AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Tensor Categories for Vertex Operator Superalgebra Extensions
About this Title
Thomas Creutzig, Shashank Kanade and Robert McRae
Publication: Memoirs of the American Mathematical Society
Publication Year:
2024; Volume 295, Number 1472
ISBNs: 978-1-4704-6724-1 (print); 978-1-4704-7772-1 (online)
DOI: https://doi.org/10.1090/memo/1472
Published electronically: March 19, 2024
Keywords: Vertex operator superalgebras,
braided tensor categories,
commutative algebra objects,
Verlinde formula,
coset vertex operator algebras
Table of Contents
Chapters
- 1. Introduction
- 2. Tensor Categories and Supercategories
- 3. Vertex Tensor Categories
- 4. Applications
Abstract
Let $V$ be a vertex operator algebra with a category $\mathcal {C}$ of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let $A$ be a vertex operator (super)algebra extension of $V$. We employ tensor categories to study untwisted (also called local) $A$-modules in $\mathcal {C}$, using results of Huang-Kirillov-Lepowsky that show that $A$ is a (super)algebra object in $\mathcal {C}$ and that generalized $A$-modules in $\mathcal {C}$ correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a $\mathcal {C}$-algebra and (under suitable conditions) of generalized $A$-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang.
Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of $V$-modules to $A$-modules is a vertex tensor functor. Two applications are given:
First, we derive Verlinde formulae for regular vertex operator superalgebras and regular $\frac {1}{2}\mathbb {Z}$-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and $\mathbb {Z}$-graded components, respectively.
Second, we analyze parafermionic cosets $C=\mathrm {Com}(V_L,V)$ where $L$ is a positive definite even lattice and $V$ is regular. If the vertex tensor category of either $V$-modules or $C$-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.
- Toshiyuki Abe, Geoffrey Buhl, and Chongying Dong, Rationality, regularity, and $C_2$-cofiniteness, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3391–3402. MR 2052955, DOI 10.1090/S0002-9947-03-03413-5
- Toshiyuki Abe, Chongying Dong, and Haisheng Li, Fusion rules for the vertex operator algebra $M(1)$ and $V^+_L$, Comm. Math. Phys. 253 (2005), no. 1, 171–219. MR 2105641, DOI 10.1007/s00220-004-1132-5
- Dražen Adamović, Representations of the $N=2$ superconformal vertex algebra, Internat. Math. Res. Notices 2 (1999), 61–79. MR 1670180, DOI 10.1155/S1073792899000033
- Dražen Adamović, Vertex algebra approach to fusion rules for $N=2$ superconformal minimal models, J. Algebra 239 (2001), no. 2, 549–572. MR 1832905, DOI 10.1006/jabr.2000.8728
- Dražen Adamović, Thomas Creutzig, Naoki Genra, and Jinwei Yang, The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$, Comm. Math. Phys. 383 (2021), no. 2, 1207–1241. MR 4239841, DOI 10.1007/s00220-021-03950-1
- Dražen Adamović and Antun Milas, Vertex operator algebras associated to modular invariant representations for $A^{(1)}_1$, Math. Res. Lett. 2 (1995), no. 5, 563–575. MR 1359963, DOI 10.4310/MRL.1995.v2.n5.a4
- Dražen Adamović and Antun Milas, On the triplet vertex algebra $\scr W(p)$, Adv. Math. 217 (2008), no. 6, 2664–2699. MR 2397463, DOI 10.1016/j.aim.2007.11.012
- Dražen Adamović and Antun Milas, Lattice construction of logarithmic modules for certain vertex algebras, Selecta Math. (N.S.) 15 (2009), no. 4, 535–561. MR 2565050, DOI 10.1007/s00029-009-0009-z
- Dražen Adamović and Antun Milas, On $W$-algebras associated to $(2,p)$ minimal models and their representations, Int. Math. Res. Not. IMRN 20 (2010), 3896–3934. MR 2738346, DOI 10.1093/imrn/rnq016
