Field Theory on Noncommutative Space-Time and the Deformed Virasoro Algebra

TL;DR

This paper explores a field theory on a noncommutative cylinder, linking it to a deformed Virasoro algebra, and clarifies the meaning of the algebra's second index as arising from space-time noncommutativity.

Contribution

It establishes a connection between noncommutative space-time models and the deformed Virasoro algebra, providing a physical interpretation for its second index.

Findings

The model on a noncommutative cylinder has a discrete-time evolution.

The Euclidean version is equivalent to a model on the complex q-plane.

The second index in the deformed Virasoro algebra is linked to space-time noncommutativity.

Abstract

We consider a field theoretical model on the noncommutative cylinder which leads to a discrete-time evolution. Its Euclidean version is shown to be equivalent to a model on the complex $q$-plane. We reveal a direct link between the model on a noncommutative cylinder and the deformed Virasoro algebra constructed earlier on an abstract mathematical background. As it was shown, the deformed Virasoro generators necessarily carry a second index (in addition to the usual one), whose meaning, however, remained unknown. The present field theoretical approach allows one to ascribe a clear meaning to this second index: its origin is related to the noncommutativity of the underlying space-time. The problems with the supersymmetric extension of the model on a noncommutative super-space are briefly discussed.