Pull back of Spin$^{\mathbb{C}}$ bundle

Question

Let $M$ be a closed $4$-d Riemannian manifold and $Z$ be its twistor space of $M$, i.e., the bundle of almost complex structures on $M$. Let $V$ be a Spin$^{\mathbb{C}}$ bundle, $V_+$ denote the positive spin bundle. We know $Z$ admits more than one almost complex structure. So it can have canonical Spin$^{\mathbb{C}}$ bundle once we fix an almost complex structure and we can have other Spin$^{\mathbb{C}}$ bundles by twisting it with a complex line bundle. Now let $\pi:Z\rightarrow M$ be the projection. Can we realize $\pi^*(V_+)$ as a sub-bundle of some Spin$^{\mathbb{C}}$ bundle of positive spinors on $Z$ and if yes how?

Answer 1

$\newcommand{\spinors}{\mathbb{S}}$The tangent bundle $TZ$ fits into an exact sequence $$ 0\to T_{/M}Z\to TZ\xrightarrow{\pi_*}\pi^*TM\to 0 $$ where the fiberwise tangent bundle $T_{/M}Z$ is two-dimensional and equipped with a canonical complex structure. Explicitly, the fiber of $Z\to M$ over a point $m$ are the (oriented) complex structures on $T_mM$, which are isomorphic to the projective line of the two-dimensional complex bundle $\spinors_+(M)\to M$ of positive spinors (which you denote by $V_+$). Note that this projective line is defined without the choice of a spin structure, as the action of $\operatorname{Spin}(4)\cong SU(2)\times SU(2)$ on $\mathbb{CP}^1$ factors through projection of the first factor to $PSU(2)$ and therefore descends to $SO(4)$. The isomorphism sends a complex structure to the line of spinors which are annihilated by Clifford multiplication with vectors $v\in T_mM\otimes\mathbb C$ which are antiholomorphic with respect to the chosen complex structure.

The exact sequence splits using the Euclidean metric on $TZ$, whose existence you take for granted when you talk about the spinor bundle of $Z$; to be precise, the splitting can be obtained via the Levi-Civita connection, with the metric on $TZ$ defined as the orthogonal sum of the induced metric on $T_{/M}Z$ and the pulled back metric on $M$. The fiberwise tangent bundle has a complex structure from the description of the fiber as the projective line of a complex vector bundle, and the pullback $\pi^*TM$ has a canonical complex structure by definition of $Z$. It follows that the sum $TZ$ has a (almost) complex structure as well.

The spinor bundle of the sum of two even-dimensional vector spaces is the tensor product of the respective spinor bundles. This shows that $\spinors_+(TZ)\cong \spinors_+(T_{/M}Z)\otimes \pi^*\spinors_+(TM)\oplus\spinors_-(T_{/M}Z)\otimes \pi^*\spinors_-(TM)$. Thus we can realize $\pi^*\spinors_+(TM)$ as a subbundle of the positive spinors on $Z$ if $\spinors_+(T_{/M}Z)$ is trivial, which we can achieve by a judicious choice of Spinᶜ-structure on the vertical tangent bundle, coming from a $PU(2)$-equivariant Spinᶜ-structure on $\mathbb{CP}^1$; it is an easy exercise that such Spinᶜ-structures are in bijection with the integers, with the positive spinor bundle given by the line bundle $\mathcal O(k)$ and the negative spinor bundle given by the line bundle $\mathcal O(k+2)$, which can be made equivariant iff $k$ is even, so that we may take $k = 0$.