- Chunrui Ai, Chongying Dong, Xiangyu Jiao, and Li Ren, The irreducible modules and fusion rules for the parafermion vertex operator algebras, Trans. Amer. Math. Soc. 370 (2018), no. 8, 5963–5981. MR 3812115, DOI 10.1090/tran/7302
- Claudia Alfes and Thomas Creutzig, The mock modular data of a family of superalgebras, Proc. Amer. Math. Soc. 142 (2014), no. 7, 2265–2280. MR 3195752, DOI 10.1090/S0002-9939-2014-11959-9
- Tomoyuki Arakawa, Representation theory of $\scr W$-algebras, Invent. Math. 169 (2007), no. 2, 219–320. MR 2318558, DOI 10.1007/s00222-007-0046-1
- Tomoyuki Arakawa, Rationality of Bershadsky-Polyakov vertex algebras, Comm. Math. Phys. 323 (2013), no. 2, 627–633. MR 3096533, DOI 10.1007/s00220-013-1780-4
- Tomoyuki Arakawa, Thomas Creutzig, and Boris Feigin, Urod algebras and translation of W-algebras, Forum Math. Sigma 10 (2022), Paper No. e33, 31. MR 4436591, DOI 10.1017/fms.2022.15
- Tomoyuki Arakawa, Thomas Creutzig, Kazuya Kawasetsu, and Andrew R. Linshaw, Orbifolds and cosets of minimal $\mathcal {W}$-algebras, Comm. Math. Phys. 355 (2017), no. 1, 339–372. MR 3670736, DOI 10.1007/s00220-017-2901-2
- Tomoyuki Arakawa, Thomas Creutzig, and Andrew R. Linshaw, Cosets of Bershadsky-Polyakov algebras and rational $\mathcal W$-algebras of type $A$, Selecta Math. (N.S.) 23 (2017), no. 4, 2369–2395. MR 3703456, DOI 10.1007/s00029-017-0340-8
- Tomoyuki Arakawa, Thomas Creutzig, and Andrew R. Linshaw, $W$-algebras as coset vertex algebras, Invent. Math. 218 (2019), no. 1, 145–195. MR 3994588, DOI 10.1007/s00222-019-00884-3
- Tomoyuki Arakawa, Ching Hung Lam, and Hiromichi Yamada, Zhu’s algebra, $C_2$-algebra and $C_2$-cofiniteness of parafermion vertex operator algebras, Adv. Math. 264 (2014), 261–295. MR 3250285, DOI 10.1016/j.aim.2014.07.021
- Jean Auger, Thomas Creutzig, and David Ridout, Modularity of logarithmic parafermion vertex algebras, Lett. Math. Phys. 108 (2018), no. 12, 2543–2587. MR 3865748, DOI 10.1007/s11005-018-1098-4
- Bojko Bakalov and Alexander Kirillov Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21, American Mathematical Society, Providence, RI, 2001. MR 1797619, DOI 10.1090/ulect/021
- Amy Barker, David Swinarski, Lauren Vogelstein, and John Wu, A new proof of a formula for the type $A_2$ fusion rules, J. Math. Phys. 56 (2015), no. 1, 011703, 10. MR 3390816, DOI 10.1063/1.4905794
- Michael Bershadsky, Conformal field theories via Hamiltonian reduction, Comm. Math. Phys. 139 (1991), no. 1, 71–82. MR 1116410
- Marcel Bischoff, Yasuyuki Kawahigashi, Roberto Longo, and Karl-Henning Rehren, Tensor categories and endomorphisms of von Neumann algebras—with applications to quantum field theory, SpringerBriefs in Mathematical Physics, vol. 3, Springer, Cham, 2015. MR 3308880, DOI 10.1007/978-3-319-14301-9
- J. Böckenhauer and D. E. Evans, Modular invariants, graphs and $\alpha$-induction for nets of subfactors. I, Comm. Math. Phys. 197 (1998), no. 2, 361–386. MR 1652746, DOI 10.1007/s002200050455
- Jens Böckenhauer and David E. Evans, Modular invariants from subfactors, Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000) Contemp. Math., vol. 294, Amer. Math. Soc., Providence, RI, 2002, pp. 95–131. MR 1907187, DOI 10.1090/conm/294/04971
- Jens Böckenhauer, David E. Evans, and Yasuyuki Kawahigashi, On $\alpha$-induction, chiral generators and modular invariants for subfactors, Comm. Math. Phys. 208 (1999), no. 2, 429–487. MR 1729094, DOI 10.1007/s002200050765
- Jonathan Brundan and Alexander P. Ellis, Monoidal supercategories, Comm. Math. Phys. 351 (2017), no. 3, 1045–1089. MR 3623246, DOI 10.1007/s00220-017-2850-9
- Scott Carnahan, Building vertex algebras from parts, Comm. Math. Phys. 373 (2020), no. 1, 1–43. MR 4050091, DOI 10.1007/s00220-019-03607-0
- S. Carnahan and M. Miyamoto, Regularity of fixed-point vertex operator algebras, arXiv:1603.05645.
- Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo, and Mihály Weiner, From vertex operator algebras to conformal nets and back, Mem. Amer. Math. Soc. 254 (2018), no. 1213, vi+85. MR 3796433, DOI 10.1090/memo/1213
- Ling Chen, On axiomatic approaches to intertwining operator algebras, Commun. Contemp. Math. 18 (2016), no. 4, 1550051, 62. MR 3493212, DOI 10.1142/S0219199715500510
- Thomas Creutzig, W-algebras for Argyres-Douglas theories, Eur. J. Math. 3 (2017), no. 3, 659–690. MR 3687436, DOI 10.1007/s40879-017-0156-2
- Thomas Creutzig, Fusion categories for affine vertex algebras at admissible levels, Selecta Math. (N.S.) 25 (2019), no. 2, Paper No. 27, 21. MR 3932636, DOI 10.1007/s00029-019-0479-6
- Thomas Creutzig, John F. R. Duncan, and Wolfgang Riedler, Self-dual vertex operator superalgebras and superconformal field theory, J. Phys. A 51 (2018), no. 3, 034001, 29. MR 3741987, DOI 10.1088/1751-8121/aa9af5
- Thomas Creutzig, Boris Feigin, and Andrew R. Linshaw, $N=4$ superconformal algebras and diagonal cosets, Int. Math. Res. Not. IMRN 3 (2022), 2180–2223. [Initially appeared as 2021, no. 24, 18768–18811]. MR 4366001, DOI 10.1093/imrn/rnaa078
- Thomas Creutzig, Jesse Frohlich, and Shashank Kanade, Representation theory of $L_k(\mathfrak {osp}(1\vert 2))$ from vertex tensor categories and Jacobi forms, Proc. Amer. Math. Soc. 146 (2018), no. 11, 4571–4589. MR 3856129, DOI 10.1090/proc/14066
- Thomas Creutzig and Davide Gaiotto, Vertex algebras for S-duality, Comm. Math. Phys. 379 (2020), no. 3, 785–845. MR 4163353, DOI 10.1007/s00220-020-03870-6
- Thomas Creutzig and Terry Gannon, Logarithmic conformal field theory, log-modular tensor categories and modular forms, J. Phys. A 50 (2017), no. 40, 404004, 37. MR 3708086, DOI 10.1088/1751-8121/aa8538
- Thomas Creutzig, Azat M. Gainutdinov, and Ingo Runkel, A quasi-Hopf algebra for the triplet vertex operator algebra, Commun. Contemp. Math. 22 (2020), no. 3, 1950024, 71. MR 4082225, DOI 10.1142/S021919971950024X
- Thomas Creutzig, Naoki Genra, and Shigenori Nakatsuka, Duality of subregular $\mathcal W$-algebras and principal $\mathcal W$-superalgebras, Adv. Math. 383 (2021), Paper No. 107685, 52. MR 4232554, DOI 10.1016/j.aim.2021.107685
- Thomas Creutzig, Yi-Zhi Huang, and Jinwei Yang, Braided tensor categories of admissible modules for affine Lie algebras, Comm. Math. Phys. 362 (2018), no. 3, 827–854. MR 3845289, DOI 10.1007/s00220-018-3217-6
- Thomas Creutzig, Shashank Kanade, and Andrew R. Linshaw, Simple current extensions beyond semi-simplicity, Commun. Contemp. Math. 22 (2020), no. 1, 1950001, 49. MR 4064909, DOI 10.1142/S0219199719500019
- T. Creutzig, S. Kanade, A. R. Linshaw, and D. Ridout, Schur-Weyl duality for Heisenberg cosets, Transform. Groups 24 (2019), no. 2, 301–354. MR 3948937, DOI 10.1007/s00031-018-9497-2
- Thomas Creutzig, Shashank Kanade, and Robert McRae, Gluing vertex algebras, Adv. Math. 396 (2022), Paper No. 108174, 72. MR 4362778, DOI 10.1016/j.aim.2021.108174
- Thomas Creutzig and Andrew R. Linshaw, The super $\mathcal {W}_{1+\infty }$ algebra with integral central charge, Trans. Amer. Math. Soc. 367 (2015), no. 8, 5521–5551. MR 3347182, DOI 10.1090/S0002-9947-2015-06214-X
- Thomas Creutzig and Andrew R. Linshaw, Cosets of affine vertex algebras inside larger structures, J. Algebra 517 (2019), 396–438. MR 3869280, DOI 10.1016/j.jalgebra.2018.10.007
- Thomas Creutzig and Andrew R. Linshaw, Trialities of $\mathcal W$-algebras, Camb. J. Math. 10 (2022), no. 1, 69–194. MR 4445343
- Thomas Creutzig, Robert McRae, and Jinwei Yang, On ribbon categories for singlet vertex algebras, Comm. Math. Phys. 387 (2021), no. 2, 865–925. MR 4315663, DOI 10.1007/s00220-021-04097-9
- Thomas Creutzig, Robert McRae, and Jinwei Yang, Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak {gl}(1|1)$, Int. Math. Res. Not. IMRN 16 (2022), 12462–12515. MR 4466006, DOI 10.1093/imrn/rnab080
- Thomas Creutzig and Antun Milas, False theta functions and the Verlinde formula, Adv. Math. 262 (2014), 520–545. MR 3228436, DOI 10.1016/j.aim.2014.05.018
