Pith. sign in

REVIEW 3 major objections 5 minor 58 references

Fusion Products of Twisted Modules in Permutation Orbifolds: II

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A single S-matrix formula determines every twisted fusion product in a cyclic permutation orbifold, with arbitrary permutations reduced to the k-cycle case.

desk verdict Real k-cycle formula, but the 'any permutation' claim rests on an unproved reduction. read the letter →

arxiv 2411.15751 v1 pith:2HRDWBBZ submitted 2024-11-24 math.QA

Chongying Dong , Feng Xu , Nina Yu This is my paper · ORCID
classification math.QA MSC 17B6981R10
keywords permutationorbifoldtwistedmodulesfusionrulesS-matrixvertexoperatoralgebraVerlindeformulamodulartensorcategorycyclic
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 determines the fusion product of two twisted modules in the orbifold of a tensor product vertex operator algebra $V^{\otimes k}$ under a cyclic permutation group. For a $k$-cycle $g$ and coprime exponents $s,r$, it proves that the fusion product of an irreducible $g^s$-twisted module and an irreducible $g^r$-twisted module is a finite sum of $g^{s+r}$-twisted modules, with coefficients expressed directly through the entries of the $S$-matrix of $V$. The same coefficient formula is extended to the $k$-fold fusion power of the canonical twisted module $T_g(V)$. Because the paper asserts that every permutation reduces to a product of disjoint cycles, the single-cycle formula is claimed to determine twisted fusion products for any permutation in $S_k$. If the claim is correct, the fusion ring of any cyclic permutation orbifold is computable from the modular data of $V$ alone.

What carries the argument

The carrying object is the category $\mathrm{Rep}(V^{\otimes k})$ of $g$-twisted modules for the cyclic permutation group $G=\langle g\rangle$, viewed as the module category of the commutative algebra $V^{\otimes k}$ inside the modular tensor category of $(V^{\otimes k})^G$-modules. The paper uses the fact that the fusion operators $T_M(N)=M\boxtimes_{V^{\otimes k}}N$ are simultaneously diagonalizable, with eigenvalues given by ratios of $S$-matrix entries $S_{j,i}/S_{0,i}$; applying this to the special modules $T_{g^s}^{0,\dots,0}$ and $T_{g^r}^{0,\dots,0}$ yields the master identity $S_{0,j}^{d+m-k}=\sum n_{t_1,\dots,t_q}S_{t_1,j}\cdots S_{t_q,j}$, which is then inverted by Verlinde-type orthogonality identities (Lemma 4.3). A separate orbit argument (Lemma 4.4) uses $\gcd(s,r)=1$ to move tensor factors between the two twisted modules, allowing the general formula to be assembled from the untwisted-with-twisted fusion products of the earlier paper [DLXY].

What would settle it

A direct check of the reduction step in a small case would settle it: take $k=4$, $s=r=2$, $V=V_{\mathbb{Z}\alpha}$ with $\alpha^2=2$, and compute $T_{g^2}^{0,0}\boxtimes T_{g^2}^{0,0}$ in two ways, once by the disjoint-cycle reduction and once by the known fusion rules of the two-cycle orbifold; a mismatch of coefficients would disprove the general-permutation claim, while exact agreement would support it.

Watch

Extended reading notes

Core claim

The central result is Theorem 4.7: with $g=(1,2,\dots,k)$, $1\le s,r<k$, $\gcd(s,r)=1$, $d=\gcd(s,k)$, $m=\gcd(r,k)$, $q=\gcd(s+r,k)$, and $g^s=\tau_1\cdots\tau_d$, $g^r=\sigma_1\cdots\sigma_m$, $g^{s+r}=\gamma_1\cdots\gamma_q$ as products of disjoint cycles, the fusion product of $T_{g^s}^{i_1,\dots,i_d}$ and $T_{g^r}^{j_1,\dots,j_m}$ equals $$ \sum_{t_1,\dots,t_q,j} \frac{S_{i_1,j}\cdots S_{i_d,j}S_{j_1,j}\cdots S_{j_m,j}S_{t'_1,j}\cdots S_{t'_q,j}}{S_{0,j}^k}\,T_{$g^{{s+r}}$}^{t_1,\dots,t_q}. $$ Here $S$ is the $S$-matrix of the original vertex operator algebra $V$. The proof obtains the coefficients by comparing eigenvalues of fusion operators in the module category of $V^{\otimes k}$ inside the orbifold category, using the permutation-orbifold $S$-matrix entries and Verlinde identities to invert the resulting equations. Section 5 gives the analogous formula for $(T_g(V))^{\boxtimes k}$, and Section 6 works out the lattice example $V=V_{\mathbb{Z}\alpha}$, $k=4$, $g=(1\,2\,3\,4)$ in full.

