On the singular sets of solutions to the Vafa-Witten and Kapustin-Witten equations on closed 4-manifolds
主 题: On the singular sets of solutions to the Vafa-Witten and Kapustin-Witten equations on closed 4-manifolds
报告人: Dr. Yuuji Tanaka (Nagoya University)
时 间: 2017-01-11 09:00-11:00
地 点: 理科1号楼1303
In this talk, we consider two kinds of gauge-theoretic equations with some "Higgs fields" on smooth four-manifolds: one is the Vafa-Witten equations; and the other is the Kapustin-Witten ones, both of which originate in topologically twisted N=4 super Yang-Mills theories. One of shared problems in the study of solutions to these equations is unboundedness of solutions in the directionof the Higgs fields. However, Taubes' recent great breakthrough in the study of Uhlenbeck type compactness theorem for SL(2,C)-connections opened up a new approach to this problem. Taubes introduces some real codimension two singular sets, outside which a sequence of "partially rescaled" SL(2,C)-connections converges after gauge transformations except bubbling out at finite set of points.
In this talk, we describe some observations on the singular sets in some cases for the Vafa-Witten equations and Kapustin-Witten equations. For the Vafa-Witten equations, we see that the singular set is empty under certain no bubbling condition. This enables us to relate the L^2 bound of the sections to the notion of irreducibility of the connections in the equations. Regarding the Kapustin-Witten equations, we focus our attention to the case that the underlying manifold X is a Kahler surface, and describe that the singular sets in this case have the structure of analytic subvariety of X.