- Thomas Creutzig, Antun Milas, and Matt Rupert, Logarithmic link invariants of $\overline U_q^H(\mathfrak {sl}_2)$ and asymptotic dimensions of singlet vertex algebras, J. Pure Appl. Algebra 222 (2018), no. 10, 3224–3247. MR 3795642, DOI 10.1016/j.jpaa.2017.12.004
- T. Creutzig, T. Quella, and V. Schomerus, Branes in the $\rm GL(1|1)$ WZNW model, Nuclear Phys. B 792 (2008), no. 3, 257–283. MR 2387340, DOI 10.1016/j.nuclphysb.2007.09.014
- Thomas Creutzig and David Ridout, Modular data and Verlinde formulae for fractional level WZW models I, Nuclear Phys. B 865 (2012), no. 1, 83–114. MR 2968504, DOI 10.1016/j.nuclphysb.2012.07.018
- Thomas Creutzig and David Ridout, Relating the archetypes of logarithmic conformal field theory, Nuclear Phys. B 872 (2013), no. 3, 348–391. MR 3048507, DOI 10.1016/j.nuclphysb.2013.04.007
- Thomas Creutzig and David Ridout, Modular data and Verlinde formulae for fractional level WZW models II, Nuclear Phys. B 875 (2013), no. 2, 423–458. MR 3093193, DOI 10.1016/j.nuclphysb.2013.07.008
- Thomas Creutzig and David Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A 46 (2013), no. 49, 494006, 72. MR 3146012, DOI 10.1088/1751-8113/46/49/494006
- Thomas Creutzig, David Ridout, and Simon Wood, Coset constructions of logarithmic $(1, p)$ models, Lett. Math. Phys. 104 (2014), no. 5, 553–583. MR 3197005, DOI 10.1007/s11005-014-0680-7
- Thomas Creutzig and Matthew Rupert, Uprolling unrolled quantum groups, Commun. Contemp. Math. 24 (2022), no. 4, Paper No. 2150023, 27. MR 4414165, DOI 10.1142/S0219199721500231
- Thomas Creutzig and Jinwei Yang, Tensor categories of affine Lie algebras beyond admissible levels, Math. Ann. 380 (2021), no. 3-4, 1991–2040. MR 4297204, DOI 10.1007/s00208-021-02159-w
- Alexei Davydov, Michael Müger, Dmitri Nikshych, and Victor Ostrik, The Witt group of non-degenerate braided fusion categories, J. Reine Angew. Math. 677 (2013), 135–177. MR 3039775, DOI 10.1515/crelle.2012.014
- P. Di Vecchia, J. L. Petersen, and H. B. Zheng, $N=2$ extended superconformal theories in two dimensions, Phys. Lett. B 162 (1985), no. 4-6, 327–332. MR 832829, DOI 10.1016/0370-2693(85)90932-3
- Chongying Dong, Vertex algebras associated with even lattices, J. Algebra 161 (1993), no. 1, 245–265. MR 1245855, DOI 10.1006/jabr.1993.1217
- Chongying Dong and Jianzhi Han, Some finite properties for vertex operator superalgebras, Pacific J. Math. 258 (2012), no. 2, 269–290. MR 2981954, DOI 10.2140/pjm.2012.258.269
- Chongying Dong and Jianzhi Han, On rationality of vertex operator superalgebras, Int. Math. Res. Not. IMRN 16 (2014), 4379–4399. MR 3250038, DOI 10.1093/imrn/rnt077
- Chongying Dong, Xiangyu Jiao, and Feng Xu, Quantum dimensions and quantum Galois theory, Trans. Amer. Math. Soc. 365 (2013), no. 12, 6441–6469. MR 3105758, DOI 10.1090/S0002-9947-2013-05863-1
- Chongying Dong, Ching Hung Lam, and Hiromichi Yamada, $W$-algebras related to parafermion algebras, J. Algebra 322 (2009), no. 7, 2366–2403. MR 2553685, DOI 10.1016/j.jalgebra.2009.03.034
- Chongying Dong, Ching Hung Lam, Qing Wang, and Hiromichi Yamada, The structure of parafermion vertex operator algebras, J. Algebra 323 (2010), no. 2, 371–381. MR 2564844, DOI 10.1016/j.jalgebra.2009.08.003
- Chongying Dong and James Lepowsky, Generalized vertex algebras and relative vertex operators, Progress in Mathematics, vol. 112, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1233387, DOI 10.1007/978-1-4612-0353-7
- Chongying Dong, Haisheng Li, and Geoffrey Mason, Compact automorphism groups of vertex operator algebras, Internat. Math. Res. Notices 18 (1996), 913–921. MR 1420556, DOI 10.1155/S1073792896000566
- Chongying Dong, Haisheng Li, and Geoffrey Mason, Regularity of rational vertex operator algebras, Adv. Math. 132 (1997), no. 1, 148–166. MR 1488241, DOI 10.1006/aima.1997.1681
- Chongying Dong and Zongzhu Lin, Induced modules for vertex operator algebras, Comm. Math. Phys. 179 (1996), no. 1, 157–183. MR 1395220
- Chongying Dong and Geoffrey Mason, On quantum Galois theory, Duke Math. J. 86 (1997), no. 2, 305–321. MR 1430435, DOI 10.1215/S0012-7094-97-08609-9