Load-bearing premise

The load-bearing premise is the unproved reduction in Section 4: when $\gcd(s,r)>1$, the fusion product $T_{\sigma^s}\boxtimes T_{\sigma^r}$ equals the tensor product of the fusion products of the disjoint cycles of $\sigma^{d'}$; the paper's claim that all permutations are covered depends on this factorization.

Editorial extensions

If this is right

  • Every twisted-twisted fusion product in a cyclic permutation orbifold is expressed by one closed formula in terms of the $S$-matrix of $V$ and the gcd combinatorics of $s,r,k$.
  • The general-permutation case is reduced to the single-cycle formula, so the fusion rules of $(V^{\otimes k})^{\langle\sigma\rangle}$ are determined for every $\sigma\in S_k$ once the single-cycle case is known.
  • The $k$-fold fusion power $(T_g(V))^{\boxtimes k}$ is computed as a sum of untwisted $V^{\otimes k}$-modules with coefficients involving $S_{0,a}^{2-2h}$, where $h=(k-1)(k-2)/2$.
  • In the rank-one lattice example with $k=4$, the formulas yield explicit rules such as $T_g^0\boxtimes T_g^0=2T_{g^2}^{0,0}+2T_{g^2}^{1,1}$ and $(T_g^0)^{\boxtimes 4}=8M^{0,0,0,0}+8M^{1,1,0,0}+8M^{0,1,1,0}+8M^{0,0,1,1}+8M^{1,0,1,0}+8M^{0,1,0,1}+8M^{1,0,0,1}+8M^{1,1,1,1}$.

Reading between the lines

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

  • One could test the unproved reduction step directly: for $k=4$, $s=2$, $r=2$ with $V=V_{\mathbb{Z}\alpha}$, compute $T_{g^2}^{0,0}\boxtimes T_{g^2}^{0,0}$ both by the disjoint-cycle reduction and by the known fusion rules of the two-cycle orbifold; agreement would confirm the reduction, disagreement would confine the formula to the coprime case.
  • The same eigenvalue-comparison mechanism may apply to cyclic orbifolds of higher-dimensional lattice vertex operator algebras or minimal models, where the $S$-matrix is known, giving concrete predictions for fusion multiplicities that can be checked numerically.
  • Read as a twisted Verlinde formula, the result suggests that the fusion ring of $\mathrm{Rep}(V^{\otimes k})$ is realised inside a tensor power of the fusion ring of $V$ with the cycle group acting by permutations; making this realisation explicit could yield a presentation of the whole orbifold fusion ring, not only of the twisted sector.
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 / 5 minor

Summary. This paper studies fusion products of twisted modules for permutation orbifolds V^{⊗k} with cyclic automorphism group generated by a permutation g. For a k-cycle g=(1,...,k) and exponents 1≤s,r<k with gcd(s,r)=1, the paper gives an explicit S-matrix formula for the fusion product of an irreducible g^s-twisted module with an irreducible g^r-twisted module (Theorem 4.7). Section 4 claims a reduction from arbitrary permutations to disjoint cycles, Section 5 derives the k-fold fusion power of T_g(V), and Section 6 works out the lattice example (V_{Zα})^{⊗4} with g=(1234).

