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On Kazama-Suzuki Duality between $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$ and $N=2$ Superconformal Vertex Algebra

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Pith's one-line read Kazama–Suzuki duality between the subregular $\mathcal{W}$-algebra $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$ and the $N=2$ superconformal vertex algebra $L^{N=2}_c$ occurs exactly at $k=-1, c=-15$ and $k=-7/3, c=1$; the paper proves…

desk verdict New KS duality at k=-1 is real and well-supported, but the module classification's completeness depends on an unproved highest-weight assertion that needs a direct check. read the letter →

arxiv 2411.08406 v3 pith:AFNM36X3 submitted 2024-11-13 math.QA

Dražen Adamović , Ana Kontrec This is my paper · ORCID
classification math.QA MSC 17B6917B6717B68
keywords vertexalgebrasubregularW-algebraN=2superconformalKazama–Suzukidualitycosetconstructionhighest-weightmoduleclassificationlatticesuperalgebra
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Kazama–Suzuki duality pairs two vertex algebras so that each is the coset (commutant) of the other after tensoring with a lattice vertex superalgebra. This paper classifies every occurrence of such a duality between the $N=2$ superconformal vertex algebra $L^{N=2}_c$ and the subregular $\mathcal{W}$-algebra $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$. The complete answer is that duality holds exactly for $k=-1$ with $c=-15$ and for $k=-7/3$ with $c=1$; explicit embeddings are constructed in both cases. The new case is then used to classify all irreducible highest-weight modules of $\mathcal{W}_{-1}(\mathfrak{sl}_4, f_{\rm sub})$, parameterized by two surfaces $S_1\cup S_2$, with one-dimensional top spaces exactly on $S_1$. A sympathetic reader should care because the duality transfers representation theory from the well-studied $N=2$ side to a less studied $\mathcal{W}$-algebra.

What carries the argument

The argument runs through three mechanisms. First, the definition of Kazama–Suzuki duality itself: injective maps $\varphi_1:V\to U\otimes F_1$ and $\varphi_2:U\to V\otimes F_{-1}$ making each algebra the Heisenberg coset of the other. Second, the classification of candidate levels uses the universal two-parameter vertex algebra $\mathcal{W}(c,\lambda)$ and its truncation-curve criterion for when two coset algebras coincide, combined with the singular-vector criterion that $(G^{+})^s$ is singular in $\mathcal{W}^k(\mathfrak{sl}_n,f_{\rm sub})$ exactly when $i(k+n-1)=s$. Third, for the new duality the explicit formulas for $\Phi$ and $\Phi_{\rm inv}$ are the load-bearing identities; they turn the $N=2$ fields into combinations of $J,L,G^{\pm},W$ and the lattice field, and vice versa. Finally, Zhu-algebra calculations convert the commutator $[G^+,G^-]$ and the field $W$ into the two polynomial equations $g_1=0$ and $g_2=0$ whose zero sets are $S_1$ and $S_2$.

What would settle it

Compute $H(0)$, $T(0)$, and the annihilation conditions $E(n-1/2)$ and $F(n+3/2)$ on $v_{x,y,z}\otimes e^{\varphi^-}$ directly from Proposition 4.4's formulas; if the weights are not exactly the stated $(h,q)$ or some annihilation condition fails, the completeness proof of Theorem 5.8 collapses. A complementary check is to enumerate irreducible highest-weight $\mathcal{W}_{-1}(\mathfrak{sl}_4,f_{\rm sub})$-modules from the Zhu algebra alone and compare with $S_1\cup S_2$.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that the classification in Theorem 3.5 is exhaustive: $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$ and $L^{N=2}_c$ are in Kazama–Suzuki duality if and only if $(k,c)=(-1,-15)$ or $(-7/3,1)$. The previously unknown case, proved in Theorem 4.6, is that $\mathcal{W}_{-1}(\mathfrak{sl}_4, f_{\rm sub})$ is the Kazama–Suzuki dual of $L^{N=2}_{-15}$; both embeddings and inverse embeddings are given explicitly, and the tensor product decomposes into spectral-flow modules. As a consequence, Theorem 5.8 classifies every irreducible highest-weight $\mathcal{W}_{-1}(\mathfrak{sl}_4,f_{\rm sub})$-module $L(x,y,z)$: the complete list is $(x,y,z)\in S_1\cup S_2$, where $S_1$ and $S_2$ are the two explicitly displayed surfaces, and $\dim L(x,y,z)_{\rm top}=1$ exactly on $S_1$ and $2$ on $S_2\setminus S_1$.

