Advertisement

NGK and HLZ: Fusion for Physicists and Mathematicians

  • Shashank KanadeEmail author
  • David Ridout
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 37)

Abstract

In this expository note, we compare the fusion product of conformal field theory, as defined by Gaberdiel and used in the Nahm–Gaberdiel–Kausch (NGK) algorithm, with the P(w)-tensor product of vertex operator algebra modules, as defined by Huang, Lepowsky and Zhang (HLZ). We explain how the equality of the two “coproducts” derived by NGK is essentially dual to the P(w)-compatibility condition of HLZ and how the algorithm of NGK for computing fusion products may be adapted to the setting of HLZ. We provide explicit calculations and instructive examples to illustrate both approaches. This document does not provide precise descriptions of all statements, it is intended more as a gentle starting point for the appreciation of the depth of the theory on both sides.

Keywords

Vertex operator algebras Conformal field theory Tensor categories Fusion 

Notes

Acknowledgements

This paper was made possible by an Endeavour Research Fellowship, ID 6127_2017, awarded to SK by the Australian Government’s Department of Education and Training. SK wishes to express sincere gratitude towards the School of Mathematics and Statistics at the University of Melbourne, where this project was undertaken, for their generous hospitality. SK is presently supported by a start-up grant provided by University of Denver. DR’s research is supported by the Australian Research Council Discovery Project DP160101520 and the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers CE140100049.

It is our privilege to thank our fellow “fusion club” members Arun Ram and Kazuya Kawasetsu for the many hours that we spent together working through the details of the approaches of NGK, HLZ, Kazhdan–Lusztig and Miyamoto. We also thank Thomas Creutzig, Hubert Saleur and Simon Wood for encouraging us to complete this article when time was lacking and deadlines were passing. We similarly thank Dražen Adamović and Paolo Papi for generous amounts of leeway in regard to this last point.

