Monthly Archives: October 2013

by lstarkston | October 28, 2013 · 10:15 am

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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by lstarkston | October 15, 2013 · 5:40 pm

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

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

and

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

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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by lstarkston | October 4, 2013 · 11:52 am

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
$\gamma(t)=\cos(t)+e_1e_2\sin(t)=-(e_1\cos(t/2)+e_2\sin(t/2))(e_1\cos(t/2)-e_2\sin(t/2))$
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.