The three‐dimensional Seiberg–Witten equations for 3/2$3/2$‐spinors: A compactness theorem

The Seiberg-Witten equations have proved to be quite powerful in studying smooth 4-manifolds since their landing in 1994. The corresponding Seiberg-Witten theory on closed 3-manifolds can either be obtained by a dimension reduction from the four-dimensional theory, or by following Floer's approach. Here we investigate the theory on 3-manifolds with boundary. The solutions to the Seiberg-Witten equations are identified with critical points to the Chern-Simons-Dirac functional, regarded as a section of the U(1) bundle over the quotient B of the configuration space. An infinite tube [0, ∞) X ∑ is added to the compact manifold and the asymptotic behavior of the solutions on the cylindrical end are studied. The moduli spaces of solutions under gauge group action are finite dimensional, compact and generically smooth. For a generic perturbation h, the moduli space M_h can be related to the moduli space M_L of the Kahler-Vortex equations on the boundary surface ∑, via a limit ing map r, which is a LagTangian immersion with respect to a canonical symplectic structure on M_L. Moreover, for a family of admissible perturbations, the moduli spaces for the perturbed Seiberg-Witten equations are mutually Legendrian cobordant.

On a system of Seiberg–Witten equations

The Seiberg-Witten equation with multiple spinors generalises the classical Seiberg-Witten equation in dimension three. In contrast to the classical case, the moduli space of solutions $\mathcal{M}$ can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections we define a signed count of points in $\mathcal{M}$ and show its invariance under small perturbations. We then study the equation on the product of a Riemann surface and a circle, describing $\mathcal{M}$ in terms of holomorphic data over the surface. We define analytic and algebro-geometric compactifications of $\mathcal{M}$, and construct a homeomorphism between them. For a generic choice of circle-invariant parameters of the equation, Fueter sections do not appear and $\mathcal{M}$ is a compact K\"ahler manifold. After a perturbation it splits into isolated points which can be counted with signs, yielding a number independent of the initial choice of the parameters. We compute this number for surfaces of low genus.

In this paper we consider twice-dimensionally reduced, generalized Seiberg-Witten (S-W) equations, defined on a compact Riemann surface. A novel feature of the reduction technique is that the resulting equations produce an extra “Higgs field”. Under suitable regularity assumptions, we show that the moduli space of gauge-equivalent classes of solutions to the reduced equations, is a smooth Kahler manifold and construct a pre-quantum line bundle over the moduli space of solutions.

In this paper, we study the Seiberg-Witten equations on the product R x Y, where Y is a compact 3-manifold with boundary. Following the approach of Salamon and Wehrheim in the instanton case, we impose Lagrangian boundary conditions for the Seiberg- Witten equations. The resulting equations we obtain constitute a nonlinear, nonlocal boundary value problem. We establish regularity, compactness, and Fredholm properties for the Seiberg- Witten equations supplied with Lagrangian boundary conditions arising from the monopole spaces studied in [20]. This work therefore serves as an analytic foundation for the construction of a monopole Floer theory for 3-manifolds with boundary.

By a theorem of McLean, the deformation space of an associative submanifold Y of an integrable G 2 manifold ( M, ϕ ) can be identified with the kernel of a Dirac operator on the normal bundle ν of Y . Here, we generalize this to the non-integrable case, and also show that the deformation space becomes smooth after perturbing it by natural parameters, which corresponds to moving Y through ‘pseudo-associative’ submanifolds. Infinitesimally, this corresponds to twisting the Dirac operator with connections A of ν . Furthermore, the normal bundles of the associative submanifolds with Spin c structure have natural complex structures, which helps us to relate their deformations to Seiberg-Witten type equations. If we consider G 2 manifolds with 2-plane fields ( M , ϕ, λ) (they always exist) we can split the tangent space TM as a direct sum of an associative 3-plane bundle and a complex 4-plane bundle. This allows us to define (almost) λ-associative submanifolds of M , whose deformation equations, when perturbed, reduce to Seiberg-Witten equations, hence we can assign local invariants to these submanifolds. Using this we can assign an invariant to ( M , ϕ, λ). These Seiberg-Witten equations on the submanifolds are restrictions of global equations on M . We also discuss similar results for the Cayley submanifolds of a Spin(7) manifold.