Load-bearing premise

The completeness half of the module classification assumes that, for a module whose top space is two-dimensional, the tensor-product vector $v_{x,y,z}\otimes e^{\varphi^-}$ is a highest weight vector for the $N=2$ action with the prescribed weights $(h,q)=(-4x+5,\ y-2x^2+4x-5/2)$; if this spectral-flow identification fails, the list $S_1\cup S_2$ could miss genuine modules or include spurious ones.

Editorial extensions

If this is right

  • Every irreducible highest-weight $\mathcal{W}_{-1}(\mathfrak{sl}_4,f_{\rm sub})$-module is realized as a subquotient of a tensor product $L^{N=2}_{-15}[h,q]\otimes F_1$, so the $N=2$ representation theory controls the $\mathcal{W}$-algebra side.
  • The Heisenberg cosets coincide, giving $\operatorname{Com}(M_J(1),\mathcal{W}_{-1}(\mathfrak{sl}_4,f_{\rm sub})) \cong \operatorname{Com}(M_H(1),L^{N=2}_{-15})$; this is an explicit parafermion/commutant isomorphism.
  • At $k=-7/3$, $c=1$, the duality reduces to the lattice-algebra identification $\mathcal{W}_{-7/3}(\mathfrak{sl}_4,f_{\rm sub})\cong F_4$ and $L^{N=2}_{1}\cong F_3$, so that case is not a source of new module data.
  • Theorem 3.5 excludes every other level and central charge, so no hidden Kazama–Suzuki dualities exist for this pair.
  • The paper conjectures the pattern extends to all $n$: $\mathcal{W}_{4-n}(\mathfrak{sl}(1|n),f_{\rm pr})$ should be Kazama–Suzuki dual to $\mathcal{W}_{-n+1}(\mathfrak{sl}_{n+2},f_{\rm sub})$, with $n=2$ being the new $c=-15$ theorem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step, not taken in the paper, is to promote the spectral-flow decompositions to a functorial equivalence of module categories; if that works, fusion rules and tensor products for the W-algebra could be computed from the N=2 side.
  • The two-surface parametrization makes character formulas directly computable; checking modular invariance of those characters would test whether $\mathcal{W}_{-1}(\mathfrak{sl}_4,f_{\rm sub})$ has a rational or $C_2$-cofinite structure.
  • If the conjecture for general $n$ holds, the truncation-curve method used here should produce an infinite family of dual pairs at negative levels, each with its own module parametrization by algebraic hypersurfaces.
  • The mechanism forcing top dimensions $\le 2$ is the singular vector $(G^+)^2=0$; at levels where a higher power $(G^+)^s$ is singular, analogous dualities with larger superconformal algebras might be found.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper classifies all possible Kazama–Suzuki dualities between the N=2 superconformal vertex algebra L^{N=2}_c and the subregular W-algebra W_k(sl4,f_sub), and claims that such a duality occurs exactly for (k=-1, c=-15) and (k=-7/3, c=1). For the new case c=-15, the authors construct explicit embeddings Φ and Φ_inv between L^{N=2}_{-15} and W_{-1}(sl4,f_sub), establish the corresponding coset identities, and use the duality to propose a complete classification of irreducible highest-weight W_{-1}(sl4,f_sub)-modules, parametrized by two algebraic surfaces S1 and S2. The main tools are the classification of coset coincidences via a two-parameter W-algebra, a singular-vector criterion for (G^+)^2, lattice vertex algebras, and Zhu algebra computations.

