Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: modular tensor category

Bottom of Page

1 to 14 of 14

- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeJan 31st 2014
- PermaLink
Author: Urs
Format: MarkdownItexthe entry _[[modular tensor category]]_ was lacking (among many things that it is still lacking) some pointers to literature that reviews the relation to QFT. I have added a handful, maybe the best one is this here: * [[Eric Rowell]], _From quantum groups to Unitary modular tensor categories_, Contemporary Mathematics 2005 ([arXiv:math/0503226](http://arxiv.org/abs/math/0503226))
the entry modular tensor category was lacking (among many things that it is still lacking) some pointers to literature that reviews the relation to QFT. I have added a handful, maybe the best one is this here:
- Eric Rowell, From quantum groups to Unitary modular tensor categories, Contemporary Mathematics 2005 (arXiv:math/0503226)
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeJul 16th 2014
- (edited Jul 16th 2014)
- PermaLink
Author: Urs
Format: MarkdownItexI see that the entry [[modular tensor category]] is full of trivial typos in the text. I am too quasi-offline now to do anything about it, though. Maybe tomorrow. That entry could generally use a bit of attention.

I see that the entry modular tensor category is full of trivial typos in the text. I am too quasi-offline now to do anything about it, though. Maybe tomorrow. That entry could generally use a bit of attention.
- CommentRowNumber3.
- CommentAuthorDavid_Corfield
- CommentTimeJul 17th 2014
- PermaLink
Author: David_Corfield
Format: MarkdownItexFixed some typos.

Fixed some typos.
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeJul 17th 2014
- PermaLink
Author: Urs
Format: MarkdownItexThanks!

Thanks!
- CommentRowNumber5.
- CommentAuthorJoe Moeller
- CommentTimeFeb 2nd 2020
- PermaLink
Author: Joe Moeller
Format: TextThe string diagram in the definition appears to be broken.
The string diagram in the definition appears to be broken.
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeMay 23rd 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Colleen Delaney]], *Lecture notes on modular tensor categories and braid group representations*, 2019 ([pdf](http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/modular+tensor+category/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+tensor+category/15">v15</a>, <a href="https://ncatlab.org/nlab/show/modular+tensor+category">current</a>
added pointer to:
- Colleen Delaney, Lecture notes on modular tensor categories and braid group representations, 2019 (pdf)
diff, v15, current
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeMay 23rd 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Eric Rowell]], [[Richard Stong]], [[Zhenghan Wang]], *On classification of modular tensor categories*, Comm. Math. Phys. 292 (2009) no. 2, 343--389 ([arXiv:0712.1377](https://arxiv.org/abs/0712.1377)) <a href="https://ncatlab.org/nlab/revision/diff/modular+tensor+category/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+tensor+category/15">v15</a>, <a href="https://ncatlab.org/nlab/show/modular+tensor+category">current</a>
added pointer to:
- Eric Rowell, Richard Stong, Zhenghan Wang, On classification of modular tensor categories, Comm. Math. Phys. 292 (2009) no. 2, 343–389 (arXiv:0712.1377)
diff, v15, current
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeMay 23rd 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to the original article: * [[Vladimir Turaev]], *Modular categories and 3-manifold invariants*, International Journal of Modern Physics B Vol. 06, No. 11n12, pp. 1807-1824 (1992) ([doi:10.1142/S0217979292000876](https://doi.org/10.1142/S0217979292000876)) <a href="https://ncatlab.org/nlab/revision/diff/modular+tensor+category/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+tensor+category/15">v15</a>, <a href="https://ncatlab.org/nlab/show/modular+tensor+category">current</a>
added pointer to the original article:
- Vladimir Turaev, Modular categories and 3-manifold invariants, International Journal of Modern Physics B Vol. 06, No. 11n12, pp. 1807-1824 (1992) (doi:10.1142/S0217979292000876)
diff, v15, current
- CommentRowNumber9.
- CommentAuthorUrs
- CommentTimeMay 14th 2022
- (edited May 14th 2022)
- PermaLink
Author: Urs
Format: MarkdownItex**Question:** It is readily plausible that > the structure of a braided tensor category (and thus also that of a modular tensor category) on [ the rep categ. of a VOA ] is entirely fixed by the genus zero conformal blocks. (here quoted from p. 36 of Ingo Runkel's "Algebra in Braided Tensor Categories and Conformal Field Theory", where the evident idea is indicated). What would be a good citation of this fact as a theorem with a proof? (I know to sift through the canonical sources on the matter, but maybe somebody knows direct pointer to volume, page and verse where this is citably proven.)

Question:

It is readily plausible that

the structure of a braided tensor category (and thus also that of a modular tensor category) on [ the rep categ. of a VOA ] is entirely fixed by the genus zero conformal blocks.

(here quoted from p. 36 of Ingo Runkel’s “Algebra in Braided Tensor Categories and Conformal Field Theory”, where the evident idea is indicated).

What would be a good citation of this fact as a theorem with a proof?

