---
tag: ZQW6M0S6M0
kind: concept
name: Donaldson theory
msc: [57R57, 57K41, 58D27]
sources:
  - url: https://en.wikipedia.org/wiki/Donaldson_theory
same_as:
  wikidata: Q5295371
  wikipedia: Donaldson theory
---

## Summary

Donaldson theory studies smooth [[4-manifolds|8H58Q6YJCH]] through moduli spaces of anti-self-dual connections, also called [[instantons|YWBTGC7MFA]], on principal $SU(2)$-bundles over them.
It was started by Simon Donaldson in the early 1980s.
Its first result, [[Donaldson's theorem|34B7K77P31]], restricts the intersection forms of smooth $4$-manifolds.
Its invariants, the Donaldson polynomial invariants, distinguish smooth structures on manifolds that are homeomorphic.

## Explanations

### L1 human

A connection is a rule for comparing directions at nearby points, and its curvature measures how much the rule twists.
On a four-dimensional space, the connections whose curvature has a special symmetry, the instantons, form a new space called a moduli space.
The shape of this moduli space reflects the smooth structure of the original space, not only its topology.
For example, it shows that the topological $4$-manifold with the $E_8$ intersection form cannot be smooth.
The [[Seiberg–Witten invariants|PRDDMSTR4X]] later gave a simpler route to many of the same results.