Semiflat Orbifold Projections

TL;DR

This paper computes the positive cone of the $K_0$-group for a specific noncommutative orbifold, classifies semiflat projections by topological genus, and shows their traces cover certain rational numbers.

Contribution

It determines the semiflat positive cone of the $K_0$-group for irrational rotation orbifolds and classifies semiflat projections by topological genus and trace values.

Findings

The semiflat positive cone is determined by positive trace classes and two topological invariants.

Semiflat orbifold projections are classified into three topological genera.

Every number in (0,1) intersecting with 2Z + 2Zθ is realized as the trace of a semiflat projection.

Abstract

We compute the semiflat positive cone $K_0^{+SF}(A_\theta^\sigma)$ of the $K_0$-group of the irrational rotation orbifold $A_\theta^\sigma$ under the noncommutative Fourier transform $\sigma$ and show that it is determined by classes of positive trace and the vanishing of two topological invariants. The semiflat orbifold projections are 3-dimensional and come in three basic topological genera: $(2,0,0)$, $(1,1,2)$, $(0,0,2)$. (A projection is called semiflat when it has the form $h + \sigma(h)$ where $h$ is a flip-invariant projection such that $h\sigma(h)=0$.) Among other things, we also show that every number in $(0,1) \cap (2\mathbb Z + 2\mathbb Z\theta)$ is the trace of a semiflat projection in $A_\theta$. The noncommutative Fourier transform is the order 4 automorphism $\sigma: V \to U \to V^{-1}$ (and the flip is $\sigma^2$: $U \to U^{-1},\ V \to V^{-1}$), where $U,V$ are the canonical unitary generators of the rotation algebra $A_\theta$ satisfying $VU = e^{2\pi i\theta} UV$.