関西ゲージ理論セミナー

過去の講演

2019年

2018年

• 族に対する10/8型不等式． 符号数$-16$のスピン多様体$M$上に$H^+(M)$ の向きを変えるなどいくつかの条件を満たす $n$ 個の可換な自己微分同相写像が存在するとき $b_+(M)\geq 3+n$.
• $M=K3\# n(S^2\times S^2)$とするとき， $T^n$ を底空間とする nonsmoothable な$M$束の構成，および$M$上の nonsmoothable な $\mathbb{Z}^n$作用の構成．

2017年

2016年

There seem to be two kinds of generalizations of the Hitchin equations on Riemann surfaces in four dimensions: one is the Vafa-Witten equations [VW]; and the other one is the Kapustin-Witten equations [KW]. Both come from topologically twisted N=4 super Yang-Mills theories. They both ask a connection A on a principal G-bundle P with G being a compact Lie group over a four-manifold X and a section B of a certain associated bundle to P over X to satisfy gauge-theoretic equations (one difference between them comes from which associated bundle one chooses in each case).

One of common obstacles to be overcome in the study of solutions to these equations is the lack of a priori compactness in the direction to the sections B. With regard to this, Taubes [Tau] made a huge breakthrough in the study of the Uhlenbeck style compactness theorem for SL(2,C) connections on a principal SU(2) or SO(3) bundle on a closed four-manifold. (A remark is that solutions to the K-W equations above are examples of Taubes' SL(2,C) connections.) 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.

This talk describes some observations on the singular sets in some cases for the Vafa-Witten equations [Tan1] and Kapustin-Witten equations [Tan2]. In the first part of this talk, we consider the Vafa-Witten equations on a closed four-manifold, and see that the singular set is empty under certain no bubbling condition. This enables us to relate the $L^2$ bound of the sections B to the reducibility of the connections A in the equations. The second part of this talk describes the case for the Kapustin-Witten equations. We assume that the underlying four-manifold X is a Kahler surface, and observe that the singular sets in this case have the structure of analytic subvarieties of X.

2015年