Significance. The explicit duality constructed in Propositions 4.1 and 4.4 is a genuine and valuable result: the embeddings are written down in closed form and the N=2 OPEs are verified by direct computation, not by an existence argument. The strategy of using Linshaw's two-parameter W-algebra to reduce coset coincidence to intersection points of truncation curves is clean and effective. The resulting parametrization of irreducible modules by the surfaces S1 and S2 is concrete and falsifiable. However, the module classification in Theorem 5.8 is not yet fully supported: its completeness direction relies on an unverified highest-weight assertion for the vector v_{x,y,z}⊗e^{φ^-}. The k=-7/3 case of the classification theorem is also asserted rather than proved in detail. These gaps are local and likely fixable, but they affect load-bearing claims.

major comments (2)
  1. [5.4, Theorem 5.8] The completeness direction of Theorem 5.8 is not proved. In the case dim L(x,y,z)_top = 2, the proof asserts that w2 = v_{x,y,z} ⊗ e^{φ^-} is a highest weight vector for the L^{N=2}_{c=-15}-action obtained from Φ in Proposition 4.1, with highest weight (h,q) = (-4x+5, y-2x^2+4x-5/2). This requires checking the twisted highest weight conditions E(r)w2 = 0 for r ≥ -1/2 and F(r)w2 = 0 for r ≥ 3/2. Using the explicit formulas for Φ, E(r) is proportional to G^+ ⊗ e^{φ^-} and F(r) is proportional to G^- ⊗ e^{-φ^-}. In particular, E(1/2)w2 receives a contribution proportional to G^+(-1)v_{x,y,z} ⊗ e^{2φ^-} from the e^{φ^-}e^{φ^-} OPE, and F(3/2)w2 receives contributions from G^-(0)v_{x,y,z} through the e^{-φ^-}e^{φ^-} OPE. Neither G^+(-1)v_{x,y,z} nor G^-(0)v_{x,y,z} is forced to vanish by Definition 2.3, so the asserted annihilation requires a nontrivial cancellation or an additional vanishing property. The proof does not provide this verification, nor does it identify the spectral-flow sector of w2 in the decomposition of Proposition 4.1(2). Since the classification of all irreducible modules and the statement 'dim L(x,y,z)_top = 2 iff (x,y,z)∈S2\S1' depend on this step, the completeness half of Theorem 5.8 is currently a gap.
  2. [3.3, Theorem 3.5] The case (k,s) = (-7/3,1) in Theorem 3.5 is disposed of with the sentence that it 'follows easily' from W_{-7/3}(sl4,f_sub) ≅ F4 and L^{N=2}_{c=1} ≅ F3. Because Theorem 1.1 and Theorem 3.5 are iff classification statements, this case is part of the central claim. The definition of Kazama–Suzuki duality requires explicit injective maps φ1: L^{N=2}_{c=1} → F4 ⊗ F_{-1} and φ2: F4 → L^{N=2}_{c=1} ⊗ F1, together with verification of the two coset identities. The cited lattice isomorphisms do not by themselves provide these maps. Please either write the embeddings and Heisenberg subalgebras, or supply a precise reference that proves this specific duality.
minor comments (3)
  1. [4.1, Claim 4.2] The symbol U is used first for the subalgebra of W generated by E,F,T,H,H⊥ and later for the extended algebra ⊕_n U(n); this makes the proof of the decomposition in Proposition 4.1(2) harder to follow. Using different letters for the two objects would improve clarity.
  2. [4.4 and Lemma 5.1] The parafermionic generator W^{N=2}_c and the W-algebra field W are both denoted W in nearby equations, which can confuse the reader. A notation such as W^N for the parafermionic generator would be helpful.
  3. [4.1, Claim 4.2] The sentence 'it is not hard to see that U contains all generators of fW ⊗ F_{-1}' is the key step in proving that the extended algebra exhausts fW ⊗ F_{-1}. Since this identification is used later in the proof of Proposition 4.1(2), a more explicit argument would make the paper more self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Kazama-Suzuki dualities are established by explicit embeddings and external coset criteria, and the module classification follows from those constructions rather than being assumed as input.

