# Duality and Self-duality of the Spin-1 Model in the Covariant Operator Formalism

## TL;DR

This paper rigorously establishes dualities in the spin-1 model in 2+1 dimensions using covariant operator quantization, revealing a physical excitation that generates Maxwell-Chern-Simons theory.

## Contribution

It applies the Kugo-Ojima-Nakanishi formalism to demonstrate dualities and self-duality in the spin-1 model within a covariant operator framework.

## Key findings

- Extraction of a massive physical excitation from the vector and B-fields.
- The excitation generates the Maxwell-Chern-Simons theory.
- Two-point functions confirm the duality structure.

## Abstract

We perform the covariant operator quantization of the spin-$1$ model in $2+1$ spacetime dimensions to rigorously establish its dualities. For this purpose, the Kugo-Ojima-Nakanishi formalism, based on an indefinite metric Hilbert space in the Heisenberg picture, is used. We show that it is possible to extract a massive physical excitation constructed from a linear combination of the vector field $A_{\mu}$ and the $B$-field. In turn, we also show that this excitation generates the Maxwell-Chern-Simons theory. This is achieved by exploring the two-point function of the vector field.