Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 21 (2025), 039, 26 pages      arXiv:2207.12969      https://doi.org/10.3842/SIGMA.2025.039

Module Categories of the Generic Virasoro VOA and Quantum Groups

Shinji Koshida
Department of Mathematics and Systems Analysis, Aalto University, Finland

Received October 25, 2024, in final form May 26, 2025; Published online June 03, 2025

Abstract
In this paper, we prove the equivalence between two ribbon tensor categories. On the one hand, we consider the category of modules of the Virasoro vertex operator algebra with generic central charge (generic Virasoro VOA) generated by those simple modules lying in the first row of the Kac table. On the other hand, we take the category of finite-dimensional type I modules of the quantum group $\mathcal{U}_q (\mathfrak{sl}_{2})$ with $q$ determined by the central charge. This is a continuation of our previous work in which we examined intertwining operators for the generic Virasoro VOA in detail. Our strategy to show the categorical equivalence is to take those results as input and directly compare the structures of tensor categories. Therefore, we are to execute the most elementary proof of categorical equivalence. We also study the category of $C_{1}$-cofinite modules of the generic Virasoro VOA. We show that it is ribbon equivalent to the category of finite-dimensional type I modules of $\mathcal{U}_q (\mathfrak{sl}_{2})\otimes \mathcal{U}_{\tilde{q}}(\mathfrak{sl}_{2})$, where $q$ and $\tilde{q}$ are again related to the central charge.

Key words: vertex operator algebra; Virasoro algebra; quantum group.

