Seiberg–Witten invariant

Seiberg-Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg-Witten equations tend to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.

pin^C-structures

The Seiberg-Witten equations depend on the choice of a complex spin structure, Spin^C, on a 4-manifold "M". In 4 dimensions the group Spin^C is

:("U"(1)×Spin(4))/(Z/2Z),

and there is a homomorphism from it to SO(4). A Spin^C-structure on "M" is a lift of the natural SO(4) structure on the tangent bundle (given by the Riemannian metric and orientation) to the group Spin^C. Every smooth compact 4-manifold "M" has Spin^C-structures (though most do not have spin structures).

eiberg-Witten equations

Fix a smooth compact 4-manifold "M", choose a spin^C-structure "s" on "M", and write "W"⁺, "W"⁻ for the associated spinor bundles, and "L" for the determinant line bundle. Write φ for a self-dual spinor field (a section of "W"⁺) and "A" for a U(1) connection on "L". The Seiberg-Witten equations for (φ,"A")are: $D^Aphi=0$ : $F^+_A=sigma(phi)$ where "D"^"A" is the Dirac operator of "A", "F"_"A" is the curvature 2-form of "A", and "F"_"A"⁺ is its self-dual part, and σ is the squaring map from "W"⁺ to imaginary self-dual 2-forms.

The solutions (φ,"A") to the Seiberg-Witten equations are called monopoles.

The moduli space of solutions

The space of solutions is acted on by the gauge group, and the quotient by this action is called the moduli space of monopoles.

The moduli space is usually a manifold. A solution is called reducible if it is fixed by some non-trivial element of the gauge group. If "b"₂⁺("M")≥1 then reducible solutions do not exist for generic metrics on "M". The moduli space is a manifold except at reducible monopoles. So if "b"₂⁺("M")≥1 then the moduli space is a (possibly empty) manifold for generic metrics. Moreover all components have dimension : $(c_1(s)^2-2chi(M)-3sign(M))/4.$

The moduli space is empty for all but a finite number of spin^C structures "s", and is always compact.

A manifold "M" is said to be of simple type if the moduli space is finite for all "s". The simple type conjecture states that if "M" is simply connected and "b"₂⁺("M")≥2 then the moduli space is finite. It is true for symplectic manifolds. If "b"₂⁺("M")=1 then there are examples of manifolds with moduli spaces of arbitrarily high dimension.

eiberg-Witten invariants

The Seiberg-Witten invariants are easiest to define for manifolds "M" of simple type. In this case the invariant is the map from spin^C structures "s" to Z taking "s" to the number of elements of the moduli space counted with signs.

If the manifold "M" has a metric of positive scalar curvature and "b"₂⁺("M")≥2 then all Seiberg-Witten invariants of "M" vanish.

If the manifold "M" is the connected sum of two manifolds both of which have "b"₂⁺≥1 then all Seiberg-Witten invariants of "M" vanish.

If the manifold "M" is simply connected and symplectic and "b"₂⁺("M")≥2 then it has a spin^C structure "s" on which the Seiberg-Witten invariant is 1. In particular it cannot be split as a connected sum of manifolds with "b"₂⁺≥1.

References

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