In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994), using the Seiberg–Witten theory studied by Nathan Seiberg and Witten (1994a, 1994b) during their investigations of Seiberg–Witten gauge theory.
They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg–Witten equations tends to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.
For detailed descriptions of Seiberg–Witten invariants see (Donaldson 1996), (Moore 2001), (Morgan 1996), (Nicolaescu 2000), (Scorpan 2005, Chapter 10).
For the relation to symplectic manifolds and Gromov–Witten invariants see (Taubes 2000).
Given a compact oriented 4 manifold, choose a smooth Riemannian metric
This reduces the structure group from the connected component GL(4)+ to SO(4) and is harmless from a homotopical point of view.
By a theorem of Hirzebruch and Hopf, every smooth oriented compact 4-manifold
Conversely such a lift determines the Spinc structure up to 2 torsion in
A spin structure proper requires the more restrictive
coming from the 2 complex dimensional positive and negative spinor representation of Spin(4) on which U(1) acts by multiplication.
comes with a graded Clifford algebra bundle representation i.e. a map
of the selfdual two forms with the traceless skew Hermitian endomorphisms of
The Clifford connection then defines a Dirac operator
acts as a gauge group on the set of all connections on
for a spinor field of positive chirality, i.e. a section of
For technical reasons, the equations are in fact defined in suitable Sobolev spaces of sufficiently high regularity.
After adding the gauge fixing condition, elliptic regularity of the Dirac equation shows that solutions are in fact a priori bounded in Sobolev norms of arbitrary regularity, which shows all solutions are smooth, and that the space of all solutions up to gauge equivalence is compact.
For generic metrics, after gauge fixing, the equations cut out the solution space transversely and so define a smooth manifold.
By the Atiyah–Singer index theorem the moduli space is finite dimensional and has "virtual dimension" which for generic metrics is the actual dimension away from the reducibles.
It means that the moduli space is generically empty if the virtual dimension is negative.
, and the harmonic part, or equivalently, the (de Rham) cohomology class of the curvature form i.e.
, the moduli space is a (possibly empty) compact manifold for generic metrics and admissible
, and hence only finitely many Spinc structures, with a non empty moduli space.
The Seiberg–Witten invariant of a four-manifold M with b2+(M) ≥ 2 is a map from the spinc structures on M to Z.
The value of the invariant on a spinc structure is easiest to define when the moduli space is zero-dimensional (for a generic metric).
In this case the value is the number of elements of the moduli space counted with signs.
The Seiberg–Witten invariant can also be defined when b2+(M) = 1, but then it depends on the choice of a chamber.
A manifold M is said to be of simple type if the Seiberg–Witten invariant vanishes whenever the expected dimension of the moduli space is nonzero.
If the manifold M has a metric of positive scalar curvature and b2+(M) ≥ 2 then all Seiberg–Witten invariants of M vanish.
If the manifold M is simply connected and symplectic and b2+(M) ≥ 2 then it has a spinc structure s on which the Seiberg–Witten invariant is 1.