Some Analytic Aspects Of Kapustin-Witten Equations On 4-manifolds

In this thesis we study two themes about gauge theory. The first theme is the Kapustin-Witten equations on 4-manifolds. The second theme is theory of the complex flat connections on higher dimensional manifolds and its applications. We organize the thesis as follows:In the Introduction, we introduce the background and the recent progress of the Kapustin-Witten equations.In Chapter 1, we introduce some basic notions about gauge theory.In Chapter 2, we consider the Kapustin-Witten equations on a closed four-manifold. We study certain analytic properties of solutions to the equations on the closed manifold-s. The main result is that there exist an L2-lower bound on the extra fields over a closed four-manifold satisfying certain conditions if the connections are not ASD connections. Furthermore, we also obtain a similar result about the Vafa-Witten equations.In Chapter 3, we recall result due to Uhlenbeck [34], which provides existence of a flat connection Г on P given a Sobolev connection on P with Lp small curvature (when 2p> dim(X)), a global gauge transformation g of A to Coulomb gauge with respect to Г and a Sobolev norm estimate for the distance between A and Г. Since Theorem 3.0.6 plays an essential role in our proof of our second main result, we include more details concerning its proof in Chapter 4.In Chapter 4, We consider a complex flat connection on a principle bundle P over a compact Riemannian manifold M= Mn, n≥5. First, we prove that the complex part of complex flat connection must with L2-bounded from below by some positive constant, if M satisfies certain conditions, unless the complex flat connection is de-coupled. Second, we observe that the complex flat connections on a compact Kahler manifold are the same as Simpson’s equations. We also prove if there is a semistable Higgs vector bundle (E,θ) on a compact Kahler-Einstein manifold with c1(TX)>0, then the vector bundle E is semistable vector bundle. If (E,θ) be a polystable Higgs vector bundle on a compact Calabi-Yau manifold, we prove that the vector bundle E is polystable.