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. 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 Seiberg–Witten moduli space of solutions of the Seiberg–Witten equations up to gauge tends to be compact, so one avoids the hard problems involved in compactifying the Yang–Mills moduli space of solutions of the Yang–Mills equations up to gauge 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). For the early history see (Jackson 1995).
Spinc structures The fourth spinc group is
Spin c ( 4 ) = ( Spin ( 4 ) × U ( 1 ) ) / Z 2 ≅ U ( 2 ) × U ( 1 ) U ( 2 ) {\displaystyle \operatorname {Spin} ^{\mathrm {c} }(4)=(\operatorname {Spin} (4)\times \operatorname {U} (1))/\mathbb {Z} _{2}\cong \operatorname {U} (2)\times _{\operatorname {U} (1)}\operatorname {U} (2)}
where the Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } acts as a sign on both factors. The group has a natural homomorphism to SO(4) = Spin(4)/±1. Given a compact oriented 4 manifold, choose a smooth Riemannian metric g {\displaystyle g} with Levi Civita connection ∇ g {\displaystyle \nabla ^{g}} . This reduces the structure group from the connected component GL(4)+ to SO(4) and is harmless from a homotopical point of view. A spinc structure or complex spin structure on M is a reduction of the structure group to Spinc, i.e. a lift of the SO(4) structure on the tangent bundle to the group Spinc. By a theorem of Hirzebruch and Hopf, every smooth oriented compact 4-manifold M {\displaystyle M} admits a Spinc structure. The existence of a Spinc structure is equivalent to the existence of a lift of the second Stiefel–Whitney class w 2 ( M ) ∈ H 2 ( M , Z / 2 Z ) {\displaystyle w_{2}(M)\in H^{2}(M,\mathbb {Z} /2\mathbb {Z} )} to a class K ∈ H 2 ( M , Z ) . {\displaystyle K\in H^{2}(M,\mathbb {Z} ).} Conversely such a lift determines the Spinc structure up to 2 torsion in H 2 ( M , Z ) . {\displaystyle H^{2}(M,\mathbb {Z} ).} A spin structure proper requires the more restrictive w 2 ( M ) = 0. {\displaystyle w_{2}(M)=0.}
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