Seiberg-Witten monopoles in three dimensions (Q675838)

Let \(X=Y\times [0,1]\) be a compact Riemannian 4-manifold with boundary, where \(Y\) is a compact Riemannian 3-manifold without boundary. Assume that \(Y\) has a spin structure which is compatible with the \(\text{Spin}_c\) structure of \(X\) endowed with the Riemannian metric \[ ds^2=dt^2+\sum^3_{i,j=1}g_{ij} dx^i dx^j, \quad t\in [0,1],\;x\in Y. \] The authors reduce the Seiberg-Witten equations on \(X\) to \(Y\) and obtain the equations of motion of a \(U(1)\)-Chern-Simons theory coupled to a massless spinorial field. Then they use topological field theory techniques to study the moduli space of gauge equivalence classes of solutions of the obtained equations. A mathematically rigorous definition of the Casson invariant is given in terms of the spectral flow of a family of self-adjoint Fredholm operators. Attempting to show that the invariant does not depend on the choice of a ``good'' metric, the authors find some connections with a result of \textit{B. L. Wang} in [`Seiberg-Witten-Floer theory for homology 3-spheres', preprint] concerning the fact that the invariant is the Euler characteristic of Seiberg-Witten-Floer homology.