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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1 Comment

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One response to “Seiberg Witten 4: Moduli spaces and invariants

  1. Laura,

    It will be great to have some applications also.
    Great posts and very useful.

    Appreciate your efforts.

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