Pith. sign in

REVIEW 3 major objections 6 minor 76 references

Tensor structure on the module category of the triplet superalgebra $\mathcal{{SW}}(m)$

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that the module category of the N=1 triplet vertex operator superalgebra SW(m) is rigid, with every dual equal to the contragredient.

desk verdict A serious and likely correct rigidity theorem for SW(m)-mod, but the no-log Fuchsian claim is underproved in the off-diagonal cases. read the letter →

arxiv 2412.20898 v1 pith:DQ6474VP submitted 2024-12-30 math.QA math-phmath.MPmath.RT

Hiromu Nakano This is my paper · ORCID
classification math.QAmath-phmath.MPmath.RT MSC 17B6917B6818M1518M05
keywords tripletvertexoperatorsuperalgebraN=1superconformalalgebralogarithmicDotsenko-FateevintegralsFuchsiandifferentialequationrigidbraidedtensorcategoryfusionringandprojectivecovers
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

This paper proves that the modules of the $N=1$ triplet vertex operator superalgebra $\mathrm{SW}(m)$ form a rigid braided tensor supercategory. In this category the tensor product $\boxtimes$ is not semisimple: the algebra is logarithmic, so projective modules are genuine summands and fusion products can be indecomposable. The main theorem is that every module is rigid and that its categorical dual $M^\vee$ coincides with the contragredient module $M^*$. The proof determines the full fusion structure from one simple module $X_2$: tensoring with $X_2$ shifts the index of simple modules by one, and on projective covers it produces neighbouring projectives (with two extra simple summands at the two ends). This yields explicit socle series for all projective covers and identifies the fusion ring of $\mathrm{SW}(m)$ as a single polynomial quotient generated by $X_2$.

What carries the argument

The load-bearing analytic object is a fourth-order Fuchsian differential equation (4.5), with regular singular points $0,1,\infty$ and Riemann scheme (4.6). Its four solutions are constructed as contour integrals of Dotsenko–Fateev integrals, that is, regularized products of power functions $(x_i)^{a_i}(x_i-1)^{b_i}(x_j-x_k)^{-2\gamma}$; the paper extends the integrals to the integer-parameter case by deforming the free-field parameters by $\epsilon$ and integrating around a small loop around $\epsilon=0$. The decisive output is the connection matrix between the fundamental systems at $z=0$ and $z=1$ (Proposition 4.22). On the categorical side, the $P(1)$-tensor product of vertex tensor supercategory theory turns four-point functions into intertwining-operator pairings, and the rigidity criterion for tensor categories converts the analytic nonvanishing of a certain combination of solutions into the self-duality of $X_2$.

What would settle it

Work with a fixed small $m$ (for instance $m=1$): compute the series expansions of the four integrals $\Psi^{\pm}_{i,j}$ around $z=0$ and $z=1$ to enough order to see whether any $z^{\rho}\log z$ or $(1-z)^{\rho}\log(1-z)$ term appears, and compare the analytically computed connection matrix with Proposition 4.22. If a logarithmic term appears or the connection matrix differs, the proof of Theorem 5.10 would need a different argument.

Watch

Extended reading notes

Core claim

The central assertion is Theorem 5.22: the braided tensor supercategory $(\mathrm{SW}(m)\text{-mod},\boxtimes)$ is rigid, and for every module $M$ its categorical dual $M^\vee$ equals its contragredient $M^*$. The route to this statement is explicit. A distinguished simple module $X_2$ is shown rigid and self-dual: its four-point correlation function satisfies a fourth-order Fuchsian equation, and the connection matrix between the two fundamental systems of Dotsenko–Fateev integral solutions forces the evaluation-and-coevaluation pair for $X_2$ to be nondegenerate. The fusion rules $X_2\boxtimes X_s=X_{s-1}\oplus X_{s+1}$ for $2\le s\le 2m$, together with $X_2\boxtimes X_{2m+1}=P_1$, then show that every simple and projective module is obtained from $X_2$ and is self-dual. A short exact-sequence argument extends rigidity to all indecomposable modules, and the paper records the socle series of all $2m$ projective covers as well as quotient presentations of the non-semisimple fusion ring and the Grothendieck ring.

