Fractional quantum Hall effect and $M$-theory

Volume 27, Issue 1 (2023), pp. 1–36

Cumrun Vafa

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Pub. online: 13 July 2023
Type: Article

Published

13 July 2023

13 July 2023

#### Abstract

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.