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Seiberg-Witten monopoles with multiple spinors on a surface times a circle. (English) Zbl 1411.14037
This paper addresses the questions of transversality and compactness for the moduli spaces of Seiberg-Witten multi-monopoles: solutions of the Seiberg-Witten equation with multiple spinors introduced in [J. A. Bryan and R. Wentworth, Turk. J. Math. 20, No. 1, 119–128 (1996; Zbl 0873.53049)]. 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 $$\mathcal{M}$$ can be non-compact due to the appearance of so-called Fueter sections. In the absence of Fueter sections the author defines a signed count of points in $$\mathcal{M}$$ and shows its invariance under small perturbations. He then studies the equation on the product of a Riemann surface and a circle, describing $$\mathcal{M}$$ in terms of holomorphic data over the surface. The author defines analytic and algebro-geometric compactifications of $$\mathcal{M}$$, and constructs 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 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. He computes this number for surfaces of low genus. This paper is organized as follows: Section 1 is an introduction to the subject. Section 2 concerns Seiberg-Witten monopoles with multiple spinors. Section 3 concerns a dimensional reduction. Section 4 deals with a holomorphic description of the moduli space. Section 5 deals with a tale of two compactifications. Section 6 concerns Fueter sections and complex geometry. Section 7 deals with moduli spaces of framed vortices. Section 8 concerns some examples and computations.
##### MSC:
 14H60 Vector bundles on curves and their moduli 14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) 57R57 Applications of global analysis to structures on manifolds 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) 35Q56 Ginzburg-Landau equations 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 32Q15 Kähler manifolds
##### Keywords:
Seiberg-Witten equation; spinors; Fueter sections
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