Load-bearing premise

The load-bearing premise is that the four regularized contour integrals $\Psi^{\pm}_{i,j}$ really give a basis of the solution space of the fourth-order equation at $z=0$ and $z=1$, with the leading powers prescribed by the Riemann scheme and no logarithmic terms; the detailed asymptotics needed for this are written out only for $i=j$, with the off-diagonal cases asserted by the same argument.

Editorial extensions

If this is right

  • Fusion of the simple modules is generated by one object: $X_2\boxtimes X_s=X_{s-1}\oplus X_{s+1}$ for $2\le s\le 2m$, and $X_2\boxtimes X_{2m+1}=P_1$ is the projective cover of $X_{2m}$.
  • Every projective cover $P_s$ has a three-step socle series: a simple bottom layer, a middle layer consisting of two copies of the other simple in the same block, and a simple top layer; in particular all projective modules are rigid and self-dual.
  • Every module in $\mathrm{SW}(m)$-mod is rigid and satisfies $M^\vee=M^*$, so the contragredient construction is the categorical dual throughout the category.
  • The non-semisimple fusion ring is $\mathbb{Z}[X]/\langle U_{4m+1}(X)-2U_{2m}(X)\rangle$ and the Grothendieck ring is $\mathbb{Z}[X]/\langle U_{2m+1}(X)-U_{2m-1}(X)-2\rangle$, where $U_n$ are Chebyshev polynomials and $X=[X_2]$.
  • As an abelian category, $\mathrm{SW}(m)$-mod is equivalent to the category of finite-dimensional modules over the small quantum group $U_q^{\mathrm{small}}(\mathfrak{sl}_2)$ at $q=e^{2\pi i/(2m+1)}$.

Reading between the lines

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

  • The monodromy data in Proposition 4.22 are a natural candidate to compare with the braiding matrices of the small quantum group at the root of unity; if they match, the paper's abelian-category equivalence would upgrade to a braided tensor superequivalence.
  • The same $\epsilon$-deformation plus contour-integral construction should apply to other fourth-order Fuchsian equations coming from superconformal four-point functions: the paper's Remark 4.21 already notes that the explicit coefficients are irrelevant, only the null-vector mechanism and the regularization matter.
  • Self-duality of all simples and the explicit projective socle series make $\mathrm{SW}(m)$-mod a plausible ribbon supercategory; constructing the twist (or proving it cannot exist) is a testable next step that is not addressed in the paper.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies the module category of the N=1 triplet vertex operator superalgebra SW(m). It constructs explicit solutions of a fourth-order Fuchsian differential equation using Dotsenko-Fateev integrals, derives a connection matrix between solutions at z=0 and z=1, and uses these analytic data together with vertex tensor supercategory methods to prove that the simple module X2 is rigid and self-dual. It then derives fusion rules X2⊠Xs = Xs−1 ⊕ Xs+1, determines the socle series of all projective covers, proves that the braided tensor supercategory (SW(m)-mod, ⊠) is rigid with M∨ = M* for every module M, and computes the non-semisimple fusion ring and Grothendieck ring. An abelian equivalence with modules over the small quantum group U^{small}_q(sl2) is also stated as a corollary.

Significance. If the main theorem is correct, this is a meaningful advance: it gives the first complete rigidity statement for the N=1 triplet superalgebra module category and matches the known Wp/quantum-group picture. The analytic construction is the technical heart and is appropriately grounded in Sussman's regularization theory and Tsuchiya-Wood's deformation method; the conformal weights fix the exponents, so no adjustable parameters enter. The paper also gives explicit projective covers and a closed-form fusion ring, which are falsifiable predictions of the category structure. The main weaknesses are omitted verifications in the analytic and structural arguments rather than conceptual errors.

