A Necessary Condition for the Existence of SWα-Monopoles

Originally, as described in [12], the SWα-equations discovered by Seiberg and Witten are 1-order partial differential equations, which solutions (A,φ), with φ 6= 0, are known as SWα-monopoles. It is known that the solutions of these 1-order SWα-equations correspond to the minimum of the functional SWα : Cα → R. However, it is not true that the minimum of SWα : Cα → R is always attained by this sort of solution. In fact, there are only a finite number of α ∈ Spin(X) such that the minimum is a SWα-monopole. We show that a necessary condition to (A,φ) ∈ Cα be a SWα-monopole is that QX(α, α) vX .(k − g,X) 4 ∈ [− 1 π , 1 4 ], where QX=intersection form of X, vX=volume of (X, g) and k − g,X is a constant depending on the scalar curvature of (X, g).