Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 6

Fractional Virasoro algebras

Pages: 1631 – 1655



Gabriele La Nave (Department of Mathematics, University of Illinois, Urbana, Il., U.S.A.)

Philip W. Phillips (Department of Physics and Institute for Condensed Matter Theory, University of Illinois, Urbana, Il., U.S.A.)


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.

The authors thank the NSF DMR-19-19143 for partial funding of this project.

Published 20 March 2020