Symmetry Algebras in Chern-Simons Theories with Boundary: Canonical Approach

TL;DR

This paper analyzes the classical symmetry algebras in boundary Chern-Simons theories using the Dirac canonical approach, revealing how boundary effects modify constraints and lead to Kac-Moody and Virasoro algebras, with implications for holography and AdS/CFT correspondence.

Contribution

It explicitly constructs the classical Kac-Moody and Virasoro algebras in boundary Chern-Simons theories via the Dirac method and clarifies their connection to symplectic reduction and holography.

Findings

Boundary effects turn Gauss law into second-class constraints.

Explicit Dirac brackets yield classical Kac-Moody and Virasoro algebras.

Both Chern-Simons and Yang-Mills-Chern-Simons theories realize holography.

Abstract

I consider the classical Kac-Moody algebra and Virasoro algebra in Chern-Simons theory with boundary within the Dirac's canonical method and Noether procedure. It is shown that the usual (bulk) Gauss law constraint becomes a second-class constraint because of the boundary effect. From this fact, the Dirac bracket can be constructed explicitly without introducing additional gauge conditions and the classical Kac-Moody and Virasoro algebras are obtained within the usual Dirac method. The equivalence to the symplectic reduction method is presented and the connection to the Ba\~nados's work is clarified. It is also considered the generalization to the Yang-Mills-Chern-Simons theory where the diffeomorphism symmetry is broken by the (three-dimensional) Yang-Mills term. In this case, the same Kac-Moody algebras are obtained although the two theories are sharply different in the canonical structures. The both models realize the holography principle explicitly and the pure CS theory reveals the correspondence of the Chern-Simons theory with boundary/conformal field theory, which is more fundamental and generalizes the conjectured anti-de Sitter/conformal field theory correspondence.