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  A generalization of the Seiberg-Witten equations

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Following the book of Friedrich "Dirac operators and riemannian geometry" (AMS, vol 25), I define the generalized Seiberg-Witten equations for $(A,A',\psi , \phi,f, g)$, with $A,A'$ two connections and $\psi, \phi$, two spinors, $f,g:M\rightarrow S^1$:

1) 

$D_A ( \psi)=0$

2)

$D_{A'} ( \phi)=0$

3) 

$F_+ (A)=-(1/4) \omega (\psi)$

4)

$F_+ (A')=-(1/4) \omega (\phi)$

5)

$f^* A= g^* A'$

6)

$f g= <\psi, \bar \phi >$

The gauge group $(h,h') \in Map(M,S^1)$ acts over the solutions of the generalized Seiberg-Witten equations:

$(h,h').(A,A',\psi,\phi,f, g)=((1/h)^* A, (1/{h'})^* A', h \psi, h' \phi,h f, h' g )$

Have we compact moduli spaces?

Moreover, the situation can perhaps be generalized to $n$ solutions of the Seiberg-Witten equations $(A_i ,\psi_i ,f_i )$

1)

$D_{A_i}( \psi_i)=0$

2)

$F_+(A_i)= -(1/4) \omega (\psi_i)$

3)

$ f_i^* A_i=B$

4)

$f_i f_j=<\psi_i , \bar \psi_j >$

asked Jul 21 in Theoretical Physics by Antoine Balan (25 points) [ revision history ]
edited 6 days ago by Antoine Balan

1 Answer

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I showed compacity at the following link: vixra.org/abs/1807.0352

answered Aug 6 by Antoine Balan (25 points) [ no revision ]

First show that $A,A'$ are compact by the curvature (following Friedrich), next show that $f/g$ is compact by 5). Finally we have $\psi,\phi, fg$ compact because finite dimensional (in the kernel of the Dirac operator.)

Another way to demonstrate it is to say that we have a compact set 1) 2) 3) 4) (because the moduli spaces of SW equations are compact) and we consider a closed set into (because of 5) and 6)) so that the moduli spaces we have are closed sets in  compact sets, so are compact.

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