full rationale

The paper's derivation chain is self-contained in the sense relevant to circularity. The classification of possible Kazama-Suzuki dualities in Theorem 3.5 uses the external coset-isomorphism criterion of Linshaw [27, Cor. 10.1], with exceptional central charges c = 0, -2 checked separately, and the two surviving candidates (k = -1, c = -15) and (k = -7/3, c = 1) are then shown to be actual dualities by explicit embeddings: Propositions 4.1 and 4.4 give the maps Phi and Phi_inv, and the OPE verifications are carried out directly in the text. No parameter is fitted to the target classification; the condition that the coset algebras coincide is a necessary condition by the definition of Kazama-Suzuki duality, not an assumption of the result. The module classification in Theorem 5.8 is derived from the constructed duality and from Zhu-algebra relations computed from the OPEs in Appendix B, so the sets S1 and S2 are outputs of the analysis rather than inputs. The one caveat is that in the completeness proof of Theorem 5.8 the statement that w2 = v_{x,y,z} tensor e^{phi^-} is a highest weight vector for the N=2 action with weights (-4x+5, y - 2x^2 + 4x - 5/2) is asserted without a displayed annihilation check; this is a potential proof gap, but it is not circular, because the asserted weights are consequences of the already-constructed embedding formulas and not assumptions used to define the classification. Self-citations such as [1] for the original N=2 / affine sl2 duality are background support, supplemented by external results ([21], [27], [11]), and they do not force the new conclusions. No fitted-input-called-prediction, self-definitional reduction, or renamed known result appears.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper depends on standard structural theorems from the vertex algebra literature, including Linshaw's criterion for quotients of W(c,lambda), Fehily's singular vector criterion, and known isomorphisms of lattice vertex algebras. No free parameters are fitted and no new entities are postulated.

assumptions (7)
  • standard math Linshaw's criterion ([27, Corollary 10.1]): simple quotients of the universal two-parameter vertex algebra W(c,lambda) at points with c not equal 0 or -2 are isomorphic only when c and lambda coincide, so coincidences must lie on intersection points of truncation curves.
    Used in Proposition 3.2 to enumerate all possible coincidences between the coset algebras N_s(sl2) and C_k; this is the load-bearing external theorem for the 'if and only if' classification.
  • standard math Fehily's criterion (Proposition 2.1, [16]): (G^+)^s is singular in W^k(sl_n,f_sub) iff i(k+n-1)=s for some i in {1,...,n-1}.
    Used in Lemma 2.2 and Theorem 3.5 to filter the candidate levels.
  • domain assumption Known isomorphisms W_{-7/3}(sl4,f_sub) congruent to F_4 (from [11]) and L^{N=2}_{c=1} congruent to F_3 (from [22]).
    Used to prove Kazama-Suzuki duality for the second candidate pair in Theorem 3.5.
  • domain assumption The universal N=2 vertex algebra V^{N=2}_{c=-15} is simple, because it is the Kazama-Suzuki dual of the simple universal affine vertex algebra V^{-5/3}(sl2) ([1], [21]).
    Used in Proposition 4.1(1) to conclude that the generators E,F,H,T generate exactly L^{N=2}_{c=-15}.
  • domain assumption The parafermion algebra N_{c=-15} is weakly generated by the fields T^perp and W^{N=2}_c, with the given formula for W^{N=2}_c (from [10,15]).
    Used in Proposition 4.4(1) to identify the W-field of W_{-1}(sl4,f_sub) with the parafermionic generator, up to normalization.
  • standard math Standard Zhu algebra formalism and commutator formula (from [30]).
    Used in Appendix A to compute [[G^+],[G^-]] and the projection of W into the Zhu algebra.
  • standard math The OPEs of W^k(sl4,f_sub) (from [17], listed in Appendix B) and of the N=2 superconformal algebra (from [24]).
    Foundation of all OPE verifications in Section 4.

