arXiv:1905.11473 [math.RT]

    work page arXiv 2023

  17. [17]

    KdV-type equations and W-algebras.Adv

    A Belavin. KdV-type equations and W-algebras.Adv. Stud. Pure Math., 19:117–125, 1989

    work page 1989

  18. [18]

    Representations of a class of lattice type vertex algebras.J

    S Berman, C Dong and S Tan. Representations of a class of lattice type vertex algebras.J. Pure Appl. Algebra, 176:27–47, 2002. arXiv:math.QA/0109215

    work page arXiv 2002

  19. [19]

    Conformal field theories via Hamiltonian reduction.Comm

    M Bershadsky. Conformal field theories via Hamiltonian reduction.Comm. Math. Phys., 139:71–82, 1991

    work page 1991

  20. [20]

    Hidden𝑆𝐿(𝑛)symmetry in conformal field theories.Comm

    M Bershadsky and H Ooguri. Hidden𝑆𝐿(𝑛)symmetry in conformal field theories.Comm. Math. Phys., 126:49–83, 1989

    work page 1989

  21. [21]

    Conditionally convergent spectral sequences.Contemporary Mathematics, 239:49–84, 1999

    J Boardman. Conditionally convergent spectral sequences.Contemporary Mathematics, 239:49–84, 1999

    work page 1999

  22. [22]

    Good grading polytopes.Proc

    J Brundan and S Goodwin. Good grading polytopes.Proc. Lon. Math. Soc., 94:155–180, 2006.arXiv:math.QA/0510205

    work page arXiv 2006

  23. [23]

    Resolving Verlinde’s formula of logarithmic CFT.arXiv:2411.11383 [math.QA]

    T Creutzig. Resolving Verlinde’s formula of logarithmic CFT.arXiv:2411.11383 [math.QA]

    work page arXiv

  24. [24]

    In preparation

    T Creutzig, J Fasquel, N Genra and D Ridout.𝔰𝔩 2|1 minimal models I: classification of irreducible modules. In preparation

    work page

  25. [25]

    On the structure of W-algebras in type A.Jpn

    T Creutzig, J Fasquel, A Linshaw and S Nakatsuka. On the structure of W-algebras in type A.Jpn. J. Math., 20:1–111, 2025. arXiv:2403.08212 [math.RT]

    work page arXiv 2025

  26. [26]

    Unitary and non-unitary $N=2$ minimal models

    T Creutzig, T Liu, D Ridout and S Wood. Unitary and non-unitary𝑁=2 minimal models.J. High Energy Phys., 1906:024, 2019. arXiv:1902.08370 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  27. [27]

    Ribbon categories of weight modules for affine𝔰𝔩 2 at admissible levels.arXiv:2411.11386 [math.QA]

    T Creutzig, R McRae and J Yang. Ribbon categories of weight modules for affine𝔰𝔩 2 at admissible levels.arXiv:2411.11386 [math.QA]

    work page arXiv

  28. [28]

    Modular Data and Verlinde Formulae for Fractional Level WZW Models I

    T Creutzig and D Ridout. Modular data and Verlinde formulae for fractional level WZW models I.Nucl. Phys., B865:83–114, 2012. arXiv:1205.6513 [hep.th]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  29. [29]

    Logarithmic Conformal Field Theory: Beyond an Introduction

    T Creutzig and D Ridout. Logarithmic conformal field theory: beyond an introduction.J. Phys., A46:494006, 2013.arXiv:1303.0847 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  30. [30]

    Modular Data and Verlinde Formulae for Fractional Level WZW Models II

    T Creutzig and D Ridout. Modular data and Verlinde formulae for fractional level WZW models II.Nucl. Phys., B875:423–458, 2013. arXiv:1306.4388 [hep.th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  31. [31]

    Classification of good gradings of simple Lie algebras

    A Elashvili and V Kac. Classification of good gradings of simple Lie algebras. InLie Groups and Invariant Theory, volume 213 ofTrans. Amer. Math. Soc., pages 85–104. American Mathematical Society, Providence, 2005

    work page 2005

  32. [32]

    Connecting affine𝑊-algebras: a case study of𝔰𝔩 4.J

    J Fasquel, Z Fehily, E Fursman and S Nakatsuka. Connecting affine𝑊-algebras: a case study of𝔰𝔩 4.J. Pure Appl. Algebra, 230:108149, 2026.arXiv:2408.13785 [math.QA]

    work page arXiv 2026

  33. [33]