- Chongying Dong and Geoffrey Mason, Quantum Galois theory for compact Lie groups, J. Algebra 214 (1999), no. 1, 92–102. MR 1684904, DOI 10.1006/jabr.1998.7694
- Chongying Dong and Li Ren, Representations of the parafermion vertex operator algebras, Adv. Math. 315 (2017), 88–101. MR 3667581, DOI 10.1016/j.aim.2017.05.016
- Chongying Dong and Qing Wang, The structure of parafermion vertex operator algebras: general case, Comm. Math. Phys. 299 (2010), no. 3, 783–792. MR 2718932, DOI 10.1007/s00220-010-1114-8
- Chongying Dong and Qing Wang, On $C_2$-cofiniteness of parafermion vertex operator algebras, J. Algebra 328 (2011), 420–431. MR 2745574, DOI 10.1016/j.jalgebra.2010.10.015
- Chongying Dong and Qing Wang, Quantum dimensions and fusion rules for parafermion vertex operator algebras, Proc. Amer. Math. Soc. 144 (2016), no. 4, 1483–1492. MR 3451226, DOI 10.1090/proc/12838
- Chongying Dong and Zhongping Zhao, Modularity in orbifold theory for vertex operator superalgebras, Comm. Math. Phys. 260 (2005), no. 1, 227–256. MR 2175996, DOI 10.1007/s00220-005-1418-2
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor categories, Mathematical Surveys and Monographs, vol. 205, American Mathematical Society, Providence, RI, 2015. MR 3242743, DOI 10.1090/surv/205
- Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581–642. MR 2183279, DOI 10.4007/annals.2005.162.581
- A. M. Gaĭnutdinov, A. M. Semikhatov, I. Yu. Tipunin, and B. L. Feĭgin, The Kazhdan-Lusztig correspondence for the representation category of the triplet $W$-algebra in logorithmic conformal field theories, Teoret. Mat. Fiz. 148 (2006), no. 3, 398–427 (Russian, with Russian summary); English transl., Theoret. and Math. Phys. 148 (2006), no. 3, 1210–1235. MR 2283660, DOI 10.1007/s11232-006-0113-6
- B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov, and I. Yu. Tipunin, Logarithmic extensions of minimal models: characters and modular transformations, Nuclear Phys. B 757 (2006), no. 3, 303–343. MR 2275182, DOI 10.1016/j.nuclphysb.2006.09.019
- Boris Feigin and Edward Frenkel, Duality in $W$-algebras, Internat. Math. Res. Notices 6 (1991), 75–82. MR 1136408, DOI 10.1155/S1073792891000119
- B. L. Feigin and A. M. Semikhatov, $\scr W^{(2)}_n$ algebras, Nuclear Phys. B 698 (2004), no. 3, 409–449. MR 2092705, DOI 10.1016/j.nuclphysb.2004.06.056
- Ingo Runkel, Jens Fjelstad, Jürgen Fuchs, and Christoph Schweigert, Topological and conformal field theory as Frobenius algebras, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 225–247. MR 2342831, DOI 10.1090/conm/431/08275
- Jens Fjelstad, Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert, TFT construction of RCFT correlators. V. Proof of modular invariance and factorisation, Theory Appl. Categ. 16 (2006), No. 16, 342–433. MR 2259258
- Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997. MR 1424041, DOI 10.1007/978-1-4612-2256-9
- Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64. MR 1142494, DOI 10.1090/memo/0494
- Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR 996026
- Jürgen Fuchs, Simple WZW currents, Comm. Math. Phys. 136 (1991), no. 2, 345–356. MR 1096120
- Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert, TFT construction of RCFT correlators. I. Partition functions, Nuclear Phys. B 646 (2002), no. 3, 353–497. MR 1940282, DOI 10.1016/S0550-3213(02)00744-7
- Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert, Ribbon categories and (unoriented) CFT: Frobenius algebras, automorphisms, reversions, Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI, 2007, pp. 203–224. MR 2342830, DOI 10.1090/conm/431/08274
- Jürgen Fuchs, Ingo Runkel, and Christoph Schweigert, The fusion algebra of bimodule categories, Appl. Categ. Structures 16 (2008), no. 1-2, 123–140. MR 2383281, DOI 10.1007/s10485-007-9102-7
- Jürgen Fuchs and Christoph Schweigert, Hopf algebras and finite tensor categories in conformal field theory, Rev. Un. Mat. Argentina 51 (2010), no. 2, 43–90. MR 2840163
- Azat M. Gaĭnutdinov, Simon D. Lentner, and Tobias Ohrmann, Modularization of small quantum groups, arXiv:1809.02116.
