论文标题
2+1维拓扑订单中的手性中央电荷与基态退化之间的关系
A relation between chiral central charge and ground state degeneracy in 2+1-dimensional topological orders
论文作者
论文摘要
$ d $二维封闭空间$σ^d $上的骨拓制顺序可能具有退化的基础状态。带有不同形状（不同指标）的空间$σ^d $形成一个模量空间$ {\ cal m} _ {σ^d} $。因此，模量空间中每个点上的退化基础状态$ {\ cal m} _ {σ^d} $在$ {\ cal m} _ {σ^d} $上形成一个复杂的向量束。有人认为，所有拓扑的$ d $尺寸的封闭空间的收集完全表征了拓扑顺序。 Using such a point of view, we propose a direct relation between two seemingly unrelated properties of 2+1-dimensional topological orders: (1) the chiral central charge $c$ that describes the many-body density of states for edge excitations (or more precisely the thermal Hall conductance of the edge), (2) the ground state degeneracy $D_g$ on closed genus $g$ surface.我们证明了$ c d_g/2 \ in \ mathbb {z}，\ g \ geq 3 $用于玻璃纤维拓扑订单。我们明确检查了该关系的有效性超过140个简单的拓扑订单。对于费米子拓扑订单，令$ d_ {g，σ}^{e} $（$ d_ {g，σ}}^{o} $）为脱发，即使（奇数）属属属$ g $表面的费用数量，带有旋转结构$σ$。然后，我们有$ 2C d_ {g，σ}^{e} \ in \ mathbb {z} $和$ 2C d_ {g，σ}^{o} \ in \ in \ mathbb {z} $ for $ g \ geq for $ g \ geq 3 $。
A bosonic topological order on $d$-dimensional closed space $Σ^d$ may have degenerate ground states. The space $Σ^d$ with different shapes (different metrics) form a moduli space ${\cal M}_{Σ^d}$. Thus the degenerate ground states on every point in the moduli space ${\cal M}_{Σ^d}$ form a complex vector bundle over ${\cal M}_{Σ^d}$. It was suggested that the collection of such vector bundles for $d$-dimensional closed spaces of all topologies completely characterizes the topological order. Using such a point of view, we propose a direct relation between two seemingly unrelated properties of 2+1-dimensional topological orders: (1) the chiral central charge $c$ that describes the many-body density of states for edge excitations (or more precisely the thermal Hall conductance of the edge), (2) the ground state degeneracy $D_g$ on closed genus $g$ surface. We show that $c D_g/2 \in \mathbb{Z},\ g\geq 3$ for bosonic topological orders. We explicitly checked the validity of this relation for over 140 simple topological orders. For fermionic topological orders, let $D_{g,σ}^{e}$ ($D_{g,σ}^{o}$) be the degeneracy with even (odd) number of fermions for genus-$g$ surface with spin structure $σ$. Then we have $2c D_{g,σ}^{e} \in \mathbb{Z}$ and $2c D_{g,σ}^{o} \in \mathbb{Z}$ for $g\geq 3$.
