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Algebra objects in direct limit completions of compact Lie group duals and the classification of c=1 vertex operator algebras

  • Sebastiano Carpi
  • Tiziano Gaudio
  • Luca Giorgetti
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Abstract

Let G be a complex reductive affine algebraic group with its symmetric tensor category π’žG of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion Ind(π’žG) of π’žG is isomorphic to the algebra π’ͺ(G/H) of regular functions on the homogeneous space G/H, for some reductive algebraic subgroup H of G. Then we apply this result to the theory of vertex operator algebra extensions. In particular, assuming the strong rationality of the $ \operatorname{A}_5$-orbifold of the vertex operator algebra Vβ„’2 associated with the rank-one root lattice β„’2:= √2β„€, we classify all the not necessarily rational simple CFT type preunitary vertex operator algebra extensions of the simple unitary Virasoro vertex operator algebra L(1,0) with central charge c=1 satisfying a certain spectrum condition. This result is the vertex operator algebra analogue of a conformal net result by Feng Xu. Every strongly rational preunitary vertex operator algebra extension of L(1,0) satisfies the above spectrum condition because of the congruence subgroup modularity property of its characters. As a consequence, we get a complete classification result for strongly rational c=1 vertex operator algebras, up to the strong rationality of Vβ„’2A5.

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Cite

BibTeXRIS
BibTeX
@misc{carpi2026algebra,
  title = {{Algebra objects in direct limit completions of compact Lie group duals and the classification of $c=1$ vertex operator algebras}},
  author = {Sebastiano Carpi and Tiziano Gaudio and Luca Giorgetti},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00347},
  eprint = {2609.00347},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00347},
}
RIS
TY  - UNPB
TI  - Algebra objects in direct limit completions of compact Lie group duals and the classification of c=1 vertex operator algebras
AU  - Sebastiano Carpi
AU  - Tiziano Gaudio
AU  - Luca Giorgetti
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00347
UR  - https://arxiv.org/abs/2609.00347
AB  - Let G be a complex reductive affine algebraic group with its symmetric tensor category π’ž_G of finite-dimensional rational representations. We prove that every simple commutative algebra object with at most countable dimension in the direct limit completion Ind(π’ž_G) of π’ž_G is isomorphic to the algebra π’ͺ(G/H) of regular functions on the homogeneous space G/H, for some reductive algebraic subgroup H of G. Then we apply this result to the theory of vertex operator algebra extensions. In particular, assuming the strong rationality of the $ \operatorname{A}_5$-orbifold of the vertex operator algebra V_β„’_2 associated with the rank-one root lattice β„’_2:= √2β„€, we classify all the not necessarily rational simple CFT type preunitary vertex operator algebra extensions of the simple unitary Virasoro vertex operator algebra L(1,0) with central charge c=1 satisfying a certain spectrum condition. This result is the vertex operator algebra analogue of a conformal net result by Feng Xu. Every strongly rational preunitary vertex operator algebra extension of L(1,0) satisfies the above spectrum condition because of the congruence subgroup modularity property of its characters. As a consequence, we get a complete classification result for strongly rational c=1 vertex operator algebras, up to the strong rationality of V_β„’_2^A_5.
ER  -