Seiberg-Witten equations in all dimensions

Abstract

Starting with an $n$-dimensional oriented Riemannian manifold with a Spin-c structure, we describe an elliptic system of equations which recover the Seiberg-Witten equations when $n=3,4$. The equations are for a U(1)-connection $A$ and spinor $\phi$, as usual, and also an odd degree form $\beta$ (generally of inhomogeneous degree). From $A$ and $\beta$ we define a Dirac operator $D_{A,\beta}$ using the action of $\beta$ and $*\beta$ on spinors (with carefully chosen coefficients) to modify $D_A$. The first equation in our system is $D_{A,\beta}(\phi)=0$. The left-hand side of the second equation is the principal part of the Weitzenb\"ock remainder for $D^*_{A,\beta}D_{A,\beta}$. The equation sets this equal to $q(\phi)$, the trace-free part of projection against $\phi$, as is familiar from the cases $n=3,4$. In dimensions $n=4m$ and $n=2m+1$, this gives an elliptic system modulo gauge. To obtain a system which is elliptic modulo gauge in dimensions $n=4m+2$, we use two spinors and two connections, and so have two Dirac and two curvature equations, that are then coupled via the form $\beta$. We also prove a collection of a priori estimates for solutions to these equations. Unfortunately they are not sufficient to prove compactness modulo gauge, instead leaving the possibility that bubbling may occur.