Study of Planar Models in Quantum Mechanics, Field theory and Gravity

TL;DR

This paper explores topological structures like instantons and monopoles in quantum mechanics, field theory, and gravity, focusing on self-dual Chern-Simons theories, duality symmetries, and higher derivative models in (2+1) dimensions.

Contribution

It extends the analysis of topological and duality properties from quantum mechanical models to (2+1)D field theories and gravity, including higher derivative Chern-Simons terms.

Findings

Demonstrated duality symmetry in quantum mechanical models.

Analyzed higher order derivative Chern-Simons terms and their effects.

Explored spin 2 tensor models in linearized gravity.

Abstract

Instantons, monopoles and vortices have become paradigms of topological structures in field theory and quantum mechanics, with important applications in particle physics, astrophysics, condensed matter physics and mathematics. We have discussed here the self-dual Chern-Simons theory specially in (2+1) dimensions. we start with a relevant topological quantum mechanical model (such as Landau problem consisting of two basic chiral oscillators) and extrapolate the analysis to (2+1)dimensional vector field theory. Aspects of selfdual symmetry in topologically massive gravity model were also considered using three different approaches. We have demonstrated how duality symmetric (or chiral) actions are already present in the quantum mechanical examples such as in usual harmonic oscillator. Using the chiral oscillator form, we will briefly develop the key concepts of the soldering mechanism. We have also discussed the non commutative property of such quantum models. Models involving higher order derivative of Abelian CS-term in (2+1)dimensions, specially the leading third order derivative Chern-Simons term were studied; Inclusion of this term with usual Maxwell term or with CS term or to both of these terms reveals many interesting observations. We have also described the spin 2 tensor models which appear in discussions of linearised gravity in (2 + 1) dimensions.