Chiral phase of the round-sphere Dirac operator ($V=D^{+}(D^{-}{D}^{+}{)}^{-1/2}$)

For the oriented round unit two-sphere, positive and negative spinor bundles have degrees $-1,+1$. Its Dirac operator has $D^2=C+1/4$ on their homogeneous sections, where $C$ is the angular-momentum Casimir operator; the eigenvalues of $D$ are the nonzero integers. Therefore the chiral phase is a unitary between the two L2 chiral spaces, commuting with $SU(2)$. For a smooth function $f$, $[D,f]=c(df)$ is bounded. The sign-operator resolvent integral expresses its phase commutator as a norm-convergent integral of compact resolvent products bounded by a constant times $(1+t^2{)}^{-1}$. Thus it intertwines multiplication by every continuous function modulo compacts. Its full graded phase is self-adjoint; the chiral phase is the component relevant to nonzero compressed indices.

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 $P{V}^{\ast }P$ 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.