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It is known that the Lagrangian for the edge states of a Chern-Simons theorydescribes a coadjoint orbit of a Kac-Moody (KM) group with its associatedKirillov symplectic form and group representation. It can also be obtained froma chiral sector of a nonchiral field theory. We study the edge states of theabelian $BF$ system in four dimensions (4d) and show the following results inalmost exact analogy: 1) The Lagrangian for these states is associated with acertain 2d generalization of the KM group. It describes a coadjoint orbit ofthis group as a Kirillov symplectic manifold and also the corresponding grouprepresentation. 2) It can be obtained from with a ``self-dual" or``anti-self-dual" sector of a Lagrangian describing a massless scalar and aMaxwell field [ the phrase ``self-dual" here being used essentially in itssense 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 thatthe edge Lagrangians for $BF$ systems are certain natural generalizations offield theory Lagrangians related to KM groups.Comment: 12 pages, SU-4240-42

EDGE STATES IN 4D AND THEIR 3D GROUPS AND FIELDS