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Compact hypergroups from discrete subfactors
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-03-18 , DOI: 10.1016/j.jfa.2021.109004
Marcel Bischoff , Simone Del Vecchio , Luca Giorgetti

Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of subfactors of this type come from taking a minimal action of a compact group on a factor and considering the fixed point subalgebra.

We show that to every irreducible local discrete subfactor NM of type III there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on M by unital completely positive (ucp) maps and which gives N as fixed points. To show this, we establish a duality pairing between the set of all N-bimodular ucp maps on M and a certain commutative unital C-algebra, whose spectrum we identify with the compact hypergroup.

If the subfactor has depth 2, the compact hypergroup turns out to be a compact group. This rules out the occurrence of compact quantum groups acting as global gauge symmetries in local conformal field theory.



中文翻译:

来自离散子因子的紧凑超群

在冯·诺依曼代数环境中,通过子因子网络描述了手性共形场理论的共形包含物,或更一般地,量子场论的包含物,如果考虑非理性理论,则可能具有无限的琼斯指数。考虑到这种情况,我们在纯子因数理论背景下研究了一类带有附加交换性约束的编织离散子因数,我们称其为局部性,它对应于量子场中像空距离处场算子之间的交换关系。理论。这种类型的子因子的示例来自对一个因子采取紧致组的最小作用并考虑了定点子代数。

我们表明,对于每个不可约局部离散子因子 ñ中号对于III型,存在一个相关的规范紧致超群(子因子的不变式)中号 通过单位完全正(ucp)映射,这给出了 ñ作为固定点。为了说明这一点,我们在所有集合之间建立了对偶配对ñ-双模ucp映射在 中号 和某个可交换的单位 C-代数,我们用紧致超群来确定其频谱。

如果子因子的深度为2,则紧凑超群证明是紧凑群。这排除了在局部共形场理论中充当整体规范对称性的紧凑量子群的出现。

更新日期:2021-03-24
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