Fractional Virasoro algebras

We show that it is possible to construct a Virasoro algebra as a central extension of the fractional Witt algebra generated by nonlocal operators of the form, $L^a_n \equiv \left ( \frac{\partial f}{\partial z} \right )^a$ where $a \in \mathbb{R}$. The Virasoro algebra is explicitly of the form,\[[ L^a_m, L^a_n ] = A_{m,n} (s) \otimes L^a_{m+n} + \delta_{m,n} h(n)cZ^a\]where $A_{m,n} (s)$ is a specific meromorphic function $c$ 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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The authors thank the NSF DMR-19-19143 for partial funding of this project.

Published 20 March 2020