Overtwisted contact structures and h-principles

Twenty-five years ago Eliashberg established a dichotomy of contact structures on 3-manifolds as either tight or overtwisted. The overtwisted manifolds were motivated by the property that they satisfy certain flexibility principals. 1. Every co-oriented 2-plane field on a 3-manifold is homotopic to an overtwisted contact structure. 2. Any two overtwisted contact structures which are homotopic as 2-plane fields are isotopic as contact structures. (More generally, there is a homotopy equivalence between the space of contact structures and the space of 2-plane fields when you restrict each space to only include those with a particular overtwisted disk.) Recently, this theorem has been generalized to higher dimensions by Borman, Eliashberg, and Murphy , who explained their work at a recent workshop. The next few posts will be an attempt to share the intuition they gave, to work through some of the steps of their paper, to provide some historical background, and to discuss some related constructions and interesting open questions.

I also want to mention, that we are happy to give author permissions to young mathematicians who want to contribute blog posts. Thanks to Roger for agreeing to contribute on this topic, and welcome to the blog.

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Graduate Student Topology and Geometry Conference

The 12th annual graduate student topology and geometry conference will be at UT Austin this April. The registration deadline is coming up on February 1. The website for registration and more information is http://ma.utexas.edu/conferences/gstgc14/. Many of the talks are given by graduate student participants, and we have lots of interesting faculty speakers in a range of topology/geometry topics. I encourage all grad students to register and submit a talk proposal (expository or original research).

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Quick note: “A symplectic prolegomenon”

I just want to draw your attention to a survey that appeared on the arXiv on January 1: A symplectic prolegomenon, by Ivan Smith.  The main point is to motivate and illustrate the Fukaya category, and to show how its algebraic structures amplify the power of Floer cohomology.  Smith uses the running examples of the nearby Lagrangian conjecture and the symplectic mapping class group to demonstrate these algebraic structures (the Oh spectral sequence, the exact triangle associated to a Dehn twist, …) in action.  There are lots of applications throughout, and one nice feature is that section 5 consists of explicit descriptions of the Fukaya categories of six (families of) symplectic manifolds.

So it’s great winter break reading, check it out!  It collects together a lot of information that had previously been scattered over a bunch of different papers.  And it includes the take-home messages of a number of rather intimidating papers.

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Seiberg Witten 4: Moduli spaces and invariants

This is my last post on defining the Seiberg-Witten equations and invariants for closed 4-manifolds based on a learning seminar at UT. Maybe I’ll post about some applications later on.

The Seiberg-Witten Configuration Space

We start with a Riemannian 4-manifold (M,g) and a Spinc structure \sigma on M. As we have seen, this data gives rise to the associated bundle S_\sigma=S_\sigma^+\oplus S_\sigma^- and the determinant line bundle det(\sigma).

Let \mathfrak{A}_\sigma(M) be the set of all Hermitian connections on det(\sigma). We have seen that such a connection gives rise to a connection on S_\sigma which is compatible with the Clifford multiplication.

The Seiberg-Witten configuration space is defined as
\mathcal{C}_\sigma(M)=\mathfrak{A}_\sigma(M)\times \Gamma(S_\sigma^+)

The Seiberg-Witten equations

The Seiberg-Witten equations take an element (A,\psi)\in \mathcal{C}_\sigma(M) as their input. We are now prepared to define these equations.

As discussed in an earlier post, a connection A on det(\sigma) gives rise to a connection on S_\sigma^+. Note that for \xi\in T^*M, c(\xi)\in c(Cl^-(4)) so c(\xi):S_\sigma^+\to S_\sigma^-. Therefore we have a Clifford structure
c: \Gamma(T^*M\otimes S_\sigma^+)\to \Gamma(S_\sigma^-)
which composes with the connection
\nabla^A: \Gamma(S_\sigma^+)\to \Gamma(T^*M\otimes S_\sigma^+)
to get a Dirac operator
D_A: \Gamma(S_\sigma^+)\to \Gamma(S_\sigma^-).

