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