$\mathbb{Z}_2$格点Gerbe理论
$\mathbb{Z}_2$ Lattice Gerbe Theory
摘要：
$d$维超立方格点上的$2$ -形式阿贝尔和非阿贝尔规范场在过去已经被许多作者讨论过，最近由Lipstein和Reid-Edwards讨论过。在这份说明中，我们记得，这种理论的一个$\mathbb{Z}_2$变体的哈密顿量是最初由韦格纳考虑的广义伊辛模型家族之一。对于这样的“$\mathbb{Z}_2$格点格贝理论”的一般参数可以用来表明，威尔逊表面的相变将发生在$d>3$之间的体积和面积缩放行为。在$3d$中，该模型在对偶下等价于无限耦合模型，并且没有看到过渡，而在$4d$中，该模型与$4d$ Ising模型对偶，并且显示连续过渡。在$5d$中，$\mathbb{Z}_2$格点格点理论在外场存在时是自对偶的，而在$6d$中，它在零外场时是自对偶的。
英文摘要：
$2$-form abelian and non-abelian gauge fields on $d$-dimensional hypercubic lattices have been discussed in the past by various authors and most recently by Lipstein and Reid-Edwards. In this note we recall that the Hamiltonian of a $\mathbb{Z}_2$ variant of such theories is one of the family of generalized Ising models originally considered by Wegner. For such "$\mathbb{Z}_2$ lattice gerbe theories" general arguments can be used to show that a phase transition for Wilson surfaces will occur for $d>3$ between volume and area scaling behaviour. In $3d$ the model is equivalent under duality to an infinite coupling model and no transition is seen, whereas in $4d$ the model is dual to the $4d$ Ising model and displays a continuous transition. In $5d$ the $\mathbb{Z}_2$ lattice gerbe theory is self-dual in the presence of an external field and in $6d$ it is self-dual in zero external field.
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