Current Algebra and Conformal Field Theory on a Figure Eight

TL;DR

This paper explores the behavior of a free scalar field on a figure eight network, revealing unique current algebra properties and a modified conformal symmetry structure distinct from traditional circle models.

Contribution

It introduces a new analysis of scalar fields on a figure eight network, deriving current algebra, and demonstrating the absence of full conformal symmetry, with a Virasoro algebra of conserved charges.

Findings

Current currents satisfy Kirchhoff's law at the junction.

Left- and right-moving currents generally do not commute quantum mechanically.

The system has a single conformal invariance, not separate for left and right movers.

Abstract

We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff's law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one dimensional spatial manifold exhibits conformal symmetry, it is natural to ask whether an analogous symmetry can be defined for the figure eight. We find that, unlike in the case of a manifold, the action plus boundary conditions for the network are not invariant under separate conformal transformations associated with left- and right-movers. Instead, the system is, at best, invariant under only a single set of transformations. Its conserved current is also found to satisfy Kirchhoff's law at the junction. We obtain the associated conserved charges, and show that they generate a Virasoro algebra. Its conformal anomaly (central charge) is computed for special values of the parameters characterizing the network.