We propose a unifying model for FQHE which on the one hand connects it to recent developments in string theory and on the other hand leads to new predictions for the principal series of experimentally observed FQH systems with filling fraction $\nu=\frac{n}{2n \pm 1}$ as well as those with $\nu=\frac{m}{m+2}$. Our model relates these series to minimal unitary models of the Virasoro and super-Virasoro algebra and is based on $SL(2,\mathbf{C})$ Chern–Simons theory in Euclidean space or $SL(2,\mathbf{R}) \times SL(2, \mathbf{R})$ Chern–Simons theory in Minkowski space. This theory, which has also been proposed as a soluble model for $2+1$ dimensional quantum gravity, and its $\mathrm{N}=1$ supersymmetric cousin, provide effective descriptions of FQHE. The principal series corresponds to quantized levels for the two $SL(2,\mathbf{R})$’s such that the diagonal $SL(2,\mathbf{R})$ has level $1$. The model predicts, contrary to standard lore, that for principal series of FQH systems the quasiholes possess non-abelian statistics. For the multi-layer case we propose that complex ADE Chern–Simons theories provide effective descriptions, where the rank of the ADE is mapped to the number of layers. Six dimensional $(2,0)$ ADE theories on the Riemann surface $\Sigma$ provides a realization of FQH systems in $\mathrm{M}$-theory. Moreover we propose that the $\mathrm{q}$-deformed version of Chern–Simons theories are related to the anisotropic limit of FQH systems which splits the zeroes of the Laughlin wave function. Extensions of the model to $3+1$ dimensions, which realize topological insulators with non-abelian topologically twisted Yang–Mills theory is pointed out.

中文翻译：

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分数量子霍尔效应和 $M$ 理论

我们提出了一个 FQHE 的统一模型，一方面将其与弦理论的最新发展联系起来，另一方面对填充分数 $\nu=\frac{n} 的实验观察到的 FQH 系统的主要系列进行新的预测{2n \pm 1}$ 以及 $\nu=\frac{m}{m+2}$ 的。我们的模型将这些级数与 Virasoro 和超 Virasoro 代数的最小酉模型联系起来，并基于欧几里德空间中的 $SL(2,\mathbf{C})$ Chern–Simons 理论或 $SL(2,\mathbf{R }) \times SL(2, \mathbf{R})$ 闵可夫斯基空间中的陈-西蒙斯理论。该理论也被提出作为 $2+1$ 维量子引力的可溶模型，及其 $\mathrm{N}=1$ 超对称表亲，提供了 FQHE 的有效描述。主序列对应于两个 $SL(2, \mathbf{R})$ 使得对角线 $SL(2,\mathbf{R})$ 具有水平 $1$。与标准知识相反，该模型预测，对于 FQH 系统的主要系列，准空洞具有非阿贝尔统计量。对于多层情况，我们建议复杂的 ADE Chern-Simons 理论提供有效的描述，其中 ADE 的秩映射到层数。黎曼曲面 $\Sigma$ 上的六维 $(2,0)$ ADE 理论提供了 $\mathrm{M}$ 理论中 FQH 系统的实现。此外，我们提出陈-西蒙斯理论的 $\mathrm{q}$ 变形版本与 FQH 系统的各向异性极限有关，该系统分裂了劳克林波函数的零点。指出该模型扩展到$3+1$维度，用非阿贝尔拓扑扭曲杨-米尔斯理论实现了拓扑绝缘体。