    Quantum Hamiltonian reductions for W-algebras.arXiv:2512.18743 [math.RT]

    J Fasquel and S Nakatsuka. Quantum Hamiltonian reductions for W-algebras.arXiv:2512.18743 [math.RT]

    work page arXiv

  34. [34]

    Orthosymplectic Feigin–Semikhatov duality.Selecta Math

    J Fasquel and S Nakatsuka. Orthosymplectic Feigin–Semikhatov duality.Selecta Math. New Ser., 31:69, 2025.arXiv:2307.14574 [math.RT]. 30 J FASQUEL, E FURSMAN AND D RIDOUT

    work page arXiv 2025

  35. [35]

    Modularity of admissible-level𝔰𝔩 3 minimal models with denominator 2.Comm

    J Fasquel, C Raymond and D Ridout. Modularity of admissible-level𝔰𝔩 3 minimal models with denominator 2.Comm. Math. Phys., 406:279, 2025.arXiv:2406.10646 [math.QA]

    work page arXiv 2025

  36. [36]

    Inverse reduction for hook-type W-algebras.Comm

    Z Fehily. Inverse reduction for hook-type W-algebras.Comm. Math. Phys., 405, 2023.arXiv:2306.14673 [math.QA]

    work page arXiv 2023

  37. [37]

    Subregular W-algebras of type𝐴.Comm

    Z Fehily. Subregular W-algebras of type𝐴.Comm. Contemp. Math., 25, 2023.arXiv:2111.05536 [math.QA]

    work page arXiv 2023

  38. [38]

    The principal W-algebra of𝔭𝔰𝔩 2|2.Symmetry Integrability Geom

    Z Fehily, C Raymond and D Ridout. The principal W-algebra of𝔭𝔰𝔩 2|2.Symmetry Integrability Geom. Methods Appl., 2026 (to appear). arXiv:2509.04795 [math.QA]

    work page arXiv 2026

  39. [39]

    Modularity of Bershadsky–Polyakov minimal models.Lett

    Z Fehily and D Ridout. Modularity of Bershadsky–Polyakov minimal models.Lett. Math. Phys., 112, 2022.arXiv:2110.10336 [math.QA]

    work page arXiv 2022

  40. [40]

    Quantization of the Drinfeld–Sokolov reduction.Phys

    B Feigin and E Frenkel. Quantization of the Drinfeld–Sokolov reduction.Phys. Lett., B246:75–81, 1990

    work page 1990

  41. [41]

    Equivalence between Chain Categories of Representations of Affine sl(2) and N=2 Superconformal Algebras

    B Feigin, A Semikhatov and I Tipunin. Equivalence between chain categories of representations of affine𝔰𝔩(2)and𝑁=2 superconformal algebras.J. Math. Phys., 39:3865–3905, 1998.arXiv:hep-th/9701043

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  42. [42]

    Logarithmic conformal field theories via logarithmic deformations

    J Fjelstad, J Fuchs, J.S Hwang, A.M Semikhatov and I.Yu Tipunin. Logarithmic conformal field theories via logarithmic deformations. Nucl. Phys., B 633:379–413, 2002

    work page 2002

  43. [43]

    Conformal invariance, supersymmetry and string theory.Nucl

    D Friedan, E Martinec and S Shenker. Conformal invariance, supersymmetry and string theory.Nucl. Phys., B271:93–165, 1986

    work page 1986

  44. [44]

    Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras.J

    V Futorny and A Tsylke. Classification of irreducible nonzero level modules with finite-dimensional weight spaces for affine Lie algebras.J. Algebra, 238:426–441, 2001

    work page 2001

  45. [45]

    Fusion rules and logarithmic representations of a WZW model at fractional level

    M Gaberdiel. Fusion rules and logarithmic representations of a WZW model at fractional level.Nucl. Phys., B618:407–436, 2001. arXiv:hep-th/0105046

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  46. [46]

    Non-Compact WZW Conformal Field Theories

    K Gawe ¸dzki. Noncompact WZW conformal field theories.NATO Science Series II: Mathematics, Physics and Chemistry, 295:247–274, 1992.arXiv:hep-th/9110076

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  47. [47]

