Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 8 / 6 / d / Solution

Amplify the chiral phase by the identity on the auxiliary ${\mathbb{C}}^2$, and let $P$ act by multiplication on each chirality. The displayed projection has image line $E_z=\mathbb{C}(1,\overline{z})$. On the chart $w=1/z$ near infinity it has frame $(\overline{w},1)$. On the equator the normalized second frame is $z$ times the first, so its clutching function has winding $+1$. Equivalently it is the conjugate of the tautological degree-$-1$ line. Thus $E$ has degree $+1$ with the complex orientation fixed in part (c).

It is equivariant for the conjugate defining action on the auxiliary ${\mathbb{C}}^2$; this representation is equivalent to the defining representation of $SU(2)$. At $z=0$ its isotropy character under $\mathrm{diag}(e^{it},e^{-it})$ is $e^{-it}$. Tensoring with the positive spinor weight $+1$ gives weight zero on $S^{+}\otimes E$, whereas tensoring with the negative weight $-1$ gives weight $-2$ on $S^{-}\otimes E$.

The compressed map isIts parametrix is $P{V}^{\ast }P$: the two products equal the respective identity $P$ modulo compacts, since every matrix entry of $P$ is a continuous function and the phase commutators in part (c) are compact. Hence this is an equivariant Fredholm operator.

Use the precisely stated homogeneous-bundle multiplicity rule from part (c). The domain, of weight zero, decomposes as $E_0\oplus E_2\oplus E_4\oplus \cdots$. The target, of weight $-2$, decomposes as $E_2\oplus E_4\oplus \cdots$. Part (b) subtracts these multiplicities, leaving only the trivial representation. ThereforeThis computes the index from the actual projection and representation weights without invoking an index theorem that merely supplies the requested number. Reversing the chiral direction gives $-1$. If $V$ instead denotes the full self-adjoint phase on $H_{+}\oplus H_{-}$, its compression is self-adjoint Fredholm and has index zero; the nonzero answer uses the standard chiral interpretation made explicit above.