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A new SL2-action on the rank-two βγ-system

  • Tomoyuki Arakawa
  • Xuanzhong Dai
  • Bailin Song
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Abstract

We construct an explicit rational SL2(ℂ)-action on the underlying vector space of the rank-two βγ-system. The construction is motivated by Creutzig's branching decomposition with respect to L−1(sl2) and a rank-one Heisenberg vertex algebra. We realize the raising and lowering operators as residues of lattice-type fields, prove that they preserve the original βγ-system despite being defined on localizations, and establish their commutation relations and integrability. The resulting action commutes with the affine L−1(sl2)-action and all nonzero Heisenberg modes, while the Heisenberg zero mode serves as its Cartan operator. Each affine multiplicity space is thereby identified with Vn⊗ℱosc, where Vn is the irreducible SL2(ℂ)-module of highest weight n and ℱosc is the Heisenberg oscillator Fock space. Thus the Heisenberg charge decomposition is realized as the weight decomposition of Vn. The infinitesimal action is not by vertex algebra derivations and does not preserve the symmetric conformal grading.

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Cite

BibTeXRIS
BibTeX
@misc{arakawa2026sl2action,
  title = {{A new $SL_2$-action on the rank-two $βγ$-system}},
  author = {Tomoyuki Arakawa and Xuanzhong Dai and Bailin Song},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08000},
  eprint = {2610.08000},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08000},
}
RIS
TY  - UNPB
TI  - A new SL_2-action on the rank-two βγ-system
AU  - Tomoyuki Arakawa
AU  - Xuanzhong Dai
AU  - Bailin Song
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08000
UR  - https://arxiv.org/abs/2610.08000
AB  - We construct an explicit rational SL_2(ℂ)-action on the underlying vector space of the rank-two βγ-system. The construction is motivated by Creutzig's branching decomposition with respect to L_−1(sl_2) and a rank-one Heisenberg vertex algebra. We realize the raising and lowering operators as residues of lattice-type fields, prove that they preserve the original βγ-system despite being defined on localizations, and establish their commutation relations and integrability. The resulting action commutes with the affine L_−1(sl_2)-action and all nonzero Heisenberg modes, while the Heisenberg zero mode serves as its Cartan operator. Each affine multiplicity space is thereby identified with V_n⊗ℱ_osc, where V_n is the irreducible SL_2(ℂ)-module of highest weight n and ℱ_osc is the Heisenberg oscillator Fock space. Thus the Heisenberg charge decomposition is realized as the weight decomposition of V_n. The infinitesimal action is not by vertex algebra derivations and does not preserve the symmetric conformal grading.
ER  -