major comments (3)
  1. [§4.4, Propositions 4.17 and 4.20] The proof of Proposition 4.17(2) is written only for i=j, and the off-diagonal cases are dismissed with "the same way". This is a load-bearing omission: Lemma 4.14(2) gives different expansions for i≠j, and the nonvanishing of the residues in (4.34) must be checked for the off-diagonal J±_{i,j} with the parameters appearing there, using the explicit formulas of Remark 4.15(2). The asymptotic statement (4.31) alone does not exclude subleading logarithmic terms, and the Riemann scheme (4.6) has exponent differences 2m−1 and 2m+1 at each singular point, so logarithms are a priori possible. Proposition 4.20's assertion that {Ψ+_{i,j}} and {Ψ−_{i,j}} are fundamental systems with the prescribed exponents and no logarithms is exactly what Theorem 5.10 needs: the inclusions (5.12), the connection formulas (5.16), and the linear-independence contradiction all presuppose that the connection matrix of Proposition 4.22 is complete. Please supply the missing off-diagonal computations or an independent no-log argument for (4.5).
  2. [§3.3, Proposition 3.13] Proposition 3.13 is stated without proof, with the remark that it can be proved "in a similar way" to [AM1, Theorem 4.4]. This proposition is used to justify the block decomposition of SW(m)-mod and is invoked again in Lemma 5.9 and Lemma 5.14. Since the block decomposition is a structural input for the classification of projective covers and for the rigidity arguments, the proof (or a precise reference covering the super case) should be included.
  3. [§5.3, proof of Theorem 5.22] The proof of Theorem 5.22 contains the assertion that, by the structure of the projective modules, every indecomposable module M that is neither simple nor projective fits into an exact sequence 0→L→M→N→0 with L and N direct sums of simple modules. This assertion is not proved or referenced, and it is not automatic for finite-length modules in an abelian category. Since it is the step that extends rigidity from simples and projectives to all modules, it needs a proof, for example by induction using the explicit socle series of the projective covers or by showing the relevant property directly.
minor comments (6)
  1. [Proposition 4.20] The phrase "fumdamental systems" should read "fundamental systems".
  2. [Corollary 5.19] The abelian equivalence with U^{small}_q(sl2)-mod is a strong claim whose proof is omitted; since it is not used for the rigidity theorem, please provide the details or state it as a conjecture or remark.
  3. [Remark 5.28] The reference to "Theorem 5.27" should be to Theorem 5.26.
  4. [Throughout] There are several typos: "arbelian" before Corollary 5.19, "monodoromy" in Remark 4.23, and "defintion" in Definitions 2.2 and 2.3.
  5. [Equation (4.31)] The constants C±_{i,j} are called "some non-zero constants"; since their nonvanishing is part of the assertion being proved, it would be clearer to record that they are explicitly computable from Remark 4.15(2).
  6. [Lemma 5.15] Lemma 5.15 is stated with no proof ("can be proved in the same way"); because it feeds into Proposition 5.17, a short indication of the modified matrix computation would improve verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular dependency: the self-duality of X2 is established from an independent Fuchsian connection computation, not from the rigidity statement.

full rationale

The central claim (Theorem 5.22) rests on Theorem 5.10, whose proof uses the connection matrix of the fourth-order Fuchsian equation (Proposition 4.22). That matrix is obtained by specializing Forrester/Sussman Dotsenko-Fateev identities (Theorems 4.11-4.12, Lemma 4.13), and the characteristic exponents in (4.6) are fixed by known conformal weights h_{r,s}. Nothing is fitted to the fusion rules or to rigidity; the no-log fundamental system (Props 4.17, 4.20) is asserted from residues of the regularized integrals, and although the off-diagonal case is underproved (the proof is written only for i=j), that is a correctness gap rather than a circular reduction: the no-log property is not assumed from the target self-duality. The only self-citation is the introductory remark that some results are 'partially based on our thesis [Nak]', which is not used as the load-bearing argument; the technical load is carried by external results [Su1, Su2, Fo1, Fo2, TW2, CMOY]. Lemmas 5.8/5.11 are representation-theoretic bounds on A0, not imported from the rigidity conclusion. The fusion ring computation (Section 5.4) follows the projective-cover structure already derived, and the abelian equivalence in Corollary 5.19 is explicitly said to be proved like [NT] using those covers; this is an omitted detail, not circularity. No equation of the paper reduces by construction to its own input, so the circularity score is 0.

