Seiberg–Witten invariants

The fourth spinc group is :\operatorname{Spin}^\mathrm{c}(4) =(\operatorname{Spin}(4)\times\operatorname{U}(1))/\Z_2 \cong\operatorname{U}(2)\times_{\operatorname{U}(1)}\operatorname{U}(2) where the \Z/2\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 with Levi Civita connection \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 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) \in H^2(M,\Z/2\Z) to a class K \in H^2(M, \Z). Conversely such a lift determines the Spinc structure up to 2 torsion in H^2(M,\Z). A spin structure proper requires the more restrictive w_2(M) = 0. A Spinc structure determines (and is determined by) a spinor bundle W = W^+ \oplus W^- coming from the 2 complex dimensional positive and negative spinor representation of Spin(4) on which U(1) acts by multiplication. We have K = c_1(W^+) = c_1(W^-). The spinor bundle W comes with a graded Clifford algebra bundle representation i.e. a map \gamma:\mathrm{Cliff}(M, g) \to \mathcal{E}\mathit{nd}(W) such that for each 1 form a we have \gamma(a):W^\pm \to W^\mp and \gamma(a)^2 = - g(a, a). There is a unique hermitian metric h on W s.t. \gamma(a) is skew Hermitian for real 1 forms a. It gives an induced action of the forms \wedge^* M by anti-symmetrising. In particular this gives an isomorphism of \wedge^+ M \cong \mathcal{E}\mathit{nd}^{sh}_0(W^+) of the selfdual two forms with the traceless skew Hermitian endomorphisms of W^+ which are then identified. ==Seiberg–Witten equations==

Let L = \det(W^+) \equiv \det(W^-) be the determinant line bundle with c_1(L) = K. For every connection \nabla_{A} = \nabla_0 + A with A \in iA^1_{\R}(M) on L, there is a unique spinor connection \nabla^{A} on W i.e. a connection such that \nabla^{A}_X(\gamma(a)) := [\nabla^{A}_X, \gamma(a)] = \gamma(\nabla^{g}_X a) for every 1-form a and vector field X. The Clifford connection then defines a Dirac operator D^A = \gamma \otimes 1 \circ \nabla^A = \gamma(dx^\mu)\nabla^A_\mu on W. The group of maps \mathcal{G} = \{u: M \to U(1)\} acts as a gauge group on the set of all connections on L. The action of \mathcal{G} can be "gauge fixed" e.g. by the condition d^*A = 0, leaving an effective parametrisation of the space of all such connections of H^1(M,\R)^{\mathrm{harm}}/H^1(M,\Z) \oplus d^* A^+_{\R}(M) with a residual U(1) gauge group action. Write \phi for a spinor field of positive chirality, i.e. a section of W^+. The Seiberg–Witten equations for (\phi, \nabla^A) are now D^A\phi=0 F^+_A=\sigma(\phi) + i\omega Here F^A \in iA^2_{\R}(M) is the closed curvature 2-form of \nabla^A, F^+_A is its self-dual part, and σ is the squaring map \phi\mapsto \left (\phi h(\phi, -) -\tfrac12 h(\phi, \phi)1_{W^+} \right) from W^+ to the a traceless Hermitian endomorphism of W^+ identified with an imaginary self-dual 2-form, and \omega is a real selfdual two form, often taken to be zero or harmonic. The gauge group \mathcal{G} acts on the space of solutions. After adding the gauge fixing condition d^*A = 0 the residual U(1) acts freely, except for "reducible solutions" with \phi = 0. For technical reasons, the equations are in fact defined in suitable Sobolev spaces of sufficiently high regularity. An application of the Weitzenböck formula {\nabla^A}^* \nabla^A \phi = \left(D^A\right)^2\phi - \left(\tfrac{1}{2} \gamma{\left(F_A^+\right)} + s\right)\phi and the identity \Delta_g \left|\phi\right|_h^2 = 2h{\left({\nabla^A}^* \nabla^A\phi, \phi\right)} - 2\left|\nabla^A\phi\right|_{g\otimes h} to solutions of the equations gives an equality \Delta\left|\phi\right|^2 + \left|\nabla^A\phi\right|^2 + \tfrac{1}{4} \left|\phi\right|^4 = (-s)\left|\phi\right|^2 - \tfrac{1}{2} h(\phi,\gamma(\omega)\phi). If |\phi|^2 is maximal \Delta|\phi|^2\ge 0, so this shows that for any solution, the sup norm \|\phi\|_\infty is a priori bounded with the bound depending only on the scalar curvature s of (M, g) and the self dual form \omega. 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 (\phi,\nabla^A) of the Seiberg–Witten equations are called monopoles, as these equations are the field equations of massless magnetic monopoles on the manifold M. ==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 \phi = 0. By the Atiyah–Singer index theorem the moduli space is finite dimensional and has "virtual dimension" :(K^2-2\chi_{\mathrm{top}}(M)-3\operatorname{sign}(M))/4 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 \omega, the reducible solutions have \phi = 0, and so are determined by connections \nabla_A = \nabla_0 + A on L such that F_0 + d A = i(\alpha + \omega) for some anti selfdual 2-form \alpha. By the Hodge decomposition, since F_0 is closed, the only obstruction to solving this equation for A given \alpha and \omega, is the harmonic part of \alpha and \omega, and the harmonic part, or equivalently, the (de Rham) cohomology class of the curvature form i.e. [F_0] = F_0^{\mathrm{harm}} = i (\omega^{\mathrm{harm}} + \alpha^{\mathrm{harm}}) \in H^2(M, \R). Thus, since the [\tfrac1{2\pi i} F_0] = K the necessary and sufficient condition for a reducible solution is : \omega^{\mathrm{harm}} \in 2\pi K + \mathcal{H}^- \in H^2(X,\R) where \mathcal{H}^- is the space of harmonic anti-selfdual 2-forms. A two form \omega is K-admissible if this condition is not met and solutions are necessarily irreducible. In particular, for b^+ \ge 1, the moduli space is a (possibly empty) compact manifold for generic metrics and admissible \omega. Note that, if b_+ \ge 2 the space of K-admissible two forms is connected, whereas if b_+ = 1 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 F^{\mathrm{harm}}. There are therefore (for fixed \omega) only finitely many K \in H^2(M,\Z), and hence only finitely many Spinc structures, with a non empty moduli space. ==Seiberg–Witten invariants==

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 Seiberg–Witten 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. The simple type conjecture states that if M is simply connected and b2+(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 b2+(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 b2+ ≥ 1 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. In particular it cannot be split as a connected sum of manifolds with b2+ ≥ 1. == See also ==

• • • • • • • • . • {{citation • {{citation|last1=Seiberg|first1= N.|author-link1=Nathan Seiberg|last2= Witten|first2= E. |author-link2=Edward Witten|title=Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD • • ==References==