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.

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2 Comments

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2 responses to “Seiberg Witten Theory 1: Spin, Spinc structures

  1. Really useful.
    I am thinking to make some diagrams along with each paragraph.
    May be I need some help here?

  2. Pingback: Seiberg-Witten Theory 2: Clifford Structures and Spinors | The Electric Handle Slide

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