Significance. If the main formula is correct, it is a substantial contribution: twisted-twisted fusion products in cyclic permutation orbifolds are expressed directly through the S-matrix of V, extending earlier untwisted-twisted results. The k-cycle proof is detailed and the lattice example is consistent, and the derivation is based on modular data rather than fitted to examples. However, the advertised 'any permutation' scope is not currently justified, because the reduction in Section 4 is asserted without proof, and the S-matrix input relies on a correction in Theorem 3.8 whose proof is only sketched.

major comments (3)
  1. [§4 (opening reduction)] The reduction from an arbitrary permutation σ to disjoint cycles is asserted without proof or reference: after writing σ^{d'}=σ_1...σ_s, the paper states T_{σ^s}⊠T_{σ^r} = (T_{σ_1^{s'}}⊠T_{σ_1^{r'}})⊗...⊗(T_{σ_s^{s'}}⊠T_{σ_s^{r'}}). This equality is load-bearing because it is the only step that extends Theorem 4.7 from k-cycles to the 'any permutation' claim of the abstract. It requires justification of (i) tensor-factorization of σ^s-twisted modules over the supports of the cycles of σ^{d'}, (ii) compatibility of the fusion product with such tensor factorization, and (iii) treatment of cycles whose lengths do not satisfy the hypotheses of Theorem 4.7, including cases where s' is congruent to 0 modulo a cycle length. As written, the general claim is conditional on these missing arguments.
  2. [§3.2, Theorem 3.8 (proof of items (1)(iv) and (2)(iv))] The manuscript announces corrections to [DXY4, Theorem 5.4] and states the resulting formulas, but the proof says only that one should 'exclude the conditions d1=f=1' from the earlier proof. No derivation is supplied. These corrected S-matrix entries are used in Lemma 4.1 and then in Lemma 4.2 and Theorems 4.6 and 4.7, so the main formula depends on them. A complete proof, or an exact statement with proof in the cited paper, is needed before the S-matrix formula can be regarded as established.
  3. [§4.2, Theorem 4.7 (scope and hypotheses)] Theorem 4.7 is stated only for 1≤s,r<k with gcd(s,r)=1, but the abstract promises fusion products for any permutation and arbitrary powers. Even for a fixed k-cycle g, the case gcd(s,r)>1 for a single cycle is not proved in the body; Lemma 4.4 uses gcd(s,r)=1 in an essential way, and the manuscript refers back to the asserted reduction in Section 4. Thus the statement that the paper determines fusion products 'for any permutation' overstates what is proved in the body.
minor comments (5)
  1. [§4 (notation)] The symbol s is used both for the exponent of σ and for the number of disjoint cycles in the reduction σ^{d'}=σ_1...σ_s; this makes the opening of Section 4 difficult to follow and should be repaired by renaming one of the two objects.
  2. [§4.1, proof of Theorem 4.6] In the q=1 case, the denominator is written as S^{k-d-m}_{j,0}, while the theorem and Lemma 4.2 use S_{0,j}; since S is symmetric this is harmless, but the notation should be unified for readability.
  3. [§4.2, Remark 4.8] Remark 4.8 states that the coefficients in Theorem 4.7 are non-negative integers, but no proof or reference is given. A one-sentence justification is needed, since non-negativity is not immediately visible from the rational expression in S-matrix entries.
  4. [§6 (example)] The worked example is computed only for n=1 in the lattice vertex operator algebra V_{Zα}; checking at least one case with n>1, where the S-matrix entries are complex phases and cancellations are less trivial, would strengthen the evidence for the formula.
  5. [References] The proof depends on [DRX3, Theorem 7.1] and [DNR, (6.3)], which are cited as arXiv preprints; please update these references to published versions if available, or state the quoted results explicitly in the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central formula is computed from the S-matrix and Verlinde-type identities, not fitted, and the cited prior results do not assume the target fusion products.

