Seiberg-Witten Theory: IV

March 19, 2022

Let \(M\) be a closed oriented smooth \(4\)-manifold with a choice of a Riemannian metric \(g\), and a spinC-structure \(\mathfrak{s}\). The spinC-structure gives rise to a Clifford representation of the complexified tangent bundle \(\rho : T^{\Bbb C} M \to \mathrm{End}(S)\) on the complex spinor bundle \(S\) which is of rank \(4 = 2^{4/2}\), admitting a chiral decomposition \(S = S^- \oplus S^+\) and associated determinant line bundle \(L = \Lambda^2 S^+ \cong \Lambda^2 S^-\) whose 1st Chern class is the integral lift of the 2nd Stiefel-Whitney class of \(M\) coming from \(\mathfrak{s}\), as discussed in Part III. We choose a connection \(A\) on \(L\) which, combined with the LC connection on \((M, g)\), determines a spin connection on \(S\). Then there is an associated Dirac operator \(\slash \!\!\! \partial_A : \Gamma^{\pm}(S) \to \Gamma^{\mp}(S)\).

Moreover, we have the Hodge star operator \(\star : \Omega^2(M) \to \Omega^2(M)\) and we shall call the eigenspaces of \(\star\) of eigenvalues \(\pm 1\) as the self-dual and anti-self-dual \(2\)-forms, giving a decomposition \(\Omega^2 = \Omega^{2,+} \oplus \Omega^{2, -}\); we shall write \(\omega^+\) to mean the self-dual part of a \(2\)-form \(\omega\). The SD-ASD decomposition mirrors the chiral decomposition \(S = S^+ \oplus S^-\). More precisely, the connection between these is explained by the spin representation \(\rho : T^{\Bbb C} M \to \mathrm{End}(S) = S \otimes S\) which may be baked to a representation

\[\displaystyle \rho : \Omega^2(M) \otimes \Bbb C = \Lambda^2 T^*M \otimes \Bbb C \to S \otimes S\]

by replacing wedge with Clifford multiplication wherever it appears. We obtain the decomposition \(S^+ \otimes S^- \cong \Omega^0 \oplus \Omega^{2,+}\), a mnemonic for which is that a left-handed and a right-handed spin-1/2 fermion can combine to give a spin 0 (which has a single state, so rank of \(\Omega^0\) is \(1\)) or a spin 1 (which has three states -1, 0, 1, so rank of \(\Omega^{2,+}\) is \(3\)) fermion; note that the projection to the first factor is simply taking trace. Whenever \(\psi\) is a positive chirality spinor (a section of \(S^+\)), we will write \((\psi \otimes \psi^\dagger)_0\) to mean the trace-0 part of the tensor, conceptualized as a self-dual 2-form.

Given all of this, we shall now write down the Seiberg-Witten equations. For a pair \((A, \psi) \in \mathcal{A}(L) \oplus \Gamma(S^+)\), these are given by

\[\displaystyle \slash \!\!\! \partial_A \psi = 0 \\ F_A^+ = (\psi \otimes \psi^\dagger)_0\]

The second equation is the quadratic part; observe both sides are \(i \Bbb R\)-valued self-dual \(2\)-forms on \(M\). We may perturb the SW equations by adding \(i \mu\) to the right hand side of the second equation, for a fixed self-dual \(2\)-form \(\mu\), which I expect is useful for transversality arguments later.

The natural gauge symmetries of these equations is recorded by the group \(\mathcal{G} = C^\infty(M, S^1)\). For \(g \in \mathcal{G}\), it acts on the complex vector bundle \(S^+\) by pointwise multiplication; thus the induced action on sections is \(\psi \mapsto g \psi\) and the induced action on the line bundle \(L\) is by \(g^2\) (see Part III); this implies the action by conjugation on the covariant derivative \(\nabla^A_L\) corresponding to the connection \(A\) is given as follows:

\[\displaystyle g^{-2} \nabla^A_L g^{2} = g^{-2} d g^{2} + g^{-2} A g^2 = -2g^{-1} dg + A\]

Therefore, the induced action on connections is given by \(A \mapsto -2g^{-1} dg + A\). It is straightforward to check that SW-equations are invariant under the action of \(\mathcal{G}\); indeed \(\slash \!\!\! \partial_{g\cdot A} (g \cdot \psi) = \gamma^\mu ( g^{-1} \nabla^A_i g )g \psi = g^{-1} \gamma^\mu \nabla^A_i \psi\) and \(F_{g\cdot A}^+ = F_A^+\) as deforming the connection by a locally exact form \(-2g^{-1} dg = -2d\log(g)\) leaves the curvature invariant. The moduli space of solutions of the SW-equations, defined by the space of solutions modded out by the action of \(\mathcal{G}\) will be denoted as \(\mathcal{M}_{\mathfrak{s}, g, \mu}\).

The main result which will be discussed in the course of time is that \(\mathcal{M}_{\mathfrak{s}, g, \mu}\) is a compact, smooth, oriented manifold of dimension \(-(2\chi(M) + 3\sigma(M))/4 + c_1(L)^2\) for a generic perturbation \(\mu\) when \(b_2^+ > 1\). In the zero dimensional case, counting the signed connected components gives us the so-called Seiberg-Witten invariants of \(M\).