Denote the curvature of the connection A by F_A. Then the curvature is a matrix of 2-forms on M, so we can consider its self-dual and anti-self dual parts F_A^+ and F_A^-.

Let (\psi\otimes \psi^*)_0 denote the traceless part of the endomorphism \psi\otimes \psi^*:S_\sigma^+\to S_\sigma^+.

Now we can define the (perturbed) Seiberg-Witten equations. Fix a closed 2-form \eta\in \Omega^2(M) (the pertubation parameter). Then the Seiberg-Witten equations are:
SW_{(\sigma,\eta)}=\begin{cases}  \frac{1}{2}c(F_A^++i\eta^+)-(\psi\otimes \psi^*)_0=0\\ {D}_A\psi = 0\end{cases}

The input to these equations is an element (A,\psi)\in \mathcal{C}_\sigma(M). The elements of \mathcal{C}_\sigma(M) which are solutions to these equations are called ((\sigma,\eta)-)monopoles.

The Gauge Action

The gauge group is \mathfrak{G}_\sigma(M)=\{\gamma: M\to U(1)| \text{ smooth}\}. It acts on \mathcal{C}_\sigma(M) by
\gamma\cdot (A,\psi) = (A-2d\gamma/\gamma, \gamma\psi)

While it seems natural enough to act on the section \psi by multiplication, why do we define the action \gamma\cdot A=A-2d\gamma/\gamma? Specifically where is the 2 coming from?

A is the connection of the determinant line bundle L of S_{\sigma}^+. We would really like to think of the gauge group as acting on S_{\sigma}^+. If g\in \mathfrak{G} acts on s\in S^+ by multiplication s \mapsto gs, then the induced action on \sigma\in L=\wedge^2 S_{\sigma}^+ is multiplication by g^2. (This goes back to the fact that in coordinate charts, the spinc structure is obtained by tensoring the spin structure with the square root of the determinant line bundle L.) Now we can look at how this acts on the covariant differentiation \nabla_A induced by the connection A on L. Here the natural action is conjugation

g^2\nabla_A(g^{-2}s)=g^2d(g^{-2})\otimes s +\nabla_As=-2g^{-1}dg\otimes s +\nabla_As

For C=(A,\psi)\in \mathcal{C}_\sigma(M) we can consider its stabilizer in \mathfrak{G}_\sigma(M). If the stabilizer of C is trivial, we say C is irreducible, otherwise we say C is reducible. It is easy to show that the reducible elements are exactly those with \psi\equiv 0, and that their stabilizers are the constant maps into S^1.

The Seiberg-Witten moduli space

The Seiberg-Witten solution space is the space of elements (A,\phi) for which the Seiberg-Witten equations are satisfied. To obtain the moduli space from this, we want to mod out by the gauge action. In order for this to be well defined, we first need to check that the space is invariant under the gauge action.

For the first equation, we can prove that F_A^+=F_{A-2g^{-1}dg}^+ because F_{A-2g^{-1}dg}^+=(F_A-d(2g^{-1}dg))^+ and d(g^{-1}dg)=0 because we can think locally that g^{-1}dg=d(log(g)) so taking its exterior derivative gives 0. Furthermore (g\phi)\otimes (g\phi)^*=gg^{-1}\phi\otimes \phi^*=\phi\otimes \phi^*, so the first equation is invariant under the gauge action.

For the second equation, D_{A-2g^{-1}dg(g\phi)} can be understood by breaking up the dirac operator into the composition of the Clifford multiplication and the connection \nabla_A on S_\sigma^+.

The discussion above about why the gauge group acts as it does on A is related to the fact that \nabla_(A-2g^{-1}dg)=\nabla_A-g^{-1}dg\otimes I_{S^+}. Applying the Clifford multiplication to this connection acting on g\phi and using the Leibniz rule for connections eventually simplifies to show that D_{A-2g^{-1}dg}(g\phi)=g D_{A}\phi so the solutions to D_A\phi=0 are invariant under the gauge action.