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

The proof does not fit parameters to data. The fixed inputs are the central charge c1,2m+1, the weights h_{r,s}, and prior structural theorems about SW(m), Fock modules, logarithmic tensor categories, and Dotsenko-Fateev integrals. No free parameters were introduced.

assumptions (7)
  • domain assumption SW(m) is C2-cofinite, has a finite classification of simple modules, and has the stated Zhu algebra (AM3, AM4).
    These facts are quoted from Adamović-Milas and are used to ensure finite length and to organize blocks and projective covers; Section 3.
  • domain assumption The category SW(m)-mod admits a braided tensor supercategory via logarithmic tensor category theory (CGNS, CKM, HLZ).
    The whole tensor product ⊠ and P(w)-intertwining formalism rests on this external theorem; Section 5.
  • domain assumption Dotsenko-Fateev integrals can be regularized and analytically continued as in Sussman (Su1, Su2).
    Used in Section 4.2-4.4 to define the solutions Ψ±_{i,j}; no proof is repeated in this paper.
  • domain assumption Fock module socle series, screening operators and their kernels have the structure given by Iohara-Koga and Tsuchiya-Kanie (IK2, TK).
    These results determine the simple module decompositions and the screening kernels used to define SW(m) and X_s; Sections 2-3.
  • domain assumption Projective covers exist and are finite-dimensional in each block because SW(m) is C2-cofinite (Huang).
    Used to define P_s and to prove the projective cover structures; Sections 5.3-5.4.
  • standard math The singular vector S2,2 and its ǫ-deformation annihilate the relevant lowest weight vectors.
    A direct calculation, partially cited to [BS, IK2]; it is the source of the fourth-order differential equation in Section 4.1 and Proposition 4.19.
  • domain assumption The classification of projective modules over the small quantum group is as stated in Küulshammer, Suter, and Xiao.
    Used only in Corollary 5.19 to identify the abelian category with the small quantum group module category.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Tensor structure on the module category of the triplet superalgebra $\mathcal{{SW}}(m)$." pith.science (2026). https://pith.science/paper/DQ6474VP

@misc{pith2026241220898,
  author       = {Pith},
  title        = {Pith review of: Tensor structure on the module category of the triplet superalgebra $\mathcalSW(m)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQ6474VP}},
  note         = {Machine review of arXiv:2412.20898}
}
abstract

We discuss the tensor structure on the category of modules of the $N=1$ triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced by Adamovi\'{c} and Milas. Based on the theory of vertex tensor supercategories, we determine the structure of fusion products between the simple and projective $\mathcal{SW}(m)$-modules and show that the tensor supercategory on $\mathcal{SW}(m)$-mod is rigid. Technically, explicit solutions of a fourth-order Fuchsian differential equation are important to show the rigidity of $\mathcal{SW}(m)$-modules. We construct solutions of this Fuchsian differential equation using the theory of the Dotsenko-Fateev integrals developed by Sussman.

Figures

Figures reproduced from arXiv: 2412.20898 by the authors.

Figure 4.1
Figure 4.1. The Pochhammer contour arround 1 and 0. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. The open subsets ± i,j and ± i,j . for x, y, z ∈ C. We define l ±,s i,j , l±,t i,j , r ±,s i,j , r ±,t i,j ∈ {−∞, 0, z1, z2, ∞}, ¯l ±,s i,j , ¯l ±,t i,j , r¯ ±,s i,j , r¯ ±,t i,j ∈ {−∞, 0, 1, ∞} as follows [PITH_FULL_IMAGE:figures/full_fig_p028_4_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

76 extracted references · 62 canonical work pages

  1. [1]

    Adamovi\' c and A

    D. Adamovi\' c and A. Milas, ``On the triplet vertex algebra W(p) ,'' Advances in Mathematics 217 (2008) 2664-2699

    work page 2008

  2. [2]

    Adamovi\' c and A

    D. Adamovi\' c and A. Milas, ``The N= 1 triplet vertex operator superalgebras: twisted sector,'' SIGMA. Symmetry, Integrability and Geometry: Methods and Applications, 4 (2008) 087

    work page 2008

  3. [3]

    Adamovi\' c and A

    D. Adamovi\' c and A. Milas, ``The N=1 triplet vertex operator superalgebras,'' Communications in Mathematical Physics, 288 (2009) 225-270

    work page 2009

  4. [4]

    Adamovi\' c and A

    D. Adamovi\' c and A. Milas, ``The structure of Zhu's algebras for certain W `- algebras,'' Advances in Mathematics 227 (2011) 2425-2456

    work page 2011

  5. [5]

