Metaplectic transformations and finite group actions on noncommutative tori

TL;DR

This paper explores how metaplectic transformations can extend K-theory classes of Heisenberg modules to noncommutative orbifolds formed by finite group actions on higher-dimensional noncommutative tori, providing new insights into their K-theory structure.

Contribution

It introduces a novel approach using metaplectic transformations to analyze K-theory classes on noncommutative tori under finite group actions, extending previous two-dimensional results to higher dimensions.

Findings

Extended K-theory classes to noncommutative orbifolds.

Described generators of K-groups for specific group actions.

Provided a framework for higher-dimensional noncommutative tori.

Abstract

In this article we describe extensions of some K-theory classes of Heisenberg modules over higher-dimensional noncommutative tori to projective modules over crossed products of noncommutative tori by finite cyclic groups, aka noncommutative orbifolds. The two dimensional case was treated by Echterhoff, L\"uck, Phillips and Walters. Our approach is based on the theory of metaplectic transformations of the representation theory of the Heisenberg group. We also describe the generators of the K-groups of the crossed products of flip actions by $\mathbb{Z}_2$ on 3-dimensional noncommutative tori.