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Pith. "Pith review of On Kazama-Suzuki Duality between $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$ and $N=2$ Superconformal Vertex Algebra." pith.science (2026). https://pith.science/paper/AFNM36X3

@misc{pith2026241108406,
  author       = {Pith},
  title        = {Pith review of: On Kazama-Suzuki Duality between $\mathcalW_k(\mathfraksl_4, f_\rm sub)$ and $N=2$ Superconformal Vertex Algebra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AFNM36X3}},
  note         = {Machine review of arXiv:2411.08406}
}
abstract

We classify all possible occurrences of Kazama-Suzuki duality between the ${N=2}$ superconformal algebra $L^{N=2}_c$ and the subregular $\mathcal{W}$-algebra $\mathcal{W}_{k}(\mathfrak{sl}_4, f_{\rm sub})$. We establish a new Kazama-Suzuki duality between the subregular $\mathcal{W}$-algebra $\mathcal{W}_k(\mathfrak{sl}_4, f_{\rm sub})$ and the $N = 2$ superconformal algebra $L^{N=2}_{c}$ for $c=-15$. As a consequence of the duality, we classify the irreducible $\mathcal{W}_{k=-1}(\mathfrak{sl}_4, f_{\rm sub})$-modules.

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Works this paper leans on

30 extracted references · 20 canonical work pages

  1. [1]

    Adamovi´ c D., Representations of the N = 2 superconformal vertex algebra, Int. Math. Res. Not. 1999 (1999), 61–79, arXiv:math.QA/9809141

    work page arXiv 1999

  2. [2]

    Algebra 239 (2001), 549–572

    Adamovi´ c D., Vertex algebra approach to fusion rules forN = 2 superconformal minimal models, J. Algebra 239 (2001), 549–572

    work page 2001

  3. [3]

    Pure Appl

    Adamovi´ c D., A construction of admissible A(1) 1 -modules of level − 4 3 , J. Pure Appl. Algebra196 (2005), 119–134, arXiv:math.QA/0401023

    work page arXiv 2005

  4. [4]

    Adamovi´ c D., Realizations of simple affine vertex algebras and their modules: the cases dsl(2) and \osp(1, 2), Comm. Math. Phys.366 (2019), 1025–1067, arXiv:1711.11342

    arXiv 2019

  5. [5]

    Bershadsky-Polyakov vertex algebras at positive integer levels and duality

    Adamovi´ c D., Kontrec A., Bershadsky–Polyakov vertex algebras at positive integer levels and duality,Trans- form. Groups 28 (2023), 1325–1355, arXiv:2011.10021

    work page Pith review arXiv 2023

  6. [6]

    Adamovi´ c D., Milas A., Wang Q., On parafermion vertex algebras of sl(2) and sl(3) at level − 3 2 , Commun. Contemp. Math. 24 (2022), 2050086, 23 paages, arXiv:2005.02631

    work page Pith review arXiv 2022

  7. [7]

    Adamovi´ c D., M¨ oseneder Frajria P., Papi P., New approaches for studying conformal embeddings and collapsing levels for W -algebras, Int. Math. Res. Not.2023 (2023), 19431–19475, arXiv:2203.08497

    arXiv 2023

  8. [8]

    Adamovi´ c D., M¨ oseneder Frajria P., Papi P., Perˇ se O., Conformal embeddings in affine vertex superalgebras, Adv. Math. 360 (2020), 106918, 50 pages, arXiv:1903.03794

    work page Pith review arXiv 2020

Show all 30 references
  1. [9]

    Adamovi´ c D., Perˇ se O., Vukorepa I., On the representation theory of the vertex algebra L−5/2(sl(4)), Commun. Contemp. Math.25 (2023), 2150104, 42 pages, arXiv:2103.02985. Kazama–Suzuki Duality between Wk(sl4, fsub) and N = 2 Superconformal Vertex Algebra 23

    2023 arXiv

  2. [10]

    Blumenhagen R., Eholzer W., Honecker A., H¨ ubel R., Hornfeck K., Coset realization of unifyingW algebras, Internat. J. Modern Phys. A10 (1995), 2367–2430, arXiv:hep-th/9406203

