Fractons emerge as charges with reduced mobility in a new class of gauge theories. Here, we generalise fractonic theories of $U(1)$ type to what we call $(k,n)$-fractonic Maxwell theory, which employs symmetric rank-$n$ tensors of $k$-forms (rank-$k$ antisymmetric tensors) as vector potentials''. The generalisation, valid in any spatial dimension $d$, has two key manifestations. First, the objects with mobility restrictions extend beyond simple charges to higher order multipoles (dipoles, quadrupoles, $\ldots$) all the way to $n^th$-order multipoles, which we call the order-$n$ fracton condition. Second, these fractonic charges themselves are characterized by tensorial densities of $(k-1)$-dimensional extended objects. For any $(k,n)$, the theory can be constructed to have a gaplessphoton modes’’ with dispersion $\omega \sim |q{|}^z$, where the integer $z$ can range from $1$ to $n$.

中文翻译：

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（k，n）-分形麦克斯韦理论

在新的量规理论中，作为流动性降低的电荷而出现了分形。在这里，我们概括了$ü（\mathrm{1个}）$ 输入我们所谓的 $（ķ，ñ）$分形麦克斯韦理论，采用对称秩$ñ$ 的张量 $ķ$-forms（等级-$ķ$反对称张量）作为vector potentials''. The generalisation, valid in any spatial dimension $d$, has two key manifestations. First, the objects with mobility restrictions extend beyond simple charges to higher order multipoles (dipoles, quadrupoles, $\ldots$) all the way to $n^th$-order multipoles, which we call the order-$n$ fracton condition. Second, these fractonic charges themselves are characterized by tensorial densities of $(k-1)$-dimensional extended objects. For any $(k,n)$, the theory can be constructed to have a gapless光子模$\omega 〜|q{|}^{ž}$，其中整数 $ž$ 范围从 $\mathrm{1个}$ 至 $ñ$。