Seiberg–Witten monopoles with multiple spinors on a surface times a circle

Abstract

The Seiberg–Witten equation with multiple spinors generalises the classical Seiberg–Witten equation in dimension 3. In contrast to the classical case, the moduli space of solutions scriptM 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 scriptM and show its invariance under small perturbations. We then study the equation on the product of a Riemann surface and a circle, describing scriptM in terms of holomorphic data over the surface. We define analytic and algebro‐geometric compactifications of scriptM, and construct a homeomorphism between them. For a generic choice of circle‐invariant parameters of the equation, Fueter sections do not appear and scriptM is a compact Kähler 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.

Adiabatic limits and Kazdan–Warner equations

“…We end this section by restating Theorem 1.4 in terms of generalised vortex equations. In [Doa,Theorem 1.8] we show that all irreducible solutions to the Seiberg-Witten equations with multipe spinors are pulled-back from solutions to (1.3) of a particular form. Using the notation of the introduction, set N = 2, k 1 = 1, k 2 = −1, and choose the bundles E 1 and E 2 to be Serre dual to each other.…”

“…The relationship between Seiberg-Witten monopoles with multiple spinors and generalised vortices (1.3) is the subject of the authors' paper [Doa] where further consequences of the results presented here are explored. In particular, Theorem 1.4 is used to construct a compactification of the moduli space of Seiberg-Witten monopoles with multiple spinors on S 1 × Σ and to compare it with a corresponding algebro-geometric moduli space [Doa,Theorem 1.5]. This, in turn, leads to the first known examples of the non-compactness phenomenon predicted by the Haydys and Walpuski's theorem [Doa,section 8].…”

The SO(3) Vortex Equations over Orbifold Riemann Surfaces

“…Subsequently, (Ψ Ψ * ) 0 denotes the component of Ψ Ψ * which is traceless only on the "spinorial" component, that is, (Ψ Ψ * ) 0 ∈ Γ (End 0 S ⊗ EndE). (3) Notice that the traceless part of the curvature equation in (29) gives us back the SO(3) monopole equations (26). It is not difficult to identify the "critical values" of τ , that is, those values where the flat connections and Seiberg-Witten solutions can appear.…”

“…Proof. We follow the proof in [68,Proposition 3.1], [26,Proposition 3.6], which applies with cosmetic changes to our situation.…”

Index theorem for $\mathbb{Z}/2$-harmonic spinors

“…All of these equations have potentially important applications. For example, Haydys and Walpuski [13] [14], conjecture a fundamental relation between the multi-spinor Seiberg-Witten equations on 3-manifolds and the spaces of G 2 -instantons on certain 7-dimensional manifolds (also see [7]). Meanwhile, Witten has conjectured [15] that spaces of solutions of the Kapustin-Witten equations can be used to compute the Jones polynomial for knots in S 3 .…”