In these notes, we carefully analyze the properties of the "ramified" Seiberg-Witten equations associated with supersymmetric configurations of the Seiberg-Witten abelian gauge theory with surface operators on an oriented closed four-manifold X. We find that in order to have sensible solutions to these equations, only surface operators with certain parameters and embeddings in X, are admissible. In addition, the corresponding "ramified" Seiberg-Witten invariants on X with positive scalar curvature and b^+_2 > 1, vanish, while if X has b^+_2 = 1, there can be wall-crossings whence the invariants will jump. In general, for each of the finite number of basic classes that corresponds to a moduli space of solutions with zero virtual dimension, the perturbed "ramified" Seiberg-Witten invariants on Kahler manifolds will depend - among other parameters associated with the surface operator - on the monopole number "l" and the holonomy parameter "alpha". Nonetheless, the (perturbed) "ramified" and ordinary invariants are found to coincide, albeit up to a sign, in some examples.

In this thesis we study a certain generalization of the gauge-theoretical Seiberg-Witten equations over a source 4-manifold X. The generalization involves a hyperKähler manifold M with certain symmetries and a nonlinear Dirac operator D acting on equivariant maps u (called spinors) with values in M.We prove a classification theorem for such hyperKähler manifolds and propose a new method for their construction. This allows us to obtain new examples of hyperKähler and closely related to them quaternionic Kähler manifolds. Our construction is quite explicit and, in some cases, this allows to obtain not only existence results but also hyper- and quaternionic- Kähler structures themselves.We also prove that harmonic spinors, i.e. solutions of the equation Du=0, are closely related to solutions of the so-called Cauchy-Riemann-Fueter equation. We then prove that solutions of the Cauchy-Riemann-Fueter equation, which are believed to be a "right" analogue of holomorphic maps in quaternionic context, are exactly those maps, whose differential has no triholomorphic component. Hence we introduce the term "aholomorphic" for such maps. It is also shown that harmonic spinors can be regarded as twisted version (in an appropriate sense) of aholomorphic maps.The last part of the thesis is devoted to the generalized Seiberg-Witten equations over Kähler surfaces. In this case we prove that the space of solutions has a holomorphic description (in the usual complex sense). Further, if X is a product of two holomorphic curves we show (modulo the adiabatic limit conjecture) that there exists a relation between holomorphic curves (in the sense of Gromov theory), the symplectic vortex equations and the generalized Seiberg-Witten equations.

We find half-BPS vortex solitons, at both weak and strong coupling, in the N=6 supersymmetric mass deformation of ABJM theory with U(N) x U(N) gauge symmetry and Chern-Simons level k. The strong coupling gravity dual is obtained by performing a Z_k quotient of the N=8 supersymmetric eleven dimensional supergravity background of Lin, Lunin and Maldacena corresponding to the mass deformed M2-brane theory. At weak coupling, the BPS vortices preserving six supersymmetries are found in the Higgs vacuum of the theory where the gauge symmetry is broken to U(1) x U(1). The classical vortex solitons break a colour-flavour locked global symmetry resulting in non-Abelian internal orientational moduli and a CP^1 moduli space of solutions. At strong coupling and large k, upon reduction to type IIA strings, the vortex moduli space and its action are computed by a probe D0-brane in the dual geometry. The mass of the D0-brane matches the classical vortex mass. However, the gravity picture exhibits a six dimensional moduli space of solutions, a section of which can be identified as the CP^1 we find classically, along with a Dirac monopole connection of strength k. It is likely that the extra four dimensions in the moduli space are an artifact of the strong coupling limit and of the supergravity approximation.

By analyzing the work of Campolattaro we arguethat the second Seiberg–Witten equation over theSpin 4 c manifold, i.e.,Fij + = 〈M,SijM〉, is the generalization ofCampolattaro's description of the electromagnetic field tensor Fμυ in thebilinear form $$F^{\mu v} = \bar \Psi S^{\mu v} \Psi $$ . It turns out thatthe Seiberg–Witten equations (also the perturbedSeiberg–Witten equations) can be well understoodfrom this point of view. We suggest that the secondSeiberg–Witten equation can be replaced by anonlinear Dirac-like equation. We also derive the spinorrepresentation of the connection on the associatedunitary line bundle over the Spin4 c manifold.

