Skip to main content
SpringerLink
Log in

NGK and HLZ: Fusion for Physicists and Mathematicians

  • Chapter
  • First Online:
Affine, Vertex and W-algebras

Part of the book series: Springer INdAM 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    A conformal field theory is said to be rational if its quantum state space is semisimple (completely reducible), decomposing into a finite direct sum of tensor products of irreducible modules. The name reflects the fact that such theories have rational central charges and conformal weights.

  2. 2.

    A conformal field theory is said to be logarithmic if its quantum state space is not completely reducible. The name reflects the fact that all known examples possess a non-diagonalisable action of the hamiltonian which leads to logarithmic singularities in certain correlation functions.

  3. 3.

    In general, one might instead have tensor products \(M \otimes \overline{M}\), where M is a V-module and \(\overline{M}\) is a module over another vertex operator algebra, \(\overline{V}\) say. In either case, M is responsible for the z-dependence of the field \(\psi (z,\overline{z})\) while the \(\overline{z}\)-dependence comes from \(\overline{M}\).

  4. 4.

    Note that a non-rational theory may also possess fields that are not generated from primary fields. The original approach to computing fusion from primary correlators will therefore produce incorrect fusion rules in general. Unfortunately, this approach is still widely employed, without comment, in the non-rational physics literature.

  5. 5.

    To be precise, Gaberdiel showed how to prove it for Virasoro vertex operator algebras in [20, Appendix B]. Coassociativity in general is stated to follow similarly in [21, Sect. 2].

  6. 6.

    This inverse limit approach to fusion has also reappeared in the work of Miyamoto [57] and Tsuchiya and Wood [70].

  7. 7.

    Because we assume that the homogeneous subspaces \(M_{[h]}\) are all finite-dimensional, we may safely ignore all questions regarding the topological nature of these duals.

  8. 8.

    This requirement will clearly need refining when it is necessary (or desirable) to include additional gradings on the V-modules.

  9. 9.

    In the physics literature, it is customary to take z and w to be the first and second insertion points, respectively, in an operator product expansion. In line with this convention, quantities like (28) naturally lead us to speak of P(w)-intertwining maps as opposed to the P(z)-intertwining maps that are ubiquitous in the mathematics literature.

  10. 10.

    We remark that it is these primary fields (and only these primary fields) that are called vertex operators in the physics literature. In the setting of (non-compact) free bosons, they are therefore not fields of the Heisenberg vertex operator algebra. The term vertex operator algebra itself presumably arose in the mathematical literature because early work concentrated on examples related to lattices (compactified free bosons) in which certain vertex operators are promoted to fields of an extended vertex operator algebra.

  11. 11.

    Here, we choose to act on \(m_2\), rather than \(m_1\), in order to keep in line with the vertex-algebraic generalisation to follow.

  12. 12.

    We recall that the action of a P(w)-intertwining map on \(\psi _1 \otimes \psi _2\) is (a projection of) \(\psi _1(w) \psi _2\), in physics notation, explaining this specialisation of insertion points.

  13. 13.

    In terms of the classification of real simple Lie algebras using involutions, the definition (51) corresponds to the split real form, while the adjoint familiar to physicists is associated with the compact real form.

  14. 14.

    The alert reader will notice that this double dual construction is overkill here because the fields Y(a, z) are independent of z. Nevertheless, we feel it helps to unpack this abstract machinery in the simplest case and see that it works. We shall consider a less straightforward example in the next section.

  15. 15.

    In fact, they took this a step further and discussed quotients by actions of fairly arbitrary subalgebras of the mode algebra of V. Appropriate filtrations by such subalgebras then lead to a consistent framework in which one can evaluate the action of any given mode. The need for quite exotic filtrations is best exemplified by referring to the rather difficult computations that arise when studying fusion products for modules over non-rational affine vertex operator algebras, see [24, 66].

  16. 16.

    We caution that despite this formula, an explicit construction of the fusion product using P(w)-compatibility conditions, among other things, is required in the work of Huang and Lepowsky [36,37,38] in order to build a braided tensor structure.

References

  1. Abe, T., Buhl, G., Dong, C.: Rationality, regularity, and \(C_2\)-cofiniteness. Trans. Am. Math. Soc. 356(8), 3391–3402 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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/9511032

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  8. Creutzig, T., Kanade, S., McRae, R.: Tensor categories for vertex operator superalgebra extensions. arXiv:1705.05017 [math.QA]

  9. Creutzig, T., Ridout, D.: Logarithmic conformal field theory: beyond an introduction. J. Phys. A 46, 494006 (2013). arXiv:1303.0847 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  11. Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Mathematics, vol. 112. Birkhäuser Boston Inc, Boston (1993)

    Book  MATH  Google Scholar 

  12. Dong, C., Li, H., Mason, G.: Vertex operator algebras and associative algebras. J. Algebra 206, 67–96 (1998). arXiv:q-alg/9612010

    Article  MathSciNet  MATH  Google Scholar 

  13. Eberle, H., Flohr, M.: Virasoro representations and fusion for general augmented minimal models. J. Phys. A 39, 15245–15286 (2006). arXiv:hep-th/0604097

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feigin, B., Nakanishi, T., Ooguri, H.: The annihilating ideals of minimal models. Int. J. Mod. Phys. A 7, 217–238 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Frenkel, E., Ben-Zvi, D.: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs, vol. 88. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  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. Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster, Pure and Applied Mathematics, vol. 134. Academic Press, Boston (1988)

    MATH  Google Scholar 

  19. Frenkel, I., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  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/9307183

