What generalizations of Seiberg-Witten theory to 4-manifolds *with boundary* do exist?


I would be especially interested in theories which "behave good" under gluing along the boundary (comparable to TQFTs).



Seiberg-Witten theory on 4-manifolds with boundary
Hi Fabian! Kronheimer and Mrowka's book [Monopoles and three-manifolds](http://books.google.com/books?id=k6AcxmuTNTIC&printsec=frontcover&dq=kronheimer+mrowka+monopoles&source=bl&ots=Epxl6Ooz7g&sig=pqvqz_jr1zJaMw4aa0_GwnMlB24&hl=en&ei=vKueS-yBFYb6NffRiI0K&sa=X&oi=book_result&ct=result&resnum=2&ved=0CAsQ6AEwAQ#v=onepage&q=&f=false) lays out comprehensively the construction of a Seiberg-Witten TQFT, called monopole Floer homology. 
It is conjectured to be isomorphic to the [Heegaard Floer homology](http://arxiv.org/abs/math/0101206) TQFT of Ozsváth-Szabó. 
There are also beautiful constructions due to [Froyshov](http://arxiv.org/abs/0809.4842) and to [Manolescu](http://arxiv.org/abs/math/0311342) 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 3-manifolds. 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 3-manifold $Y$, 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 Chern-Simons-Dirac 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 semi-infinite Morse homology. $\overline{HM}\_\bullet$ is the part coming from the restriction of $CSD$ to the $U(1)$-fixed-points, 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 4-manifold $X$ in any of the three theories is... zero. The famous SW invariant of a 4-manifold 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 TQFT-map $\widehat{HM}\_\bullet(S^3) \to \widehat{HM}\_\bullet(S^3)$ 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.


