Discrete C*-Frobenius Algebras and Infinite-Index Extensions in Algebraic Quantum Field Theories
Abstract
We introduce discrete C*-Frobenius algebras as categorical data for constructing infinite-index extensions of Möbius covariant nets. The underlying representation is a countable direct sum of simple dualized modules, while the direct sum itself is not necessarily dualizable. Accordingly, rather than requiring a single bounded multiplication morphism, we work with compatible families of bounded left and right multiplication morphisms. Using Bin Gui's theory of categorical extensions, we associate to every discrete C*-Frobenius algebra two (generally non-local) Möbius covariant extensions generated by left and right charged fields, respectively. If the discrete C*-Frobenius algebra is commutative, the two extensions coincide and yield a local Möbius covariant extension. As applications, for every C*-tensor category generated by an invertible object with trivial self-braiding such that its fusion product rule is given by ℤ, we construct two classes of examples: the simple current extension by ℤ and the discrete Longo-Rehren construction.
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BibTeX
@misc{xu2026discrete,
title = {{Discrete $C^*$-Frobenius Algebras and Infinite-Index Extensions in Algebraic Quantum Field Theories}},
author = {Ziyun Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14961},
eprint = {2609.14961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14961},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discrete C^*-Frobenius Algebras and Infinite-Index Extensions in Algebraic Quantum Field Theories
AU - Ziyun Xu
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14961
UR - https://arxiv.org/abs/2609.14961
AB - We introduce discrete C^*-Frobenius algebras as categorical data for constructing infinite-index extensions of Möbius covariant nets. The underlying representation is a countable direct sum of simple dualized modules, while the direct sum itself is not necessarily dualizable. Accordingly, rather than requiring a single bounded multiplication morphism, we work with compatible families of bounded left and right multiplication morphisms. Using Bin Gui's theory of categorical extensions, we associate to every discrete C^*-Frobenius algebra two (generally non-local) Möbius covariant extensions generated by left and right charged fields, respectively. If the discrete C^*-Frobenius algebra is commutative, the two extensions coincide and yield a local Möbius covariant extension. As applications, for every C^*-tensor category generated by an invertible object with trivial self-braiding such that its fusion product rule is given by ℤ, we construct two classes of examples: the simple current extension by ℤ and the discrete Longo-Rehren construction.
ER -