full rationale

The derivation chain for the main formula is self-contained once the cited S-matrix and modular-tensor-category results are accepted. Lemma 4.1 records S-matrix entries quoted from [DXY4] and [DRX2]; these are parameter-free expressions in terms of the S-matrix of V and do not presuppose the twisted-twisted fusion products computed in this paper. Lemma 4.2 starts from the assumed expansion T^0_{g^s} ⊠ T^0_{g^r} = sum n T^{...}_{g^{s+r}} and uses the eigenvalue identity from [DRX3, Theorem 7.1] to obtain a linear equation for the unknown multiplicities n. Theorem 4.6 then solves that equation by Verlinde orthogonality and the elementary identities in Lemma 4.3; no coefficient is fitted to the output, and the lattice example in Section 6 is a direct specialization, not a source of fitted parameters. Theorem 4.7 reduces the general twisted-twisted case to Theorem 4.6 using the untwisted-with-twisted fusion products from [DLXY, Theorems 3.9 and 3.10], which are independent of the target formula. The self-citations are numerous, but the load-bearing ones are parameter-free theorems whose stated assumptions do not include the fusion products being derived, so they count as independent support. The only notable gap is the reduction at the start of Section 4 from an arbitrary permutation sigma to disjoint cycles, where the equality T_{sigma^s} ⊠ T_{sigma^r} = (T_{sigma_1^{s'}} ⊠ T_{sigma_1^{r'}}) tensor ... tensor (T_{sigma_s^{s'}} ⊠ T_{sigma_s^{r'}}) is asserted without proof or reference; this is an omitted justification affecting the abstract's 'any permutation' phrasing, but it is not a circular reduction because the equality is not derived from, and does not identify, the target coefficients with an input of the calculation. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted free parameters and no new postulated entities. Its central claim rests on standard hypotheses on V (rationality, C2-cofiniteness, CFT type, self-duality, conformal weight positivity), on the classification and S-matrix results from prior literature, and on the recent algebraic framework of [DRX3] and [DNR]. The heaviest assumptions are the unproved [DRX3]/[DNR] framework and the terse reduction to disjoint cycles for general permutations.

assumptions (6)
  • domain assumption V is a simple, rational, C2-cofinite vertex operator algebra of CFT type, and V is self-dual (V ≅ V')
    Assumed in Section 2.1 and used for the Verlinde formula and modular tensor category structure (Theorems 2.15 and 2.23). The abstract omits self-duality, but the proofs require it.
  • domain assumption The conformal weight of any irreducible g-twisted V-module is nonnegative, and zero if and only if the module is isomorphic to V
    Stated in Section 2.1 after Definition 2.13; needed so the orbifold V^G is rational and C2-cofinite and quantum dimensions are finite.
  • standard math Rep(V) and Rep(V^{⊗k}) are fusion categories, and V^{⊗k} is a commutative associative algebra in the modular tensor category C_{(V^{⊗k})^G}
    Invoked from [H3], [KO], [CKM]; used throughout Section 4 to apply fusion-ring and Verlinde-type reasoning.
  • standard math The S-matrix entries of the permutation orbifold are as stated in Theorem 3.8, including the corrections to cases (1)(iv) and (2)(iv) made in this paper
    Taken from [DXY4] and corrected here; these entries are the numerical input for Lemma 4.1 and Theorem 4.7.
  • domain assumption The algebraic framework of [DRX3, Theorem 7.1] and [DNR, (6.3)] holds, namely that the fusion operators T_j and T_λ can be simultaneously diagonalized with the stated eigenvalues and a common eigenvector basis exists
    Invoked in the proof of Lemma 4.2 without proof; these are recent preprints by overlapping authors and the framework is essential for deriving the coefficient identities.
  • standard math Every irreducible g^s-twisted module of V^{⊗k} with g a k-cycle is of the form T_{τ_1}(M^{i_1}) ⊗ ... ⊗ T_{τ_d}(M^{i_d}) where τ_i are the disjoint cycles of g^s
    From [BDM]; used to parametrize all twisted modules throughout.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fusion Products of Twisted Modules in Permutation Orbifolds: II." pith.science (2026). https://pith.science/paper/2HRDWBBZ

@misc{pith2026241115751,
  author       = {Pith},
  title        = {Pith review of: Fusion Products of Twisted Modules in Permutation Orbifolds: II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2HRDWBBZ}},
  note         = {Machine review of arXiv:2411.15751}
}
abstract

