Edge States in 4D and their 3D Groups and Fields

TL;DR

This paper explores the structure of edge states in 4D abelian and nonabelian BF systems, revealing their relation to generalized Kac-Moody groups and coadjoint orbits, extending known 3D results.

Contribution

It introduces a 2D generalization of Kac-Moody groups for 4D BF systems and links edge state Lagrangians to these groups, extending the understanding of boundary theories.

Findings

Edge states form coadjoint orbits of a 2D generalization of Kac-Moody groups.

Edge Lagrangians derive from self-dual sectors of scalar and Maxwell fields.

Similar structures are found in both abelian and nonabelian BF systems.

Abstract

It is known that the Lagrangian for the edge states of a Chern-Simons theory describes a coadjoint orbit of a Kac-Moody (KM) group with its associated Kirillov symplectic form and group representation. It can also be obtained from a chiral sector of a nonchiral field theory. We study the edge states of the abelian $BF$ system in four dimensions (4d) and show the following results in almost exact analogy: 1) The Lagrangian for these states is associated with a certain 2d generalization of the KM group. It describes a coadjoint orbit of this group as a Kirillov symplectic manifold and also the corresponding group representation. 2) It can be obtained from with a ``self-dual" or ``anti-self-dual" sector of a Lagrangian describing a massless scalar and a Maxwell field [ the phrase ``self-dual" here being used essentially in its sense in monopole theory]. There are similar results for the nonabelian $BF$ system as well. These shared features of edge states in 3d and 4d suggest that the edge Lagrangians for $BF$ systems are certain natural generalizations of field theory Lagrangians related to KM groups.