Fractional Virasoro Algebras

Abstract

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}+δ_{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.

Cited by in corpus (1)

• Renormalization and conformal invariance of non-local quantum electrodynamics