Seiberg Witten Theory 1: Spin, Spinc structures

Seiberg-Witten theory has been one of the most useful tools for understanding exotic 4-manifolds. It has been around for awhile now, but it involves a lot of geometric definitions and analytic proofs so it is difficult to approach as a grad student. Ciprian Manolescu’s recent disproof of the triangulation conjecture has brought Seiberg-Witten Floer homology into the spotlight again, which has convinced a group of us at UT Austin to go through and learn this stuff carefully from the beginning. Since this seems potentially useful to a wider audience, I’m posting some of what we have gone through. The first 2 or 3 posts will be loosely based on talks in our learning seminar given by Cagri, Richard, and me, and the written reference we have relied most on is a book by Nicolaescu called Notes on Seiberg-Witten Theory.

We started from the beginning defining curvature and connections on vector bundles and principal bundles. That part is a bit definitional/computational and not too blog friendly, so I’ll just include a link: ConnectionsCurvature. Here I’ll start with spin and spinc structures from a fairly topological perspective. In the next post I’ll talk about their relations to Clifford structures and Dirac operators, and eventually get to connections on these bundles and various associated bundles, so we can define the Seiberg Witten invariants.

Spin and Spinc structures on 3 and 4-manifolds

We can specify any vector bundle , (specifically we will be concerned with the tangent bundle) over a manifold M with fiber $V^n$ uniquely up to isomorphism by cocycle data $(\{U_{\alpha}\}, \{g_{\beta \alpha}\})$ , where $\{U_\alpha\}$ is an open cover of M, and $g_{\beta \alpha}: U_\alpha \cap U_\beta \to GL(V^n)$ satisfy the “cocycle condition”
$g_{\alpha \beta}(x)g_{\beta \gamma}(x)g_{\gamma \alpha}(x)=I_V$

The vector bundle is formed by taking the disjoint union over all $\alpha$ of the trivial bundles $U_\alpha \times V$ and quotienting out by identifications of the fibers $x\times V\subset U_\alpha\times V$ and $x\times V \subset U_\beta \times V$ above a point $x\in U_\alpha \cap U_\beta$ by the isomorphism $g_{\beta \alpha}(x)$ . The cocycle condition ensures that everything glues up coherently to a global vector bundle. Similarly a principal G-bundle can be specified by cocycle data $(\{U_\alpha\}, \{g_{\beta \alpha}\})$ where now $g_{\beta \alpha}: U_\alpha\cap U_\beta \to G$ and $g_{\alpha \beta}(x)g_{\beta \gamma}(x)g_{\gamma \alpha}(x)=1_G$ for $x\in U_\alpha\cap U_\beta$ . It is frequently useful for computations to think about vector bundles and principal bundles through these local trivializations, keeping track of the gluing maps.

Given an orientable n-manifold, we consider its tangent bundle described by gluing maps $\{g_{\alpha\beta}: U_\alpha\cap U_\beta\to GL(n,\mathbb{R})\}$ . By choosing a metric and orientation, we can reduce its structure group to $SO(n)$ , meaning we can assume the maps $g_{\alpha\beta}$ have image in $SO(n)$ . We can use these gluing maps to construct a principal $SO(n)$ bundle (the bundle of orthonormal frames) whose associated bundle is the tangent bundle. Spin and Spinc structures are types of lifts of this principal bundle.

For all $n\geq 3$ , $\pi_1(SO(n))=\mathbb{Z}/2$ so $SO(n)$ has a double cover, which conveniently, is also a Lie group called $Spin(n)$ .