Therefore we can mod out the Seiberg Witten solution space by the gauge action to get a well-defined space.

Properties of the Seiberg Witten moduli space

The reason the Seiberg-Witten equations are so useful is that the moduli space is actually a compact smooth manifold in many cases. When there are no reducible solutions to the equations, the moduli space defined by a generic perturbation is a smooth manifold (one needs to show that the linearization of a map defined by the Seiberg Witten equations and the gauge action is Fredholm and then use Sard-Smale to show that generic perturbations correspond to regular values).

Compactness of the manifold requires some analytic estimates. The Weitzenbock forumla is the main tool in obtaining bounds on solutions to the Seiberg-Witten equations.

After going through hard work to show these properties, which I am avoiding here, one just needs to worry about reducible solutions. Notice that if there are reducible solutions (A,0) then they satisfy F_A^+=\eta for our chosen perturbation. Since both of these forms are closed, they represent cohomology classes. The cohomology class of the curvature [F_A^+]=-2\pi ic_1(L)^+ is independent of A, so we only have reducible solutions when [\eta]=-2\pi ic_1(L)^+. When the dimension of the positive second homology is at least 1, then a generic perturbation will avoid this phenomenon.

The Seiberg-Witten invariant of a 4-manifold is given by the homology class of the moduli space of solutions in the configuration space. This configuration space is homotopy equivalent to \mathbb{CP}^\infty so its cohomology has a canonical generator in even degrees. By evaluating this generator against the homology class of the Seiberg-Witten moduli space we obtain an integer SW_{M,g,\eta}(L)\in \mathbb{Z}.

A priori this integer depends on the metric and perturbation, but when b_2^+>1, the subspace of perturbations which allows for reducible solutions (bad perturbations) is codimension 2. Since the space of metrics on a manifold is convex, we can find a path through the space of metrics and good perturbations connecting any two pairs (g_1,\eta_1), (g_2,\eta_2) which lifts to a cobordism between the moduli space at (g_1,\eta_1) and the moduli space at (g_2,\eta_2). Therefore SW gives a diffeomorphism invariant of the 4-manifold, and it has been used very effectively to distinguish many homeomorphic but not diffeomorphic 4-manifolds (exotic pairs).

When b_2^+=1, there is a codimension 1 space of bad perturbations which forms a wall between two chambers. Within each chamber SW_{M,g,\eta}(L) stays constant, and there is a well-understood wall-crossing formula describing the difference of SW in the two different chambers. By keeping track of a little more information, it is still possible to use information from the Seiberg-Witten invariants to distinguish exotic pairs (this has been used a lot for finding exotic \mathbb{CP}^2\#N\overline{\mathbb{CP}^2}).

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Seiberg Witten 3: Dirac operators, Spin and Spinc connections

This is my third post on the set-up for the Seiberg-Witten invariants of 4-manifolds. The next post will finally define the Seiberg-Witten equations and invariants, so this is the last bit of background.

Symbols, generalized Laplacians, and Dirac operators

In order to define the Seiberg-Witten equations, we need to understand certain partial differential operators called Dirac operators. If you don’t know the formal definition of partial differential operators and their symbols, here is a link with some definitions and examples.

The class of all second order partial differential operator with the same symbol as the usual Laplacian: \sigma_L(\xi)=-|\xi|^2I\in End(E,E) are called generalized Laplacians. Note that the symbol \sigma_L(\xi): E_x\to E_x of a generalized Laplacian is an isomorphism on each fiber for \xi\neq 0, which means generalized Laplacians are elliptic operators. An elliptic operator L is good because there are estimates on the norms of solutions to equations of the form Lu=v. This allows us to use Fredholm theory to describe the space of solutions to equations using elliptic operators. (In particular the linearization of an elliptic operator is Fredholm, i.e. has finite dimensional kernel and cokernel).

