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 $\ddot{I}•$, as usual, and also an odd degree form ${\hat{I}}^2$ (generally of inhomogeneous degree). From $A$ and ${\hat{I}}^2$ we define a Dirac operator $D_{A,{\hat{I}}^2}$ using the action of ${\hat{I}}^2$ and $\ast {\hat{I}}^2$ on spinors (with carefully chosen coefficients) to modify $D_A$. The first equation in our system is $D_{A,{\hat{I}}^2}(\ddot{I}•)=0$. The left-hand side of the second equation is the principal part of the Weitzenböck remainder for $D_{A,{\hat{I}}^2}^{\ast }{D}_{A,{\hat{I}}^2}$. The equation sets this equal to $q(\ddot{I}•)$, the trace-free part of projection against $\ddot{I}•$, 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 ${\hat{I}}^2$. 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.

34 pages. v2 added section 6, showing solutions to the equations are absolute minima of a certain energy functional