Lagrangian correspondences and Donaldson's TQFT construction of the Seiberg–Witten invariants of $3$–manifolds

Abstract

Using Morse–Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected $3$–manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed intersections of such Lagrangians, we supply the analytic details of Donaldson’s TQFT construction of the Seiberg–Witten invariants of a closed $3$–manifold.