    Allen, S

    R. Allen, S. Lentner, C. Schweigert and S. Wood, ``Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson,'' (2021), https://arxiv.org/abs/2107.05718 arXiv:2107.05718

    arXiv 2021

  6. [6]

    K. D. Barron, ``N=1 Neveu-Schwarz vertex operator superalgebras over Grassmann algebras and with odd formal variables,'' (1999), arXiv:math/9910007[math.QA] https://arxiv.org/abs/math/9910007

    arXiv 1999

  7. [7]

    V. A. Belavin, ``On the N= 1 super Liouville four-point functions,'' Nuclear Physics B, 798 (2008) 423-442

    work page 2008

  8. [8]

    A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, ``Infinite conformal symmetry in two-dimensional quantum field theory,'' Nuclear Physics B, 241 (1984) 333-380

    work page 1984

Show all 76 references
  1. [9]

    Benoit and Y

    L. Benoit and Y. Saint-Aubin, ``Fusion and the Neveu-Schwarz singular vectors,'' International Journal of Modern Physics A 9 (1994) 547

    1994

  2. [10]

    Blondeau-Fournier, P

    O. Blondeau-Fournier, P. Mathieu, D. Ridout and S. Wood, ``Superconformal minimal models and admissible Jack polynomials,'' Advances in Mathematics 314 (2017) 71-123

    2017

  3. [11]

    Canagasabey, J

    M. Canagasabey, J. Rasmussen and D. Ridout, ``Fusion rules for the N=1 superconformal logarithmic minimal models I: The Neveu-Schwarz sector,'' Journal of Physics A: Mathematical and Theoretical, 48 (2015) 415402

    2015

  4. [12]

    Creutzig, N

    T. Creutzig, N. Genra, S. Nakatsuka and R. Sato, ``Correspondences of categories for subregular W-algebras and principal W-superalgebras,'' Communications in Mathematical Physics, 393 (2022) 1-60

    2022

  5. [13]

    Creutzig, S

    T. Creutzig, S. Kanade and A. R. Linshaw, ``Simple current extensions beyond semi-simplicity,'' Commun. Contemp. Math. 22 (2020), no. 1, 1950001, 49 pp

    2020

  6. [14]

    Creutzig, S

    T. Creutzig, S. Kanade and R. McRae, ``Tensor categories for vertex operator superalgebra extensions,'' (2017), arXiv:1705.05017[math.QA] https://arxiv.org/abs/1705.05017

    2017 arXiv

  7. [15]

    Creutzig, S

    T. Creutzig, S. Lentner and M. Rupert, ``An algebraic theory for logarithmic Kazhdan-Lusztig correspondences,'' (2023), arXiv:2306.11492[math.QA] https://arxiv.org/abs/2306.11492

    2023 arXiv

  8. [16]

    Creutzig, R

    T. Creutzig, R. McRae and J. Yang, ``On ribbon categories for singlet vertex algebras,'' Communications in Mathematical Physics, 387 (2021) 865-925

    2021

  9. [17]

    Creutzig, R

    T. Creutzig, R. McRae and J. Yang, ``Tensor structure on the Kazhdan–Lusztig category for affine gl (1|1) ,'' International Mathematics Research Notices, 16 (2022) 12462-12515

    2022

  10. [18]

    Creutzig, R

    T. Creutzig, R. McRae, F. Orosz Hunziker and J. Yang, `` N=1 super Virasoro tensor categories,'' (2024), arXiv:2412.18127[math.QA] https://arxiv.org/abs/2412.18127

    2024

  11. [19]

    Creutzig and D

    T. Creutzig and D. Ridout, ``Logarithmic conformal field theory: Beyond an introduction,'' J. Phys. A46 (2013) 494006

    2013

  12. [20]

    V. S. Dotsenko and V. A. Fateev, ``Conformal algebra and multipoint correlation functions in 2D statistical models,'' Nucl. Phys. B240 (1984) 312-348

    1984

  13. [21]

    V. S. Dotsenko and V. A. Fateev, ``Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge c 1 ,'' Nucl. Phys. B251 (1985) 691-734

    1985

  14. [22]

    Etingof, G

    P. Etingof, G. Shlomo, D. Nikshych and V. Ostrik, Tensor Categories. Number volume 205 in Mathematical Surveys and Monographs. American Mathematical Society 2015

