We show that it is possible to construct a Virasoro algebra as a central extension of the fractional Witt algebra generated by non-local operators of the form, $L_n^a\equiv\left(\frac{\partial f}{\partial z}\right)^a$ where $a\in {\mathbb R}$. The Virasoro algebra is explicitly of the form, \beq [L^a_m,L_n^a]=A_{m,n}L^a_{m+n}+\delta_{m,n}h(n)cZ^a \eeq where $c$ is the central charge (not necessarily a constant), $Z^a$ is in the center of the algebra and $h(n)$ obeys a recursion relation related to the coefficients $A_{m,n}$. In fact, we show that all central extensions which respect the special structure developed here which we term a multimodule Lie-Algebra, are of this form. This result provides a mathematical foundation for non-local conformal field theories, in particular recent proposals in condensed matter in which the current has an anomalous dimension.

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分数维拉索罗代数

我们表明，可以将 Virasoro 代数构造为由以下形式的非局部运算符生成的分数维特代数的中心扩展，$L_n^a\equiv\left(\frac{\partial f}{\partial z }\right)^a$ 其中 $a\in {\mathbb R}$。Virasoro 代数的形式明确为 \beq [L^a_m,L_n^a]=A_{m,n}L^a_{m+n}+\delta_{m,n}h(n)cZ^a \eeq 其中 $c$ 是中心电荷（不一定是常数），$Z^a$ 位于代数的中心，$h(n)$ 服从与系数 $A_{m,n 相关的递归关系}$。事实上，我们证明了所有尊重这里开发的特殊结构的中心扩展，我们称之为多模李代数，都是这种形式。这一结果为非局部共形场理论提供了数学基础，