- Davide Gaiotto and Miroslav Rapčák, Vertex algebras at the corner, J. High Energy Phys. 1 (2019), 160, front matter+85. MR 3919335, DOI 10.1007/jhep01(2019)160
- César Galindo, Crossed product tensor categories, J. Algebra 337 (2011), 233–252. MR 2796073, DOI 10.1016/j.jalgebra.2011.04.012
- Gerald Höhn, Genera of vertex operator algebras and three-dimensional topological quantum field theories, Vertex operator algebras in mathematics and physics (Toronto, ON, 2000) Fields Inst. Commun., vol. 39, Amer. Math. Soc., Providence, RI, 2003, pp. 89–107. MR 2029792
- Yi-Zhi Huang, Two-dimensional conformal geometry and vertex operator algebras, Progress in Mathematics, vol. 148, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1448404
- Yi-Zhi Huang, A theory of tensor products for module categories for a vertex operator algebra. IV, J. Pure Appl. Algebra 100 (1995), no. 1-3, 173–216. MR 1344849, DOI 10.1016/0022-4049(95)00050-7
- Yi-Zhi Huang, Virasoro vertex operator algebras, the (nonmeromorphic) operator product expansion and the tensor product theory, J. Algebra 182 (1996), no. 1, 201–234. MR 1388864, DOI 10.1006/jabr.1996.0168
- Yi-Zhi Huang, Generalized rationality and a “Jacobi identity” for intertwining operator algebras, Selecta Math. (N.S.) 6 (2000), no. 3, 225–267. MR 1817614, DOI 10.1007/PL00001389
- Yi-Zhi Huang, Differential equations and intertwining operators, Commun. Contemp. Math. 7 (2005), no. 3, 375–400. MR 2151865, DOI 10.1142/S0219199705001799
- Yi-Zhi Huang, Vertex operator algebras and the Verlinde conjecture, Commun. Contemp. Math. 10 (2008), no. 1, 103–154. MR 2387861, DOI 10.1142/S0219199708002727
- Yi-Zhi Huang, Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10 (2008), no. suppl. 1, 871–911. MR 2468370, DOI 10.1142/S0219199708003083
- Yi-Zhi Huang, Cofiniteness conditions, projective covers and the logarithmic tensor product theory, J. Pure Appl. Algebra 213 (2009), no. 4, 458–475. MR 2483831, DOI 10.1016/j.jpaa.2008.07.016
- Yi-Zhi Huang, Two constructions of grading-restricted vertex (super)algebras, J. Pure Appl. Algebra 220 (2016), no. 11, 3628–3649. MR 3506472, DOI 10.1016/j.jpaa.2016.05.004
- Yi-Zhi Huang, On the applicability of logarithmic tensor category theory, arXiv:1702.00133.
- Yi-Zhi Huang, Alexander Kirillov Jr., and James Lepowsky, Braided tensor categories and extensions of vertex operator algebras, Comm. Math. Phys. 337 (2015), no. 3, 1143–1159. MR 3339173, DOI 10.1007/s00220-015-2292-1
- Yi-Zhi Huang and Liang Kong, Open-string vertex algebras, tensor categories and operads, Comm. Math. Phys. 250 (2004), no. 3, 433–471. MR 2094470, DOI 10.1007/s00220-004-1059-x
- Yi-Zhi Huang and Liang Kong, Full field algebras, Comm. Math. Phys. 272 (2007), no. 2, 345–396. MR 2300247, DOI 10.1007/s00220-007-0224-4
- Yi-Zhi Huang and Liang Kong, Modular invariance for conformal full field algebras, Trans. Amer. Math. Soc. 362 (2010), no. 6, 3027–3067. MR 2592945, DOI 10.1090/S0002-9947-09-04933-2
- Yi-Zhi Huang and James Lepowsky, Tensor products of modules for a vertex operator algebra and vertex tensor categories, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 349–383. MR 1327541, DOI 10.1007/978-1-4612-0261-5_{1}3
- Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, I, Selecta Math. (N. S.) 1 (1995), no. 4, 699–756.
- Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra, II, Selecta Math. (N. S.) 1 (1995), no. 4, 757–786.
- Yi-Zhi Huang and James Lepowsky, A theory of tensor products for module categories for a vertex operator algebra. III, J. Pure Appl. Algebra 100 (1995), no. 1-3, 141–171. MR 1344848, DOI 10.1016/0022-4049(95)00049-3
- Yi-Zhi Huang and James Lepowsky, Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke Math. J. 99 (1999), no. 1, 113–134. MR 1700743, DOI 10.1215/S0012-7094-99-09905-2
- Yi-Zhi Huang and James Lepowsky, Tensor categories and the mathematics of rational and logarithmic conformal field theory, J. Phys. A 46 (2013), no. 49, 494009, 21. MR 3146015, DOI 10.1088/1751-8113/46/49/494009
- Yi-Zhi Huang, James Lepowsky, and Lin Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: introduction and strongly graded algebras and their generalized modules, Conformal field theories and tensor categories, Math. Lect. Peking Univ., Springer, Heidelberg, 2014, pp. 169–248. MR 3585368
- Y.-Z. Huang, J. Lepowsky, and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, II: Logarithmic formal calculus and properties of logarithmic intertwining operators, arXiv:1012.4196.
- 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, arXiv:1012.4197.
- 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, arXiv:1012.4198.
- 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, arXiv:1012.4199.
- 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, arXiv:1012.4202.
- 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, arXiv:1110.1929.
- 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, arXiv:1110.1931.