    Reduction by stages for affine W-algebras.arXiv:2501.04501 [math.RT]

    N Genra and T Juillard. Reduction by stages for affine W-algebras.arXiv:2501.04501 [math.RT]

    work page arXiv

  48. [48]

    Quantum Reduction for Affine Superalgebras

    V Kac, S Roan and M Wakimoto. Quantum reduction for affine superalgebras.Comm. Math. Phys., 241:307–342, 2003. arXiv:math-ph/0302015

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  49. [49]

    Quantum Reduction and Representation Theory of Superconformal Algebras

    V Kac and M Wakimoto. Quantum reduction and representation theory of superconformal algebras.Adv. Math., 185:400–458, 2004. arXiv:math-ph/0304011

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  50. [50]

    Relaxed highest-weight modules I: rank $1$ cases

    K Kawasetsu and D Ridout. Relaxed highest-weight modules I: Rank 1 cases.Comm. Math. Phys., 368:627–663, 2019. arXiv:1803.01989 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  51. [51]

    Relaxed highest-weight modules II: classifications for affine vertex algebras.Comm

    K Kawasetsu and D Ridout. Relaxed highest-weight modules II: classifications for affine vertex algebras.Comm. Contemp. Math., 24:2150037, 2022.arXiv:1906.02935 [math.RT]

    work page arXiv 2022

  52. [52]

    An admissible-level𝔰𝔩 3 model.Lett

    K Kawasetsu, D Ridout and S Wood. An admissible-level𝔰𝔩 3 model.Lett. Math. Phys., 112:96, 2022.arXiv:2107.13204 [math.QA]

    work page arXiv 2022

  53. [53]

    Fusion rules and (sub)modular invariant partition functions in nonunitary theories.Phys

    I Koh and P Sorba. Fusion rules and (sub)modular invariant partition functions in nonunitary theories.Phys. Lett., B215:723–729, 1988

    work page 1988

  54. [54]

    Symmetric invariant bilinear forms on vertex operator algebras.J

    H Li. Symmetric invariant bilinear forms on vertex operator algebras.J. Pure Appl. Algebra, 96:279–297, 1994

    work page 1994

  55. [55]

    The physics superselection principle in vertex operator algebra theory.J

    H Li. The physics superselection principle in vertex operator algebra theory.J. Algebra, 196:436–457, 1997

    work page 1997

  56. [56]

    Abelianizing vertex algebras.Comm

    H Li. Abelianizing vertex algebras.Comm. Math. Phys., 259:391–411, 2005.arXiv:math.QA/0409140

    work page arXiv 2005

  57. [57]

    Secondary Quantum Hamiltonian Reduction

    J Madsen and E Ragoucy. Secondary quantum Hamiltonian reductions.Comm. Math. Phys., 185:509–541, 1997. arXiv:hep-th/9503042

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  58. [58]

    Strings in AdS_3 and the SL(2,R) WZW Model. Part 1: The Spectrum

    J Maldacena and H Ooguri. Strings in𝐴𝑑𝑆 3 and the SL (2, 𝑅) WZW model. Part 1: The spectrum.J. Math. Phys., 42:2929–2960, 2001. arXiv:hep-th/0001053

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  59. [59]

    On rationality for𝐶 2-cofinite vertex operator algebras.arXiv:2108.01898 [math.QA]

    R McRae. On rationality for𝐶 2-cofinite vertex operator algebras.arXiv:2108.01898 [math.QA]

    work page arXiv

  60. [60]

    Fusion rules and rigidity for weight modules over the simple admissible affine𝔰𝔩 (2) andN=2 superconformal vertex operator superalgebras.arXiv:2411.11387 [math.QA]

    H Nakano, F Orosz Hunziker, A Ros Camacho and S Wood. Fusion rules and rigidity for weight modules over the simple admissible affine𝔰𝔩 (2) andN=2 superconformal vertex operator superalgebras.arXiv:2411.11387 [math.QA]

    work page arXiv

  61. [61]

    Quantum geometry of bosonic strings.Phys

    A Polyakov. Quantum geometry of bosonic strings.Phys. Lett. B, 103:207–210, 1981

    work page 1981

  62. [62]

    Quantum geometry of fermionic strings.Phys

    A Polyakov. Quantum geometry of fermionic strings.Phys. Lett. B, 103:211–213, 1981

    work page 1981

  63. [63]

