L5b: Differential forms as spinors.

We are going to rebrand the differential forms as spinors. In L4b we saw how $\textrm{Spin}(V\oplus V^*)$ sits inside $Cl(V\oplus V^*)$. In other words, the Clifford product is giving a representation $\textrm{Spin}(V\oplus V^*) \to \textrm{End}(Cl(V\oplus V^*))$. One can prove that this representation decomposes into $2^n$ subrepresentations which are all isomorphic to $$Cl(V^*)\det(V) = \wedge^\bullet V^* \det(V)\subset Cl(V\oplus V^*).$$

Each of these is the spin representation of the spin group, a representation that does not descend to $\textrm{SO}(V\oplus V^*)$ (recall that $\textrm{SO}(V\oplus V^*)$ is not simply connected!). The elements of the spin representation are called spinors. Moreover, the spin representation splits into two irreducible half-spin representations: $\wedge^{ev} V^* \det(V)$, $\wedge^{od} V^* \det(V)$, of even and odd spinors.

 Also in L4b, we related the action of $V \oplus V^*$ on $\wedge^\bullet V^*$ with the Clifford product of $Cl(V\oplus V^*)$ on $Cl(V^*)\det(V) \subset Cl(V\oplus V^*)$. Having this in mind, we will call the differential forms (without the factor $\det(V)$ spinors.

One last but very important thing about spinors is that they come equipped with a pairing. For differential forms this map $$(,) : \wedge^\bullet V^* \otimes \wedge^\bullet V^* \to det V$$ is given by $$(\phi, \psi) = (\phi^T\wedge \psi)_{top},$$ where $\phantom{.}^T$ is the anti-automorphism of the Clifford algebra extending the map $(v_1 \ldots v_n)^T=v_n\ldots v_1$ and $top$ denotes the top-degree component of the differential form. Moreover, this pairing is $\textrm{Spin}_0(V\oplus V^*)$-invariant.