Clifford Algebras and Spinors, Part II: Spin Structures and Dirac Operators

A very good reference for today's material is Dan Freed's (unpublished) notes on Dirac operators, available here.

Spin(n)

Consider the Clifford algebra $Cl(\mathbb E^n)$ as constructed in yesterday's post. Define maps $t, \beta: Cl(\mathbb E^n) \to Cl(\mathbb E^n)$ via
$$(e_1 \cdots e_k)^t = e_k \cdots e_1, \ \beta(e_1 \dots e_k) = (-1)^k e_k \dots e_2 e_1$$
 There is a natural inclusion $\mathbb E^n \hookrightarrow Cl(\mathbb E^n)$. Given $x \in Cl(\mathbb E^n)$ and $v \in \mathbb E^n$, we can consider the product $x v x^t$. In general, this might not be contained in $\mathbb E^n \subset Cl(\mathbb E^n)$.

Definition. We define the group $Pin(n)$ to consist of all those $g \in Cl(\mathbb E^n)$ such that
$$g \beta(g) = 1, \ \ g v \beta(g) \subset \mathbb E^n \ \forall\ v \in \mathbb E^n.$$
Similarly, we define the group $Spin(n)$ to be the subgroup of $Pin(n)$ such that $gg^t = 1$.

Theorem. The natural action of $Pin(n)$ on $\mathbb E^n$ is by othogonal transformations, giving a natural map $Pin(n) \to O(n)$. This map is a double cover. Similarly, $Spin(n)$ is a double cover of $SO(n)$. If $n \geq 2$, $Spin(n)$ is simply connected.

The importance of the spin groups is due to the following basic fact. Suppose that $G$ is a Lie group with Lie algebra $\mathfrak{g}$. Any representation of $G$ induces a representation of $\mathfrak{g}$. However,  given a representation of $\mathfrak{g}$, it is not always possible to integrate it to a representation of $G$. But it is always possible to integrate a representation of $\mathfrak{g}$ to produce a representation of the universal cover of $G$. For $n \geq 2$, $Spin(n)$ is the universal cover of $SO(n)$.

Spin Structures

Let $(M^n, g)$ be a Riemannian manifold. Recall that the frame bundle $O(M)$ is the manifold consisting of pairs $(x, \mathbb{e})$ where $x \in M$ and $\mathbb{e} = \{e_1, \dots, e_n\}$ is an orthonormal frame in $T_x M$. Since the orthogonal group $O(n)$ acts on the set of orthonormal frames, this makes $F(M)$ into a principal $O(n)$ bundle over $M$. Let us assume that $M$ is oriented, so that we may reduce its structure group to $SO(n)$.

Suppose that $V$ is a representation of $SO(n)$. Then we may form the associated bundle $SO(M) \times_{SO(n)} V$, which is a vector bundle over $M$ with structure group $SO(n)$. If we take the defining representation then we obtain the tangent bundle, but of course there are many others. Unfortunately, since $SO(n)$ is not simply connected, not every representation of $\mathfrak{so}_n$ can be integrated to a representation of $SO(n)$. At the level of geometry, this means that in a certain sense there are certain vector bundles over $M$ that are "missing"! Even more disturbing, is that these "missing" bundles appear to be necessary to describe many of the fundamental particles that appear in the standard model--so this has real world consequences. The solution is to equip $M$ with a spin structure.

Definition. A spin structure on $M$ is a principal $Spin(n)$-bundle $Spin(M)$ over $M$ together with a bundle morphism $Spin(M) \to SO(M)$ which is a reduction of structure (i.e., satisfies the obvious axioms).

As you might expect, not every manifold admits a spin structure, and spin structures may not be unique. Loosely speaking, a spin structure is a slightly stronger notion of orientability. Spin structures may always be chosen locally, and the obstruction to consistent gluing is not too difficult to characters as a certain $\mathbb Z_2$ cohomology class, called the second Stiefel-Whitney class.

Spin Connection and Dirac Operators

The reduction of structure $Spin(M) \to SO(M)$ allows us to pull back the Levi-Civita connection on $SO(M)$ to obtain a connection on $Spin(M)$, called the spin connection. Let $S_0$ be the spinor module described in the previous post. Then we may construct the associated bundle
$$S = Spin(M) \times_{Spin(n)} S_0$$
which is called the spinor bundle. Moreover, since $S_0$ is a Clifford module, there is well-defined notion of Clifford multiplication on sections of $S$. We may then define the Dirac operator $\mathcal{D}$ by
$$\mathcal{D} = \sum_{a=1}^n c(e_a) \nabla_{e_a}$$
where $\{e_a\}$ is any orthonormal frame, $\nabla$ is the spin connection, and $c$ denotes Clifford multiplication.

Next time: the Weitzenböck formula, and maybe a vanishing theorem.