Semiflat Orbifold Projections

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 $VU = e^{2\pi i\theta} UV$.