One can show explicitly that
$Spin(3)=SU(2)$
$Spin(4)=SU(2)\times SU(2)$
by constructing 2-fold covers $SU(2)\to SO(3)$ and $SU(2)\times SU(2)\to SO(4)$ using the following idea. Identify $\mathbb{R}^4$ with the quaternions. Observe that $SU(2)\cong S^3$ can be identified with the unit quaternions, and that the unit quaternions act by conjugation on the imaginary quaternions in a norm-preserving way. This action by conjugation induces a map from $SU(2)$ to $SO(3)$ whose kernel is $\pm 1$ . Similarly an action of $SU(2)\times SU(2)$ on the quaternions can be defined by $(P,Q)\cdot X = PXQ^{-1}$ . Again one can check this action is orthogonal so there is an induced map $SU(2)\times SU(2)\to SO(4)$ whose kernel is two points.

A spin structure on an n-manifold M is a $Spin(n)$ bundle over M which lifts the principal $SO(n)$ bundle associated to $TM$ .

We can also define $Spin^c(n)=(U(1)\times Spin(n)/\{\pm 1\}\to SO(n)$ , and similarly define $Spin^c$ structures on a manifold.

On a 4-manifold, a spin structure gives rise to two rank 2 complex associated bundles as follows. $Spin(4)=SU(2)\times SU(2)$ has two natural projection maps onto $SU(2)$ , $pr_1,pr_2$ . These can be viewed as representations, so if $\widetilde{P}$ is a $Spin$ structure on $M^4$ , we obtain two complex rank 2 associated bundles
$S^\pm = \widetilde{P}\times_{pr_{1,2}}\mathbb{C}^2$
We will see these representations again in the context of Clifford structures, when we discuss how $Spin(n)$ sits inside a Clifford algebra.

If $M^4$ admits a $spin^c$ structure, we also have two projections:
$pr_{1,2}: Spin^c(4)=U(1)\times SU(2)\times SU(2)/\{\pm 1\}\to U(2)$
These similarly admit two complex rank two associated bundles $W^{\pm}=\widetilde{P}\times_{pr_{1,2}}\mathbb{C}^2$ . In this case $\bigwedge^2W^+=\bigwedge^2W^-$ .

Obstructions to Spin and Spinc structures:

The obstruction to a $Spin$ structure is the Stiefel-Whitney class $w_2(M)$ , which can be viewed as a Cech cohomology class as follows. If $\{g_{\alpha\beta}: U_\alpha\cap U_\beta\to SO(n)\}$ are the gluing maps for the $SO(n)$ bundle defining the structure group for $TM$ , each map $g_{\alpha\beta}$ has exactly two lifts to maps $h_{\alpha\beta}: U_{\alpha}\cap U_\beta\to Spin(n)$ . Then
$w_{\alpha\beta\gamma}:=h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}\in \mathbb{Z}/2=\ker(Spin(n)\to SO(n))$
This collection $\{w_{\alpha\beta\gamma}\}$ is a Cech 2-cycle and so it represents a Cech cohomology class, called the second Stiefel-Whitney class $w_2(M)$ . When $h_{\alpha\beta}$ satisfy the cocycle condition: $w_{\alpha\beta\gamma}=1$ , this cohomology class vanishes and the $h_{\alpha\beta}$ define a spin structure.

Because $Spin^c(n)=(Spin(n)\times S^1)/\{\pm (1,1)\}$ , a $Spin^c$ structure can be specified by cocycle data relating to the gluing maps $\{g_{\alpha\beta}\}$ for the $SO(n)$ structure bundle of the tangent bundle. This cocycle data is given by a collection of maps
$h_{\alpha\beta}: U_\alpha\cap U_\beta \to Spin(n)$
$z_{\alpha\beta}: U_\alpha\cap U_\beta \to S^1$
satisfying two requirements

(1) $\rho(h_{\alpha\beta})=g_{\alpha\beta}$ (the $Spin^c$ structure is a cover of the $SO(n)$ bundle)
(2) $(h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}, z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha})\in\{(1,1),(-1,-1)\}$ (cocycle condition)

