2024/11/14 by Fine, Joel, Ghosh, Partha
#53C21 #53C27 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.09348
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 DA,β using the action of β and *β on spinors (with carefully chosen coefficients) to modify DA. The first equation in our system is DA,β(ϕ)=0. The left-hand side of the second equation is the principal part of the Weitzenböck remainder for D^*A,βDA,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.