Dirac operators are 1st order partial differential operators which square to a generalized Laplacian. Dirac operators inherit many of the nice properties of Laplacians, specifically they are also elliptic (though in a weaker sense than the Laplacian–my vague understanding is that the bounds we get from ellipticity of the Laplacian are uniform, whereas the bounds we get from ellipticity of a Dirac operator depend on the point in the manifold; in the case of compact manifolds these coincide).

Dirac Operators and Clifford multiplication

We mentioned above that the symbol of a generalized Laplacian, (which is the square of a Dirac operator) is \sigma_L(\xi)=-|\xi|^2I, for \xi\in \Gamma(T^*M). Additionally, one can show that the symbol of a Dirac operator (which squares to a generalized Laplacian), is the square root of the symbol of the generalized Laplacian. Therefore (\sigma_D(\xi))^2=-|\xi|^2I so \sigma_D gives us a Clifford multiplication. In conclusion, a Dirac operator give rise to a Clifford structures by taking its symbol.

Conversely, given a Clifford structure, c: \Gamma(T^*M)\to \Gamma(End(E)) (equivalently c: \Gamma(T^*M\otimes E)\to \Gamma(E)) and a connection \nabla: \Gamma(E)\to \Gamma(T^*M\otimes E) we can compose them

D:\Gamma(E)\xrightarrow{\nabla}\Gamma(T^*M\otimes E)\xrightarrow{c}\Gamma(E)

and the resulting operator is a Dirac operator.

Spin connections

A Riemannian manifold M has a distinguished connection, the Levi-Civita connection \nabla^M, which has nice properties namely it preserves the metric g (this can be phrased either as \nabla^Mg=0 or \nabla(g(X,Y))=g(\nabla X,Y)+g(X,\nabla(Y))), and it is torsion free meaning \nabla_XY-\nabla_YX-[X,Y]=0. Basically, this is a natural connection on TM when a Riemannian metric g is given.

Using the metric and orientation on M, the structure bundle of TM reduces to an SO(n)-bundle. Namely, we can find gluing maps defining the tangent bundle that map into SO(n): \{g_{\alpha\beta}: U_\alpha\cap U_\beta \to SO(n)\} which define a principal SO(n)-bundle P_{SO(n)}\to M. The Levi-Civita connection on TM induces a principal SO(n)-connection on P_{SO(n)} specified locally by

\omega_{\alpha}\in \Omega^1(U_{\alpha})\otimes \mathfrak{so}(n).

We have the double cover map \tau: Spin(n)\to SO(n), which induces, by differentiating at 1, an isomorphism \tau_*: \mathfrak{spin}(n)\to \mathfrak{so}(n).

If we have a Spin structure on M, this means there are lifts \widetilde{g}_{\alpha\beta}: U_\alpha\cap U_\beta\to Spin(n) such that \tau\circ \widetilde{g}_{\alpha\beta}=g_{\alpha\beta}. These define a principal Spin(n) bundle P_{Spin(n)}. In this case, the Levi-Civita connection on P_{SO(n)} induces a connection \widetilde{\nabla}^M on P_{Spin(n)} which is locally defined by

\tau_*^{-1}\omega_{\alpha}\in \Omega^1(U_{\alpha})\otimes \mathfrak{spin}(n).

So Riemannian manifolds with spin structures have a distinguished connection on the Spin(n) bundle.

The representations \rho_\pm: Spin(4)\to Aut(\mathbb{S}^\pm), and \rho=(\rho_+,\rho_-) give rise to an associated bundle S_0=P_{Spin}\times_\rho \mathbb{S}. The spin connection on M induces a connection \nabla^{S_0} on S_0 whose local matrix valued 1-forms are defined by

\rho_*\tau_*^{-1}\omega_{\alpha}\in \Omega^1(U_\alpha)\otimes End(\mathbb{S}).

Recall that T^*M acts on S_0 by the Clifford multiplication c: Cl(TM)\otimes \mathbb{C}\to End(S_0). The composition of the Clifford multiplication with the induced connection on S_0 yields a Dirac operators D_0.

