open problems in Seiberg-Witten Theory on 4-Manifolds

Question

What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or review of them would be welcome.

Answer 1

One basic structural problem about the SW invariants is the question of simple type: suppose that $X$ is a simply connected 4-manifold with $b^+>1$, and $\mathfrak{s}$ a $\mathrm{Spin}^c$-structure such that $SW_X(\mathfrak{s})\neq 0$. Must $\mathfrak{s}$ arise from an almost complex structure? This is true when $X$ is symplectic (Taubes in "$SW\Rightarrow Gr$") but open in general.

The 11/8-conjecture (that for a closed Spin 4-manifold $X$ of signature $\sigma$, one has $b_2(X)\geq 11|\sigma|/8$) is open. SW theory has yielded strong results in this direction (Furuta's 10/8 theorem); proving the conjecture via SW theory is very hard but might be possible.

Essentially all of the fundamental questions about the classification of smooth 4-manifolds, or about the existence and uniqueness of symplectic structures on them, are open. We do not know how much Seiberg-Witten theory sees. For instance:

Suppose $X$ is a closed 4-manifold with an almost complex structure $J$. Let $w\in H^2(X;\mathbb{R})$ be a class with $w^2>0$. Is there a symplectic form $\omega$ with compatible almost complex structure homotopic to $J$ and symplectic class $w$? The "Taubes constraints" are the following necessary conditions, which constrain the SW invariants in terms of $w$ and $c=c_1(TX,J)$ (see e.g. Donaldson's survey on the SW equations): (i) $SW(\mathfrak{s}_{can})=\pm 1$ (the sign can be made precise) where $\mathfrak{s}_{can}$ is the $\mathrm{Spin}^c$-structure arising from $J$; (ii) $-c\cdot w\geq 0$; and (iii) if $SW(\mathfrak{s})\neq 0$ then $|c_1(\mathfrak{s})\cdot [\omega]| \leq -c \cdot [\omega]$, with equality iff $\mathfrak{s}$ is isomorphic to $\mathfrak{s}_{can}$ or its conjugate. The question is: if $X$ is simply connected, are these sufficient conditions? (Example: Fintushel-Stern knot surgery on an elliptically fibered K3 surface along a knot with monic Alexander polynomial.)

Answer 2

One basic problem is determining the relationship between Seiberg-Witten invariants and Donaldson invariants of $4$-manifolds. Witten himself proposed the precise relationship between the two in the original paper Monopoles and 4-Manifolds, but as far as I know the relationship has not been proven in general. Witten's conjecture has been proven in many cases, however. See the answer to this question for a good overview of the current status of this problem.

Answer 3

It might be useful to generalize a theorem of Donaldson and Sullivan, that the Donaldson invariants are defined for quasi-conformal 4-manifolds, to the category of Seiberg-Witten invariants. More generally, one would like to know which smooth invariants of 4-manifolds are defined for quasi-conformal 4-manifolds.