For a spin-c manifold $M$ of four dimension, we can define the Einstein-Seiberg-Witten equations $ESW(g,A,\psi)$, with $g$ a metric, $A$ a connection for a line bundle and $\psi$ a spinor:

$$(1): {\cal D}_A(\psi)=0$$

with ${\cal D}_A$ the Dirac operator of the spin-c structure.

$$(2): F(A)_+ (X,Y)= <Ric(X).Ric(Y).\psi,\psi> + g(Ric (X),Ric (Y))<\psi,\psi>$$

with $F(A)_+$ the self-dual part of the curvature of $A$ and $Ric$ the Ricci curvature as an endomorphism of the tangent bundle.

If the manifold is Einstein $ESW(g,A,\psi)=SW(A,\psi )$.

We have the gauge group:

$$ {\cal G}={\cal C}^{\infty}(S^1). Diff(M)$$

with $Diff(M)$ the group of diffeomorphisms of $M$ acting on the ESW equations. Then the moduli space is:

$${\cal M}(M)= ESW (g,A,\psi)/{\cal G}$$

Can we define good ESW invariants?