Hamiltonian analysis of the noncommutative Chern-Simons theory

TL;DR

This paper performs a Hamiltonian analysis of noncommutative Chern-Simons theory, revealing its topological phase space structure, the quantization condition of the coupling, and the form of the vacuum state related to noncommutative Wess-Zumino-Witten action.

Contribution

It provides a detailed Hamiltonian and geometric quantization analysis of noncommutative Chern-Simons theory, highlighting its topological features and vacuum structure.

Findings

Classical phase space has nontrivial topology

Quantization of the symplectic structure requires Chern-Simons coefficient quantization

The physical Hilbert space is one-dimensional with an explicit vacuum wavefunction

Abstract

In this paper the hamiltonian analysis of the pure Chern-Simons theory on the noncommutative plane is performed. We use the techniques of geometric quantization to show that the classical reduced phase space of the theory has nontrivial topology and that quantization of the symplectic structure on this space is possible only if the Chern-Simons coefficient is quantized. Also we show that the physical Hilbert space of the theory is one dimensional and give an explicit expression for the vacuum wavefunction. This vacuum state is found to be related to the noncommutative Wess-Zumino-Witten action.