Gauge theoretical equivariant Gromov-Witten invariants and the full Seiberg-Witten invariants of ruled surfaces
It was first observed by \textit{I. Mundet i Riera} in [Oxford University Ph.D. thesis (1999), Topology 42, 525--553 (2003; Zbl 1032.53079)] and by \textit{K. Cieliebak}, \textit{A. R. Gaio} and \textit{D. A. Salamon} [Int. Math. Res. Not. 2000, No. 16, 831--882 (2000; Zbl 1083.53084)] that the machinery developed for the gauge theoretic equations known as the vortex equations can (in principle) be adapted to produce an interesting generalization of the Gromov-Witten invariants for symplectic manifolds. The novelty comes from replacing a symplectic manifold by a symplectic (in fact Kähler) fibration over a Kähler manifold. The invariants are defined for data sets \((\hat{K},\hat{P},F)\), where \(\hat{K}\) is a compact Lie group, \(\hat{P}\) is a principal \(\hat{K}\)-bundle over a closed Riemann surface, and \(F\) is a symplectic manifold with a Hamiltonian \(\hat{K}\)-action. Like all gauge-theoretic invariants, they are extracted from a moduli space of solutions to a set of non-linear partial differential equations. In this case the equations are a generalization of the vortex equations. In this paper the authors discuss a further generalization in which in addition to the group \(\hat{K}\), a normal subgroup \(K\subset\hat{K}\) is specified. They mostly consider the special case in which \(\hat{K}=U(r)\times U(r_0)\), \(K=U(r_0)\), and \(F=Hom(\mathbb{C}^r\times \mathbb{C}^{r_0})\). Moreover, the case \(r=1\) is the only case in which any detailed computation of resulting invariants is attempted. There is a comparison with Seiberg-Witten type invariants of a ruled surface defined by the authors in [Int. J. Math. 7, 811--832 (1996; Zbl 0959.57029)].
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