Seiberg-Witten equations in all dimensions
Joel Fine, Partha Ghosh
TL;DR
This work introduces a unified elliptic system of Seiberg–Witten type on Spin$^c$ manifolds across all dimensions by perturbing the Dirac operator with an odd-degree form $\beta$, and prescribing a curvature equation whose principal part matches the Weitzenböck remainder. In dimensions $n=4m$ and $n=4m-2$ the system is constructed with one Dirac operator (and two spinors/curves in the latter case) so that ellipticity modulo gauge is achieved, and explicit index formulas are derived; in odd dimensions a single Dirac operator $D_{A,\beta}$ is used with a sum of $d\beta$ and $d^*\beta$-type terms. The paper provides detailed Weitzenböck identities for these perturbed operators, analyzes zeroth-order contributions $Q(\beta)$ across dimensions, and proves robust a priori estimates and an energy identity (showing solutions are absolute minima of a natural functional in 8D). However, while a priori bounds are established, compactness modulo gauge is not guaranteed in higher dimensions due to possible bubbling, signaling rich analytic structure and potential applications to higher-dimensional geometric problems.
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.