\mathbf{Spin^c} connections

Remember, a Spin^c(n)-bundle is specified by gluing data

\{(h_{\alpha\beta}, z_{\alpha\beta}): U_{\alpha}\cap U_{\beta} \to Spin(n)\times U(1)\}
satisfying the cocycle condition

(h_{\alpha\beta}h_{\beta\gamma}h_{\gamma\alpha}, z_{\alpha\beta}z_{\beta\gamma}z_{\gamma\alpha})=\pm (1,1).
We want to understand Spin^c structures for M and their connections. Let \sigma be a Spin^c structure on M given by the Spin^c(4) bundle P_{Spin^c}.

Letting \rho^c=(\rho^c_+,\rho^c_-), the associated spinor bundle to \sigma is S_\sigma=P_{Spin(4)}\times_{\rho^c} \mathbb{S}, which splits into S^\pm_\sigma = P_{Spin(4)}\times_{\rho^c_\pm}\mathbb{S}^{\pm}. A connection on the Spin^c bundle will induce a connection on S_\sigma, S^+_\sigma,S^-_\sigma. Also note that S_\sigma has a Clifford structure, inherited from the map c: Cl(V)\otimes \mathbb{C}\to End(\mathbb{S}).

In the case that M has a spin structure, P_{Spin^c}=P_{Spin}\otimes (det\sigma)^{1/2} and S_\sigma = S_0\otimes (det\sigma)^{1/2}.

In the general case, we will construct connections on the associated bundles using the Levi-Civita connection on M, and a choice of connection on the determinant line bundle of \sigma.

In the case that TM is the trivial bundle, the determinant line bundle has a square root, and P_{Spin^c}=P_{Spin}\otimes (det\sigma) and S_\sigma=S_0\otimes (det\sigma)^{1/2}. We have the natural lift \widetilde{\nabla}^M of the Levi-Civita connection to P_{Spin}. This induces a natural connection \nabla^{S_0} on the associated bundle S_0, which we can tensor with any connection on the line bundle (det\sigma)^{1/2} to get a connection on S_{\sigma}=S_0\otimes (det\sigma)^{1/2}.

Remember that S_0 had a Clifford structure c as well as a natural connection S_0 which together give rise to a Dirac operator. We obtain a similar structure on S_\sigma by twisting the triple (S_0,\nabla^{S_0}, c) with a line bundle with connection (L,\nabla^L) to obtain a triple (S_0\otimes L, \nabla, c_L) where

\nabla(s\otimes x) = \nabla^{S_0}s\otimes x +s\otimes \nabla^Lx


c_L: \Omega^*M \xrightarrow{c}End(S_0)\xrightarrow{\cdot \otimes I_L} End(S_0\otimes L)

Therefore over trivial charts, a choice of connection A on (det\sigma)^{1/2} gives rise to a Dirac triple (S_\sigma,\nabla_A, c_\sigma).

In general the determinant line bundle does not have a global square root, though over any trivial chart it does. When the determinant line bundle has a square root, the connections on det(\sigma) are related to the connection on (det(\sigma))^{1/2} as follows. If the connection on det(\sigma) is defined by

\{\omega_\alpha \in \Omega^1(U_\alpha)\otimes \mathfrak{u}(1)\}

then the induced connection on (det(\sigma))^{1/2} is defined by

\{\frac{1}{2}\omega_\alpha \in \Omega^1(U_\alpha)\otimes \mathfrak{u}(1)\}.

We can always choose a connection on det(\sigma). This induces a connection over each trivial chart on (det(\sigma))^{1/2}. Then we can twist this in to the locally defined Dirac triples (S_0,\nabla^{S_0},c), to obtain (S_\sigma, \nabla, c) on each trivial chart U_\alpha. Finally, one can use a partition of unity to glue all these pieces back together to a global Dirac triple (S_\sigma,\nabla, c).

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Seiberg-Witten Theory 2: Clifford Structures and Spinors

Here is the second post on setting up the Seiberg-Witten equations on a 4-manifold, based on our learning seminar at UT Austin. The first post is here.

