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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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