arXiv 1301.34802013-01-15math-phhep-thmath.MP

非对易几何中的规范网络

Gauge networks in noncommutative geometry

Matilde Marcolli, Walter D. van Suijlekom

备注：: 30 pages
DOI：: 10.1016/j.geomphys.2013.09.002

摘要：

我们引入规范网络作为自旋网络和格点规范场到几乎对易流形的推广。研究了有限谱三元组范畴中的规范表示（模等价）的位形空间;规范网络在相应的希尔伯特空间中作为标准正交基出现。我们给出了许多规范网络的例子，也超出了众所周知的自旋网络的例子。我们发现这个希尔伯特空间上的哈密顿算子，诱导规范网络对应的C*-代数上的时间演化。给定嵌入自旋流形中的π的谱三元组范畴中的表示，我们在π上定义离散化的Dirac算子。我们计算了这个Dirac算符在四维格点上的谱作用量，发现它在格点规范理论和Higgs场格点系统中都约化为Wilson作用量。因此，在连续极限中，它简化为杨-米尔斯-希格斯系统。对于三维的情况下，我们涉及的光谱作用泛函的Kogut-Susskind哈密顿量。

英文摘要：

We introduce gauge networks as generalizations of spin networks and lattice gauge fields to almost-commutative manifolds. The configuration space of quiver representations (modulo equivalence) in the category of finite spectral triples is studied; gauge networks appear as an orthonormal basis in a corresponding Hilbert space. We give many examples of gauge networks, also beyond the well-known spin network examples. We find a Hamiltonian operator on this Hilbert space, inducing a time evolution on the C*-algebra of gauge network correspondences. Given a representation in the category of spectral triples of a quiver embedded in a spin manifold, we define a discretized Dirac operator on the quiver. We compute the spectral action of this Dirac operator on a four-dimensional lattice, and find that it reduces to the Wilson action for lattice gauge theories and a Higgs field lattice system. As such, in the continuum limit it reduces to the Yang-Mills-Higgs system. For the three-dimensional case, we relate the spectral action functional to the Kogut-Susskind Hamiltonian.

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