Conformal Edge Currents in Chern-Simons Theories

Abstract

We develop elementary canonical methods for the quantization of abelian and nonabelian Chern-Simons actions using well known ideas in gauge theories and quantum gravity. Our approach does not involve choice of gauge or clever manipulations of functional integrals. When the spatial slice is a disc, it yields Witten's edge states carrying a representation of the Kac-Moody algebra. The canonical expression for the generators of diffeomorphisms on the boundary of the disc are also found, and it is established that they are the Chern-Simons version of the Sugawara construction. This paper is a prelude to our future publications on edge states, sources, vertex operators, and their spin and statistics in 3d and 4d topological field theories.

Three-dimensional quantum geometry and black holes

“…Other methods yielding the same result are the W ZW approach followed in [3], and the symplectic method [32]. The idea of looking at first class quantities and their algebra in Chern-Simons theory was first discussed in [28]. The presence of central terms in the algebra of global charges in Chern-Simons theory was first discussed in [33].…”

Absence of the holographic principle in noncommutative Chern-Simons theory

“…In the commutative limit θ → 0, (2.14) are edge variables as q(Ξ) and q(Ξ ) for two distributions Ξ and Ξ with the same boundary values are weakly equivalent (i.e., they only differ by a Gauss law constraint). [6] This is however not the case for nonzero θ. Up to order θ (2.15) where ∆ = Ξ − Ξ , with Ξ and Ξ having the same boundary values.…”

“…(We recall that in commutative Chern-Simons theory (θ = 0) with a boundary, it is sufficient to have gauge transformations vanish at the boundary for gauge invariance, and there is no need to impose conditions on its derivatives. [6])…”

“…Following [6], for the constraint to have meaning when the spatial domain has a boundary one should introduce a distribution function Λ and write it instead according to…”

Global charges in Chern-Simons theory and the 2+1 black hole

“…[3]. In that reference, the authors studied the differentiability properties of the generators of gauge transformations.…”