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My question is: I am request for the reference that Is there any relationship between the Seiberg-Witten Invariant and Donaldson's Invariant? Or the relationship between Seiberg-Witten Moduli Space and Yang-Mills Moduli space?

My question is a reference request.

My question is based on the following observation: For given compact smooth manifold M,

Consider the Seiberg-Witten Equation without perturbation, $F^+_A+(\phi \phi^*)_0$=0 $D_A^+ \phi =0$. Then every solution of the Seiberg-Witten Equation is a pair $(A, \phi)$, A is connection and $\phi$ is a spinor. However, for the reducible solution of Seiberg-Witten equation(the solution that $\phi=0$), we get the equation that $F_A^+$=0 An anti-selfdual equation of the curvature. When we change the orientation of the manifold M, all the Yang-Mills connection can be the antiself-dual. So I think in some way the Yang-Mills connection is the reducible solution of the Seiberg-Witten equation. Therefore, I believe there must be some strictly relationship between two manifold.

However, I ignore the difference of the $Spin^c$ structure and $SU(2)$, but I think with a proper choose of a Lie group homomorphism $f:SU(2)\rightarrow Spin^c$, we can map the Yang-Mills Moduli space into Seiberg-Witten Moduli space.

Maybe there exist some related questions in MO, but not what I want. Evans has asked a similar question:Is there a Seiberg-Witten version of Donaldson-Thomas theory?

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1 Answer

up vote 11 down vote accepted

This was conjectured by Witten in his paper in his paper Monopoles and four-manifolds. The conjecture says that if $X$ has Seiberg-Witten simple type (meaning that $SW_X(\mathfrak{s})$ is nonzero only when the moduli space associated to $X$ is 0-dimensional) and satisfies some mild homological conditions (including $b_1(X)=0$ and $b^+(X)>1$ odd), then $X$ has Donaldson simple type, its Donaldson and Seiberg-Witten basic classes coincide, and its Donaldson series has the form

${\mathcal D}^w_X(h) = 2^{2+(7\chi(X)+11\sigma(X))/4}e^{Q(h)/2}\sum_{\mathfrak{s}} (-1)^{w^2+c_1(\mathfrak{s})\cdot w} SW_X(\mathfrak{s})e^{c_1(\mathfrak{s})\cdot h}$

where $\mathfrak{s}$ ranges over Spin^c structures on $X$ and $Q(h)$ is the intersection form on $X$. Recall that the Donaldson series is defined as a formal power series by the sum $\sum_i D^w_X((1+\frac{x}{2})\frac{h^i}{i!})$, where $x$ is a point of $X$ and $h$ is an element of $H_2(X)$; this definition is originally due to Kronheimer and Mrowka, in Embedded surfaces and the structure of Donaldson's polynomial invariants.

There is a series of papers by Feehan and Leness proving many cases of the conjecture, although it has not been established in full. For an overview of this program, originally proposed by Pidstrigach and Tyurin, you might try their survey article PU(2) monopoles and relations between four-manifold invariants. Notably, their most recent paper, Witten's conjecture for many four-manifolds of simple type, proves it assuming that $c_1^2(X) \geq \chi_h(X)-3$, and before that, A general SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants was used by Kronheimer-Mrowka to prove enough cases of the conjecture to establish the Property P conjecture, in Witten's conjecture and Property P.

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@Sivek, Thank you very much! You answer is excellent! I can not access to the internet yesterday~ just sorry for that –  Siqi He Nov 14 '12 at 23:36
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