# Fractional Virasoro Algebras

## TL;DR

This paper constructs a fractional Virasoro algebra as a central extension of a fractional Witt algebra generated by non-local operators, providing a mathematical foundation for non-local conformal field theories with anomalous dimensions.

## Contribution

It introduces a new fractional Virasoro algebra framework based on non-local operators, extending classical conformal symmetry structures.

## Key findings

- Explicit form of the fractional Virasoro algebra with non-local operators
- Central charge not necessarily constant, with a recursive structure
- Framework applicable to non-local conformal field theories in condensed matter

## 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}+\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.