Let $V$ be a simple, rational, $C_{2}$-cofinite vertex operator algebra of CFT type, and let $k$ be a positive integer. In this paper, we determine the fusion products of twisted modules for $V^{\otimes k}$ and $G = \left\langle g \right\rangle$ generated by any permutation $g \in S_{k}$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 53 canonical work pages

  1. [1]

    Abe, C_ 2 -cofiniteness of the 2 -cycle permutation orbifold models of minimal Virasoro vertex operator algebras, Commun

    T. Abe, C_ 2 -cofiniteness of the 2 -cycle permutation orbifold models of minimal Virasoro vertex operator algebras, Commun. Math. Phys. 303 (2011), 825--844

    work page 2011

  2. [2]

    Abe, C_ 2 -cofiniteness of 2-cyclic permutation orbifold models, Commun

    T. Abe, C_ 2 -cofiniteness of 2-cyclic permutation orbifold models, Commun. Math. Phys. 317 (2013), 425--445

    work page 2013

  3. [3]

    T. Abe, G. Buhl, C. Dong, Rationality, Regularity, and C_ 2 -cofiniteness, Trans. Amer. Math. Soc. 356 (2004), 3391--3402

    work page 2004

  4. [4]

    Barron, C

    K. Barron, C. Dong, G. Mason, Twisted sectors for tensor product vertex operator algebras associated to permutation groups, Commun. Math. Phys. 227 (2002), no. 2, 349--384

    work page 2002

  5. [5]

    Bakalov, J

    B. Bakalov, J. Elsinger, V.G. Kac, I. Todorov, Orbifolds of lattice vertex algebras, Jpn. J. Math. 18 (2023), no. 2, 169--274

    work page 2023

  6. [6]

    Barron, Y.-Z

    K. Barron, Y.-Z. Huang, J. Lepowsky, An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras, J. Pure Appl. Algebra 210 (2007), no. 3, 797--826

    work page 2007

  7. [7]

    Bakalov, A

    B. Bakalov, A. Kirillov, Lectures on Tensor Categories and Modular Functors, University Lecture Series, 21, 2000

    work page 2000

  8. [8]

    R. E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Natl. Acad. Sci. USA 83 (1986), 3068--3071

    work page 1986

Show all 58 references
  1. [9]

    Creutzig, S

    T. Creutzig, S. Kanade, R. McRae, Tensor categories for vertex operator superalgebra extensions, Creutzig, Mem. Amer. Math. Soc. 295 (2024), no. 1472, vi+181 pp

    2024

  2. [10]

    Carnahan, M

    S. Carnahan, M. Miyamoto, Regularity of fixed-point vertex operator subalgebras, arXiv:1603.05645

    arXiv

  3. [11]

    Dong, Vertex algebras associated with even lattices, J

    C. Dong, Vertex algebras associated with even lattices, J. Algebra 160 (1993), 245--265

    1993

  4. [12]

    C. Dong, J. Lepowsky, Generalized Vertex Algebras and Relative Vertex Operators, Progress in Math., Vol. 112, Birkhauser, Boston, 1993

    1993

  5. [13]

    C. Dong, X. Lin, Twisted Verlinde formula for vertex operator algebras, arXiv:2310.15563

    arXiv

  6. [14]

    C. Dong, H. Li, G. Mason, Simple currents and extensions of vertex operator algebras, Commun. Math. Phys. 180 (1996), 671--707

    1996

  7. [15]

    C. Dong, H. Li, G. Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), 571--600

    1998

  8. [16]

    C. Dong, H. Li, G. Mason, Modular invariance of trace functions in orbifold theory and generalized moonshine, Commun. Math. Phys. 214 (2000), 1--56

    2000

  9. [17]

    C. Dong, X. Lin, S. Ng, Congruence property in conformal field theory, Algebra Number Theory 9 (2015), 2121--2166

    2015

  10. [18]

