Edge observables of the Maxwell-Chern-Simons theory

TL;DR

This paper investigates the boundary properties and edge observables of the Maxwell-Chern-Simons theory, revealing its algebraic structure and quantization without gauge fixing, especially on a 2-disk.

Contribution

It provides a detailed analysis of boundary constraints, identifies edge observables with a $U(1)$ Kac-Moody algebra, and explores their quantum properties without gauge fixing.

Findings

Edge observables form a $U(1)$ Kac-Moody algebra

Explicit solutions on a 2-disk are obtained

Quantum edge states are characterized in the Fock space

Abstract

We analyze the Lagrangian and Hamiltonian formulations of the Maxwell-Chern-Simons theory defined on a manifold with boundary for two different sets of boundary equations derived from a variational principle. We pay special attention to the identification of the infinite chains of boundary constraints and their resolution. We identify edge observables and their algebra [which corresponds to the well-known $U(1)$ Kac-Moody algebra]. Without performing any gauge fixing, and using the Hodge-Morrey theorem, we solve the Hamilton equations whenever possible. In order to give explicit solutions, we consider the particular case in which the fields are defined on a $2$-disk. Finally, we study the Fock quantization of the system and discuss the quantum edge observables and states.