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]

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$]$

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)

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

]]>**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.)

]]>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)

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)

added pointer to:

- Colleen Delaney,
*Lecture notes on modular tensor categories and braid group representations*, 2019 (pdf)

Thanks!

]]>Fixed some typos.

]]>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.

]]>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)