What generalizations of SeibergWitten theory to 4manifolds with boundary do exist?
I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to TQFTs).
What generalizations of SeibergWitten theory to 4manifolds with boundary do exist? I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to TQFTs). 


Hi Fabian! Kronheimer and Mrowka's book Monopoles and threemanifolds lays out comprehensively the construction of a SeibergWitten TQFT, called monopole Floer homology. It is conjectured to be isomorphic to the Heegaard Floer homology TQFT of OzsváthSzabó. There are also beautiful constructions due to Froyshov and to Manolescu which do not apply in quite so much generality. The structure of monopole Floer homology is as follows. The TQFT is a functor on the cobordism category $COB_{3+1}$ whose objects are connected, smooth, oriented 3manifolds. In fact, the TQFT consists of a trio of functors, denoted $\widehat{HM}_{\bullet}$, $\overline{HM_\bullet}$ and $\check{HM}_{\bullet}$ (the last of these is such a sophisticated invariant that you have to download a special LaTeX package just to typeset it properly). These are $\mathbb{Z}[U]$modules; there's a story about gradings that's too long to be worth summarising here. There are natural transformations which, for any connected 3manifold $Y$, define the maps in a long exact sequence $$ \cdots\to \widehat{HM} _{\bullet}(Y) \to \overline{HM_\bullet}(Y)\to \check{HM}_{\bullet}(Y) \to \widehat{HM}_{\bullet}(Y) \to \cdots $$ Why this structure? Well, the theory is based on the ChernSimonsDirac functional $CSD$ on a global Coulomb gauge slice through a space of (connection, spinor) pairs. $CSD$ is a $U(1)$equivariant functional, and $\check{HM}_{\bullet}$ is, philosophically, its $U(1)$equivariant semiinfinite Morse homology. $\overline{HM}_\bullet$ is the part coming from the restriction of $CSD$ to the $U(1)$fixedpoints, and $\widehat{HM}_\bullet$ is the equivariant homology relative to the fixed point set. Now here's a subtlety for the TQFT enthusiasts out there to get your teeth into (axiomatize, explain...)! The invariant of a closed 4manifold $X$ in any of the three theories is... zero. The famous SW invariant of a 4manifold with $b_+>0$ comes about via a secondary operation, not part of the TQFT itself. Delete two balls from $X$ to get a cobordism from $S^3$ to itself. When $b_+(X)>0$, there are generically no reducible SW monopoles on this cobordism, and this implies that the TQFTmap $\widehat{HM}_\bullet(S^3) \to \widehat{HM}_\bullet(S^3)$ lifts canonically to a map $\widehat{HM}_\bullet(S^3) \to \check{HM}_\bullet(S^3)$; it is this lift that carries the SW invariant. 