    2015

  15. [23]

    B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov and I. Y. Tipunin, ``The Kazhudan-Lusztig correspondence for the representation category of the triplet W -algebra in logarithmic conformal field theories,'' Theoret. and Math. Phys. 148 (2006) 1210-1235

    2006

  16. [24]

    B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov and I. Y. Tipunin, ``Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center,'' Comm. Math. Phys. 265 (2006) 47–93

    2006

  17. [25]

    B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov and I. Y. Tipunin, ``Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT,'' Theor.Math.Phys. 148 (2006) 1210-1235; Teor.Mat.Fiz. 148 (2006) 398-427

    2006

  18. [26]

    B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov and I. Y. Tipunin, ``Logarithmic extensions of minimal models: characters and modular transformation,'' Nucl. Phys. B757 (2006) 303-343

    2006

  19. [27]

    Felder, ``BRST approach to minimal models,'' Nuclear Phys

    G. Felder, ``BRST approach to minimal models,'' Nuclear Phys. B317 (1989) 215-236

    1989

  20. [28]

    Felder and R

    G. Felder and R. Silvotti, ``Conformal blocks of minimal models on a Riemann surface,'' Commun. Math. Phys. 144 (1992) 17-40

    1992

  21. [29]

    Fjelstad, J

    J. Fjelstad, J. Fuchs, S. Hwang, A. M. Semikhatov and I. Y. Tipunin, ``Logarithmic conformal field theories via logarithmic deformation,'' Nucl. Phys. B633 (2002) 379-413

    2002

  22. [30]

    P. J. Forrester, ``Integration formulas and exact calculations in the Calogero–Sutherland model,'' Modern Physics Letters B, 9 (1995) 359-371

    1995

  23. [31]

    P. J. Forrester, Log-gases and random matrices (LMS-34). Princeton University Press 2010

    2010

  24. [32]

    Frenkel, J

    I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the Monster. Academic press 1989

    1989

  25. [33]

    M. R. Gaberdiel and H. G. Kausch, ``Indecomposable fusion products,'' Nucl. Phys. B477 (1996) 293-318

    1996

  26. [34]

    M. R. Gaberdiel and H. G. Kausch, ``A Local Logarithmic Conformal Field Theory,'' Nucl. Phys. B538 (1999) 631-658

    1999

  27. [35]

    Gannon and C

    T. Gannon and C. Negron, ``Quantum SL (2) and logarithmic vertex operator algebras at (p,1) -central charge,'' (2021), arXiv:2104.12821[math.QA] https://arxiv.org/abs/2104.12821

    2021 arXiv

  28. [36]

    Gurarie, ``Logarithmic Operators in Conformal Field Theory,'' Nucl

    V. Gurarie, ``Logarithmic Operators in Conformal Field Theory,'' Nucl. Phys. B410 (1993) 535-549

    1993

  29. [37]

    Haraoka, Linear Differential Equations in the Complex Domain

    Y. Haraoka, Linear Differential Equations in the Complex Domain. Lecture Notes in Mathematics Vol. 2271, Springer, New York 2020

    2020

  30. [38]

    Y. Z. Huang, ``Cofiniteness conditions, projective covers and the logarithmic tensor product theory,'' J. Pure Appl. Algebra, 213 (2009) 458-475

    2009

  31. [39]

    Y. Z. Huang, J. Lepowsky and L. Zhang, ``Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules,'' Proceedings of a Workshop Held at Beijing International Center for ...

    2014

  32. [40]

    Y. Z. Huang, J. Lepowsky and L. Zhang, ``Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I -1pt I: Logarithmic formal calculus and properties of logarithmic intertwining operators,'' (2010), arXiv:1012.4196[math.QA] https://arxiv.org/...

    2010 arXiv

  33. [41]

    Y. Z. Huang, J. Lepowsky and L. Zhang, ``Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, III: Intertwining maps and tensor product bifunctors,'' (2010), arXiv:1012.4197[math.QA] https://arxiv.org/abs/1012.4197

    2010 arXiv

  34. [42]

    Y. Z. Huang, J. Lepowsky and L. Zhang, ``Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, IV: Constructions of tensor product bifunctors and the compatibility conditions,'' (2010), arXiv:1012.4198[math.QA] https://arxiv.org/abs/1012.4198

    2010 arXiv

  35. [43]

    Y. Z. Huang, J. Lepowsky and L. Zhang, ``Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, V: Convergence condition for intertwining maps and the corresponding compatibility condition,'' (2010), arXiv:1012.4199[math.QA] https://arxiv.or...