    C. Dong, H. Li, F. Xu, N. Yu, Fusion products of twisted modules in permutation orbifolds, Trans. Am. Math. Soc. 377 (2024), 1717--1760

    2024

  11. [19]

    C. Dong, X. Jiao, F. Xu, Quantum dimensions and quantum Galois theory, Trans. Am. Math. Soc. 365 (2013), 6441--6469

    2013

  12. [20]

    C. Dong, R. Ng, L. Ren, Orbifolds and minimal modular extensions, arXiv:2108.05225

    arXiv

  13. [21]

    C. Dong, L. Ren, Congruence property in orbifold theory, Proc. Amer. Math. Soc. 146 (2018), 497--506

    2018

  14. [22]

    C. Dong, L. Ren, F. Xu, On orbifold theory, Adv. Math. 321 (2017), 1--30

    2017

  15. [23]

    C. Dong, L. Ren, F. Xu, S-matrix in orbifold theory, J. Algebra 568 (2021), 139--159

    2021

  16. [24]

    C. Dong, L. Ren, F. Xu, Coset constructions and Kac-Wakimoto Hypothesis, arXiv:2404.00778

    arXiv

  17. [25]

    C. Dong, F. Xu, N. Yu, 2-cyclic permutations of lattice vertex operator algebras, Proc. Amer. Math. Soc. 144 (2016), 3207--3220

    2016

  18. [26]

    C. Dong, F. Xu, N. Yu, 2-permutations of lattice vertex operator algebras: higher rank, J. Algebra 476 (2017), 1--25

    2017

  19. [27]

    C. Dong, F. Xu, N. Yu, The 3-permutation orbifold of a lattice vertex operator algebra, J. Pure Appl. Algebra 222 (2018), no. 6, 1316--1336

    2018

  20. [28]

    C. Dong, F. Xu, N. Yu, S-matrix in permutation orbifolds, J. Algebra 606 (2022), 851--876

    2022

  21. [29]

    Etingof, S

    P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor Categories, 205 Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2015

    2015

  22. [30]

    Etingof, D

    P. Etingof, D. Nikshych, V. Ostrik, On fusion categories, Ann. of Math. 162 (2005), 581--642

    2005

  23. [31]

    Frenkel, Y.-Z

    I. Frenkel, Y.-Z. Huang, J. Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993)

    1993

  24. [32]

    I. B. Frenkel, J. Lepowsky, A. Meurman, Vertex Operator Algebras and the Monster, Pure and Applied Math. 134, Academic Press, Massachusetts, 1988

    1988

  25. [33]

    Gui, Genus-zero permutation-twisted conformal blocks for tensor product vertex operator algebras: The tensor-factorizable case

    B. Gui, Genus-zero permutation-twisted conformal blocks for tensor product vertex operator algebras: The tensor-factorizable case. arXiv:2111.04662

  26. [34]

    Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J

    Y.-Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Algebra 100 (1995), 173--216

    1995

  27. [35]

    Huang, Vertex operator algebras and the Verlinde conjecture, Commun

    Y.-Z. Huang, Vertex operator algebras and the Verlinde conjecture, Commun. Contemp. Math. 10 (2008), no. 1, 103--154

    2008

  28. [36]

    Huang, Rigidity and modularity of vertex tensor categories, Commun

    Y.-Z. Huang, Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10 (2008), suppl. 1, 871--911

    2008

  29. [37]

    Huang, J

    Y.-Z. Huang, J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, I, II, Selecta Mathematica, New Series 1 (1995), 699--756; 757--786

    1995

  30. [38]

    Huang, J

    Y.-Z. Huang, J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, III, J. Pure Appl. Algebra 100 (1995), 141--171

    1995

  31. [39]

    Kirillov Jr., Modular categories and orbifold models II, 2001, arXiv:math/0110221

    A. Kirillov Jr., Modular categories and orbifold models II, 2001, arXiv:math/0110221

    2001 arXiv

  32. [40]

    Kirillov Jr., Modular categories and orbifold models, Commun

    A. Kirillov Jr., Modular categories and orbifold models, Commun. Math. Phys. 229 (2002), 309--335

    2002

  33. [41]

