# Constructing symplectomorphisms between symplectic torus quotients

## TL;DR

This paper explores the relationships between certain singular symplectic quotients derived from torus representations, establishing conditions for symplectomorphism and classifying their equivalence classes with explicit examples.

## Contribution

It identifies a family of torus representations where symplectic quotients are graded regularly symplectomorphic to circle quotients and provides explicit classifications and examples.

## Key findings

- Certain symplectic quotients are graded regularly symplectomorphic to circle quotients.
- Explicit descriptions of symplectic quotients as Poisson differential spaces are provided.
- Examples show that algebraic isomorphisms do not imply symplectomorphism.

## Abstract

We identify a family of torus representations such that the corresponding singular symplectic quotients at the $0$-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic.