Seiberg–Witten invariants

Definition

Let $M$ be a closed, oriented, smooth 4-manifold with $b_2^+(M) \ge 2$, where $b_2^+(M)$ is the number of positive eigenvalues of its intersection form. Fix a Riemannian metric $g$ on $M$ and an orientation of $H^1(M; \R) \oplus H^2_+(M; \R)$. Let $\mathfrak{s}$ be a spin$^c$ structure on $M$, a generalization of a spin structure, with spinor bundles $W^\pm$ and determinant line bundle $L$. For a $U(1)$-connection $A$ on $L$, let $D_A \colon \Gamma(W^+) \to \Gamma(W^-)$ be the associated Dirac operator.

1. Let $F_A^+$ be the self-dual part of the curvature of $A$. Let $\sigma(\psi)$ be the trace-free part of the endomorphism $\psi \otimes \psi^*$ of $W^+$, viewed as a self-dual $2$-form by Clifford multiplication. Let $\eta$ be a generic self-dual $2$-form. The Seiberg–Witten equations for a pair $(A, \psi)$ with $\psi \in \Gamma(W^+)$ are $$D_A \psi = 0, \qquad F_A^+ = \sigma(\psi) + i\eta.$$
2. The gauge group $\mathrm{Map}(M, S^1)$ acts by $u \cdot (A, \psi) = (A - 2u^{-1} du, u \psi)$, and the moduli space $\mathcal{M}(\mathfrak{s})$ is the set of solutions modulo this action.
3. Let $c_1(L)$ be the first Chern class of $L$, $\chi(M)$ the Euler characteristic and $\sigma(M)$ the signature. The moduli space is a compact smooth oriented manifold of dimension $$d(\mathfrak{s}) = \frac{c_1(L)^2 - 2\chi(M) - 3\sigma(M)}{4}.$$

The Seiberg–Witten invariant $SW_M(\mathfrak{s}) \in \Z$ is defined as follows. If $d(\mathfrak{s}) = 0$, it is the signed number of points of $\mathcal{M}(\mathfrak{s})$. If $d(\mathfrak{s}) > 0$ is even, it is the integral over $\mathcal{M}(\mathfrak{s})$ of a power of a natural degree-$2$ class. In the other cases it is $0$.