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The coupled Seiberg-Witten equations, vortices, and moduli spaces of stable pairs. (English) Zbl 0846.57013
Recently, Seiberg and Witten introduced new invariants of 4-manifolds which are defined by counting solutions of certain nonlinear differential equations [E. Witten, Monopoles and four-manifolds, Math. Res. Lett. 1, No. 6, 769-796 (1994)]. In the present paper, the authors generalize the Seiberg-Witten equations by coupling them to connections in unitary vector bundles. More precisely, fix a $$\text{Spin}^c$$-structure on a Riemannian 4-manifold $$X$$, and denote by $$\sum^\pm$$ the associated spinor bundles. The equations, considered in the paper, are $\begin{cases} \mathbb{D}_{A,b} \psi = 0 \\ \Gamma (F^+_{A,b}) = (\psi \overline \psi)_0. \end{cases}$ This is a system of equations for a pair $$(A, \psi)$$ consisting of a unitary connected in a unitary bundle $$E$$ over $$X$$, and a positive spinor $$\psi \in A^0 (\sum^+ \otimes E)$$. The symbol $$b$$ denotes a connection in the determinant line bundle of the spinor bundles $$\sum^\pm$$ and $$\mathbb{D}_{A,b} : A^0 (\sum^+ \otimes E) \to A^0 (\sum^= \otimes E)$$ is the Dirac operator obtained by coupling the connection in $$\sum^+$$ defined by $$b$$ (and by the Levi-Civita connection in the tengent bundle) with the variable connection $$A$$ in $$E$$. These equations specialize to the original Seiberg-Witten equations if $$E$$ is a line bundle. The main result of the paper is the following: Let $$(X,g)$$ be a Kähler surface of total scalar curvature $$\sigma_g$$, and let $$\sum$$ be the canonical $$\text{Spin}^c$$-structure with associated Chern connection $$c$$. Fix a unitary vector bundle $$E$$ of rank $$r$$ over $$X$$, and define $$\mu_g (\sum^+ \otimes E) = (\deg_g (E)/r) + \sigma_g$$. Then for $$\mu_g < 0$$, the space of solutions of the coupled Seiberg-Witten equations is isomorphic, as a real analytic space, to the moduli space of stable pairs of TOP type $$E$$ with parameter $$(- 1/4 \pi) \sigma_g$$.

##### MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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