---
tag: PRDDMSTR4X
kind: concept
name: Seiberg–Witten invariants
msc: [57R57, 57K41, 53C27]
sources:
  - url: https://en.wikipedia.org/wiki/Seiberg%E2%80%93Witten_invariants
same_as:
  wikidata: Q7446569
  wikipedia: Seiberg–Witten invariants
---

## Summary

The Seiberg–Witten invariants are integer invariants of smooth closed oriented [[4-manifolds|8H58Q6YJCH]], one for each spin$^c$ structure.
They count solutions of the Seiberg–Witten equations, which couple a $U(1)$-connection with a spinor field through a Dirac operator.
They were introduced by Edward Witten in 1994, following work of Nathan Seiberg and Witten in physics.
They recover many results of [[Donaldson theory|ZQW6M0S6M0]] with much simpler analysis, because the moduli spaces are compact.

## Explanations

### L1 human

The Seiberg–Witten equations are a system of differential equations that one can write on any smooth four-dimensional space with a little extra structure.
One counts their solutions, with signs, and the count does not depend on the choices made along the way.
This count is a number attached to the smooth space, and spaces that look the same to topology can have different counts.
For example, take a K3 surface blown up at one point, and the connected sum of $3$ complex projective planes with $20$ reversed ones.
These two spaces are homeomorphic, but their counts differ, so they are not diffeomorphic.