Free Field Realization of 𝒲-Algebra Associated with Exceptional Lie Algebras

Abstract

We study the free field realization of the 𝒲-algebra associated with the exceptional Lie algebras E₆, E₇, E₈, and F₄. We develop a recursive construction in which a 𝒲-algebra of rank r is obtained from a 𝒲-algebra of rank r−1 together with a free boson. The 𝒲-currents are constructed from the zero commutation relation with the screening charges. The 𝒲E₆/𝒲E₇ algebra is constructed from the 𝒲D₅/𝒲D₆ algebra and is shown to be the same as that realized from the 𝒲A₅/𝒲E₆ algebra, up to a change of the free field basis. The spin-$8$ generator of the 𝒲E₈ algebra is built from the 𝒲D₇ algebra. The recursive construction of the 𝒲BC_r algebras is also studied. We then realize the 𝒲F₄ algebra based on the 𝒲BC₃ algebra. Furthermore, the 𝒲-charges of the generators of the 𝒲E_6,7, 𝒲BC_2,3, and 𝒲F₄ algebras are calculated and expressed in terms of the Casimir invariants.

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BibTeX

@misc{ide2026free,
  title = {{Free Field Realization of $\mathcalW$-Algebra Associated with Exceptional Lie Algebras}},
  author = {Daichi Ide and Katsushi Ito and Shigeki Miyazaki},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14345},
  eprint = {2609.14345},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14345},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Free Field Realization of 𝒲-Algebra Associated with Exceptional Lie Algebras
AU  - Daichi Ide
AU  - Katsushi Ito
AU  - Shigeki Miyazaki
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.14345
UR  - https://arxiv.org/abs/2609.14345
AB  - We study the free field realization of the 𝒲-algebra associated with the exceptional Lie algebras E_6, E_7, E_8, and F_4. We develop a recursive construction in which a 𝒲-algebra of rank r is obtained from a 𝒲-algebra of rank r−1 together with a free boson. The 𝒲-currents are constructed from the zero commutation relation with the screening charges. The 𝒲E_6/𝒲E_7 algebra is constructed from the 𝒲D_5/𝒲D_6 algebra and is shown to be the same as that realized from the 𝒲A_5/𝒲E_6 algebra, up to a change of the free field basis. The spin-$8$ generator of the 𝒲E_8 algebra is built from the 𝒲D_7 algebra. The recursive construction of the 𝒲BC_r algebras is also studied. We then realize the 𝒲F_4 algebra based on the 𝒲BC_3 algebra. Furthermore, the 𝒲-charges of the generators of the 𝒲E_6,7, 𝒲BC_2,3, and 𝒲F_4 algebras are calculated and expressed in terms of the Casimir invariants.
ER  -

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