    Gauge transformations and diffeomorphisms.Int

    A Polyakov. Gauge transformations and diffeomorphisms.Int. J. Mod. Phys., A5:833–842, 1990

    work page 1990

  64. [64]

    sl^(2)_{-1/2}: A Case Study

    D Ridout.𝔰𝔩(2) −1/2: A case study.Nucl. Phys., B814:485–521, 2009.arXiv:0810.3532 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  65. [65]

    Fusion in Fractional Level sl^(2)-Theories with k=-1/2

    D Ridout. Fusion in fractional level b𝔰𝔩(2)-theories with𝑘=− 1 2 .Nucl. Phys., B848:216–250, 2011.arXiv:1012.2905 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  66. [66]

    Bosonic Ghosts at $c=2$ as a Logarithmic CFT

    D Ridout and S Wood. Bosonic ghosts at𝑐=2 as a logarithmic CFT.Lett. Math. Phys., 105:279–307, 2015.arXiv:1408.4185 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  67. [67]

    From Jack polynomials to minimal model spectra

    D Ridout and S Wood. From Jack polynomials to minimal model spectra.J. Phys., A48:045201, 2015.arXiv:1409.4847 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  68. [68]

    Relaxed singular vectors, Jack symmetric functions and fractional level $\widehat{\mathfrak{sl}}(2)$ models

    D Ridout and S Wood. Relaxed singular vectors, Jack symmetric functions and fractional level b𝔰𝔩(2)models.Nucl. Phys., B894:621–664, 2015.arXiv:1501.07318 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  69. [69]

    The Verlinde formula in logarithmic CFT

    D Ridout and S Wood. The Verlinde formula in logarithmic CFT.J. Phys. Conf. Ser., 597:012065, 2015.arXiv:1409.0670 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  70. [70]

    Comments on the𝑁=2,3,4 superconformal algebras in two dimensions.Phys

    A Schwimmer and N Seiberg. Comments on the𝑁=2,3,4 superconformal algebras in two dimensions.Phys. Lett. B, 184:191–196, 1987

    work page 1987

  71. [71]

    Inverting the Hamiltonian Reduction in String Theory

    A Semikhatov. Inverting the Hamiltonian reduction in string theory. In28th International Symposium on Particle Theory, Wendisch-Rietz, Germany, pages 156–167, 1994.arXiv:hep-th/9410109. REDUCTION AND INVERSE-REDUCTION FUNCTORS I: STANDARDV k (𝔰𝔩 2 )-MODULES 31

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  72. [72]

    The MFF Singular Vectors in Topological Conformal Theories

    A Semikhatov. The MFF singular vectors in topological conformal theories.Modern Phys. Lett., A9:1867–1896, 1994. arXiv:hep-th/9311180

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  73. [73]

    Representation theory of the principal equivariant affineW-algebra and Langlands duality.arXiv:2510.06990 [math.RT]

    D Simon. Representation theory of the principal equivariant affineW-algebra and Langlands duality.arXiv:2510.06990 [math.RT]

    work page arXiv

  74. [74]

    PhD thesis, University of Melbourne, 2019

    S Siu.Singular vectors for theW 𝑁 algebras and the BRST cohomology for relaxed highest-weight𝐿𝑘 (𝔰𝔩 2 )-modules. PhD thesis, University of Melbourne, 2019

    work page 2019

  75. [75]

    On structure constants and fusion rules in the $SL(2,\BC)/SU(2)$ WZNW model

    J Teschner. On structure constants and fusion rules in the𝑆𝐿(2,ℂ)/𝑆𝑈(2)WZNW model.Nucl. Phys., B546:390–422, 1999. arXiv:hep-th/9712256

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  76. [76]

    Rationality of Virasoro vertex operator algebras.Int

    W Wang. Rationality of Virasoro vertex operator algebras.Int. Math. Res. Not., 1993:197–211, 1993

    work page 1993

  77. [77]

    Cambridge University Press, Cambridge, 1994

    C Weibel.An Introduction to Homological Algebra, volume 38 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1994. (Justine Fasquel)Universit ´e Bourgogne Europe, CNRS, IMB UMR 5584, 21000 Dijon, France. Email address:justine.fasquel@u-bourgogne.fr (Ethan Fursman)School of Mathematics and Statistics, University of Melbour...

    work page 1994