Focusing on the maps $z_{\alpha\beta}$ , we almost get a $U(1)$ bundle except that instead of the cocycle condition we have that $z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha}=\pm 1$ . Because $U(1)=S^1$ is abelian, by looking instead at $\lambda_{\alpha\beta}=z_{\alpha\beta}^2$ , we obtain gluing maps satisfying the cocycle condition, so they form a genuine $U(1)$ bundle, or equivalently a complex line bundle L. We can calculate $c_1(L)$ as follows. Write $\lambda_{\alpha\beta}=e^{i\theta_{\alpha\beta}}$ . Then
$n_{\alpha\beta\gamma}=\frac{1}{2\pi}(\theta_{\alpha\beta}+\theta_{\beta\gamma}+\theta_{\gamma\alpha})$
is an integer (since $e^{i(\theta_{\alpha\beta}+\theta_{\beta\gamma}+\theta_{\gamma\alpha})})=1$ ), and the $n_{\alpha\beta\gamma}$ define the cocycle representing $c_1(L)\in H^2(M;\mathbb{Z})$ .

There is a relationship between $w_{\alpha\beta\gamma}:= h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}$ given by requirement (2) above. Namely,
$w_{\alpha\beta\gamma} = sign(z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha}) = n_{\alpha\beta\gamma} \mod 2$
(where $w_{\alpha\beta\gamma}\in \{-1,1\}=\mathbb{Z}/2$ where the group structure is multiplication, and $n_{\alpha\beta\gamma}\in \{0,1\}=\mathbb{Z}/2$ where the group structure is addition.) Since $\{w_{\alpha\beta\gamma}\}$ represents $w_2(M)$ , and $\{n_{\alpha\beta\gamma}\}$ represents $c_1(L)$ , the existence of a $Spin^c$ structure implies $w_2(M)\equiv c_1(L) \mod 2$ .

For any $Spin^c$ structure $\sigma$ the associated line bundle L constructed above is called $det(\sigma)$ . It is not obvious from this definition that this is the determinant of any vector bundle, but in fact it will be the determinant of the spinor bundle associated to the Spinc structure via the spinor representation we will discuss in the next post.

Note that the set of line bundles over M acts on the set of $Spin^c$ structures as follows. For a line bundle L defined by gluing maps $\zeta_{\alpha\beta}: U_{\alpha}\cap U_\beta \to S^1$ , and a $Spin^c$ structure $\sigma$ defined by gluing maps $[h_{\alpha\beta}, z_{\alpha\beta}]$ , $\sigma\otimes L$ is defined by $[h_{\alpha\beta}, z_{\alpha\beta}\zeta_{\alpha\beta}]$ . Observe that $det(\sigma\otimes L)=det(\sigma)\otimes L^2$ so $c_1(\sigma\otimes L)=c_1(det(\sigma))+2c_1(L)$ . One can prove that the action of the line bundles on the set of $Spin^c$ structures is free and transitive.

For any manifold with a spin structure, there is a canonical $Spin^c$ structure $\sigma_0$ , obtained by composing the maps $h_{\alpha\beta}:U_{\alpha}\cap U_\beta \to Spin(n)$ with the obvious map $Spin(n)\to Spin(n)\times S^1/\{\pm 1\}$ sending $g\in Spin(n)$ to $[(g,1)]$ where $[\cdot]$ denotes the equivalence class by modding out by $\pm 1$ . Given this canonical $Spin^c$ structure, any other is represented by gluing cocycles $\{(h_{\alpha\beta},z_{\alpha\beta})\}$ where $\{z_{\alpha\beta}\}$ satisfy the cocycle condition (since $\{h_{\alpha\beta}\}$ do). In other words the $\{z_{\alpha\beta}\}$ define a complex line bundle L. Any $Spin^c$ structure is given by $\sigma_0\otimes L$ , and the associated line bundle is $det(\sigma_0\otimes L)=L^{\otimes 2}$ . Therefore any $Spin$ structure canonically determines a square root of $det(\sigma)$ for any $Spin^c$ structure $\sigma$ .