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References

  1. Bakalov B., Kirillov Jr. A., Lectures on tensor categories and modular functors, Univ. Lecture Ser., Vol. 21, American Mathematical Society, Providence, RI, 2001.
  2. Bauer M., Bernard D., Conformal field theories of stochastic Loewner evolutions, Comm. Math. Phys. 239 (2003), 493-521, arXiv:hep-th/0210015.
  3. Belavin A.A., Polyakov A.M., Zamolodchikov A.B., Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333-380.
  4. Cardy J., Conformal field theory and statistical mechanics, in Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, Oxford University Press, Oxford, 2010, 65-98, arXiv:0807.3472.
  5. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1995.
  6. Creutzig T., Jiang C., Orosz Hunziker F., Ridout D., Yang J., Tensor categories arising from the Virasoro algebra, Adv. Math. 380 (2021), 107601, 35 pages, arXiv:2002.03180.
  7. Creutzig T., Lentner S., Rupert M., An algebraic theory for logarithmic Kazhdan-Lusztig correspondences, arXiv:2306.11492.
  8. Creutzig T., Lentner S., Rupert M., Characterizing braided tensor categories associated to logarithmic vertex operator algebras, arXiv:2104.13262.
  9. Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Grad. Texts Contemp. Phys., Springer, New York, 1997.
  10. Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), American Mathematical Society, Providence, RI, 1987, 798-820.
  11. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457.
  12. Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Math. Surveys Monogr., Vol. 205, American Mathematical Society, Providence, RI, 2015.
  13. Feigin B.L., Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu., Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center, Comm. Math. Phys. 265 (2006), 47-93, arXiv:hep-th/0504093.
  14. Felder G., Wieczerkowski C., Topological representations of the quantum group $U_q({\rm sl}_2)$, Comm. Math. Phys. 138 (1991), 583-605.
  15. Finkelberg M., An equivalence of fusion categories, Geom. Funct. Anal. 6 (1996), 249-267, Erratum, Geom. Funct. Anal. 23 (2013), 249-267.
  16. Frenkel E., Ben-Zvi D., Vertex algebras and algebraic curves, 2nd ed., Math. Surveys Monogr., Vol. 88, American Mathematical Society, Providence, RI, 2004, arXiv:math.QA/0007054.
  17. Frenkel I., Khovanov M.G., Canonical bases in tensor products and graphical calculus for $U_q({\mathfrak s}{\mathfrak l}_2)$, Duke Math. J. 87 (1997), 409-480.
  18. Frenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure Appl. Math., Vol. 134, Academic Press, Boston, MA, 1988.
  19. Frenkel I., Zhu M., Vertex algebras associated to modified regular representations of the Virasoro algebra, Adv. Math. 229 (2012), 3468-3507, arXiv:1012.5443.
  20. Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu., Feigin B.L., The Kazhdan-Lusztig correspondence for the representation category of the triplet $W$-algebra in logorithmic CFT, Theoret. and Math. Phys. 148 (2006), 1210-1235, arXiv:math.QA/0512621.
  21. Gannon T., Negron C., Quantum $\mathrm{SL}(2)$ and logarithmic vertex operator algebras at $(p,1)$-central charge, J. Eur. Math. Soc. (JEMS),to appear, arXiv:2104.12821.
  22. Gómez C.G., Ruiz-Altaba M., Sierra G., Quantum groups in two-dimensional physics, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 1996.
  23. Green M.B., Schwarz J.H., Witten E., Superstring theory I, II, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 1987.
  24. Hansson T.H., Hermanns M., Simon S.H., Viefers S.F., Quantum hall physics: Hierarchies and conformal field theory techniques, Rev. Modern Phys. 89 (2017), 025005, 61 pages, arXiv:1601.01697.
  25. Huang Y.-Z., A theory of tensor products for module categories for a vertex operator algebra. IV, J. Pure Appl. Algebra 100 (1995), 173-216, arXiv:q-alg/9505019.
  26. Huang Y.-Z., Two-dimensional conformal geometry and vertex operator algebras, Progr. Math., Vol. 148, Birkhäuser, Boston, MA, 1997.
  27. Huang Y.-Z., Differential equations and intertwining operators, Commun. Contemp. Math. 7 (2005), 375-400, arXiv:math.QA/0206206.
  28. Huang Y.-Z., Kirillov Jr. A., Lepowsky J., Braided tensor categories and extensions of vertex operator algebras, Comm. Math. Phys. 337 (2015), 1143-1159, arXiv:1406.3420.
  29. Huang Y.-Z., Lepowsky J., Toward a theory of tensor products for representations of a vertex operator algebra, in Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991), World Scientific Publishing, River Edge, NJ, 1992, 344-354.
  30. Huang Y.-Z., Lepowsky J., Tensor products of modules for a vertex operator algebra and vertex tensor categories, in Lie Theory and Geometry, Progr. Math., Vol. 123, Birkhäuser, Boston, MA, 1994, 349-383.
