---
tag: DRNH767W84
kind: definition
name: Donaldson theory
msc: [57R57, 57K41, 58D27]
defines: ZQW6M0S6M0
uses: [8H58Q6YJCH, F45VQ47GPV, Q61VFHCH2J, 8E37E74XVP, NJ21PN5ZGJ]
sources:
  - url: https://en.wikipedia.org/wiki/Donaldson_theory
---

## Definition

Let $M$ 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.
Let $P \to M$ be a [[principal bundle|8E37E74XVP]] with structure group $SU(2)$ and second Chern class $c_2(P) = k$.
For a [[connection|NJ21PN5ZGJ]] $A$ on $P$ with curvature $F_A$, write $F_A = F_A^+ + F_A^-$ with $\ast F_A^{\pm} = \pm F_A^{\pm}$.

1. The connection $A$ is *anti-self-dual* if $F_A^+ = 0$.
2. The gauge group $\mathcal{G}$ of bundle automorphisms of $P$ acts on anti-self-dual connections.
3. The *moduli space* $\mathcal{M}_k(g)$ is the set of anti-self-dual connections on $P$ modulo $\mathcal{G}$.

*Donaldson theory* is the study of the spaces $\mathcal{M}_k(g)$ and of the invariants of $M$ obtained from them.
The main such invariants are the Donaldson polynomial invariants, defined by integrating cohomology classes over $\mathcal{M}_k(g)$.

## Remarks

Let $b_1(M)$ be the first Betti number and $b_2^+(M)$ the number of positive eigenvalues of the intersection form.
For a generic metric, the irreducible part of $\mathcal{M}_k(g)$ is a smooth manifold of dimension $8k - 3(1 - b_1(M) + b_2^+(M))$.
For 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]].
A 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.
Anti-self-dual connections are the absolute minima of the Yang–Mills functional in their bundle.
The polynomial invariants are defined when $b_2^+(M) \ge 2$ and do not depend on the metric.