References

  1. 1.
    Abe, T., Buhl, G., Dong, C.: Rationality, regularity, and \(C_2\)-cofiniteness. Trans. Am. Math. Soc. 356(8), 3391–3402 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Adamović, D., Milas, A.: Lattice construction of logarithmic modules for certain vertex algebras. Selecta Math. New Ser. 15, 535–561 (2009). arXiv:0902.3417 [math.QA]MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Astashkevich, A.: On the structure of Verma modules over Virasoro and Neveu-Schwarz algebras. Commun. Math. Phys. 186, 531–562 (1997). arXiv:hep-th/9511032MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Canagasabey, M., Rasmussen, J., Ridout, D.: Fusion rules for the logarithmic \(N=1\) superconformal minimal models I: the Neveu-Schwarz sector. J. Phys. A 48, 415402 (2015). arXiv:1504.03155 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Canagasabey, M., Ridout, D.: Fusion rules for the logarithmic \(N=1\) superconformal minimal models II: including the Ramond sector. Nucl. Phys. B 905, 132–187 (2016). arXiv:1512.05837 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Creutzig, T., Huang, Y.Z., Yang, J.: Braided tensor categories of admissible modules for affine Lie algebras. Commun. Math. Phys. 362, 827–854 (2018). arXiv:1709.01865 [math.QA]MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Creutzig, T., Kanade, S., Linshaw, A., Ridout, D.: Schur-Weyl duality for Heisenberg cosets. Transform. Groups 24(2), 301–354 (2019). arXiv:1611.00305 [math.QA]MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Creutzig, T., Kanade, S., McRae, R.: Tensor categories for vertex operator superalgebra extensions. arXiv:1705.05017 [math.QA]
  9. 9.
    Creutzig, T., Ridout, D.: Logarithmic conformal field theory: beyond an introduction. J. Phys. A 46, 494006 (2013). arXiv:1303.0847 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Creutzig, T., Ridout, D.: Relating the archetypes of logarithmic conformal field theory. Nucl. Phys. B 872, 348–391 (2013). arXiv:1107.2135 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Mathematics, vol. 112. Birkhäuser Boston Inc, Boston (1993)zbMATHCrossRefGoogle Scholar
  12. 12.
    Dong, C., Li, H., Mason, G.: Vertex operator algebras and associative algebras. J. Algebra 206, 67–96 (1998). arXiv:q-alg/9612010MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Eberle, H., Flohr, M.: Virasoro representations and fusion for general augmented minimal models. J. Phys. A 39, 15245–15286 (2006). arXiv:hep-th/0604097MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Feigin, B., Fuchs, D.: Cohomology of some nilpotent subalgebras of the Virasoro and Kac-Moody lie algebras. J. Geom. Phys. 5, 209–235 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Feigin, B., Nakanishi, T., Ooguri, H.: The annihilating ideals of minimal models. Int. J. Mod. Phys. A 7, 217–238 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, vol. 88. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  17. 17.
    Frenkel, I., Huang, Y.Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc.104, viii+64 (1993)Google Scholar
  18. 18.
    Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, vol. 134. Academic Press, Boston (1988)zbMATHGoogle Scholar
  19. 19.
    Frenkel, I., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gaberdiel, M.: Fusion in conformal field theory as the tensor product of the symmetry algebra. Int. J. Mod. Phys. A 9, 4619–4636 (1994). arXiv:hep-th/9307183MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gaberdiel, M.: Fusion rules of chiral algebras. Nucl. Phys. B 417, 130–150 (1994). arXiv:hep-th/9309105MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Gaberdiel, M.: Fusion of twisted representations. Int. J. Mod. Phys. A 12, 5183–5208 (1997). arXiv:hep-th/9607036MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Gaberdiel, M.: An introduction to conformal field theory. Rep. Prog. Phys. 63, 607–667 (2000). arXiv:hep-th/9910156CrossRefGoogle Scholar
  24. 24.
    Gaberdiel, M.: Fusion rules and logarithmic representations of a WZW model at fractional level. Nucl. Phys. B 618, 407–436 (2001). arXiv:hep-th/0105046MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Gaberdiel, M., Kausch, H.: Indecomposable fusion products. Nucl. Phys. B477, 293–318 (1996). arXiv:hep-th/9604026MathSciNetCrossRefGoogle Scholar
  26. 26.
    Gaberdiel, M., Kausch, H.: A rational logarithmic conformal field theory. Phys. Lett. B 386, 131–137 (1996). arXiv:hep-th/9606050MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gaberdiel, M., Runkel, I., Wood, S.: Fusion rules and boundary conditions in the \(c=0\) triplet model. J. Phys. A 42, 325403 (2009). arXiv:0905.0916 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Gainutdinov, A., Jacobsen, J., Read, N., Saleur, H., Vasseur, R.: Logarithmic conformal field theory: a lattice approach. J. Phys. A 46, 494012 (2013). arXiv:1303.2082 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Gainutdinov, A., Vasseur, R.: Lattice fusion rules and logarithmic operator product expansions. Nucl. Phys. B 868, 223–270 (2013). arXiv:1203.6289 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Gurarie, V.: Logarithmic operators in conformal field theory. Nucl. Phys. B 410, 535–549 (1993). arXiv:hep-th/9303160MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Huang, Y.Z.: On the applicability of logarithmic tensor category theory. arXiv:1702.00133 [math.QA]
  32. 32.
    Huang, Y.Z.: Two-Dimensional Conformal Geometry and Vertex Operator Algebras. Progress in Mathematics, vol. 148. Birkhäuser, Boston (1997)zbMATHGoogle Scholar
  33. 33.
    Huang, Y.Z.: Cofiniteness conditions, projective covers and the logarithmic tensor product theory. J. Pure Appl. Algebra 213, 458–475 (2009). arXiv:0712.4109 [math.QA]MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Huang, Y.Z., Jr, A.K., Lepowsky, J.: Braided tensor categories and extensions of vertex operator algebras. Commun. Math. Phys. 337, 1143–1159 (2015). arXiv:1406.3420 [math.QA]MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Huang, Y.Z., Lepowsky, J.: Tensor products of modules for a vertex operator algebra and vertex tensor categories. In: Lie Theory and Geometry, Progress in Mathematics, vol. 123, pp. 349–383. Birkhäuser, Boston (1994). arXiv:hep-th/9401119zbMATHCrossRefGoogle Scholar
  36. 36.
