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arXiv 2504.03391hep-thmath-phmath.MP
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论从规范联络构型空间上的几何构造中涌现的几乎交换谱三元组

On the emergence of an almost-commutative spectral triple from a geometric construction on a configuration space

Johannes Aastrup, Jesper M. Grimstrup

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20 pages
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AI总结：

本文证明在半经典极限下，规范联络构型空间上的几何构造可涌现出几乎交换谱三元组，该结构涉及HD-代数、带有双重费米子结构的Hilbert空间及空间Dirac算子。

摘要：

我们证明了几乎交换谱三元组的结构在半经典极限下，从规范联络构型空间上的几何构造中涌现出来。该几何构造类似于一个带有 Dirac 算子的谱三元组，该 Dirac 算子在构型空间上与所谓的 $\mathbf{HD}$-代数相互作用。$\mathbf{HD}$-代数是构型空间上的算子值函数代数，由底层流形上向量场流沿路的平行移动生成。在半经典极限下，$\mathbf{HD}$-代数产生了一个几乎交换代数，其中有限因子取决于 $\mathbf{HD}$-代数的表示以及半经典状态所局域化的构型空间上的点。有趣的是，我们发现几乎交换代数作用的 Hilbert 空间带有一个双重费米子结构，这类似于在标准模型的非交换表述中发现的费米子加倍现象。最后，涌现出的几乎交换代数与在半经典极限中涌现的空间 Dirac 算子发生相互作用。这种相互作用涉及几乎交换代数的两个因子。

英文摘要：

We show that the structure of an almost-commutative spectral triple emerges in a semi-classical limit from a geometric construction on a configuration space of gauge connections. The geometric construction resembles that of a spectral triple with a Dirac operator on the configuration space that interacts with the so-called $\mathbf{HD}$-algebra, which is an algebra of operator-valued functions on the configuration space, and which is generated by parallel-transports along flows of vector-fields on the underlying manifold. In a semi-classical limit the $\mathbf{HD}$-algebra produces an almost-commutative algebra where the finite factor depends on the representation of the $\mathbf{HD}$-algebra and on the point in the configuration space over which the semi-classical state is localized. Interestingly, we find that the Hilbert space, in which the almost-commutative algebra acts, comes with a double fermionic structure that resembles the fermionic doubling found in the noncommutative formulation of the standard model. Finally, the emerging almost-commutative algebra interacts with a spatial Dirac operator that emerges in the semi-classical limit. This interaction involves both factors of the almost-commutative algebra.

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