The purpose of this paper is to explain the equivariant Euler class associated to an oriented G-equivariant Fredholm section S : B → E of a Hilbert space bundle over a Hilbert manifold. The key hypotheses are that the Lie group G is compact, the isotropy subgroups are finite, and the zero set of the section is compact. The present paper is motivated by our joint work with Gaio [9] on invariants of Hamiltonian group actions. In this work the Fredholm section arises from a version of the vortex equations, where the target space is a symplectic manifold with a Hamiltonian G-action [8, 19, 20]. In many interesting cases the resulting moduli spaces are compact and so the results of the present paper can be applied. Other examples of Fredholm sections with compact zero sets are the Seiberg–Witten equations over a four-manifold [25] or the harmonic map equations when the target space is a negatively curved manifold (see e.g. [14]). This is in sharp contrast to the Gromov–Witten invariants of general (compact) symplectic manifolds [11, 16, 17, 22] and to the Donaldson invariants of smooth four-manifolds [10], where the moduli spaces are noncompact and the compactifications are the source of some major difficulties of the theory. Since the unperturbed moduli space is compact, our framework is considerably simpler than the one required for the construction of the Gromov–Witten invariants. Our exposition follows closely the work of Li–Robbin–Ruan [16]. In the case G = {1l} similar results were proved in [6, 12, 21]. In [12] Fulton proved that, if B is a finite dimensional complex manifold, E → B is a holomorphic vector bundle, and S : B → E is a holomorphic section, then the zero set M := S−1(0) carries a fundamental cycle (in singular homology) which is Poincare dual to the Euler class. This was extended to the infinite ∗Supported by National Science Foundation grant DMS-0072267

We prove an abstract compactness theorem for a family of generalized Seiberg-Witten equations in dimension three. This result recovers Taubes' compactness theorem for stable flat $\mathbf{P}\mathrm{SL}_2(\mathbf{C})$-connections as well as the compactness theorem for Seiberg-Witten equations with multiple spinors. Furthermore, this result implies a compactness theorem for the ADHM$_{1,2}$ Seiberg-Witten equation, which partially verifies a conjecture by Doan and Walpuski.

In this paper, we study the Seiberg-Witten equations on a compact 3-manifold with boundary. Solutions to these equations are called monopoles. Under some simple topological assumptions, we show that the solution space of all monopoles is a Banach manifold in suitable function space topologies. We then prove that the restriction of the space of monopoles to the boundary is a submersion onto a Lagrangian submanifold of the space of connections and spinors on the boundary. Both these spaces are infinite dimensional, even modulo gauge, since no boundary conditions are specified for the Seiberg-Witten equations on the 3-manifold. We study the analytic properties of these monopole spaces with an eye towards developing a monopole Floer theory for 3-manifolds with boundary, which we pursue in Part II.

In this paper we deal with the (2 + 1)-dimensional Higgs model governed by the Ginzburg–Landau Lagrangian. The static solutions of this model, called otherwise vortices, are described by the theorem of Taubes. This theorem gives, in particular, an explicit description of the moduli space of vortices (with respect to gauge transforms). However, much less is known about the moduli space of dynamical solutions. A description of slowly moving solutions may be given in terms of the adiabatic limit. In this limit the dynamical Ginzburg–Landau equations reduce to the adiabatic equation coinciding with the Euler equation for geodesics on the moduli space of vortices with respect to the Riemannian metric (called T-metric) determined by the kinetic energy of the model. A similar adiabatic limit procedure can be used to describe approximately solutions of the Seiberg–Witten equations on 4-dimensional symplectic manifolds. In this case the geodesics of T-metric are replaced by the pseudoholomorphic curves while the solutions of Seiberg–Witten equations reduce to the families of vortices defined in the normal planes to the limiting pseudoholomorphic curve. Such families should satisfy a nonlinear ∂-equation which can be considered as a complex analogue of the adiabatic equation. Respectively, the arising pseudoholomorphic curves may be considered as complex analogues of adiabatic geodesics in (2 + 1)-dimensional case. In this sense the Seiberg–Witten model may be treated as a (2 + 1)-dimensional analogue of the (2 + 1)-dimensional Abelian Higgs model2.

In the first part of this chapter, we define the Chern-Simons action for U(1)-bundles, following [Man98]. We describe the moduli space of gauge equivalence classes of flat U(1)-bundles, which turn out to be the critical points of the action functional. However, for our purposes this ’classical’ moduli space of solutions is not suitable, but instead we have to take the derived moduli space.