# Seiberg-Witten theory on 4-manifolds with boundary

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).

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I added the TQFT tag. –  Chris Schommer-Pries Mar 16 '10 at 3:15

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.
If what you say is true, then these can't be TQFTs is the sense of the Atiyah-Segal axioms (i.e. as functors from $COB_{3+1}$ to Vect). It follows from the axioms that for any such TQFT and any 3-manifold X the value of the TQFT on the 4-manifold $X \times S^1$ is the dimension of the vector space associated to X. If this is zero for all X, then the TQFT is the zero TQFT =><=. My recollection is that what you get is similar to an Atiyah-Segal TQFT, but not quite defined on all bordisms. –  Chris Schommer-Pries Mar 16 '10 at 3:12