    Karel, H

    M. Karel, H. Li, Certain generating subspaces for vertex operator algebras, J. Algebra 217 (1999), 393--421

    1999

  34. [42]

    V. Kac, R. Longo, F. Xu, Solitons in affine and permutation orbifolds, Commun. Math. Phys. 253 (2005), no. 3, 723--764

    2005

  35. [43]

    Kirillov Jr., V

    A. Kirillov Jr., V. Ostrik, On a q -analogue of the McKay correspondence and the ADE classification of sl_2 conformal field theories, Adv. Math. 171 (2002), 183--227

    2002

  36. [44]

    Lepowsky, Calculus of twisted vertex operators, Proc

    J. Lepowsky, Calculus of twisted vertex operators, Proc. Natl. Acad. Sci. USA 82 (1985), 8295--8299

    1985

  37. [45]

    Li, An analogue of the Hom functor and a generalized nuclear democracy theorem, Duke Math

    H. Li, An analogue of the Hom functor and a generalized nuclear democracy theorem, Duke Math. J. 93 (1998), no. 1, 73--114

    1998

  38. [46]

    Li, Some finiteness properties of regular vertex operator algebras, J

    H. Li, Some finiteness properties of regular vertex operator algebras, J. Algebra 212 (1999), 495--514

    1999

  39. [47]

    Lusztig, Leading coefficients of character values of Hecke algebras, Proc

    G. Lusztig, Leading coefficients of character values of Hecke algebras, Proc. Symp. Pure Math. 47 (1987), 235--262

    1987

  40. [48]

    Miyamoto, A Z _3 -orbifold theory of lattice vertex operator algebra and Z _3 -orbifold constructions, Springer Proc

    M. Miyamoto, A Z _3 -orbifold theory of lattice vertex operator algebra and Z _3 -orbifold constructions, Springer Proc. Math. Stat. 40, 319--344, 2013

    2013

  41. [49]

    Miyamoto, C_2 -cofiniteness of cyclic-orbifold models, Commun

    M. Miyamoto, C_2 -cofiniteness of cyclic-orbifold models, Commun. Math. Phys. 335 (2015), 1279--1286

    2015

  42. [50]

    Milas, M

    A. Milas, M. Penn, C. Sadowski, Permutation orbifolds of Virasoro vertex algebras and W-algebras, J. Algebra 570 (2021), 267--296

    2021

  43. [51]

    Milas, M

    A. Milas, M. Penn, C. Sadowski, S_3 -permutation orbifolds of Virasoro vertex algebras, J. Pure Appl. Algebra 227 (2023), no. 10, Paper No. 107378, 25 pp

    2023

  44. [52]

    Milas, M

    A. Milas, M. Penn, H. Shao, Permutation orbifolds of the Heisenberg vertex algebra H(3) , J. Math. Phys. 60 (2019), no. 2, 021703, 17 pp

    2019

  45. [53]

    Mueger, Modular Categories, arXiv:1201.6593

    M. Mueger, Modular Categories, arXiv:1201.6593

    arXiv

  46. [54]

    Serre, A Course in Arithmetic, Grad

    J.-P. Serre, A Course in Arithmetic, Grad. Texts in Math., vol. 7, Springer-Verlag, 1973

    1973

  47. [55]

    Verlinde, Fusion rules and modular transformation in 2D conformal field theory, Nucl

    E. Verlinde, Fusion rules and modular transformation in 2D conformal field theory, Nucl. Phys. B 300 (1988), 360--376

    1988

  48. [56]

    Xu, Some computations in the cyclic permutations of completely rational nets, Commun

    F. Xu, Some computations in the cyclic permutations of completely rational nets, Commun. Math. Phys. 267 (2006), no. 3, 757--782

    2006

  49. [57]

    Xu, Intertwining operators for twisted modules of a colored vertex operator superalgebra, J

    X. Xu, Intertwining operators for twisted modules of a colored vertex operator superalgebra, J. Algebra 175 (1995), no. 1, 241--273

    1995

  50. [58]

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

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

    1996

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.