Clifford Algebras and Structures

For a vector space V with inner product g, its Clifford Algebra is defined as the tensor algebra of V modded out by all relations generated by setting v\otimes v=-g(v,v)1,

Cl(V)= \otimes V/\langle v\otimes v +g(v,v)1: v\in V\rangle.

For a vector bundle E\to M, any map c: T^*M\to End(E) satisfying c(u)c(v)+c(v)c(u)=-2g(u,v)I_E for all u,v\in \Gamma(T^*M) (equivalently satisfying c(v)^2=-|v|I_E for all v) extends to a representation
c:Cl(T^*M)\to End(E)
Such a map is called a Clifford structure.

There are two reasons we are interested in Clifford algebras and Clifford structures for Seiberg-Witten theory. The first is their relation to Spin and Spinc structures. The second is their relation to Dirac operators. In this post we will focus on their relation to Spin and Spinc structures, and discuss Dirac operators next.

Clifford algebras and Spin

Let Cl(n) denote the Clifford algebra of \mathbb{R}^n with its standard inner product. Let (e_1,\cdots, e_n) denote the standard orthonormal basis for \mathbb{R}^n. Consider the multiplicative subgroup of Cl(n) generated by unit vectors of \mathbb{R}^n. This is called Pin(n).

There is a natural \mathbb{Z}/2 grading on Cl(n) induced by a bijection Cl(n)\leftrightarrow \bigwedge^* \mathbb{R}^n identifying e_{i_1}\cdots e_{i_k} \leftrightarrow e_{i_1}\wedge \cdots \wedge e_{i_k}. The integer grading on the exterior power reduces to a \mathbb{Z}/2 grading (even/odd) on the Clifford algebra. This yields a splitting Cl(n)= Cl^+(n)\oplus Cl^-(n). Define Spin(n) to be the intersection of Pin(n) with the even summand Cl^+(n).

Before, we defined Spin(n) to be the universal double cover of SO(n). We can show this new definition of Spin agrees with the old definition, by explicitly constructing a double cover map from this subset of Cl(n) to SO(n).

There is an action of Cl(n) on \mathbb{R}^n given by signed conjugation (using the multiplicative structure of the Clifford algebra). If v\in \mathbb{R}^n is a unit vector (i.e. a generator of Pin(n)) then for any x\in \mathbb{R}^n
-vxv^{-1} = vxv = x-2\langle x,v \rangle v
Here we have used the fact that for unit vectors -vv=1 so v^{-1}=-v, and the relation vx+xv=-2g(v,x) in the Clifford algebra. This can be interpreted geometrically: the action (v,x)\mapsto -vxv^{-1} is the reflection of x over the hyperplane orthogonal to v.

The group of orthogonal transformations is generated by reflections over hyperplanes, so we have a representation called the twisted adjoint representation:
\rho: Pin(k)\to O(k)
defined by \rho(y)x = yx\varepsilon(y^{-1}) where \varepsilon(Cl^\pm(n))=\pm 1 (extend linearly). Restricting this to Spin(k) this is just usual conjugation, which corresponds to an even number of reflections so the image lies in SO(k):
\rho: Spin(k)\to SO(k)
This map is a surjective group homomorphism, and by studying the elements of Spin(k) which lie in the center of Cl(k), we see that the kernel of \rho: Spin(k)\to SO(k) is two elements \{\pm 1\}. Because these are nice smooth compact Lie groups, this implies that \rho is a covering map. To check it is not the trivial double cover, we can find a path in Spin(k) between -1 and 1 given by
for t\in [-\pi,\pi] [observe this path is a product of two unit vectors at each t and is thus in Spin(n)].
Therefore this definition of Spin(n) agrees with the previous one.

