Seiberg–Witten invariants
In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by , using the Seiberg–Witten theory studied by 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 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 , , , , . For the relation to symplectic manifolds and Gromov–Witten invariants see . For the early history see . Spin-structures. The Spin group (in dimension 4) is where the formula_2 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 formula_3 with Levi Civita connection formula_4. This reduces the structure group from the connected component GL(4) to SO(4) and is harmless from a homotopical point of view. A Spin-structure or complex spin  structure on "M" is a reduction of the structure group to Spin, i.e. a lift of the SO(4) structure on the tangent bundle to the group Spin. By a theorem of Hirzebruch and Hopf, every smooth oriented compact 4-manifold formula_5 admits a Spin structure. The existence of a Spin structure is equivalent to the existence of a lift of the second Stiefel-Whitney class formula_6 to a class formula_7 Conversely such a lift determines the Spin structure up to 2 torsion in formula_8 A spin structure proper requires the more restrictive formula_9 A Spin structure determines (and is determined  by) a spinor bundle formula_10 coming from the 2 complex dimensional positive and negative spinor representation of Spin(4) on which U(1) acts by multiplication. We have formula_11. The spinor bundle formula_12 comes with a graded Clifford algebra bundle representation i.e. a map formula_13 such that for each 1 form formula_14 we have formula_15 and formula_16. There is a unique hermitian metric formula_17 on formula_12 s.t. formula_19 is skew Hermitian for real 1 forms formula_14. It gives an induced action of the forms formula_21 by anti-symmetrising. In particular this gives an isomorphism of formula_22 of the selfdual two forms with  the traceless skew Hermitian endomorphisms of formula_23 which are then identified. Seiberg–Witten equations. Let formula_24 be the determinant line bundle with formula_25. For every connection formula_26 with formula_27 on formula_28, there is a unique spinor connection formula_29 on formula_12 i.e. a connection such that formula_31 for every 1-form formula_14 and vector field formula_33. The Clifford connection then defines a Dirac operator formula_34 on formula_12. The group of maps formula_36 acts as a gauge group on the set of all connections on formula_28. The action of formula_38 can be "gauge fixed" e.g. by the condition formula_39, leaving an effective  parametrisation of the space of all such connections of formula_40 with a residual formula_41 gauge group action. Write formula_42 for a spinor field of positive chirality, i.e. a section of formula_23. The Seiberg–Witten equations for formula_44 are now Here formula_47 is the closed curvature 2-form of formula_48, formula_49 is its self-dual part, and σ is the squaring map formula_50 from formula_23 to the a traceless Hermitian endomorphism of formula_23 identified with an imaginary self-dual 2-form, and formula_53 is a real selfdual two form, often taken to be zero or harmonic. The gauge group formula_38 acts  on the space of solutions. After adding the gauge fixing condition formula_39 the residual U(1) acts freely, except for "reducible solutions" with formula_56. For technical reasons, the equations are in fact defined in suitable Sobolev spaces of sufficiently high regularity. An application of the Weitzenböck formula and the identity to solutions of the equations gives an equality If formula_60 is maximal formula_61, so this shows that for any solution, the sup norm formula_62 is "a priori" bounded with the bound depending only on the scalar curvature formula_63 of formula_64 and the self dual form formula_53. 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. The solutions formula_66 of the Seiberg–Witten equations are called monopoles, as these equations are the field equations of massless magnetic monopoles on the manifold formula_5. 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. For generic metrics, after gauge fixing, the equations cut out the solution space transversely and so define a smooth manifold. The residual U(1) "gauge fixed" gauge group U(1) acts freely except at reducible monopoles i.e. solutions with formula_56. 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. For a self dual 2 form formula_53, the reducible solutions have formula_56, and so are determined  by connections formula_72 on formula_28 such that formula_74 for some anti selfdual 2-form formula_75. By the Hodge decomposition, since formula_76 is closed, the only obstruction to solving this equation for formula_77 given formula_75 and formula_53, is the harmonic part of formula_75 and formula_53, and the harmonic part, or equivalently, the (de Rham) cohomology class of the curvature form i.e. formula_82. Thus, since the formula_83 the necessary and sufficient condition for a reducible solution is where formula_85 is the space of harmonic anti-selfdual 2-forms. A two form formula_53 is formula_87-admissible if this condition is "not" met and  solutions are necessarily irreducible. In particular, for formula_88, the moduli space is a (possibly empty) compact manifold for generic metrics and admissible formula_53. Note that, if formula_90 the space of formula_87-admissible two forms is connected, whereas if formula_92 it has two connected components (chambers). The moduli space can be given a natural orientation from an orientation on the space of positive harmonic 2 forms, and the first cohomology. The "a priori" bound on the solutions, also gives "a priori" bounds on formula_93. There are therefore (for fixed formula_53) only finitely many formula_95, and hence only finitely many Spin structures , with a non empty moduli space. Seiberg–Witten invariants. The Seiberg–Witten invariant of a four-manifold "M" with "b"("M") ≥ 2 is a map from the spin structures on "M" to Z. The value of the invariant on a spin 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 "b"("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. The simple type conjecture states that if "M" is simply connected and "b"("M") ≥ 2 then the manifold is of simple type. This is true for symplectic manifolds. If the manifold "M" has a metric of positive scalar curvature and "b"("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" ≥ 1 then all Seiberg–Witten invariants of "M" vanish. If  the manifold "M" is simply connected and symplectic and "b"("M") ≥ 2 then it has a spin structure "s" on which the Seiberg–Witten invariant is 1. In particular it cannot be split as a connected sum of manifolds with "b" ≥ 1.