    1995 arXiv

  3. [11]

    Creutzig T., Fasquel J., Linshaw A.R., Nakatsuka S., On the structure of W-algebras in type A, Jpn. J. Math. 20 (2025), 1–111, arXiv:2403.08212

    2025 arXiv

  4. [12]

    Creutzig T., Genra N., Nakatsuka S., Duality of subregular W-algebras and principal W-superalgebras, Adv. Math. 383 (2021), 107685, 52 pages, arXiv:2005.10713

    2021 arXiv

  5. [13]

    Math., Vol

    Creutzig T., Linshaw A.R., Cosets of the W k(sl4, fsubreg)-algebra, in Vertex Algebras and Geometry, Con- temp. Math., Vol. 711, American Mathematical Society, Providence, RI, 2018, 105–117, arXiv:1711.11109

    2018 arXiv

  6. [14]

    Creutzig T., Linshaw A.R., Trialities of W-algebras, Camb. J. Math.10 (2022), 69–194, arXiv:2005.10234

    2022 arXiv

  7. [15]

    Algebra 322 (2009), 2366– 2403, arXiv:0809.3630

    Dong C., Lam C.H., Yamada H., W-algebras related to parafermion algebras, J. Algebra 322 (2009), 2366– 2403, arXiv:0809.3630

    2009 arXiv

  8. [16]

    Fehily Z., Subregular W-algebras of type A, Commun. Contemp. Math. 25 (2023), 2250049, 44 pages, arXiv:2111.05536

    2023 arXiv

  9. [17]

    B 698 (2004), 409–449, arXiv:math.QA/0401164

    Feigin B.L., Semikhatov A.M., W (2) n algebras, Nuclear Phys. B 698 (2004), 409–449, arXiv:math.QA/0401164

    2004

  10. [18]

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

    1998 arXiv

  11. [19]

    (N.S.)23 (2017), 2157–2202, arXiv:1606.0096

    Genra N., Screening operators for W-algebras, Selecta Math. (N.S.)23 (2017), 2157–2202, arXiv:1606.0096

    2017

  12. [20]

    Goddard P., Kent A., Olive D., Virasoro algebras and coset space models, Phys. Lett. B152 (1985), 88–92

    1985

  13. [21]

    Gorelik M., Kac V., On simplicity of vacuum modules, Adv. Math. 211 (2007), 621–677, arXiv:math- ph/0606002

    2007

  14. [22]

    Lecture Ser., Vol

    Kac V., Vertex algebras for beginners, 2nd ed., Univ. Lecture Ser., Vol. 10, American Mathematical Society, Providence, RI, 1998

    1998

  15. [23]

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

    2003 arXiv

  16. [24]

    B 321 (1989), 232–268

    Kazama Y., Suzuki H., New N = 2 superconformal field theories and superstring compactification, Nuclear Phys. B 321 (1989), 232–268

    1989

  17. [25]

    Math., Vol

    Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progr. Math., Vol. 227, Birkh¨ auser, Boston, MA, 2004

    2004

  18. [26]

    Li H., Certain extensions of vertex operator algebras of affine type, Comm. Math. Phys.217 (2001), 653–696, arXiv:math.QA/0003038

    2001

  19. [27]

    , N), Compos

    Linshaw A.R., Universal two-parameter W∞-algebra and vertex algebras of type W(2, 3, . . . , N), Compos. Math. 157 (2021), 12–82, arXiv:1710.02275

    2021 arXiv

  20. [28]

    B835 (2010), 314–342, arXiv:1001.3960

    Ridout D., bsl(2)−1/2 and the triplet model, Nuclear Phys. B835 (2010), 314–342, arXiv:1001.3960

    2010 arXiv

  21. [29]

    Wang W., W1+∞ algebra, W3 algebra, and Friedan–Martinec–Shenker bosonization, Comm. Math. Phys. 195 (1998), 95–111, arXiv:q-alg/9708008

    1998 arXiv

  22. [30]

    Zhu Y., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc.9 (1996), 237–302

    1996

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