    Huang, Y.Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra I. Selecta Math. New Ser. 1(4), 699–756 (1995). arXiv:hep-th/9309076MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Huang, Y.Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra II. Selecta Math. New Ser. 1(4), 757–786 (1995). arXiv:hep-th/9309159MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Huang, Y.Z., Lepowsky, J.: A theory of tensor products for module categories for a vertex operator algebra III. J. Pure Appl. Algebra 100(1–3), 141–171 (1995). arXiv:hep-th/9505018MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Huang, Y.Z., Lepowsky, J.: Tensor categories and the mathematics of rational and logarithmic conformal field theory. J. Phys. A 46, 494009 (2013). arXiv:1304.7556 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Huang, Y.Z., Lepowsky, J., Li, H., Zhang, L.: On the concepts of intertwining operator and tensor product module in vertex operator algebra theory. J. Pure Appl. Algebra 204, 507–535 (2006). arXiv:math.QA/0409364MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Huang, Y.Z., Lepowsky, J., Zhang, L.: Logarithmic tensor product theory I–VIII. arXiv:1012.4193 [math.QA], arXiv:1012.4196 [math.QA], arXiv:1012.4197 [math.QA], arXiv:1012.4198 [math.QA], arXiv:1012.4199 [math.QA], arXiv:1012.4202 [math.QA], arXiv:1110.1929 [math.QA], arXiv:1110.1931 [math.QA]
  42. 42.
    Huang, Y.Z., Yang, J.: Logarithmic intertwining operators and associative algebras. J. Pure Appl. Algebra 216, 1467–1492 (2012). arXiv:1104.4679 [math.QA]MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Jr, A.K., Ostrik, V.: On a \(q\)-analogue of the McKay correspondence and the ADE classification of \(\mathfrak{sl}_2\) conformal field theories. Adv. Math. 171, 183–227 (2002). arXiv:math.QA/0101219
  44. 44.
    Kac, V.: Vertex Algebras for Beginners. University Lecture Series, vol. 10. American Mathematical Society, Providence (1996)zbMATHGoogle Scholar
  45. 45.
    Kazhdan, D., Lusztig, G.: Affine Lie algebras and quantum groups. Int. Math. Res. Not. 1991, 21–29 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras I. J. Am. Math. Soc. 6, 905–947 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras II. J. Am. Math. Soc. 6, 949–1011 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras III. J. Am. Math. Soc. 7, 335–381 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras IV. J. Am. Math. Soc. 7, 383–453 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Kytölä, K., Ridout, D.: On staggered indecomposable Virasoro modules. J. Math. Phys. 50, 123503 (2009). arXiv:0905.0108 [math-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and their Representations. Progress in Mathematics, vol. 227. Birkhäuser, Boston (2004)zbMATHCrossRefGoogle Scholar
  52. 52.
    Li, H.: Representation theory and tensor product theory for vertex operator algebras. Ph.D. Thesis, Rutgers University (1994). arXiv:hep-th/9406211
  53. 53.
    Li, H.: An analogue of the Hom functor and a generalized nuclear democracy theorem. Duke Math. J. 93, 73–114 (1998). arXiv:q-alg/9706012MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Mathieu, P., Ridout, D.: From percolation to logarithmic conformal field theory. Phys. Lett. B 657, 120–129 (2007). arXiv:0708.0802 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Mathieu, P., Ridout, D.: Logarithmic \(M \left(2, p \right)\) minimal models, their logarithmic couplings, and duality. Nucl. Phys. B 801, 268–295 (2008). arXiv:0711.3541 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Milas, A.: Weak modules and logarithmic intertwining operators for vertex operator algebras. In: Recent developments in infinite-dimensional Lie algebras and conformal field theory, Contemporary Mathematics, vol. 297, pp. 201–225. American Mathematical Society (2002). arXiv:math.QA/0101167
  57. 57.
    Miyamoto, M.: \(C_1\)-cofiniteness and fusion products for vertex operator algebras. In: Conformal field theories and tensor categories, Mathematical Lectures from Peking University, pp. 271–279. Springer, Heidelberg (2014). arXiv:1305.3008 [math.QA]Google Scholar
  58. 58.
    Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B 212, 451–460 (1988)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Morin-Duchesne, A., Rasmussen, J., Ridout, D.: Boundary algebras and Kac modules for logarithmic minimal models. Nucl. Phys. B 899, 677–769 (2015). arXiv:1503.07584 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Nahm, W.: Quasirational fusion products. Int. J. Mod. Phys. B 8, 3693–3702 (1994). arXiv:hep-th/9402039zbMATHCrossRefGoogle Scholar
  62. 62.
    Pearce, P., Rasmussen, J., Zuber, J.B.: Logarithmic minimal models. J. Stat. Mech. 0611, P11017 (2006). arXiv:0607232 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Rasmussen, J.: Classification of Kac representations in the logarithmic minimal models \(LM \left(1, p \right)\). Nucl. Phys. B 853, 404–435 (2011). arXiv:1012.5190 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Read, N., Saleur, H.: Associative-algebraic approach to logarithmic conformal field theories. Nucl. Phys. B 777, 316–351 (2007). arXiv:hep-th/0701117MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Ridout, D.: On the percolation BCFT and the crossing probability of Watts. Nucl. Phys. B 810, 503–526 (2009). arXiv:0808.3530 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Ridout, D.: Fusion in fractional level \(\widehat{\mathfrak{sl}} \left(2 \right)\)-theories with \(k=-\tfrac{1}{2}\). Nucl. Phys. B 848, 216–250 (2011). arXiv:1012.2905 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Ridout, D., Wood, S.: Bosonic ghosts at \(c=2\) as a logarithmic CFT. Lett. Math. Phys. 105, 279–307 (2015). arXiv:1408.4185 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Ridout, D., Wood, S.: The Verlinde formula in logarithmic CFT. J. Phys. Conf. Ser. 597, 012065 (2015). arXiv:1409.0670 [hep-th]CrossRefGoogle Scholar
  69. 69.
    Rohsiepe, F.: On reducible but indecomposable representations of the Virasoro algebra. arXiv:hep-th/9611160
  70. 70.
    Tsuchiya, A., Wood, S.: The tensor structure on the representation category of the \(\cal{W}_p\) triplet algebra. J. Phys. A 46, 445203 (2013). arXiv:1201.0419 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Wood, S.: Fusion rules of the \(W \left( p, q \right)\) triplet models. J. Phys. A 43, 045212 (2010). arXiv:0907.4421 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Zhang, L.: Vertex tensor category structure on a category of Kazhdan-Lusztig. New York J. Math. 14, 261–284 (2008). arXiv:math.QA/0701260
  73. 73.
    Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9, 237–302 (1996)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DenverDenverUSA
  2. 2.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia

Personalised recommendations