- Yi-Zhi Huang and Antun Milas, Intertwining operator superalgebras and vertex tensor categories for superconformal algebras. I, Commun. Contemp. Math. 4 (2002), no. 2, 327–355. MR 1901149, DOI 10.1142/S0219199702000622
- Yi-Zhi Huang and Antun Milas, Intertwining operator superalgebras and vertex tensor categories for superconformal algebras. II, Trans. Amer. Math. Soc. 354 (2002), no. 1, 363–385. MR 1859279, DOI 10.1090/S0002-9947-01-02869-0
- Kenji Iohara and Yoshiyuki Koga, Representation theory of the Virasoro algebra, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2011. MR 2744610, DOI 10.1007/978-0-85729-160-8
- Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto, Quantum reduction for affine superalgebras, Comm. Math. Phys. 241 (2003), no. 2-3, 307–342. MR 2013802, DOI 10.1007/s00220-003-0926-1
- Victor G. Kac and Minoru Wakimoto, Quantum reduction and representation theory of superconformal algebras, Adv. Math. 185 (2004), no. 2, 400–458. MR 2060475, DOI 10.1016/j.aim.2003.12.005
- Victor G. Kac and Minoru Wakimoto, On rationality of $W$-algebras, Transform. Groups 13 (2008), no. 3-4, 671–713. MR 2452611, DOI 10.1007/s00031-008-9028-7
- Victor G. Kac and Minoru Wakimoto, Representations of affine superalgebras and mock theta functions, Transform. Groups 19 (2014), no. 2, 383–455. MR 3200431, DOI 10.1007/s00031-014-9263-z
- Victor G. Kac and Minoru Wakimoto, Representations of affine superalgebras and mock theta functions II, Adv. Math. 300 (2016), 17–70. MR 3534829, DOI 10.1016/j.aim.2016.03.015
- V. G. Kac and M. Wakimoto, Representations of affine superalgebras and mock theta functions. III, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 4, 65–122; English transl., Izv. Math. 80 (2016), no. 4, 693–750. MR 3535359, DOI 10.4213/im8408
- Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2
- H. G. Kausch, Extended conformal algebras generated by a multiplet of primary fields, Phys. Lett. B 259 (1991), no. 4, 448–455. MR 1107489, DOI 10.1016/0370-2693(91)91655-F
- Yasuyuki Kawahigashi, Conformal field theory, tensor categories and operator algebras, J. Phys. A 48 (2015), no. 30, 303001, 57. MR 3367967, DOI 10.1088/1751-8113/48/30/303001
- Kazuya Kawasetsu, $\mathcal W$-algebras with non-admissible levels and the Deligne exceptional series, Int. Math. Res. Not. IMRN 3 (2018), 641–676. MR 3801442, DOI 10.1093/imrn/rnw240
- Alexander Kirillov Jr., Modular categories and orbifold models, Comm. Math. Phys. 229 (2002), no. 2, 309–335. MR 1923177, DOI 10.1007/s002200200650
- A. Kirillov, Jr., On $G$-equivariant modular categories, arXiv:math/0401119.
- Alexander Kirillov Jr. and Viktor Ostrik, On a $q$-analogue of the McKay correspondence and the ADE classification of $\mathfrak {sl}_2$ conformal field theories, Adv. Math. 171 (2002), no. 2, 183–227. MR 1936496, DOI 10.1006/aima.2002.2072
- Hiroki Kondo and Yoshihisa Saito, Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to ${\mathfrak {sl}}_2$, J. Algebra 330 (2011), 103–129. MR 2774620, DOI 10.1016/j.jalgebra.2011.01.010
- Liang Kong, Full field algebras, operads and tensor categories, Adv. Math. 213 (2007), no. 1, 271–340. MR 2331245, DOI 10.1016/j.aim.2006.12.007
- Matthew Krauel and Geoffrey Mason, Jacobi trace functions in the theory of vertex operator algebras, Commun. Number Theory Phys. 9 (2015), no. 2, 273–306. MR 3361295, DOI 10.4310/CNTP.2015.v9.n2.a2
- James Lepowsky and Haisheng Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2023933, DOI 10.1007/978-0-8176-8186-9
- James Lepowsky and Robert Lee Wilson, A new family of algebras underlying the Rogers-Ramanujan identities and generalizations, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 12, 7254–7258. MR 638674, DOI 10.1073/pnas.78.12.7254
- Hai Sheng Li, Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994), no. 3, 279–297. MR 1303287, DOI 10.1016/0022-4049(94)90104-X
- Wanglai Li, Representations of vertex operator superalgebras and abelian intertwining algebras, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick. MR 2696773
- Xingjun Lin, Mirror extensions of rational vertex operator algebras, Trans. Amer. Math. Soc. 369 (2017), no. 6, 3821–3840. MR 3624394, DOI 10.1090/tran/6749
- R. Longo and K.-H. Rehren, Nets of subfactors, Rev. Math. Phys. 7 (1995), no. 4, 567–597. Workshop on Algebraic Quantum Field Theory and Jones Theory (Berlin, 1994). MR 1332979, DOI 10.1142/S0129055X95000232
- Robert McRae, Twisted modules and $G$-equivariantization in logarithmic conformal field theory, Comm. Math. Phys. 383 (2021), no. 3, 1939–2019. MR 4244264, DOI 10.1007/s00220-020-03882-2
- Antun Milas, Weak modules and logarithmic intertwining operators for vertex operator algebras, Recent developments in infinite-dimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000) Contemp. Math., vol. 297, Amer. Math. Soc., Providence, RI, 2002, pp. 201–225. MR 1919819, DOI 10.1090/conm/297/05099
- Antun Milas, Logarithmic intertwining operators and vertex operators, Comm. Math. Phys. 277 (2008), no. 2, 497–529. MR 2358293, DOI 10.1007/s00220-007-0375-3
- M. Miyamoto, Flatness and semi-rigidity of vertex operator algebras, arXiv:1104.4675.
