= Degree-one compression of a chiral sphere phase
{title2=$\operatorname{ind}(PVP)=1$}
Let $P$ project onto a degree-one equivariant line bundle on the complex-oriented round two-sphere, and let $V$ be its positive-to-negative chiral <Dirac operator> phase. The compressed map has parametrix $PV^*P$ modulo compacts. Its domain bundle has degree zero and its target degree two. Their <homogeneous line-bundle representations on the two-sphere> are respectively $E_0\oplus E_2\oplus E_4\oplus\cdots$ and $E_2\oplus E_4\oplus\cdots$, so the <equivariant Fredholm index> is the trivial representation and the ordinary index is one. Reversing chirality negates the index. Compressing the full self-adjoint phase instead gives ordinary index zero.
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