    2010 arXiv

  36. [44]

    Y. Z. Huang, J. Lepowsky and L. Zhang, ``Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms,'' (2010), arXiv:1012.4202[math....

    2010 arXiv

  37. [45]

    Y. Z. Huang, J. Lepowsky and L. Zhang, ``Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps,'' (2011), arXiv:1110.1929[math.QA] https://arxiv.o...

    2011 arXiv

  38. [46]

    Y. Z. Huang, J. Lepowsky and L. Zhang, ``Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra,'' (2011), arXiv:1110.1931[math.QA] h...

    2011 arXiv

  39. [47]

    Y. Z. Huang and A. Milas, ``Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, I,'' Communications in Contemporary Mathematics, 4 (2002) 327-355

    2002

  40. [48]

    Iohara and Y

    K. Iohara and Y. Koga, ``Representation theory of Neveu-Schwarz and Ramond algebras I: Verma modules,'' Adv. Math., 178 (2003) 1-65

    2003

  41. [49]

    Iohara and Y

    K. Iohara and Y. Koga, ``Representation theory of Neveu-Schwarz and Ramond Algebra I -1pt I: Fock modules,'' Ann. Inst. Fourier, Grenoble, 53, 6 (2003) 1755-1818

    2003

  42. [50]

    M. Jeng, G. Piroux and P. Ruelle, ``Height variables in the Abelian sandpile model: scaling fields and correlations,'' J. Stat. Mech, 0610 (2006) P105

    2006

  43. [51]

    Joyal and R

    A. Joyal and R. Street, ``Braided tensor categories,'' Adv. Math. 102 (1993) 20-78

    1993

  44. [52]

    Kac and W

    V. Kac and W. Wang, ``Vertex Operator Superalgebras and Their,'' Mathematical aspects of conformal and topological field theories and quantum groups, 175, 161 (1994)

    1994

  45. [53]

    Kanade and D

    S. Kanade and D. Ridout, ``NGK and HLZ: Fusion for physicists and mathematicians,'' Affine, Vertex and W-algebras (2019) 135-181

    2019

  46. [54]

    H. G. Kausch, ``Extended conformal algebras generated by multiplet of primary fields,'' Phys. Lett. B, 259 (1991) 448-455

    1991

  47. [55]

    Kazhdan and G

    D. Kazhdan and G. Lusztig, ``Tensor structures arising from affine Lie algebras. IV,'' Journal of the American Mathematical Society, 7 (1994) 383-453

    1994

  48. [56]

    u lshammer, ``Representation type and Auslander-Reiten theory of Frobenius-Lusztig kernels,'' Ph.D. thesis, Christian-Albrechts-Universit\

    J. K\" u lshammer, ``Representation type and Auslander-Reiten theory of Frobenius-Lusztig kernels,'' Ph.D. thesis, Christian-Albrechts-Universit\" a t zu Kiel (2012)

    2012

  49. [57]

    Li, ``The physics superselection principle in vertex operator algebra theory,'' Journal of Algebra 196 (1997), no

    H. Li, ``The physics superselection principle in vertex operator algebra theory,'' Journal of Algebra 196 (1997), no. 2, 436-457

    1997

  50. [58]

    Mathieu and D

    P. Mathieu and D. Ridout, ``From Percolation to Logarithmic Conformal Field Theory,'' Phys. Lett. B657 (2007) 120-129

    2007

  51. [59]

    McRae and J

    R. McRae and J. Yang, ``Structure of Virasoro tensor categories at central charge 13-6p-6p^ -1 for integers p>1 ,'' (2021), arXiv:2011.02170[math.QA] https://arxiv.org/abs/2011.02170

    2021

  52. [60]

    Nagatomo and A

    K. Nagatomo and A. Tsuchiya, ``The Triplet Vertex Operator Algebra W(p) and Restricted Quantum Group at Root of Unity,'' Exploring new structures and natural constructions in mathematical physics. Vol. 61. Mathematical Society of Japan, (2011) 1-50

