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  "body": "\n## Summary\n\nThe Seiberg–Witten invariants are integer invariants of smooth closed oriented [[4-manifolds|8H58Q6YJCH]], one for each spin$^c$ structure.\nThey count solutions of the Seiberg–Witten equations, which couple a $U(1)$-connection with a spinor field through a Dirac operator.\nThey were introduced by Edward Witten in 1994, following work of Nathan Seiberg and Witten in physics.\nThey recover many results of [[Donaldson theory|ZQW6M0S6M0]] with much simpler analysis, because the moduli spaces are compact.\n\n## Explanations\n\n### L1 human\n\nThe Seiberg–Witten equations are a system of differential equations that one can write on any smooth four-dimensional space with a little extra structure.\nOne counts their solutions, with signs, and the count does not depend on the choices made along the way.\nThis count is a number attached to the smooth space, and spaces that look the same to topology can have different counts.\nFor example, take a K3 surface blown up at one point, and the connected sum of $3$ complex projective planes with $20$ reversed ones.\nThese two spaces are homeomorphic, but their counts differ, so they are not diffeomorphic.\n",
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