The spinor representation

We have already seen that Spin(4)\cong SU(2)\times SU(2) so Spin(4) and Spin^c(4) naturally admit two complex rank two representations coming from the projections onto the two factors of SU(2). However, it is useful to understand these representations from the Clifford algebra perspective so that the representations carry the additional information of a Clifford structure. In fact, there is a complex representation of the entire (complexified) Clifford algebra Cl(4) which splits into a direct sum of two complex rank two representations, which behave nicely with respect to the \mathbb{Z}/2 grading on the Clifford algebra. More specifically:

Theorem: There is a complex vector space \mathbb{S}=\mathbb{S}^+\oplus \mathbb{S}^- with \dim_{\mathbb{C}}\mathbb{S}^+=\dim_{\mathbb{C}}\mathbb{S}^-=2, and an \mathbb{C}-linear isomorphism
c: Cl(4)\otimes \mathbb{C}\to End(\mathbb{S})
such that c(Cl^+(4))\cong End(\mathbb{S}^+)\oplus End(\mathbb{S}^-) and c(Cl^-(4)\cong Hom(\mathbb{S}^+,\mathbb{S}^-)\oplus Hom(\mathbb{S}^-,\mathbb{S}^+).

To prove this, we have to define \mathbb{S}, \mathbb{S}^\pm, and the map c, and then verify that c is an algebra isomorphism satisfying the specified properties. There are a lot of things to check so I will define everything, and say a few things about how the map c works which hopefully make it more believable that c is an algebra isomorphism.

Let V=\mathbb{R}^4 with standard coordinates and standard almost complex structure J. This almost complex structure gives rise to a splitting of V\otimes \mathbb{C} = V^{1,0}\oplus V^{0,1}, where V^{1,0} is the i-eigenspace of J and V^{0,1} is the -i-eigenspace of J. We have orthonormal bases for these pieces given by:
V^{1,0}=span\left(\varepsilon_1 := \frac{1}{\sqrt{2}}(e_1-if_1), \varepsilon_2 := \frac{1}{\sqrt{2}}(e_2-if_2)\right)
V^{0,1}=span\left(\overline{\varepsilon}_1 := \frac{1}{\sqrt{2}}(e_1+if_1), \overline{\varepsilon}_2 := \frac{1}{\sqrt{2}}(e_2+if_2)\right)

Define \mathbb{S}:= \bigwedge^* V^{1,0}, and its splitting by \mathbb{S}^+ := \bigwedge^{even}V^{1,0} and \mathbb{S}^- := \bigwedge^{odd}V^{1,0}.

Now we need to define c: Cl(V)\otimes \mathbb{C} \to End (\mathbb{S}) with the properties specified in the theorem. We will define c on elements of V\otimes \mathbb{C} and then extend this to a map on the Clifford algebra by setting c(e_{i_1}\cdots e_{i_k})=c(e_{i_1})\cdot \cdots \cdot c(e_{i_k}) and extending complex linearly. To specify c on V\otimes \mathbb{C}, it suffices to say what c does to vectors in V^{1,0} and V^{0,1}.

For v\in V^{1,0}, c(v) is the endomorphism of \mathbb{S} obtained by wedging with v:
c(v)(u_1\wedge \cdots u_k)=\sqrt{2}v\wedge u_1\wedge \cdots \wedge u_k

For \overline{v}\in V^{0,1} c(\overline{v}) is contraction with \overline{v}:
c(\overline{v})(u_1\wedge \cdots u_k) = \sqrt{2}\sum_{j=1}^k (-1)^j g(v,u_j)u_1\wedge \cdots \wedge \widehat{u_j} \wedge \cdots u_k

One needs to check that this respects the Clifford algebra structure, and is an isomorphism. Initially, this may look wrong because for example when v\in V^{1,0}
c(v)^2(u_1\wedge \cdots \wedge u_k) = v\wedge v\wedge u_1\wedge \cdots \wedge u_k=0
and it seems like we should have c(v)^2=-|v|^2I. However, the algebra structure we want to preserve is complex linear on Cl(V)\otimes \mathbb{C} and has the Clifford structure only on the Cl(V) piece. Therefore, for example when v=e_j-if_j\in V^{1,0},
0=c(e_j-if_j)^2 = (c(e_j)-ic(f_j))^2 = (c(e_j))^2-ic(e_j)c(f_j)-ic(f_j)c(e_j)-(c(f_j))^2 = |e_j|^2-i2g(e_j,f_j)-|f_j|^2

