# The symplectic Seiberg-Witten equations

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The Heisenberg algebra is defined as $xy-yx=iw(x,y)$, with $w$ the symplectic form. For a symplectic manifold, we can define an infinite dimensional representation of the Heisenberg algebra, the Weil representation, in the case of a metaplectic structure. We define the C-metaplectic group as:

$$C-Mp(2n)= Mp(2n)\times S^1 /\{ 1, -1 \}$$

with $Mp(2n)$ the metaplectic group, a two fold covering space of the symplectic group $Sp(2n)$.

Then, we can define the symplectic Seiberg-Witten equations with help of the symplectic Dirac operator as defined by Habermann (Lecture Notes in Mathematics 1887).

$$D_w^A( \psi) =0$$

$$F(A) (x,y)=i w(x,y) < \psi , \psi >$$

with $D_w^A$ the symplectic Dirac operator and $A$, the connection of the line bundle. $F(A)$ is the curvature of the connection $A$.

Can we define symplectic Seiberg-Witten invariants?

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