(I know to sift through the canonical sources on the matter, but maybe somebody knows direct pointer to volume, page and verse where this is citably proven.)
- CommentRowNumber10.
- CommentAuthorUrs
- CommentTimeMay 15th 2022
- PermaLink
Author: Urs
Format: MarkdownItexadded a brief mentioning of this statement to the entry ([here](https://ncatlab.org/nlab/show/modular+tensor+category#RelationToConformalBlocks)) <a href="https://ncatlab.org/nlab/revision/diff/modular+tensor+category/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+tensor+category/16">v16</a>, <a href="https://ncatlab.org/nlab/show/modular+tensor+category">current</a>

added a brief mentioning of this statement to the entry (here)

diff, v16, current
- CommentRowNumber11.
- CommentAuthorUrs
- CommentTimeMay 16th 2022
- PermaLink
Author: Urs
Format: MarkdownItexadded pointers to * [[Yi-Zhi Huang]], *Vertex operator algebras, the Verlinde conjecture and modular tensor categories*, Proc. Nat. Acad. Sci. **102** (2005) 5352-5356 $[$[arXiv:math/0412261](http://arxiv.org/abs/math/0412261), [doi:10.1073/pnas.0409901102](https://doi.org/10.1073/pnas.0409901102)$]$ and * [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], Section 8.27.6 of: *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) $[$[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205)$]$ and used these to slightly expand the paragraph on VOAs ([here](https://ncatlab.org/nlab/show/modular+tensor+category#RepCategoriesOfVOAs)) <a href="https://ncatlab.org/nlab/revision/diff/modular+tensor+category/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+tensor+category/18">v18</a>, <a href="https://ncatlab.org/nlab/show/modular+tensor+category">current</a>
added pointers to
- Yi-Zhi Huang, Vertex operator algebras, the Verlinde conjecture and modular tensor categories, Proc. Nat. Acad. Sci. 102 (2005) 5352-5356 $[$ arXiv:math/0412261, doi:10.1073/pnas.0409901102 $]$
and
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Section 8.27.6 of: Tensor Categories, AMS Mathematical Surveys and Monographs 205 (2015) $[$ ISBN:978-1-4704-3441-0 $]$
and used these to slightly expand the paragraph on VOAs (here)

diff, v18, current
- CommentRowNumber12.
- CommentAuthorUrs
- CommentTimeJun 10th 2022
- PermaLink
Author: Urs
Format: MarkdownItexadded further original references on the construction of MTCs as VOA-representation categories: * [[Yi-Zhi Huang]], *Rigidity and modularity of vertex tensor categories*, Communications in Contemporary Mathematics **10** supp01 (2008) 871-911 $[$[arXiv:math/0502533](https://arxiv.org/abs/math/0502533), [doi:10.1142/S0219199708003083](https://doi.org/10.1142/S0219199708003083)$]$ * [[Yi-Zhi Huang]], [[James Lepowsky]], [[Lin Zhang]], *Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules*, In: Bai, Fuchs, Huang, Kong, Runkel, Schweigert (eds.), *Conformal Field Theories and Tensor Categories* Mathematical Lectures from Peking University. Springer (2014) $[$[arXiv:1012.4193](https://arxiv.org/abs/1012.4193), [doi:10.1007/978-3-642-39383-9_5](https://doi.org/10.1007/978-3-642-39383-9_5)$]$ <a href="https://ncatlab.org/nlab/revision/diff/modular+tensor+category/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+tensor+category/21">v21</a>, <a href="https://ncatlab.org/nlab/show/modular+tensor+category">current</a>
added further original references on the construction of MTCs as VOA-representation categories:
- Yi-Zhi Huang, Rigidity and modularity of vertex tensor categories, Communications in Contemporary Mathematics 10 supp01 (2008) 871-911 $[$ arXiv:math/0502533, doi:10.1142/S0219199708003083 $]$
- Yi-Zhi Huang, James Lepowsky, Lin Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, In: Bai, Fuchs, Huang, Kong, Runkel, Schweigert (eds.), Conformal Field Theories and Tensor Categories Mathematical Lectures from Peking University. Springer (2014) $[$ arXiv:1012.4193, doi:10.1007/978-3-642-39383-9_5 $]$
diff, v21, current
- CommentRowNumber13.
- CommentAuthorUrs
- CommentTimeDec 17th 2022
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Bojko Bakalov]], [[Alexander Kirillov]], *Modular Tensor Categories*, Ch. 3 in: *Lectures on tensor categories and modular functors*, University Lecture Series **21**, Amer. Math. Soc. (2001) [[pdf](https://ncatlab.org/nlab/files/BakalovKirillov-Ch3-ModularTensorCategories.pdf), [web](http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html), [ams:ulect/21](https://bookstore.ams.org/view?ProductCode=ULECT/21)] <a href="https://ncatlab.org/nlab/revision/diff/modular+tensor+category/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+tensor+category/23">v23</a>, <a href="https://ncatlab.org/nlab/show/modular+tensor+category">current</a>
added pointer to:
- Bojko Bakalov, Alexander Kirillov, Modular Tensor Categories, Ch. 3 in: Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001) [pdf, web, ams:ulect/21]
diff, v23, current
- CommentRowNumber14.
- CommentAuthorUrs
- CommentTimeMay 7th 2023
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Orit Davidovich]], [[Tobias Hagge]], [[Zhenghan Wang]], _On Arithmetic Modular Categories_, arXiv preprit, 2013 [[arXiv:1305.2229](https://arxiv.org/abs/1305.2229)] <a href="https://ncatlab.org/nlab/revision/diff/modular+tensor+category/25">diff</a>, <a href="https://ncatlab.org/nlab/revision/modular+tensor+category/25">v25</a>, <a href="https://ncatlab.org/nlab/show/modular+tensor+category">current</a>
added pointer to:
- Orit Davidovich, Tobias Hagge, Zhenghan Wang, On Arithmetic Modular Categories, arXiv preprit, 2013 [arXiv:1305.2229]
diff, v25, current

1 to 14 of 14

nForum

Discussion Feed

Not signed in

Discussion Tag Cloud

nLab > Latest Changes: modular tensor category