For basis elements, the map c is a sum of the exterior and interior products. To compute for example, c(e_j) we split this into the V^{1,0} and V^{0,1} parts, so
c(e_j)=c\left(\frac{1}{2}(e_j-if_j)+\frac{1}{2}(e_j+if_j)\right)=\frac{\sqrt{2}}{2}\left((e_j-if_j)\wedge\cdot +\iota_{e_j-if_j} \right)
If you want to be slightly more convinced without completing the proof that c(v)^2=-|v|^2I for real elements of Cl(V) it is fairly easy at this point to check that c(e_j)^2=-I at least on the \bigwedge^0V^{1,0} part of \mathbb{S}=\bigwedge V^{1,0} (since any map that starts with contraction vanishes and \iota_x(y\wedge f)=-fg(x,y) for f\in \bigwedge^0V^{1,0} a complex number, and x,y\in V\subset Cl(V)\otimes 1).

We get the last property in the theorem easily from the definition of c. For v\in V\otimes \mathbb{C}, c(v) either raises or lowers by 1, wedge power of an element of \mathbb{S}=\bigwedge V^{1,0}. Therefore c(v) sends \mathbb{S}^+ to \mathbb{S}^- and vice versa. Extending this over the entire Clifford algebra, we see that the endomorphisms in c(Cl^+(4)) preserve \mathbb{S}^+ and \mathbb{S}^- (since they switch between \mathbb{S}^\pm an even number of times) and c(Cl^-(4)) sends \mathbb{S}^\pm to \mathbb{S}^\mp.

Note: We can rewrite the isomorphism c: Cl(4)\otimes \mathbb{C}\to End(\mathbb{S}) as a map
c: Cl(4)\otimes \mathbb{C}\otimes \mathbb{S}\to \mathbb{S}.
This will be useful when we use this representation to form associated bundles and consider sections of those bundles and maps between the spaces of sections.

This theorem generalizes for Cl(2n), producing a complex vector space \bigwedge V^{1,0} which splits where dim(V)=2n, whose endomorphisms are isomorphic to Cl(2n)\otimes \mathbb{C}, where Cl^+ preserves the splitting and Cl^- switches the components. In the odd dimensional case, the situation is slightly different, but reduces to the even case by showing that Cl(2n-1)\cong Cl^+(2n). For the purposes of Seiberg-Witten Floer homology, it will be useful to know Cl(3)\cong Cl^+(4) which implies Cl(3)\otimes \mathbb{C}\cong End(\mathbb{S}^+)\oplus End(\mathbb{S}^-).

Spinor bundles

Now that we have this representation of the complexification of the Clifford algebra, we can restrict to get a representation of Spin. Because Spin(4)\subset Cl^+(4), and c(Cl^+(4)) preserves the splitting \mathbb{S}=\mathbb{S}^+\oplus \mathbb{S}^-, we get two representations
\rho_{\pm}: Spin(4)\to Aut(\mathbb{S}^\pm)
Note the image of Spin(4) lands in automorphisms instead of only endomorphisms because elements of Spin(4) are invertible in Cl(4). These two representations correspond to the same ones we obtain by identifying Spin(4)\cong SU(2)\times SU(2) and projecting onto one component.

We can extend these maps to Spin^c by defining
\rho^c_{\pm}: Spin^c(4)\to Aut(\mathbb{S}^\pm)
by \rho^c_\pm((g,z))=z\rho_{\pm}(g) for g\in Spin(4), z\in U(1).

Note this is well defined since \rho^c_\pm((-g,-z))=\rho^c_\pm((g,z)).

Given a Spin or Spinc structure on a manifold, these representations give rise to associated bundles S^\pm \to M. These bundles show up in the set-up for the Seiberg-Witten configuration space, which I will get to in another post.

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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(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
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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