NGK and HLZ: Fusion for Physicists and Mathematicians

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


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.


Vertex operator algebras Conformal field theory Tensor categories Fusion 



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.


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Authors and Affiliations

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

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