- Gregory Moore and Nathan Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), no. 2, 177–254. MR 1002038
- Michael Müger, Galois extensions of braided tensor categories and braided crossed $G$-categories, J. Algebra 277 (2004), no. 1, 256–281. MR 2059630, DOI 10.1016/j.jalgebra.2004.02.026
- Michael Müger, Conformal orbifold theories and braided crossed $G$-categories, Comm. Math. Phys. 260 (2005), no. 3, 727–762. MR 2183964, DOI 10.1007/s00220-005-1291-z
- Cris Negron, Log-modular quantum groups at even roots of unity and the quantum Frobenius I, Comm. Math. Phys. 382 (2021), no. 2, 773–814. MR 4227163, DOI 10.1007/s00220-021-04012-2
- Victor Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003), no. 2, 177–206. MR 1976459, DOI 10.1007/s00031-003-0515-6
- Victor Ostrik and Michael Sun, Level-rank duality via tensor categories, Comm. Math. Phys. 326 (2014), no. 1, 49–61. MR 3162483, DOI 10.1007/s00220-013-1869-9
- Bodo Pareigis, On braiding and dyslexia, J. Algebra 171 (1995), no. 2, 413–425. MR 1315904, DOI 10.1006/jabr.1995.1019
- A. M. Polyakov, Gauge transformations and diffeomorphisms, Internat. J. Modern Phys. A 5 (1990), no. 5, 833–842. MR 1035397, DOI 10.1142/S0217751X90000386
- Thomas J. Robinson, On replacement axioms for the Jacobi identity for vertex algebras and their modules, J. Pure Appl. Algebra 214 (2010), no. 10, 1740–1758. MR 2608103, DOI 10.1016/j.jpaa.2009.12.018
- R. Sato, Equivalences between weight modules via $N=2$ coset constructions, arXiv:1605.02343.
- Akihiro Tsuchiya and Yukihiro Kanie, Fock space representations of the Virasoro algebra. Intertwining operators, Publ. Res. Inst. Math. Sci. 22 (1986), no. 2, 259–327. MR 849260, DOI 10.2977/prims/1195178069
- Yukihiro Kanie and Akihiro Tsuchiya, Fock space representations of Virasoro algebra and intertwining operators, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), no. 1, 12–15. MR 839795
- Akihiro Tsuchiya and Simon Wood, The tensor structure on the representation category of the $\scr W_p$ triplet algebra, J. Phys. A 46 (2013), no. 44, 445203, 40. MR 3120909, DOI 10.1088/1751-8113/46/44/445203
- Akihiro Tsuchiya and Simon Wood, On the extended $W$-algebra of type ${\mathfrak {sl}}_2$ at positive rational level, Int. Math. Res. Not. IMRN 14 (2015), 5357–5435. MR 3384444, DOI 10.1093/imrn/rnu090
- V. G. Turaev, Quantum invariants of knots and 3-manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 1994. MR 1292673
- Vladimir Turaev, Crossed group-categories, Arab. J. Sci. Eng. Sect. C Theme Issues 33 (2008), no. 2, 483–503 (English, with English and Arabic summaries). MR 2500054
- Erik Verlinde, Fusion rules and modular transformations in $2$D conformal field theory, Nuclear Phys. B 300 (1988), no. 3, 360–376. MR 954762, DOI 10.1016/0550-3213(88)90603-7
- Minoru Wakimoto, Infinite-dimensional Lie algebras, Translations of Mathematical Monographs, vol. 195, American Mathematical Society, Providence, RI, 2001. Translated from the 1999 Japanese original by Kenji Iohara; Iwanami Series in Modern Mathematics. MR 1793723, DOI 10.1090/mmono/195
- M. Wakimoto, Fusion rules for $N = 2$ superconformal modules, arXiv:hep-th/9807144.
- Feng Xu, New braided endomorphisms from conformal inclusions, Comm. Math. Phys. 192 (1998), no. 2, 349–403. MR 1617550, DOI 10.1007/s002200050302
- Xiaoping Xu, Intertwining operators for twisted modules of a colored vertex operator superalgebra, J. Algebra 175 (1995), no. 1, 241–273. MR 1338977, DOI 10.1006/jabr.1995.1185
- H. Yamauchi, A Theory of Simple Current Extensions of Vertex Operator Algebras and Applications to the Moonshine Vertex Operator Algebra, Ph.D. thesis, University of Tsukuba, 2004.
- Hiroshi Yamauchi, Module categories of simple current extensions of vertex operator algebras, J. Pure Appl. Algebra 189 (2004), no. 1-3, 315–328. MR 2038578, DOI 10.1016/j.jpaa.2003.10.006
- Hiroshi Yamauchi, Extended Griess algebras and Matsuo-Norton trace formulae, Conformal field theory, automorphic forms and related topics, Contrib. Math. Comput. Sci., vol. 8, Springer, Heidelberg, 2014, pp. 75–107. MR 3559202
- Yongchang Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237–302. MR 1317233, DOI 10.1090/S0894-0347-96-00182-8