    Article  MathSciNet  MATH  Google Scholar 

  21. Gaberdiel, M.: Fusion rules of chiral algebras. Nucl. Phys. B 417, 130–150 (1994). arXiv:hep-th/9309105

    Article  MathSciNet  MATH  Google Scholar 

  22. Gaberdiel, M.: Fusion of twisted representations. Int. J. Mod. Phys. A 12, 5183–5208 (1997). arXiv:hep-th/9607036

    Article  MathSciNet  MATH  Google Scholar 

  23. Gaberdiel, M.: An introduction to conformal field theory. Rep. Prog. Phys. 63, 607–667 (2000). arXiv:hep-th/9910156

    Article  Google Scholar 

  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/0105046

    Article  MathSciNet  MATH  Google Scholar 

  25. Gaberdiel, M., Kausch, H.: Indecomposable fusion products. Nucl. Phys. B477, 293–318 (1996). arXiv:hep-th/9604026

    Article  MathSciNet  Google Scholar 

  26. Gaberdiel, M., Kausch, H.: A rational logarithmic conformal field theory. Phys. Lett. B 386, 131–137 (1996). arXiv:hep-th/9606050

    Article  MathSciNet  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  30. Gurarie, V.: Logarithmic operators in conformal field theory. Nucl. Phys. B 410, 535–549 (1993). arXiv:hep-th/9303160

    Article  MathSciNet  MATH  Google Scholar 

  31. Huang, Y.Z.: On the applicability of logarithmic tensor category theory. arXiv:1702.00133 [math.QA]

  32. Huang, Y.Z.: Two-Dimensional Conformal Geometry and Vertex Operator Algebras. Progress in Mathematics, vol. 148. Birkhäuser, Boston (1997)

    MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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/9401119

    Chapter  MATH  Google Scholar 

  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/9309076

    Article  MathSciNet  MATH  Google Scholar 

  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/9309159

    Article  MathSciNet  MATH  Google Scholar 

  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/9505018

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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/0409364

    Article  MathSciNet  MATH  Google Scholar 

  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. Huang, Y.Z., Yang, J.: Logarithmic intertwining operators and associative algebras. J. Pure Appl. Algebra 216, 1467–1492 (2012). arXiv:1104.4679 [math.QA]

    Article  MathSciNet  MATH  Google Scholar 

  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. Kac, V.: Vertex Algebras for Beginners. University Lecture Series, vol. 10. American Mathematical Society, Providence (1996)

    MATH  Google Scholar 

  45. Kazhdan, D., Lusztig, G.: Affine Lie algebras and quantum groups. Int. Math. Res. Not. 1991, 21–29 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras I. J. Am. Math. Soc. 6, 905–947 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras II. J. Am. Math. Soc. 6, 949–1011 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras III. J. Am. Math. Soc. 7, 335–381 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kazhdan, D., Lusztig, G.: Tensor structures arising from affine Lie algebras IV. J. Am. Math. Soc. 7, 383–453 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kytölä, K., Ridout, D.: On staggered indecomposable Virasoro modules. J. Math. Phys. 50, 123503 (2009). arXiv:0905.0108 [math-ph]

    Article  MathSciNet  MATH  Google Scholar 

  51. Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and their Representations. Progress in Mathematics, vol. 227. Birkhäuser, Boston (2004)

    Book  MATH  Google Scholar 

  52. Li, H.: Representation theory and tensor product theory for vertex operator algebras. Ph.D. Thesis, Rutgers University (1994). arXiv:hep-th/9406211

  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/9706012

    Article  MathSciNet  MATH  Google Scholar 

  54. Mathieu, P., Ridout, D.: From percolation to logarithmic conformal field theory. Phys. Lett. B 657, 120–129 (2007). arXiv:0708.0802 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B 212, 451–460 (1988)

    Article  MathSciNet  Google Scholar 

  59. Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123, 177–254 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  61. Nahm, W.: Quasirational fusion products. Int. J. Mod. Phys. B 8, 3693–3702 (1994). arXiv:hep-th/9402039

    Article  MATH  Google Scholar 

  62. Pearce, P., Rasmussen, J., Zuber, J.B.: Logarithmic minimal models. J. Stat. Mech. 0611, P11017 (2006). arXiv:0607232 [hep-th]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  64. Read, N., Saleur, H.: Associative-algebraic approach to logarithmic conformal field theories. Nucl. Phys. B 777, 316–351 (2007). arXiv:hep-th/0701117

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  68. Ridout, D., Wood, S.: The Verlinde formula in logarithmic CFT. J. Phys. Conf. Ser. 597, 012065 (2015). arXiv:1409.0670 [hep-th]

    Article  Google Scholar 

  69. Rohsiepe, F.: On reducible but indecomposable representations of the Virasoro algebra. arXiv:hep-th/9611160

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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]

    Article  MathSciNet  MATH  Google Scholar 

  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. Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Am. Math. Soc. 9, 237–302 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Denver, Denver, CO, 80208, USA

    Shashank Kanade

  2. School of Mathematics and Statistics, University of Melbourne, Parkville, VIC, 3010, Australia

    David Ridout

Authors
  1. Shashank Kanade
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Ridout
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shashank Kanade .

Editor information

Editors and Affiliations

  1. Faculty of Science and Department of Mathematics, University of Zagreb, Zagreb, Croatia

    Dražen Adamović

  2. Department of Mathematics, Sapienza University of Rome, Rome, Italy

    Paolo Papi

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kanade, S., Ridout, D. (2019). NGK and HLZ: Fusion for Physicists and Mathematicians. In: Adamović, D., Papi, P. (eds) Affine, Vertex and W-algebras. Springer INdAM Series, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-030-32906-8_7

Download citation

Publish with us

Policies and ethics

Search

Navigation