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  "body": "\n## Definition\n\nLet $M$ be a closed, oriented, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] with $b_2^+(M) \\ge 2$, where $b_2^+(M)$ is the number of positive eigenvalues of its [[intersection form|WK8GKRKAVD]].\nFix a [[Riemannian metric|Q61VFHCH2J]] $g$ on $M$ and an orientation of $H^1(M; \\R) \\oplus H^2_+(M; \\R)$.\nLet $\\mathfrak{s}$ be a spin$^c$ structure on $M$, a generalization of a [[spin structure|RSPZ9TCFQ8]], with spinor bundles $W^\\pm$ and determinant line bundle $L$.\nFor a $U(1)$-connection $A$ on $L$, let $D_A \\colon \\Gamma(W^+) \\to \\Gamma(W^-)$ be the associated [[Dirac operator|3DXGJTEQZH]].\n\n1. Let $F_A^+$ be the self-dual part of the curvature of $A$.\nLet $\\sigma(\\psi)$ be the trace-free part of the endomorphism $\\psi \\otimes \\psi^*$ of $W^+$, viewed as a self-dual $2$-form by Clifford multiplication.\nLet $\\eta$ be a generic self-dual $2$-form.\nThe *Seiberg–Witten equations* for a pair $(A, \\psi)$ with $\\psi \\in \\Gamma(W^+)$ are\n$$D_A \\psi = 0, \\qquad F_A^+ = \\sigma(\\psi) + i\\eta.$$\n2. 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.\n3. Let $c_1(L)$ be the first Chern class of $L$, $\\chi(M)$ the Euler characteristic and $\\sigma(M)$ the signature.\nThe moduli space is a compact smooth oriented manifold of dimension\n$$d(\\mathfrak{s}) = \\frac{c_1(L)^2 - 2\\chi(M) - 3\\sigma(M)}{4}.$$\n\nThe *Seiberg–Witten invariant* $SW_M(\\mathfrak{s}) \\in \\Z$ is defined as follows.\nIf $d(\\mathfrak{s}) = 0$, it is the signed number of points of $\\mathcal{M}(\\mathfrak{s})$.\nIf $d(\\mathfrak{s}) \u003e 0$ is even, it is the integral over $\\mathcal{M}(\\mathfrak{s})$ of a power of a natural degree-$2$ class.\nIn the other cases it is $0$.\n\n## Remarks\n\nWhen $b_2^+(M) \\ge 2$, the number $SW_M(\\mathfrak{s})$ does not depend on $g$ or $\\eta$, so it is an invariant of $M$ with its orientation data.\nFor example, a closed symplectic $4$-manifold with $b_2^+ \\ge 2$ has $SW = \\pm 1$ on its canonical spin$^c$ structure, by a theorem of Clifford Taubes.\nIn contrast, a connected sum $M_1 \\# M_2$ with $b_2^+(M_i) \\ge 1$ for both summands has all Seiberg–Witten invariants zero.\nWhen $b_2^+(M) = 1$ the count can jump as $g$ and $\\eta$ vary, so the definition then depends on a chamber.\nNormalizations of $\\sigma(\\psi)$ and of the gauge action differ by constants between authors.\n",
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