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  "body": "\n## Definition\n\nLet $M$ be a closed, oriented, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] with a [[Riemannian metric|Q61VFHCH2J]] $g$, and let $\\ast$ be the Hodge star of $g$ on $2$-forms.\nLet $P \\to M$ be a [[principal bundle|8E37E74XVP]] with structure group $SU(2)$ and second Chern class $c_2(P) = k$.\nFor a [[connection|NJ21PN5ZGJ]] $A$ on $P$ with curvature $F_A$, write $F_A = F_A^+ + F_A^-$ with $\\ast F_A^{\\pm} = \\pm F_A^{\\pm}$.\n\n1. The connection $A$ is *anti-self-dual* if $F_A^+ = 0$.\n2. The gauge group $\\mathcal{G}$ of bundle automorphisms of $P$ acts on anti-self-dual connections.\n3. The *moduli space* $\\mathcal{M}_k(g)$ is the set of anti-self-dual connections on $P$ modulo $\\mathcal{G}$.\n\n*Donaldson theory* is the study of the spaces $\\mathcal{M}_k(g)$ and of the invariants of $M$ obtained from them.\nThe main such invariants are the Donaldson polynomial invariants, defined by integrating cohomology classes over $\\mathcal{M}_k(g)$.\n\n## Remarks\n\nLet $b_1(M)$ be the first Betti number and $b_2^+(M)$ the number of positive eigenvalues of the intersection form.\nFor a generic metric, the irreducible part of $\\mathcal{M}_k(g)$ is a smooth manifold of dimension $8k - 3(1 - b_1(M) + b_2^+(M))$.\nFor example, for $k = 1$ and $M$ simply connected with negative definite form, the dimension is $8 - 3 = 5$; this is the moduli space used in the proof of [[Donaldson's theorem|34B7K77P31]].\nA connection with $F_A^- = 0$ but $F_A^+ \\ne 0$ is self-dual, not anti-self-dual, and so does not lie in $\\mathcal{M}_k(g)$; reversing the orientation of $M$ exchanges the two notions.\nAnti-self-dual connections are the absolute minima of the Yang–Mills functional in their bundle.\nThe polynomial invariants are defined when $b_2^+(M) \\ge 2$ and do not depend on the metric.\n",
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