  31. Huang Y.-Z., Lepowsky J., A theory of tensor products for module categories for a vertex operator algebra. I, Selecta Math. (N.S.) 1 (1995), 699-756, arXiv:hep-th/9309076.
  32. Huang Y.-Z., Lepowsky J., A theory of tensor products for module categories for a vertex operator algebra. II, Selecta Math. (N.S.) 1 (1995), 757-786, arXiv:hep-th/9309076.
  33. Huang Y.-Z., Lepowsky J., A theory of tensor products for module categories for a vertex operator algebra. III, J. Pure Appl. Algebra 100 (1995), 141-171, arXiv:q-alg/9505018.
  34. Huang Y.-Z., Lepowsky J., Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke Math. J. 99 (1999), 113-134, arXiv:q-alg/9706028.
  35. Iohara K., Koga Y., Representation theory of the Virasoro algebra, Springer Monogr. Math., Springer, London, 2011.
  36. Kac V., Vertex algebras for beginners, 2nd ed., Univ. Lecture Ser., Vol. 10, American Mathematical Society, Providence, RI, 1998.
  37. Kassel C., Quantum groups, Grad. Texts in Math., Vol. 155, Springer, New York, 1995.
  38. Kazhdan D., Lusztig G., Tensor structures arising from affine Lie algebras. I, J. Amer. Math. Soc. 6 (1993), 905-947.
  39. Kazhdan D., Lusztig G., Tensor structures arising from affine Lie algebras. II, J. Amer. Math. Soc. 6 (1993), 949-1011.
  40. Kazhdan D., Lusztig G., Tensor structures arising from affine Lie algebras. III, J. Amer. Math. Soc. 7 (1994), 335-381.
  41. Kazhdan D., Lusztig G., Tensor structures arising from affine Lie algebras. IV, J. Amer. Math. Soc. 7 (1994), 383-453.
  42. Kondo H., Saito Y., 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, arXiv:0901.4221.
  43. Koshida S., Kytölä K., The quantum group dual of the first-row subcategory for the generic Virasoro VOA, Comm. Math. Phys. 389 (2022), 1135-1213, arXiv:2105.13839.
  44. Kytölä K., Peltola E., Conformally covariant boundary correlation functions with a quantum group, J. Eur. Math. Soc. (JEMS) 22 (2020), 55-118, arXiv:1408.1384.
  45. Lentner S.D., A conditional algebraic proof of the logarithmic Kazhdan-Lusztig correspondence, arXiv:2501.10735.
  46. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progr. Math., Vol. 227, Birkhäuser, Boston, MA, 2004.
  47. Ludwig W.W., Methods of conformal field theory in condensed matter physics, in Low-Dimensional Quantum Field Theories for Condensed Matter Physicists, Ser. Modern Condensed Matter Phys., Vol. 6, World Scientific Publishing, 1995, 389-455.
  48. Lusztig G., Introduction to quantum groups, Progr. Math., Vol. 110, Birkhäuser, Boston, MA, 1993.
  49. Masbaum G., Vogel P., $3$-valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994), 361-381.
  50. McRae R., Non-negative integral level affine Lie algebra tensor categories and their associativity isomorphisms, Comm. Math. Phys. 346 (2016), 349-395, arXiv:1506.00113.
  51. McRae R., Yang J., An $\mathfrak{sl}_2$-type tensor category for the Virasoro algebra at central charge 25 and applications, Math. Z. 303 (2023), 32, 40 pages, arXiv:2202.07351.
  52. Moore G., Reshetikhin N., A comment on quantum group symmetry in conformal field theory, Nuclear Phys. B 328 (1989), 557-574.
  53. Mussardo G., Statistical field theory, Oxf. Grad. Texts, Oxford University Press, Oxford, 2010.
  54. Nagatomo K., Tsuchiya A., The triplet vertex operator algebra $W(p)$ and the restricted quantum group $\overline U_q(sl_2)$ at $q={\rm e}^{\frac{\pi i}{p}}$, in Exploring New Structures and Natural Constructions in Mathematical Physics, Adv. Stud. Pure Math., Vol. 61, Math. Soc. Japan, Tokyo, 2011, 1-49, arXiv:0902.4607.
  55. Nakano H., Hunziker F.O., Camacho A.R., Wood S., Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}_{2}$ and ${\mathcal{N}=2}$ superconformal vertex operator superalgebras, arXiv:2411.11387.
  56. Ostrik V., Module categories over representations of ${{\rm SL}_q (2)}$ in the non-semisimple case, arXiv:math.QA/0509530.
  57. Pasquier V., Saleur H., Common structures between finite systems and conformal field theories through quantum groups, Nuclear Phys. B 330 (1990), 523-556.
  58. Ramírez C., Ruegg H., Ruiz-Altaba M., The contour picture of quantum groups: conformal field theories, Nuclear Phys. B 364 (1991), 195-233.
  59. Schechtman V.V., Varchenko A.N., Quantum groups and homology of local systems, in Algebraic Geometry and Analytic Geometry (Tokyo, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, 182-197.
  60. Schramm O., Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221-288, arXiv:math.PR/9904022.
  61. Tsuchiya A., Wood S., The tensor structure on the representation category of the $\mathcal{W}_p$ triplet algebra, J. Phys. A 46 (2013), 445203, 40 pages, arXiv:1201.0419.
  62. Varchenko A., Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, Adv. Ser. Math. Phys., Vol. 21, World Scientific Publishing Co., River Edge, NJ, 1995.

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