    2011

  53. [61]

    Nahm, ``Quasirational fusion products,'' Int

    W. Nahm, ``Quasirational fusion products,'' Int. J. Mod. Phys. B8 (1994) 3693-3702

    1994

  54. [62]

    Nakano, ``The category of modules of the triplet W-algebras associated to the Virasoro minimal models,'' the Doctor Thesis, Mathematical Institute, Tohoku University (2023)

    H. Nakano, ``The category of modules of the triplet W-algebras associated to the Virasoro minimal models,'' the Doctor Thesis, Mathematical Institute, Tohoku University (2023)

    2023

  55. [63]

    Nigro, ``Integrals of Motion for Critical Dense Polymers and Symplectic Fermions,'' J

    A. Nigro, ``Integrals of Motion for Critical Dense Polymers and Symplectic Fermions,'' J. Stat. Mech. 0910 (2009) P10007

    2009

  56. [64]

    P. A. Pearce and J. Rasmussen, ``Solvable critical dense polymers,'' J. Stat. Mech. 0702 (2007) P015

    2007

  57. [65]

    Read and H

    N. Read and H. Saleur, ``Associative-algebraic approach to logarithmic conformal field theories,'' Nucl. Phys. B777 (2007) 316-351

    2007

  58. [66]

    Ridout, ``On the Percolation BCFT and the Crossing Probability of Watts,'' Nucl

    D. Ridout, ``On the Percolation BCFT and the Crossing Probability of Watts,'' Nucl. Phys, B810 (2009) 503-526

    2009

  59. [67]

    Ridout and S

    D. Ridout and S. Wood, ``Modular transformations and Verlinde formulae for logarithmic (p_+, p_-) -models,'' Nucl. Phys. B880 (2014) 175-202

    2014

  60. [68]

    Sussman, ``The singularities of Selberg--and Dotsenko--Fateev-like integrals,'' (2023), arXiv:2301.03750[math.QA] https://arxiv.org/abs/2301.03750

    E. Sussman, ``The singularities of Selberg--and Dotsenko--Fateev-like integrals,'' (2023), arXiv:2301.03750[math.QA] https://arxiv.org/abs/2301.03750

    2023 arXiv

  61. [69]

    Sussman, ``The regularization of Dotsenko–Fateev integrals,'' Letters in Mathematical Physics, 113 (2023) 1-29

    E. Sussman, ``The regularization of Dotsenko–Fateev integrals,'' Letters in Mathematical Physics, 113 (2023) 1-29

    2023

  62. [70]

    Suter, ``Modules over U _ q ( sl _2) ,'' Communications in Mathematical Physics, 163 (1994) 359-393

    R. Suter, ``Modules over U _ q ( sl _2) ,'' Communications in Mathematical Physics, 163 (1994) 359-393

    1994

  63. [71]

    Tsuchiya and Y

    A. Tsuchiya and Y. Kanie, ``Fock space representations of the Virasoro algebra - Intertwining operators,'' Publ. RIMS, Kyoto Univ. 22 (1986) 259-327

    1986

  64. [72]

    Tsuchiya and S

    A. Tsuchiya and S. Wood, ``The tensor structure on the representation category of the W _ p triplet algebra,'' J. Phys. A 46 (2013) 445203

    2013

  65. [73]

    Tsuchiya and S

    A. Tsuchiya and S. Wood, ``On the extended W-algebra of type sl_2 at positive rational level,'' International Mathematics Research Notices 2015.14 (2015) 5357-5435

    2015

  66. [74]

    Xiao, ``Restricted representations of U(sl(2)) -quantizations,'' Algebra Colloquium, 1 (1994) 56-66

    J. Xiao, ``Restricted representations of U(sl(2)) -quantizations,'' Algebra Colloquium, 1 (1994) 56-66

    1994

  67. [75]

    X. P. Xu, ``Intertwining operators for twisted modules of a colored vertex operator superalgebra,'' Journal of Algebra, 175 (1995) 241-273

    1995

  68. [76]

    Zhu, ``Modular invariance of characters of of vertex operator algebras,'' J

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

    1996

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.