What are some of the open problems in SeibergWitten Theory on 4Manifolds.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.

2$\begingroup$ Since you are interested in Ncg, I want to make the remark that a former fellow PhD student of mine  Vadim Alekseev  has studied a noncommutative generalization of SeibergWitten invariants, i.e., defined in the context of spectral triples. Here is the link to his PhD thesis: webdoc.sub.gwdg.de/diss/2011/alekseev $\endgroup$– Marc PalmFeb 5, 2013 at 5:29
3 Answers
One basic structural problem about the SW invariants is the question of simple type: suppose that $X$ is a simply connected 4manifold 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 11/8conjecture (that for a closed Spin 4manifold $X$ 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 4manifolds, or about the existence and uniqueness of symplectic structures on them, are open. We do not know how much SeibergWitten theory sees. For instance:
Suppose $X$ is a closed 4manifold 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: FintushelStern knot surgery on an elliptically fibered K3 surface along a knot with monic Alexander polynomial.)

$\begingroup$ I omitted to say that the Taubes constraints apply when $b^+>1$, and the question I mentioned at the end concerns that case. $\endgroup$ Feb 15, 2013 at 2:36
One basic problem is determining the relationship between SeibergWitten invariants and Donaldson invariants of $4$manifolds. Witten himself proposed the precise relationship between the two in the original paper Monopoles and 4Manifolds, 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.

$\begingroup$ it's just amazing to see the witten's works.thanks for the link. $\endgroup$– KoushikFeb 5, 2013 at 3:25
It might be useful to generalize a theorem of Donaldson and Sullivan, that the Donaldson invariants are defined for quasiconformal 4manifolds, to the category of SeibergWitten invariants. More generally, one would like to know which smooth invariants of 4manifolds are defined for quasiconformal 4manifolds.