Symplectic fibrations and the Abelian vortex equations
Let \(X\), \(S\) be smooth, oriented manifolds with \(\dim(X)- \dim(S)= 2\). Consider a proper submersion \(\pi: X\to S\). Then, \(\pi\) is a smooth fibre bundle, and its typical fibre is a compact orientable surface \(\Sigma\). The \(r\)th symmetric product bundle \(\pi_r: \text{Sym}^r_S(X)\to S\), is defined to be the quotient by the symmetric group \(S_r\) of the fibre product \(X^{\times r}_S= \{(x_1,x_2,\dots, x_r)\in X^{\times r}\mid\pi(x_1)= \pi(x_2)=\cdots= \pi(x_r)\}\) with its natural projection to \(S\). To make \(\text{Sym}^r_S(X)\) a smooth manifold one should choose a complex structure \(j\) on the vertical tangent bundle \(T^v X= \ker(D\pi)\subset TX\) compatibly with the orientations. The plain result is the following. Let \((X,\pi,\Omega)\) be a proper, locally Hamiltonian surface-fibration over a manifold \(S\), and \(r\) a positive integer. Choose {\parindent=7mm \begin{itemize}\item[(1)] an \(\Omega\)-positive complex structure \(j\) on \(T^v X\); \item[(2)] a Hermitian line bundle \((L,|\cdot|)\) over \(X\) such that \(L|_{X_s}\) has degree \(r\) for each \(s\in S\), and a unitary connection \(A_{\text{ref}}\) on \(L\); \item[(3)] a real parameter \(\tau\). We require \(\tau> 2\pi ra^{-1}\), where \(a\) is a symplectic area of the fibre. \end{itemize}} There is a procedure which associates to these data a closed two-form \(\nu(\Omega,\tau,L)\) on \(\text{Sym}^r_S(X; j)\) (\(j\) is a complex structure) which makes it a locally Hamiltonian fibration. This procedure is compatible with restriction of the base \(S\). The form \(\nu(\Omega, \tau, L)\) restricts on each fibre \(\text{Sym}^r(X_s)\) to the canonical Kähler form arising from the abelian vortex equations with parameter \(\tau\). Its cohomology class is determined and does not depend on the line bundle \(L\). When \(S\) is a point the above gives a new proof of the \textit{N. S. Manton} and \textit{S. M. Nasir} formula [Commun. Math. Phys. 199, No. 3, 591--604 (1999; Zbl 0920.58069)]. At the article the reader can find some relations between Seiberg-Witten theory on 3-manifolds fibred over the circle and symplectic Floer homology. The paper under review is a part of the doctoral thesis written under advice of S. Donaldson.
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